Astroyte CA2+ activity and astrocyte-neuron interaction investigated through experimental measurements and mathermatical modeling

Astrocytes and neurons, the major cell types in the mammalian brain, communicate bidirectionally with one another. Astrocytes respond to neuronal activity primarily through G-protein coupled receptor (GPCR) activation and subsequent intracellular Ca2+ concentration elevation. These glial cells also play a major role in K+ buffering and neurotransmitter uptake, which are linked to Ca2+ elevations. These and other potential Ca2+-dependent functions (e.g. release of neuroactive compounds, known as gliotransmission), have the ability to considerably modify neuronal activity and brain function. Unsurprisingly, astrocyte Ca2+ activity is implicated in several diseases and behaviors, such as locomotion. Recent advances in Ca2+ imaging have enabled the study of astrocyte Ca2+ signaling. However, astrocyte Ca2+ activity is highly variable (e.g. between experimental trials) and has complex spatiotemporal patterns. Moreover, basic principles of astrocyte Ca2+ activity are unclear, such as the degree of its heterogeneity and its reliability in responding to ongoing stimulation. In addition, there are very few computational models of astrocyte Ca2+ activity that are based on experimental measurements, most of which do not consider Ca2+ heterogeneity. Such models would allow us to interpret experimental findings, evaluate existing hypotheses, generate new testable hypotheses, and guide future experiments in the astrocyte field. To address these issues, we integrated two-photon Ca2+ imaging experiments and computational modeling. We first developed a biophysical, mechanistic model of astrocyte Ca2+ activity based on our experimental measurements. We found that Ca2+ responses have complex, but informative, relationships with their underlying cellular mechanisms, resulting in response variability among different experimental trials, different cells, and different regions within one cell. Our model also identified important roles for various cellular mechanisms in generating a variety of Ca2+ response patterns. We also developed a data-driven probabilistic model of astrocyte Ca2+ activity and used it to find stimulation frequency-dependent response patterns in astrocytes. We characterized mechanisms that allow astrocytes to respond in two opposing manners to the same agonist, depending on the stimulation frequency. These collective findings provide new perspectives on interpreting astrocyte Ca2+ dynamics under various experimental conditions. Additionally, the biophysical and probabilistic models we developed provide valuable tools for future studies of astrocyte activity.

ASTROCYTE CA2+ ACTIVITY AND ASTROCYTE-NEURON INTERACTION INVESTIGATED THROUGH EXPERIMENTAL MEASUREMENTS AND MATHEMATICAL MODELING by Marsa Taheri A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Biomedical Engineering The University of Utah May 2019 Copyright © Marsa Taheri 2019 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Marsa Taheri has been approved by the following supervisory committee members: John A. White , Chair 12/17/2018 Date Approved Alla Borisyuk , Member 12/17/2018 Date Approved David Krizaj , Member 12/17/2018 Date Approved Richard D. Rabbitt , Member 12/17/2018 Date Approved Alan D. Dorval II , Member 12/17/2018 Date Approved and by David W. Grainger the Department/College/School of , Chair/Dean of Biomedical Engineering and by David B. Kieda, Dean of The Graduate School. ABSTRACT Astrocytes and neurons, the major cell types in the mammalian brain, communicate bidirectionally with one another. Astrocytes respond to neuronal activity primarily through G-protein coupled receptor (GPCR) activation and subsequent intracellular Ca2+ concentration elevation. These glial cells also play a major role in K+ buffering and neurotransmitter uptake, which are linked to Ca2+ elevations. These and other potential Ca2+-dependent functions (e.g. release of neuroactive compounds, known as gliotransmission), have the ability to considerably modify neuronal activity and brain function. Unsurprisingly, astrocyte Ca2+ activity is implicated in several diseases and behaviors, such as locomotion. Recent advances in Ca2+ imaging have enabled the study of astrocyte Ca2+ signaling. However, astrocyte Ca2+ activity is highly variable (e.g. between experimental trials) and has complex spatiotemporal patterns. Moreover, basic principles of astrocyte Ca2+ activity are unclear, such as the degree of its heterogeneity and its reliability in responding to ongoing stimulation. In addition, there are very few computational models of astrocyte Ca2+ activity that are based on experimental measurements, most of which do not consider Ca2+ heterogeneity. Such models would allow us to interpret experimental findings, evaluate existing hypotheses, generate new testable hypotheses, and guide future experiments in the astrocyte field. To address these issues, we integrated two-photon Ca2+ imaging experiments and computational modeling. We first developed a biophysical, mechanistic model of astrocyte Ca2+ activity based on our experimental measurements. We found that Ca2+ responses have complex, but informative, relationships with their underlying cellular mechanisms, resulting in response variability among different experimental trials, different cells, and different regions within one cell. Our model also identified important roles for various cellular mechanisms in generating a variety of Ca2+ response patterns. We also developed a data-driven probabilistic model of astrocyte Ca2+ activity and used it to find stimulation frequency-dependent response patterns in astrocytes. We characterized mechanisms that allow astrocytes to respond in two opposing manners to the same agonist, depending on the stimulation frequency. These collective findings provide new perspectives on interpreting astrocyte Ca2+ dynamics under various experimental conditions. Additionally, the biophysical and probabilistic models we developed provide valuable tools for future studies of astrocyte activity. iv “I am among those who think that science has great beauty. A scientist in his laboratory is not only a technician: he is also a child placed before natural phenomena which impress him like a fairy tale.” - Marie Curie TABLE OF CONTENTS ABSTRACT .................................................................................................................. iii LIST OF TABLES ....................................................................................................... viii ACKNOWLEDGMENTS .............................................................................................. ix Chapters 1 INTRODUCTION........................................................................................................ 1 1.1 Astrocyte Ca2+ Signaling: A Brief History ........................................................ 1 1.2 Astrocytes’ Diverse Roles in the Brain.............................................................. 2 1.3 Astrocyte-Neuron Interactions .......................................................................... 4 1.4 Controversies and Experimental Limitations ..................................................... 6 1.5 Mathematical Models of Astrocyte Ca2+ Signaling ............................................ 8 1.6 References ...................................................................................................... 12 2 DIVERSITY OF EVOKED ASTROCYTE CA2+ DYNAMICS QUANTIFIED THROUGH EXPERIMENTAL MEASUREMENTS AND MATHEMATICAL MODELING ................................................................................................................. 22 2.1 Abstract .......................................................................................................... 23 2.2 Introduction .................................................................................................... 23 2.3 Materials and Methods.................................................................................... 24 2.4 Results ............................................................................................................ 30 2.5 Discussion ...................................................................................................... 37 2.6 References ...................................................................................................... 39 2.7 Supplementary Figures ................................................................................... 42 3 MATHEMATICAL INVESTIGATION OF IP3-DEPENDENT CALCIUM DYNAMICS IN ASTROCYTES .................................................................................. 43 3.1 Abstract .......................................................................................................... 44 3.2 Introduction .................................................................................................... 44 3.3 The Mathematical Model ................................................................................ 46 3.4 Results ............................................................................................................ 47 3.5 Discussion ...................................................................................................... 54 3.6 Appendix: Mathematical Model ...................................................................... 57 3.7 References ...................................................................................................... 60 4 CA2+ MEASUREMENTS AND PROBABILISTIC MODELING REVEAL STIMULATION FREQUENCY-DEPENDENT RESPONSE PATTERNS IN ASTROCYTES ............................................................................................................. 74 4.1 Abstract .......................................................................................................... 74 4.2 Introduction .................................................................................................... 75 4.3 Results ............................................................................................................ 78 4.4 Discussion .................................................................................................... 111 4.5 Materials and Methods.................................................................................. 121 4.6 Supplementary Figures ................................................................................. 128 4.7 References .................................................................................................... 131 5 CONCLUSION ........................................................................................................ 138 5.1 Major Findings and Implications................................................................... 138 5.2 Future Directions and Preliminary Results .................................................... 144 5.3 References .................................................................................................... 151 vii LIST OF TABLES Tables 2.1 Model parameters .................................................................................................... 28 2.2 Experimentally observed variability in Ca2+ dynamics ............................................. 31 3.1. Model parameters ................................................................................................... 73 4.1 Estimated model transition rates .............................................................................. 91 4.2 Effects of transition rates corresponding to spontaneous or evoked activity ............. 93 ACKNOWLEDGMENTS This dissertation would not have been possible without the guidance and gracious support from my mentors, colleagues, and family members. I am indebted to my advisor, Dr. John White, for extending me the freedom and resources to explore research questions and approaches that I am most passionate about. The knowledge and experience he has shared with me will guide me throughout my career. I will remain ever grateful for his kind support and for all he has taught me. I would like to express my sincerest gratitude to Dr. Fernando Fernandez and Dr. Nathan Smith: I thank Fernando for always making himself available to provide invaluable feedback on my work and Nathan for training me on multiple experimental techniques and on astrocyte research. I am grateful to them both for their continuous encouragement and guidance. A special thanks goes to Dr. Alla Borisyuk, our collaborator and a member of my supervisory committee: the discussions we had, the feedback she offered, and the mathematical approaches she taught me have been truly instrumental to my research and I cannot thank her enough. I am also deeply thankful to Dr. Alan (Chuck) Dorval, Dr. David Krizaj, and Dr. Rick Rabbitt— the rest of my committee members—as well as Dr. Karen Wilcox for their invaluable feedback and guidance on my research. I thank Greg Handy for being an amazing collaborator and for playing a significant role in my research, Dr. Mike Gee for introducing me to astrocyte research and for his encouragements, and Feliks Royzen for patiently helping me carry out the “isolated hippocampal preparation” experiments. I would also like to thank other past and present members of the White lab, for I learned so much about science and life from each and every one of them. Finally, I cannot thank my family enough for being a constant source of motivation, for all the sacrifices they made, and for their relentless support and help throughout the years. x CHAPTER 1 INTRODUCTION 1.1 Astrocyte Ca2+ Signaling: A Brief History Astrocytes, unlike neurons, are not electrically excitable and their membrane currents are passive (Kuffler and Potter, 1964). Hence, when they were initially studied in the mid-twentieth century, they were thought of as silent support cells in the brain (Dennis and Gerschenfeld, 1969). More than a decade later, it was discovered that cultured astrocytes express a variety of metabotropic and ionotropic receptors (Kettenmann and Schachner, 1985; Murphy and Pearce, 1987) as well as neurotransmitter uptake mechanisms (Henn, 1976; Schousboe, 1981). However, it was also argued that cultured astrocytes were highly sensitive to culturing conditions and, thus, may not represent astrocytes in vivo (Juurlink and Hertz, 1985). A breakthrough in glial biology occurred after the development of cell-permeable Ca2+ indicator dyes, such as fura-2 and fluo-3: It was shown that neurotransmitters induce intracellular Ca2+ elevations in astrocytes (Enkvist et al., 1989) and that Ca2+ waves propagate throughout astrocyte networks (Cornell-Bell et al., 1990). These findings suggested that astrocytes are capable of responding to neuronal activity and long-range signaling in the brain. It was later shown that, following such Ca2+ elevations, cultured astrocytes release glutamate and alter neuronal activity through NMDA receptors 2 (Parpura et al., 1994). This led to the concept of gliotransmission and the tripartite synapse model (Araque et al., 1999; Bezzi et al., 1998; Haydon, 2001). The tripartite synapse model proposes that the synapse consist of three major units: the presynaptic neuron, the postsynaptic neuron, and the astrocyte process (branch), which can release gliotransmitters and modulate synaptic activity (Araque et al., 1999). With advances in Ca2+ imaging, including the development of geneticallyencoded Ca2+ indicators (GECIs), such as GCaMP, and higher resolution microscopy techniques, additional intriguing roles for astrocyte Ca2+ activity have been revealed. Studies also began focusing more on astrocyte activity in acute slices and in vivo, strengthening the notion that astrocyte Ca2+ signaling plays a role under physiological conditions. 1.2 Astrocytes’ Diverse Roles in the Brain Studies in recent decades have revealed that astrocytes play much more diverse roles in the central nervous system (CNS) than traditionally thought, many of which are Ca2+-dependent. Astrocyte Ca2+ activity has been associated with K+ buffering (Filosa et al., 2006; Larsen et al., 2014; Wang et al., 2012), neurotransmitter uptake regulation (e.g. glutamate and GABA; (Anderson and Swanson, 2000; Shigetomi et al., 2011; Yu et al., 2018; Zhou and Danbolt, 2013)), the release of neuroactive compounds (e.g. glutamate, D-Serine, and adenosine 5′-triphosphate (ATP); (Anderson and Swanson, 2000; Bezzi et al., 1998; Haydon, 2001; Liu et al., 2005; Newman, 2003; Wang et al., 2012)), blood flow regulation (Attwell et al., 2010; Seidel et al., 2015; Verkhratsky et al., 2012), synaptic modulation (Covelo and Araque, 2018; Di Castro et al., 2011; Kang et al., 1998; 3 Liu et al., 2005; Wang et al., 2012), and more (Khakh and Sofroniew, 2015). Furthermore, astrocyte Ca2+ elevations can propagate through multiple subcompartments of one astrocyte (i.e. its soma and numerous large and small processes) or between multiple astrocytes, often referred to as Ca2+ waves (Haydon, 2001). This allows for Ca2+ signaling to be used as a method for inter- and intracellular communication. Through such functions, astrocytes could modulate neuronal activity on the level of a single synapse as well as large neural networks. Indeed, astrocytes have been shown to contribute to long-term potentiation (LTP) and long-term depression (LTD) in the hippocampus (Adamsky et al., 2018; Gerlai et al., 1995; Khakh and Sofroniew, 2015; Nishiyama et al., 2002; Skucas et al., 2011; Suzuki et al., 2011), neocortex (Pankratov and Lalo, 2015), and cerebellum (Shibuki et al., 1996). Moreover, studies have proposed that astrocytes are involved in learning and memory through modulating various brain rhythms. For instance, Lee et al. (2014) and Sakatani et al. (2008) have shown that astrocytes play a role in gamma oscillations. Others have proposed the involvement of astrocytes in hippocampal theta rhythms (Hassanpoor et al., 2014; Haydon, 2001; Mishima and Hirase, 2010; Sibille et al., 2015a). Astrocyte Ca2+ signaling has also been associated with specific behavioral outcomes. For instance, widespread Ca2+ activity has been observed during locomotion (Paukert et al., 2014; Srinivasan et al., 2015), sensory stimulation (Ding et al., 2013), and startle response (Ding et al., 2013; Srinivasan et al., 2015). On the other hand, decreasing astrocyte Ca2+ activity has led to increased obsessive-compulsive-like behavior (Yu et al., 2018), decreased food intake (Chen et al., 2016), and disrupted cortical plasticity (Chen et al., 2012). 4 Finally, astrocytes are thought to be involved in several neurological diseases, possibly playing both neuro-protective and neuro-damaging roles (Verkhratsky et al., 2012). Astrocytes undergo morphological, phenotypical, and functional changes in epilepsy (Álvarez-Ferradas et al., 2015; Ding et al., 2007; Stewart et al., 2010), Huntington’s disease (Jiang et al., 2016; Khakh and Sofroniew, 2014; Tong et al., 2014), Alzheimer’s disease (Delekate et al., 2014; Kuchibhotla et al., 2009; Mattson and Chan, 2003; Orr et al., 2015; Ricci et al., 2009), and Parkinson’s disease (McGeer and McGeer, 2008; Mena and García de Yébenes, 2008). More specifically, these altered astrocytes (often known as reactive astrocytes) exhibit altered Ca2+ dynamics which could have a range of downstream effects and lead to changes in neuronal function (Álvarez-Ferradas et al., 2015; Delekate et al., 2014; Ding et al., 2007; Jiang et al., 2016; Kuchibhotla et al., 2009). 1.3 Astrocyte-Neuron Interactions Astrocytes express a variety of functional receptors, most of them being metabotropic GPCRs (Porter and McCarthy, 1997). These are major avenues of communication from neurons to astrocytes. Activation of such receptors leads to increases in intracellular Ca2+ in astrocytes through different pathways, including through inositol (1,4,5)-trisphosphate (IP3) production. The type of GPCR expressed can change during mouse development (Sun et al., 2013) and is likely circuit-specific (Khakh and Sofroniew, 2015). For example, in the hippocampus, studies have shown that minimal stimulation of Schaffer collaterals (projection from CA3 region of hippocampus to CA1 region) induces astrocyte Ca2+ responses in the CA1 stratum radiatum (mediated by the 5 metabotropic glutamate receptor mGluR5, a GPCR) and suggested that these responses regulate the probability of neurotransmitter release in neurons (Khakh and Sofroniew, 2015; Panatier et al., 2011). In contrast, CA3 stratum lucidum astrocytes only respond during intense action potential bursts of mossy fiber axons; these responses are mediated by the GPCR mGluR2/3 (Haustein et al., 2014). Therefore, CA3 astrocytes do not respond to sparse neuronal input but may, instead, control neuronal network synchronization through their global Ca2+ transients (Sasaki et al., 2014). As described earlier, other astrocyte mechanisms, such as neurotransmitter transporters, may also affect Ca2+ activity (Khakh and Sofroniew, 2015; Schummers et al., 2008). These Ca2+ elevations, in turn, can have several downstream effects. Astrocytes modulate neuronal activity through five major pathways, many of which are Ca2+-dependent : (1) through NCX and NKA pumps (Ca2+ transients cause NCX to pump Na+ into the cell, which then activates NKA to pump this Na+ out and, in return, K+ is pumped in, decreasing extracellular K+ briefly (Wang et al., 2012); (2) through glutamate transporters, since astrocytic glutamate transport is driven by the coupled transport of K+ and Na+ down their respective concentration gradients (Anderson and Swanson, 2000; Sibille et al., 2014); (3) through GABA transporters in a similar fashion (Anderson and Swanson, 2000; Sibille et al., 2014); (4) through inward-rectifying K+ channels expressed in astrocytes (Kir4.1 channels), which are affected by neuronal activity (Sibille et al., 2014); (5) lastly, through Ca2+-dependent gliotransmission (Anderson and Swanson, 2000; Bezzi et al., 1998; Haydon, 2001; Kang et al., 1998; Newman, 2003), which is possible but controversial (explained next, in Section 1.4). Importantly, because astrocyte Ca2+ activity is affected by multiple pathways, it is 6 complex and difficult to interpret. On the other hand, given the diversity of Ca2+dependent astrocyte functions, it is likely that distinct Ca2+ signals lead to distinct downstream effects (Khakh and Sofroniew, 2015). Hence, identifying informative features of astrocyte Ca2+ dynamics and relating specific cellular mechanisms with these features is essential to understanding astrocyte function, and is a major goal of this dissertation. 1.4 Controversies and Experimental Limitations While the study of astrocytes has caused great interest in recent years, leading to many new discoveries, there is much that is not known about astrocyte function (Araque et al., 2014; Khakh and Sofroniew, 2015; Nedergaard and Verkhratsky, 2012). The investigation of the bidirectional communication between astrocytes and neurons is still in its early stages and, despite significant research efforts, there is little consensus on the role of astrocyte signaling in the brain, with many studies remaining controversial. For example, the concept of gliotransmission and the details of how this gliotransmission may occur are the subject of great debate (Araque et al., 2014). Similarly, the tripartite synapse model hypothesis has been challenged, with many proposing that it may only exists in astrocyte culture and pathological conditions (Hamilton and Attwell, 2010; Petravicz et al., 2008). Overall, much of this controversy in results can be traced back to the fact that astrocytes are very heterogeneous. Not only are they extremely diverse from one brain region to another (Chai et al., 2017; Oberheim et al., 2012), and possibly even within one brain region, but they are also functionally different when cultured than in acute slice or 7 in vivo (Cahoy et al., 2008a; Pivneva et al., 2008). Therefore, the results from one study and experimental setup often do not carry over to another setup. Other controversies in the field are due to poor Ca2+ imaging methods that existed up until more recently, particularly with Ca2+ dye loading techniques which limit the experiments to culture or young and healthy animals (Peterlin et al., 2000; Reeves et al., 2011). This is an important issue, since it has now been shown that astrocyte Ca2+ activity differs drastically between the young and adult brain (Sun et al., 2013). More recently, it was also shown that the use of fluorescent dyes, but not GECIs, is actually toxic to the astrocyte and disrupts is functions (Smith et al., 2018). Once again, these imply that many earlier findings in the astrocyte field are not applicable to astrocytes in vivo. Finally, Ca2+ dyes do not readily diffuse into astrocyte processes (which are thought to be the actual sites of interaction with neuronal synapses) (Reeves et al., 2011). That, and the unavailability of higher resolution microscopy techniques, until more recently, means that many previous studies of astrocytes were limited to the somas and, possibly, some thick processes. More recently, the availability of new transgenic mice, improved GECIs, and high spatial and temporal resolution two-photon microscopes have allowed scientists to more accurately monitor astrocyte Ca2+ activity. However, there are still limitations in studying astrocyte function. In particular, the complexity and variability of astrocyte Ca2+ signals, examined in this dissertation, provide a challenge in interpreting and analyzing experimental data. Additionally, selectively manipulating the pathways leading to Ca2+ transients in astrocytes is a challenge, but is an area of active research (Bang et al., 2016; Khakh and McCarthy, 2015; Xie et al., 2015a). 8 1.5 Mathematical Models of Astrocyte Ca2+ Signaling As a consequence of the many controversies, unknowns, and limitations that exist in the field of astrocyte function, astrocytes and their effects on neuronal activity have largely been ignored in computational neuroscience. Few modeling studies have attempted to examine astrocyte Ca2+ dynamics, whether in the context of astrocyteastrocyte communication or astrocyte-neuron interactions (Manninen et al., 2018; Volman et al., 2012). Below is a summary of some major computational models (both biophysical and statistical) in the astrocyte field, including some limitations of these models. Few models of astrocytes (as well as other cell types) have taken a phenomenological, statistical modeling approach to describe Ca2+ events directly. Such models include those by Skupin and Falcke (2007, 2010) for astrocytes, and Tilūnaitė et al. (2017) for HEK293T cells. These statistical models incorporate stochasticity of Ca2+ events; but to do so, they simplify Ca2+ activity to a point process by considering it as a sequence of Ca2+ “spikes”, ignoring the event kinetics. After characterizing Ca2+ interspike intervals, these studies found that astrocyte Ca2+ spike generation is Poissonian. However, they focus on activity in astrocyte somas and do not consider the dynamic time course of stimulation that cells are exposed to under physiological conditions (except Tilūnaitė et al. who examined HEK293T cells). A second type of astrocyte Ca2+ activity models, which are the more common type, are biophysical, mechanistic models consisting of several differential equations. These models have several parameters, which are often not derived from astrocyte data, and mostly focus on one specific aspect of astrocyte activity. For example, Ullah et al. 9 (2006) showed that IP3 diffusion through gap junctions allows for anti-phase synchronization of Ca2+ oscillations in neighboring astrocytes, as was observed in experiments from cultured astrocytes. However, this was a close-cell model (disregarding known intracellular Ca2+ fluctuations arising from the activity of plasma membrane channels and pumps). Other models (De Pittà et al., 2009; Di Garbo et al., 2007) investigated the role of astrocyte Ca2+ signals in synaptic activity using a detailed model of IP3 creation and degradation. Di Garbo et al. (2007) also included additional Ca2+ channels located on the plasma membrane, a capacitive Ca2+ entry channel, and the ionotropic purinergic receptor P2X to explore the mechanisms of one specific type of Ca2+ response to bath applications of ATP in cultured astrocytes. This deterministic model was extended by Toivari et al. (2011) who included stochastic elements, such as opening and closing of the IP3 receptor and Brownian motion through Ca2+ channels, in order to investigate stochastic mechanisms involved in Ca2+ response generation. Although this model explores sources of response variability observed experimentally, it relies on data from cultured astrocytes and bath applications of agonists, which is not physiological. On the other hand, another model by Lavrentovich and Hemkin (2008) showed that spontaneous, and presumably stochastic, Ca2+ oscillations can be achieved in an entirely deterministic, singlecompartment Ca2+ and IP3 model by tuning it into a chaotic regime. Work has also been done on modeling the full tripartite synapse: neuronal activity modulates astrocytes which, in turn, affect the neurons. The majority of these models assume that astrocytes affect neurons only through gliotransmission, when their Ca2+ levels reach an arbitrary threshold (De Pittà et al., 2011; Postnov et al., 2007; Volman et 10 al., 2007). However, whether gliotransmission occurs in astrocytes, particularly in noncultured astrocytes under physiological conditions, is highly debated, as mentioned in Section 1.4. Even if gliotransmission does occur, it is not the only effect of astrocyte Ca2+ activity onto neurons (see Sections 1.3 and 1.4). Also, how potential gliotransmission relates to astrocyte Ca2+ dynamics (e.g. whether it occurs after a certain Ca2+ amplitude is reached) is unknown. A few models consider the role of other astrocyte functions on neurons: either astrocyte K+ buffering or neurotransmitter uptake, but not both. These models also fully ignore the role of astrocyte Ca2+ dynamics. Wei et al. (2014) focused on the fact that astrocytes are known to buffer K+ out of the synaptic cleft. By considering an extended Hodgkin-Huxley-like model of a neuron, they accounted for astrocyte influences on nearby neurons by modeling extracellular K+ concentrations. Scimemi et al. (2009) and Allam et al. (2012) both presented models that allowed for detailed diffusion of glutamate within the synaptic cleft, and suggested that astrocytes play an active role in synaptic plasticity via glutamate uptake. Further, Sibille et al. (2015) specifically modeled K+ uptake in astrocytes by including the Kir4.1 channel and NKA pump. They showed that K+ uptake in astrocytes had the ability to regulate neuronal excitability through a pathway that is independent of gliotransmission. Their study also suggested that astrocytes might be involved in regulating theta rhythm activity, though this was not studied in detail. A few studies have also investigated the role of astrocyte function on a network of neurons. For example, Wade et al. (2011) showed that astrocytes allowed for dynamic coordination between nearby or remote synapses, facilitating synaptic plasticity. Amiri et al. (2012) found that after including astrocytes in a network, unwanted network 11 synchronization emerges. Finally, Reato et al. (2012) demonstrated that astrocyte activity can lower seizure threshold in a network of neurons. It needs to be emphasized again that these three models assumed that the astrocyte to neuron feedback occurred via gliotransmission. Overall, astrocyte mathematical models are often non-astrocyte-specific, meaning that the data and parameters used were obtained from other cell types that also have Ca2+ dynamics (e.g. smooth muscle cells) rather than from astrocyte experiments. When experiments are used in these models, they are often from cultured astrocytes and focusing on the activity in astrocyte somas (rather than fine processes). When stimulation of astrocytes is examined, it is through bath applications of agonists, which is less physiological than other methods, e.g. focal, brief pulses of agonists or direct neuronal stimulation. Additionally, most studies have ignored astrocyte Ca2+ variability (e.g. trialto-trial) and focused solely on one small aspect of astrocyte activity, such as spontaneous Ca2+ events or gliotransmission. Finally, some of these models are closed-cell models and do not consider the known intracellular Ca2+ fluctuations that occur through plasma membrane channels and pumps. This gap between mathematical modeling and astrocyte experiments has not only prevented the field from effectively studying astrocyte-neuron interactions through computational models, but has also been a barrier to examining experimental astrocyte Ca2+ dynamics on a more mathematical and quantitative level. A mathematical model of astrocyte Ca2+ activity based on novel experimental data can be substantially useful for testing hypotheses that exist in the field, making experimentally verifiable predictions, identifying areas where more research and experimentation is needed, and investigating 12 some of the ongoing controversies in the field. Such a model could also be used to test how changes in astrocyte function could lead to changes in neuronal function, or vice versa, during neurological diseases such as epilepsy. 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The original publication appeared in Frontiers in Systems Neuroscience on October 23, 2017 (Volume 11, Issue 79). The contributing authors to this work are myself, Gregory Handy, Alla Borisyuk, and John A. White. I performed all the experiments and analyzed all the experimental data. GH and I built the mathematical model together, with equal contribution. Together with GH, I ran the model simulations, analyzed the simulation results, and wrote the initial draft of the manuscript (which was then edited by all co-authors). 23 ORIGINAL RESEARCH published: 23 October 2017 doi: 10.3389/fnsys.2017.00079 Diversity of Evoked Astrocyte Ca2+ Dynamics Quantified through Experimental Measurements and Mathematical Modeling Marsa Taheri 1† , Gregory Handy 2† , Alla Borisyuk 2* and John A. White 3* 1 Department of Bioengineering, University of Utah, Salt Lake City, UT, United States, 2 Department of Mathematics, University of Utah, Salt Lake City, UT, United States, 3 Department of Biomedical Engineering, Boston University, Boston, MA, United States Edited by: Charles J. Wilson, University of Texas at San Antonio, United States Reviewed by: James Sneyd, University of Auckland, New Zealand David Terman, Ohio State University Columbus, United States *Correspondence: Alla Borisyuk [e-mail] John A. White [e-mail] † Co-first authors. Received: 01 August 2017 Accepted: 04 October 2017 Published: 23 October 2017 Citation: Taheri M, Handy G, Borisyuk A and White JA (2017) Diversity of Evoked Astrocyte Ca2+ Dynamics Quantified through Experimental Measurements and Mathematical Modeling. Front. Syst. Neurosci. 11:79. doi: 10.3389/fnsys.2017.00079 Astrocytes are a major cell type in the mammalian brain. They are not electrically excitable, but generate prominent Ca2+ signals related to a wide variety of critical functions. The mechanisms driving these Ca2+ events remain incompletely understood. In this study, we integrate Ca2+ imaging, quantitative data analysis, and mechanistic computational modeling to study the spatial and temporal heterogeneity of cortical astrocyte Ca2+ transients evoked by focal application of ATP in mouse brain slices. Based on experimental results, we tune a single-compartment mathematical model of IP3 -dependent Ca2+ responses in astrocytes and use that model to study response heterogeneity. Using information from the experimental data and the underlying bifurcation structure of our mathematical model, we categorize all astrocyte Ca2+ responses into four general types based on their temporal characteristics: Single-Peak, Multi-Peak, Plateau, and Long-Lasting responses. We find that the distribution of experimentally-recorded response types depends on the location within an astrocyte, with somatic responses dominated by Single-Peak (SP) responses and large and small processes generating more Multi-Peak responses. On the other hand, response kinetics differ more between cells and trials than with location within a given astrocyte. We use the computational model to elucidate possible sources of Ca2+ response variability: (1) temporal dynamics of IP3 , and (2) relative flux rates through Ca2+ channels and pumps. Our model also predicts the effects of blocking Ca2+ channels/pumps; for example, blocking store-operated Ca2+ (SOC) channels in the model eliminates Plateau and Long-Lasting responses (consistent with previous experimental observations). Finally, we propose that observed differences in response type distributions between astrocyte somas and processes can be attributed to systematic differences in IP3 rise durations and Ca2+ flux rates. Keywords: glia, calcium imaging, GPCR, IP3 , computational neuroscience INTRODUCTION Astrocytes are a major glial cell type (Verkhratsky et al., 2012b), playing many key roles in the mammalian brain. Astrocytes are involved in the uptake of neurotransmitters (e.g., glutamate and GABA; Anderson and Swanson, 2000; Zhou and Danbolt, 2013); the release of ′ neuroactive compounds including glutamate, D-Serine, and adenosine 5 -triphosphate (ATP) Frontiers in Systems Neuroscience | www.frontiersin.org 1 October 2017 | Volume 11 | Article 79 24 Diversity of Evoked Astrocyte Ca2+ Dynamics Taheri et al. of Ca2+ between the cell and the extracellular space (ECS), to fit the evoked responses. This approach is more realistic (Dupont and Croisier, 2010) and leads to important modeling results. Third, we examine astrocyte response heterogeneity across trials, cells, and subcompartments (i.e., soma and processes) within each cell, and propose a new classification of responses into four types directly motivated by our bifurcation analysis of our mathematical model (Handy et al., 2017): Single-Peak, Multi-Peak, Plateau, and Long-Lasting responses. Experimentally, we find that SP Ca2+ response kinetics do not vary consistently between different subcompartments of one astrocyte, but rather vary between cells and trials. In contrast, the frequency of occurrences of observed Ca2+ response types varies between astrocyte subcompartments. Using our mathematical model, we explore underlying mechanisms of Ca2+ responses and their heterogeneity by varying IP3 temporal dynamics and Ca2+ channel/pump flux rates. We predict the IP3 time course and composition of flux rates that can reproduce the response type distributions observed experimentally for each astrocyte subcompartment, without requiring feedback-induced oscillations in the IP3 waveform (Politi et al., 2006). Our model provides a tool to study mechanisms and generate predictions that can be applied to other experimental data. (Bezzi et al., 1998; Anderson and Swanson, 2000; Haydon, 2001; Newman, 2003; Liu et al., 2004; Wang et al., 2013); regulation of blood flow (Verkhratsky et al., 2012b; Seidel et al., 2015); and K+ buffering (Wallraf et al., 2006; Wang et al., 2013; Larsen et al., 2014). Many of these functions are regulated in a Ca2+ -dependent manner (Kang et al., 1998; Anderson and Swanson, 2000; Haydon, 2001; Wang et al., 2012, 2013; Khakh and Sofroniew, 2015), though the exact mechanisms are still being investigated. Astrocytes express a variety of functional receptors, mostly metabotropic G-protein-coupled receptors (GPCRs), enabling major communication avenues between neurons and astrocytes. Activation of GPCRs leads to increases in intracellular Ca2+ in astrocytes, primarily through the release of inositol (1, 4, 5)-trisphosphate (IP3 ) into the cytosol, which subsequently opens intracellular Ca2+ stores (Haydon, 2001). Ca2+ increase, in turn, has many effects including, directly or indirectly, driving other transporters and exchangers (e.g., the Na+ /Ca2+ exchanger and Na+ /K+ ATPase pump) (Anderson and Swanson, 2000; Wang et al., 2012) and, possibly, releasing biologically active compounds called gliotransmitters (Bezzi et al., 1998; Haydon, 2001; Liu et al., 2004; Wang et al., 2013). Furthermore, astrocyte Ca2+ elevations can propagate through multiple astrocyte subcompartments or multiple astrocytes (Haydon, 2001), enabling inter- and intra-cellular communication. While Ca2+ activity is a major method of astrocyte signaling, there is little consensus on its downstream effects and how it may encode information (Pasti et al., 1995; Nedergaard and Verkhratsky, 2012; Evans and Blackwell, 2015). Moreover, the extent of Ca2+ response heterogeneity in astrocytes is not wellcharacterized, though temporal and spatial heterogeneity of Ca2+ responses have been addressed by some groups (Verkhratsky and Kettenmann, 1996; Xie et al., 2012; Bonder and McCarthy, 2014; Tang et al., 2015; Jiang et al., 2016). Computational models of IP3 -mediated Ca2+ responses in astrocytes have been used to investigate Ca2+ oscillations (Politi et al., 2006; Ullah et al., 2006; Lavrentovich and Hemkin, 2008; De Pittà et al., 2009; Skupin et al., 2010), influences of astrocytes on neuronal activity (Di Garbo et al., 2007), and the role of intrinsic and extrinsic stochastic events in creating Ca2+ response heterogeneity (Toivari et al., 2011). However, many such Ca2+ models are closed-cell: they disregard fluctuations in total intracellular Ca2+ levels resulting from the activity of plasma membrane channels and pumps. Others models were created based on data from either cell types other than astrocytes or from cultured astrocytes (which are thought to be astrocytelike cells and not represent in vivo astrocytes, Cahoy et al., 2008). Moreover, many models are based on experiments in which agonists were bath-applied to cultured astrocytes, which is far from a physiological stimulation. Here, we present an integrated, experimental and computational study of ATP-evoked Ca2+ transients in astrocytes, with several novel aspects. First, rather than fitting data from bath application of ATP, we use brief focal ATP pulses to astrocytes in acute brain slices that better mimic physiological conditions (Pasti et al., 1995). Second, we use an open-cell mathematical model, accounting for the exchange Frontiers in Systems Neuroscience | www.frontiersin.org MATERIALS AND METHODS Ca2+ Imaging All procedures were in accordance with the NIH Guide for the Care and Use of Laboratory Animals and approved by the University of Utah Institutional Animal Care and Use Committee. The data were obtained from targeted reporter mice (PC::G5-tdT) crossed with the GFAP-CreER mouse line, and thus express the GCaMP5G genetically-encoded Ca2+ indicator in astrocytes (Gee et al., 2014). Cre recombination in GFAP-CreER crosses was induced by a single intraperitoneal injection of tamoxifen in peanut oil (225 mg/kg). All mice were female and were 5–8 weeks old. To extract the brain, the mice were anesthetized in a closed chamber with isoflurane (1.5%) and decapitated. The brains were then rapidly removed and immersed in ice-cold cutting solution that contained 230 mM sucrose, 1 mM KCl, 0.5 mM CaCl2 , 10 mM MgSO4 , 26 mM NaHCO3 , 1.25 mM NaH2 PO4 , 0.04 mM Na-Ascorbate, and 10 mM glucose (pH = 7.2–7.4). Coronal slices (400 µmthick) were cut using a VT1200 Vibratome (Leica Microsystems, Wetzlar, Germany) and transferred to oxygenated artificial cerebrospinal fluid (aCSF) that contained 124 mM NaCl, 2.5 mM KCl, 2 mM CaCl2 , 2 mM MgSO4 , 26 mM NaHCO3 , 1.25 mM NaH2 PO4 , 0.004 mM Na-Ascorbate, and 10 mM glucose (pH = 7.27.4; osmolarity = 310 mOsm). Slices were allowed to recover in oxygenated aCSF at room temperature for 1 h before experiments. During the recordings, the slices were placed in a perfusion chamber and superfused with aCSF gassed with 95% O2 and 5% CO2 at room temperature for the duration of the experiment. To evoke Ca2+ responses, 500 µM ATP (Tocris Bioscience, Bristol, UK, catalog no. 3245; similar concentration used by Otsu et al., 2015; Kim et al., 2016) dissolved in aCSF was delivered locally via a glass pipette (10 2 October 2017 | Volume 11 | Article 79 25 Diversity of Evoked Astrocyte Ca2+ Dynamics Taheri et al. psi; ranging between 16 and 250 ms with exact values specified in each section or figure) using a Picospritzer III (Parker Instrumentation, Chicago, IL) (see Figure 1A). The pipette also included 5 µM Alexa Fluor 594 so that it could be visualized more readily. Two-photon imaging was performed using a Prairie twophoton microscope with a mode-locked Ti:Sapphire laser source emitting 140 fs pulses at an 80 MHz repetition rate with a wavelength adjustable from 690 to 1,040 nm (Chameleon Ultra I; Coherent, Santa Clara, CA). We used laser emission wavelengths of 920 nm to excite GCaMP5G or 1,040 nm to excite tdTomato. A 20 × 0.95 NA water-immersion objective was used for all the Ca2+ images (Olympus, Tokyo, Japan). Astrocyte Ca2+ signaling was recorded at a frame rate of 1 Hz. All imaging was done on cortical astrocytes in the primary somatosensory cortex. in MATLAB and were defined as a !F/F 0 exceeding n standard deviations above the baseline value (where n was 7 for the soma and 6 for the processes) and having a value of at least 40%. Only those ROI traces were included in our analyses that responded to the agonist and that had minimal or no spontaneous activity before the agonist application, in order to avoid mistaking spontaneous Ca2+ activity for evoked activity. To calculate Ca2+ response durations, the trough before (Trough1) the first response peak and after (Trough2) the last peak were automatically detected. Trough1 (or Trough2) was defined as the last (or first) data point, before (or after) the peak that had a value <3.3 standard deviations above the baseline. The time between the two troughs determined the duration of the experimental Ca2+ response. Due to the absence of noise in the mathematical model (described below in section Mathematical Model), simulated responses were determined as the first or last points where the Ca2+ concentration reached a value >40% of the baseline. In Figure 2A, the circles on the traces mark the first and last troughs found using this algorithm for both experimentally recorded and simulated traces. The rise and decay times were calculated from the 10–90% points of the rising and falling phases, respectively, of the Ca2+ responses. To calculate the experimental normalized average Ca2+ trace (Figure 3A), only those SP Ca2+ responses were selected that had a short-duration (<25 s) and were from ROIs which had not responded to an agonist application for at least 3 min. The !F/F 0 was then normalized such that the maximum peak value for each trace was 1 and the peaks of all traces were aligned (hence, removing any variability in latency from the stimulus time). Finally, the average and standard deviation of the traces were calculated. Similar to !F/F 0 traces, the model Ca2+ trace (Figure 3B) was normalized and its baseline adjusted such that the trace has a range of 0–1. We used the same duration calculation method for the normalized traces of the experiments and model: the response onset and offset are the first and last points, respectively, that are >20% the baseline value. Data Analysis The two-photon images were processed and analyzed using custom-written MATLAB (2014a, 2015b; MathWorks, Natick, MA) scripts. Each time-lapse image was first processed with a 3 × 3 median filter (using MATLAB’s medfilt2 command). For each pixel, the median fluorescence of 20 frames preceding the agonist application was used to calculate the baseline fluorescence (F 0 ). Using this baseline, the percent change in fluorescence (100 ∗ !F/F 0 ) of each pixel was calculated throughout the timelapse image. Upon displaying the maximum !F/F 0 projections, ROIs were manually selected for the soma, large processes, and small processes. Large processes (also known as branches, defined in Khakh and Sofroniew, 2015) were chosen at a distance of ∼3–7 µm from the soma perimeter (surrounded with a 2 × 2 µm box). Small processes (also known as branchlets, defined in Khakh and Sofroniew, 2015) were chosen at about 12–20 µm from the soma perimeter (1 × 1 µm box). Finally, the !F/F 0 trace for each selected ROI was filtered with an order-3 onedimensional median filter (using medfilt1 in MATLAB). The Ca2+ response peaks were found using the findpeaks command FIGURE 1 | Astrocyte Ca2+ imaging and observed response variability. (A) (Left) A 12 µm z-stack (centered around the imaging plane) of tdTomato indicating the location of three recorded astrocytes relative to the pipette (shown with arrow), which is also visible since it contains Alexa Fluor 594. The cell numbers correspond to those in (B). (Right) Time series of intracellular Ca2+ responses to focal application of ATP (application duration, 63 ms) from a glass pipette (white dotted line) in a mouse acute brain slice. Examples of selected ROIs are shown for the soma, large processes, and small processes (cyan lines in the 20 s frame). These ROIs correspond to the traces of cell 1, trial 1 in (B). Scale bars, 20 µm. (B) Example traces of simultaneous responses from different subcompartments of three cells (same cells shown in A), in two different ATP application trials. Frontiers in Systems Neuroscience | www.frontiersin.org 3 October 2017 | Volume 11 | Article 79 26 Diversity of Evoked Astrocyte Ca2+ Dynamics Taheri et al. FIGURE 2 | Astrocyte Ca2+ response types and kinetics. (A) Four categories of astrocyte Ca2+ responses observed experimentally (first and second columns) and in the model (third column). Experimental data show astrocyte Ca2+ responses to a focal application of ATP. Each of the two experimental columns contain one example of each response type to show the variability within each response category. The model traces are chosen to reproduce responses similar to the first column. For parameter values, see Figure 4A. The circles for each trace show the response onset and offset (calculation described in section Materials and Methods). Horizontal scale bars, 10 s; vertical scale bars, 100% !F/F for the experiment and 0.5 µM [Ca2+ ]cyt for the model. (B) Flow chart summarizing the algorithm for Ca2+ response type categorization (details in section Materials and Methods). (C) The rise times, decay times, durations, and amplitudes of SP responses in the soma and a large process of the same astrocyte (N = 9, stimulus durations 16–250 ms), paired with one another. Only amplitude results were significantly different (*p < 0.05; see text), suggesting that there is more variability in SP kinetics between cells and trials than between astrocyte subcompartments. (D) The distribution of observed Ca2+ response types varies between the somas, large processes, and small processes (N = 3 mice, 3 trials per mouse, 3–5 cells per trial, up to 10 ROIs per cell; only responsive cells with minimal spontaneous activity were included here; stimulus durations 30–63 ms). Mathematical Model mice). IP3 then binds to IP3 receptors (IP3 Rs) on the endoplasmic reticulum (ER) membrane, which consequently opens IP3 R channels and allows Ca2+ to exit the ER and enter the cytosol. When there is significant depletion of Ca2+ from the ER, storeoperated Ca2+ (SOC) channels are activated, and allow for additional Ca2+ to flow from the ECS into the cytosol. Sustained, elevated levels of cytosolic Ca2+ inactivate IP3 Rs, and activate sarco/endoplasmic reticulum Ca2+ ATPase (SERCA) pumps and plasma membrane Ca2+ ATPase (PMCA) pumps, which then transfer the cytosolic Ca2+ back to the ER and ECS, respectively. After the degradation of IP3 , these pumps and channels return the system to steady state. Unlike many previous studies on astrocyte Ca2+ dynamics (Ullah et al., 2006; De Pittà et al., 2009), our model is an opencell model in which the total intracellular Ca2+ levels can change. Additionally, in this work, we directly control the time course of Because fluorescent signals from GCaMP5G in this animal model are both linear and very fast compared with evoked Ca2+ transients (Gee et al., 2014), model results are directly comparable with experimental data, particularly if we compare them independent of response amplitude and latency. For this comparison, we model astrocyte Ca2+ activity in a single compartment, which may represent any one functional subcompartment of an astrocyte. Its parameters can be adjusted to a specific dataset (e.g., the soma or large processes, as we do in Figure 7A). Figure 3B shows a simplified schematic of the Ca2+ activity components in our model: activation of a metabotropic Gprotein-coupled receptor (GPCR; e.g., P2Y receptors, commonly found on astrocytes) leads to IP3 production (We ignore the P2X ionotropic receptor, not found in cortical astrocytes from Frontiers in Systems Neuroscience | www.frontiersin.org 4 October 2017 | Volume 11 | Article 79 27 Diversity of Evoked Astrocyte Ca2+ Dynamics Taheri et al. FIGURE 3 | Short-duration SP Ca2+ traces in experiments and the model. (A) Normalized average short-duration (< 25 s) SP Ca2+ traces of the soma (N = 8, mean ± standard deviation, stimulus durations 16–250 ms). The response amplitude and latency were ignored by normalizing the amplitudes and aligning the response peaks. (B) Simplified schematic of an astrocyte and its major Ca2+ components incorporated in the model (except for GPCR dynamics, which is replaced with IP3 as the input). Additionally, leak terms between the cytosol and extracellular space (JECS_add ) and between the cytosol and the ER (JER_leak ) are included in the model. Arrows show the direction of Ca2+ flux. The model has only one compartment, tuned to represent data from different astrocyte subcompartments (see section Variability between Astrocyte Subcompartments: Model). (C) Normalized SP response in the model, fitted to match the experimental trace in (A). IP3 parameters (A, drise , r rise , ddec ): 0.25, 12, 0.002, 40. The experimental and model Ca2+ rise times (calculated between the black star markers), decay times (between the black circles), and durations (between the first and last red circles) are as follows: 2.90, 3.90, and 14.7 s (for experiments), and 3.19, 4.38, and 14.05 s (for the model). (D) Simulated IP3 dynamics and Ca2+ fluxes corresponding to (C). IP3 dynamics, treating IP3 as the effective representation of the agonist influence on the cell. This allows us to make predictions as to how the IP3 time course affects the resulting Ca2+ responses. The general shape of IP3 time course is based on previous experimental and modeling work (described below). Our resulting model consists of a system of three differential equations. Two of those equations model the changes in cytosolic Ca2+ concentration, c, and total intracellular Ca2+ concentration, ctot , as follows: the leak between the ER and the cytosol, J SERCA captures the flux due to the SERCA pump, J ECS_add is the leak between the ECS and the cytosol as well as additional plasma membrane fluxes not explicitly modeled, J PMCA represents the PMCA pump, and J SOC is the flux through the SOC channels. Further, γ is ratio of the volume of the cytosol to the volume of the ER, and δ is the ratio of membrane transport to ER transport. The remaining differential equation tracks the deactivation rate of the IP3 R, dh h∞ (c, p) − h = , dt τh (c, p) ! " # $ dc = JIP3R c, cER , p + JER_leak (c, cER ) − JSERCA (c) dt ! $ + δ JECS_add (c) − JPMCA (c) + JSOC (cER ) , (1) dctot = δ[JECS_add (c) − JPMCA (c) + JSOC (cER )], dt where h∞ is the equilibrium binding probability and τh is the time constant. We define these quantities more specifically in the following subsections. (2) IP3 Receptor Model and the concentration in the ER is given by cER = (ctot – c)γ . We will also denote IP3 concentration as p. The Ji ’s in these equations represent fluxes through the pumps and channels that are shown schematically in Figure 3B. J IP3R represents the flux from the ER to the cytosol through the IP3 R channel, J ER_leak is Frontiers in Systems Neuroscience | www.frontiersin.org (3) We use the Li-Rinzel IP3 receptor (IP3 R) model to capture the Ca2+ dynamics through the IP3 R channel (Li and Rinzel, 1996). Their model accounts for three binding sites on the receptor: a binding site for activating Ca2+ , n(c), deactivating Ca2+ , h(c,p), and IP3 , m(p). They take the fast variables, the 5 October 2017 | Volume 11 | Article 79 28 Diversity of Evoked Astrocyte Ca2+ Dynamics Taheri et al. binding of activating Ca2+ and IP3 , to be in quasi-steady state, and they model the slow variable, the binding of deactivating Ca2+ , explicitly. When open, the flux through the channel is determined by the concentration gradient of Ca2+ between the cytosol and the ER. In total, the equations governing this model are ! "3 JIP3R = vIP3R m∞ p n∞ (c)3 h3 (cER − c) , TABLE 1 | Model parameters. Parameter Description where p c , n∞ (c) = , p + d1 c + d5 ! " ! " Q2 (p) 1 h∞ c, p = , τh c, p = , Q2 (p) + c a2 (Q2 (p) + c) # $ ! " p + d1 Q2 p = d2 , p + d3 ! " m∞ p = and the dynamics of h are governed by Equation (3). The values for the constants can be found in Table 1. This model from Li and Rinzel is a simplification of the one by De Young and Keizer (1992), which was based on data collected on Purkinje neurons. While the structure of the IP3 receptor found in astrocytes is most likely very similar to ones found here, the rate constants determining the open probability of receptor are likely to be different. However, little experimental data is available for comparison, and current astrocyte models use constants provided by Li and Rinzel (Ullah et al., 2006; Lavrentovich and Hemkin, 2008; De Pittà et al., 2009). The fast and slow components of this receptor result in excitable behavior. When enough IP3 enters the system, Ca2+ is released from ER and has a positive feedback onto the receptor, allowing for additional Ca2+ release, before the deactivating components of the receptor are bound with Ca2+ . Frontiers in Systems Neuroscience | www.frontiersin.org 0.222 s−1 νER_leak Cytosol to ER leak 0.002 s−1 νin Rate of leak into Cytosol from Plasma Membrane 0.05 µM s−1 k out Rate of leak out of Cytosol from Plasma Membrane 1.2 s−1 νSERCA Max SERCA Flux 0.9 µM s−1 k SERCA Half-Saturation for SERCA 0.1 µM νPMCA Max PMCA Flux 10 µM s−1 k PMCA Half-Saturation for PMCA 2.5 µM νSOC Max SOC channels Flux 1.57 µM s−1 k SOC Half-Saturation for SOC channels 90 µM δ Scale Factor (ratio of membrane transport to ER transport) 0.2 d1 Dissociation constant for IP3 0.13 µM d2 Dissociation constant for Ca2+ inhibition 1.049 µM d3 Receptor dissociation constant for IP3 0.9434 µM d5 Ca2+ activation constant 0.08234 µM a2 Ca2+ inhibition constant 0.04 µM-1 s-1 r rise Rate of Exponential Growth [0.002–12] s−1 ddecay Duration of IP3 decline [15–220] s drise Duration of IP3 increase [1–41] s A Max amplitude of IP3 transient [0.2–0.9] µM JSOC (cER ) = vSOC k2SOC . 2 2 kSOC + cER Additional Membrane Fluxes We fixed the Hill coefficient for this model to be 1.75, as used in Cao et al. (2014), and the default max flow rate (vSERCA ) and the dissociation constant (kSERCA ) were used in De Pittà et al. (2009), and listed in Table 1. While the SERCA pump is found on the ER, the PMCA pump is found on the plasma membrane and has the ability to pump 2+ Ca from the cytosol into the ECS. Both pumps require ATP to function, and are believed to have similar dynamics. As a result, we chose to use the following equation for the PMCA pump, as used in Croisier et al. (2013): c2 + k2PMCA 5.4054 Max IP3 Receptor Flux Although the molecular mechanism of SOC channels in astrocytes is actively debated, it has been shown that they open when ER Ca2+ is depleted, letting Ca2+ flow from the ECS into the cytosol (Verkhratsky et al., 2012b). To model this mathematically, we follow Croisier et al. (2013) and set c1.75 . c1.75 + k1.75 SERCA JPMCA (c) = vPMCA (Cyt vol)/(ER vol) νIP3R SOC Channels In the paper by MacLennan et al. (1997), Ca2+ flow associated with the SERCA pump was shown to depend sigmoidally on the intracellular Ca2+ concentration. We assume that the dependence on Ca2+ concentration has a similar form in astrocytes and can be modeled with the following Hill function, c2 γ γ is from Ullah et al. (2006), kserca and vserca are from De Pittà et al. (2009), νin is from Lavrentovich and Hemkin (2008), νsoc are found in Croisier et al. (2013), and d1 , d2 , d3 , and d5 are from the model developed by De Young and Keizer (1992). Values for νIP3R , νleak , νPMCA , kout , kPMCA , kSOC , δ, and a2 were fitted to our data in section Single-Peak Responses and Resulting Model and values for νSOC , νPMCA , νSERCA were further tuned to our data in section Variability between Astrocyte Subcompartments: Model. SERCA and PMCA Pumps JSERCA (c) = vSERCA Value/Units The model also accounts for an IP3 R-independent Ca2+ leak term between the cytosol and the ER. This term is driven by the concentration gradient between the pools of Ca2+ and are a linear approximation of various channels and pumps not modeled explicitly, and is given by the following equation JER_leak (c, cER ) = vER_leak (cER − c) . We also account for additional fluxes across the plasma membrane with the equation . JECS_add (c) = vin − kout c, 6 October 2017 | Volume 11 | Article 79 29 Diversity of Evoked Astrocyte Ca2+ Dynamics Taheri et al. where the first term represents the constant leak of Ca2+ into the cytosol from the ECS and second term captures additional Ca2+ extrusion not explicitly modeled, such as Ca2+ extrusion from the Na+ /Ca2+ exchanger (Höfer et al., 2002; Ullah et al., 2006; Keener and Sneyd, 2009; Verkhratsky et al., 2012a). glutamate applications in cultured astrocytes by Pasti et al. (1995) and accounted for the observed discrepancies by adjusting model parameters, to obtain a reasonable range of IP3 amplitudes, total durations, rise durations, and decay durations. The complete set of IP3 parameters we chose to use in our model is as follows: A = (0.2, 0.375, 0.55, 0.725, 0.9), rrise = (0.002, 0.04, 0.07, 0.09, 0.12, 0.15, 0.3, 0.44, 0.8, 1, 1.6, 12), ddec = (15, 56, 97, 138, 179, 220), and drise = (1, 11, 21, 31, 41) (resulting in a total of 600 IP3 traces; only a subset of rrise value were used for each drise -value, in order to avoid creating repetitions in the IP3 time courses). All 600 IP3 traces were used throughout this paper, unless otherwise noted in the figure captions and text. IP3 Dynamics As discussed earlier, IP3 is produced as a result of GPCR activation. After diffusing in the cytosol and reacting with IP3 Rs, it diffuses through intercellular gap junctions and/or degrades (Höfer et al., 2002). This biochemical pathway has been studied in astrocytes (Fiacco and McCarthy, 2004; Haydon and Carmignoto, 2006; Petravicz et al., 2008) and included in complex biophysical models (Di Garbo et al., 2007; De Pittà et al., 2009). However, rather than including the production and degradation of IP3 explicitly, we opted to treat IP3 waveforms as inputs to our model, generated by a simple equation and ignoring possible feedback of Ca2+ on IP3 (Höfer et al., 2002; Politi et al., 2006), for two reasons. First, this simpler approach made it easier for us to explore how IP3 kinetics affect the shape of Ca2+ events in the absence of feedback. Second, because IP3 production and degradation have not been measured in astrocytes, treating IP3 as an input reduces the number of free parameters in the model substantially. Our simple model makes the following assumptions: following a pulse of ATP, IP3 exponentially saturates to a level denoted as s∞ , and then exponentially decays back to steady state: p(t) = 0 t < t∗ ∗ s∞ · (1 − e−rrise (t−t ) ) t ∗ ≤ t < t ∗ + drise ⎩ ∗ A · e−rdec ·(t−[t +drise ]) t ∗ + drise ≤ t ⎧ ⎨ Parameter Fitting While the mechanisms behind the Ca2+ fluxes, and hence the mathematical form of these fluxes, are similar between cell types, it is known that the specific dynamics and time scales of these channels can vary. Further, many of these channels and pumps have not been investigated in astrocytes with sufficient detail to capture specific parameter values. As a result, we fitted several parameters (vip3r , vleak , vpmca , kout , kpmca , ksoc , δ, a2 ) of our model in order to match the kinetics of an average, shortduration (<25 s) experimental Ca2+ transient (Figure 3A). We first established a reasonable IP3 transient to act as a driving force for short-duration Ca2+ transients, and then fitted parameters accordingly by hand. In addition to fitting the experimental data, we also required the model to have a realistic resting astrocyte Ca2+ concentration in the cytosol (∼0.1 µM) and the ER (∼200 µM) (Verkhratsky and Butt, 2007). All fitted parameters are similar in magnitudes as those found in the literature and the complete list of parameter values can be found in Table 1. The fitted, normalized model simulation can be seen in Figure 3B. (4) where s∞ = A 1 − e−rrise ·drise , rdec = − 1 ddec log $ % 0.005 , A Monte Carlo Simulations t ∗ is the time of stimulus, A is the max amplitude, rrise and rdec are the rate of rise and decay respectively, and drise and ddec are the durations (0–100% and vice-versa) of the rising and decaying phases, respectively. To determine a reasonable range of the IP3 parameters A, rrise , ddec , and drise for use in our simulations, we examined and compared data from previous experimental (Pasti et al., 1995; Tanimura et al., 2009; Nezu et al., 2010) and modeling (De Pittà et al., 2009) studies. Tanimura et al. (2009) and Nezu et al. (2010) had imaged IP3 dynamics during ATP-induced (bath applied) Ca2+ responses of COS-7 and HSY-EA1 cells. While they found differences in peak IP3 concentrations within one cell type, between the two cell types, and with different ATP concentrations (Tanimura et al., 2009; Nezu et al., 2010), we used their results and IP3 traces to estimate a range of IP3 amplitudes, rise durations, and decay durations. We also ran simulations using the detailed GPCR model developed by De Pittà et al. (2009) to generate IP3 dynamics and Ca2+ responses to different glutamate concentrations applied for short durations (<5 s). We compared these results with results from local, brief (<100 ms) Frontiers in Systems Neuroscience | www.frontiersin.org To account for experimental variability between astrocyte subcompartments (soma, small, and large processes; Figure 2D), we considered a broader parameter space than what is found in Table 1. We ran 30 simulations for each IP3 transient (each IP3 parameter set in Table 1), choosing vpmca , vserca , and vsoc from a uniform distribution centered at the default value found in Table 1, with a maximum set at 150% of this value and minimum set at 50%. The system was first allowed to equilibrate to steady state with these new parameters, and then the IP3 stimulus was applied and Ca2+ response recorded. Once this data set was created, we separated the three dimensional parameter space into 27 subspaces (based on the values of vpmca , vserca , and vsoc ), and examined the distribution of Ca2+ response types in each subspace. We also examined these distributions while limiting the ranges of IP3 drise parameters, which were divided into increasingly larger ranges: drise -values from 1 to 11, 1 to 21, 1 to 31, and 1 to 41 s (the full range of drise , as listed in Table 1). After examining the response type distributions, Figure 7 was created by choosing the parameter subspace, or subspaces, that best matched the experimental data in Figure 2D. 7 October 2017 | Volume 11 | Article 79 30 Diversity of Evoked Astrocyte Ca2+ Dynamics Taheri et al. Categorization of Ca2+ Response Types large enough to be a MP response, we checked if its height was >15% of the adjacent peak height (rather than >5% as for the model traces). After examining the astrocyte literature, our experimental data, and the model simulations, we categorized all Ca2+ responses into four major categories based on their duration and shape: Single-Peak (SP), Plateau (PL), Multi-Peak (MP), and LongLasting (LL). A general definition for each response type is as follows (with corresponding examples and flowchart in Figures 2A,B): Statistics A signal that has more than one peak in succession (for experimental data, ≤16 s gap between each consecutive response), with at least one trough reaching <50% of the maximum adjacent peak height. Only those peaks are considered that have heights >5% of the adjacent peak height. The paired sample t-test was used to compare SP kinetics between the soma and large processes. Pearson’s chi-square test was used to compare the distributions of experimental Ca2+ response types recorded from the soma, large processes, and small processes. Fisher’s exact 2-tail test was used to compare the frequencies of specific Ca2+ response types recorded in these subcompartments. The Kolmogorov-Smirnov test was used to compare the distribution of Ca2+ durations observed experimentally and generated in the model using different subsets of IP3 transient parameters. Long-Lasting RESULTS Multi-Peak A signal that stays elevated continuously, without returning close to baseline (i.e., <15% of the maximum adjacent height) or having additional peaks (with troughs reaching <50% of the maximum adjacent height) for more than 70 s. When there are multiple peaks, if the signal remains elevated for >70 s between any two adjacent troughs, it will be considered a LL response; otherwise, it will be a MP response. Experimentally-Observed Variability in Evoked Ca2+ Responses We applied brief (16–250 ms) local pulses of ATP and examined Ca2+ responses in three subcompartments of each imaged astrocyte: the soma, large processes, and small processes (Figure 1A; regions of interest are marked for one cell at t = 20 s). In Figure 1B, we plot Ca2+ traces vs. time for the three cells, with cell 1 being the astrocyte from the time-lapse image in Figure 1A. In such data, we observed variability among simultaneously recorded responses of different cells, simultaneous responses of different subcompartments within a given cell, and between trials in a given subcompartment. In the first trial (black traces), simultaneous responses of three cells differed substantially, and they each displayed differences among their subcompartments (soma, large processes, and small processes). In the second trial (gray traces; with 5 min of rest between trials and the same agonist concentration, agonist application duration, and pipette location as in trial 1), cell 1’s soma and large process (same process as in trial 1) responded with a smaller amplitude, duration, and latency. On the other hand, the soma and the same large process of cell 2 failed to respond to the second stimulus. In contrast, cell 3 responded consistently between the two trials. Such variability between trials, subcompartments, and cells was not uncommon in our experiments. Even when agonists are bath applied and the spatial variability of the agonist concentration is smaller than with our ATP pulse experiments, responses in different cells or astrocyte subcompartments vary greatly in their shapes and durations (e.g., Xie et al., 2012). Our goal in this paper is to characterize these different forms of response variability (summarized in Table 2) in more detail, to develop a mathematical model that generates the variety of observed Ca2+ responses (i.e., with a variety of temporal features, similar to those seen experimentally), and to use the model to examine the sources of these forms of response variability (Table 2), particularly the spatial variability among different astrocyte subcompartments. In order to quantify diversity in astrocyte Ca2+ signaling, we divided all Ca2+ responses into four main categories according to their shape and duration: Single-Peak (SP), Plateau (PL), Single-Peak A signal with one clear peak, without any subsequent major oscillations or bumps. A major oscillation/bump is one with a sufficiently large height (>5% of the adjacent peak height) or sufficiently long duration (lasting >50% of the main peak’s duration). Plateau A signal with one main peak and subsequent bump or oscillation that either has a sufficiently long duration (>50% of the main peak’s duration) or is elevated with its troughs >50% of the peak heights. In our mathematical model, when changing the IP3 time course, the simulated Ca2+ responses transition between response types as a continuum. As a result, to automatically determine the cytosolic Ca2+ response types of the mathematical model simulations, we developed an extensive MATLAB script to implement the classification procedure described above (more details and the MATLAB scripts can be found in the ModelDB database; Hines et al., 2004; http://senselab.med.yale. edu/modeldb/default.asp; Model no. 189344). It is also worth noting that Ca2+ responses that had amplitudes too small to be detected experimentally (<0.4 µM) or had amplitudes too high to be biologically reasonable (>3.5 µM) were not included in our analyses. For experimental traces, Ca2+ response types were determined using a similar algorithm; however, due to inherent noise in experimental signals, the algorithm was manually implemented rather than using the MATLAB script. Moreover, the following step was modified to avoid mistaking inherent experimental noise for additional peaks in the Ca2+ traces: in the case of a second peak or oscillation, to determine if the peak was Frontiers in Systems Neuroscience | www.frontiersin.org 8 October 2017 | Volume 11 | Article 79 31 Diversity of Evoked Astrocyte Ca2+ Dynamics Taheri et al. TABLE 2 | Experimentally observed variability in Ca2+ dynamics. Forms of Ca2+ response variability different trials (cf. the black and gray traces in Figure 1B). For two reasons, we believe that variability in the spatiotemporal synthesis and degradation of IP3 are the main contributors to trial to trial variability. First, it seems unlikely that GPCR and Ca2+ channel/pump properties change on the short time scale of our experiments (<20 min). Second, IP3 uncaging experiments provide direct evidence that IP3 is the sole source of trial to trial variability. Fiacco and McCarthy (2004) found that astrocyte Ca2+ response kinetics to multiple IP3 uncaging trials were consistent in any one cell, suggesting that trial to trial variability in agonist application experiments stems mainly from factors upstream of the IP3 waveform. Interestingly, Fiacco and McCarthy (2004) did observe variability in response duration from cell to cell in their IP3 uncaging experiments. This finding supports the hypothesis that different cells are likely to have inherently diverse properties downstream of IP3 dynamics. Compatible with this hypothesis, cell to cell variability is also seen in our data (Figure 1B) and in response to bath-applied agonists (e.g., Xie et al., 2012). We speculate that cell to cell variability is dominated by different distributions and properties of Ca2+ channels and pumps, in addition to differences in GPCR expression levels (and subsequent differences in IP3 kinetics). We explore this idea in the simulations described later. Likely source of variability IP3 dynamics* Ca2+ channels/pumps Among different astrocytes Yes Yes Among different subcompartments of one astrocyte Yes Yes From trial to trial (same ROI and agonist amount) Yes No *The differences in IP3 dynamics may be a consequence of either differences in GPCR expression or functional properties, spatial diffusion of IP3 and consequent interactions, or stochasticity in the production and degradation of IP3 , downstream of GPCR activation. Multi-Peak (MP), and Long-Lasting (LL). This response type categorization is based on our observed experimental data, the astrocyte literature (Verkhratsky and Kettenmann, 1996; Xie et al., 2012; Bonder and McCarthy, 2014), and the mathematical structures underlying our model dynamics, described in section Model Verification and in Handy et al. (2017). In Figure 2A (first and second columns), we show example traces of cytosolic Ca2+ elevations elicited by 30–63 ms applications of ATP. For each response type, two examples of experimental recordings are shown in order to illustrate the variability of observed Ca2+ responses within each response category. In the third column of Figure 2A, we show a stereotypical response for each class, generated by our model, chosen to match the experimental responses in the first column (for model details see section Model Verification). Figure 2B shows a flow chart summarizing the categorization algorithm (see section Materials and Methods for additional details on response type definitions). A similar classification of responses was proposed by Xie et al. (2012), where astrocyte Ca2+ responses to agonist-bath applications (with durations of tens of seconds) were categorized into three classes based only on the response shape. We also categorized responses based on duration, particularly separating those responses that lasted 70 s or more (described in Khakh and Sofroniew, 2015) into a separate category. We altered some definitions of the three response types proposed by Xie et al. (2012) in order to incorporate this fourth response type, as well as to describe the variability not only in our experimental data, but also in our model simulations using the same algorithm. Variability between Astrocyte Subcompartments Previous studies (e.g., Tang et al., 2015) have reported differences in Ca2+ kinetics between the astrocyte soma and processes in response to neural stimulation. Using ATP application, we investigated whether the kinetics of SP responses (the most commonly observed response type in our experiments, as described below) varied between the soma and large processes of individual astrocytes. To control for trial to trial and cell to cell variability, we examined pairs of somas and large processes of the same astrocyte that responded to the same trial of ATP application with a SP response (N = 9). While we found that the Ca2+ amplitude is greater in large processes than in somas (paired t-test, p = 0.023), we found no significant differences between the SP durations (p = 0.059), rise times (p = 0.586), or decay times (p = 0.367) (Figure 2C). The difference in amplitudes could be a result of differences in ROI sizes and consequent !F/F0 calculations, which involve averaging over the ROI area, as opposed to intrinsic differences in response kinetics. While we did not find significant differences between the paired SP response kinetics of the somas and large processes, we did note differences in the likelihood of observing certain types of responses in each of these subcompartments. In Figure 2D, we plotted the distribution of response types observed in the somas, large processes, and small processes in our experiments over nine trials (three mice). Our results indicate that SP responses are the most common response type in the soma, while MP responses are rarest. However, the finer the astrocyte processes, the more likely they are to exhibit MP responses instead of SP transients (Fisher’s exact 2-tail test, between the MP responses of somas and large processes p = 0.026, and of the somas and small processes p = 0.0016). Further, PL and LL responses were observed at a lower rate in all three subcompartments. The following are the Forms of Ca2+ Response Variability and their Sources Having established the presence of a variety of responses in the data, we next sought mechanisms that underlie the observed evoked Ca2+ response variability under different contexts (summarized in Table 2). In subsequent sections, these mechanisms will be explored in detail in our mathematical model. Trial to Trial and Cell to Cell Variability Our results show that the same recording site (region of interest, ROI) can respond differently to identical agonist pulses in Frontiers in Systems Neuroscience | www.frontiersin.org 9 October 2017 | Volume 11 | Article 79 32 Diversity of Evoked Astrocyte Ca2+ Dynamics Taheri et al. percentages of observed response types (in order, SP, PL, MP, LL) in each astrocyte subcompartment: 63.64, 18.18, 0.0, 18.18% (somas); 47.06, 13.73, 33.33, 5.88% (large processes); 31.67, 10.0, 51.67, 6.67% (small processes). In comparing the overall response type distributions, we found the largest differences to be between the distributions of the somas and small processes (Pearson’s chi-squared test, p = 0.016). The variability in response type distributions between subcompartments could be the result of differences in both the IP3 dynamics and the functional properties of Ca2+ channels and pumps. As above, the differences in IP3 dynamics within different astrocyte subcompartments could arise from differences in GPCR expression levels or functional properties, differences in agonist binding probability or IP3 diffusion within or between cells (due to differences in subcompartment size and shape), or differences in regulatory mechanisms downstream of GPCR activation. Moreover, differences in functional properties of Ca2+ channels and pumps in different astrocyte subcompartments could also arise from differences in subcompartment size and shape, differences in expression levels and spatial distributions of these Ca2+ components, or simply different activity levels of these components. To summarize, we suggest that the mechanisms behind the observed variability in all three cases (Table 2) stem mainly from differences in one or both of the following: (1) IP3 dynamics, (2) Ca2+ fluxes through various Ca2+ handling mechanisms (e.g., SOC channels). We will next examine the plausibility and consequences of each of these mechanisms in our mathematical model (some other potential sources of variability, e.g., the volume ratio of the cytosol to the ER, have also been examined in our model (Handy et al., 2017) but were found to not have a major effect on Ca2+ responses). variable and noisy, we chose to fit our model parameters to the average trace from the soma (Figure 3A). Because short-duration SP responses are the briefest observed astrocyte Ca2+ responses, we used an IP3 input within the lower end of the parameter range in Table 1 (specific values in Figure 3 caption) to generate a SP response in the model. The fitted SP simulation, along with the corresponding IP3 dynamics and Ca2+ fluxes driving this response, is shown in Figures 3C,D. Before the IP3 stimulus, the simulated system is at steady-state, with the Ca2+ fluxes governed by the ER leak and SERCA pump balanced. When IP3 is released into the cytosol, the IP3 R becomes activated and Ca2+ flows from the ER into the cytosol. Depletion of ER Ca2+ causes SOC channel activation, though this flux remains small for the duration of this response. The increase in cytosolic Ca2+ also quickly activates the SERCA pumps, allowing for Ca2+ to be pumped back into the ER. As the cytosolic Ca2+ concentration continues to rise, the PMCA pumps also become activated and some Ca2+ is lost to the ECS. Additional Ca2+ is also released into the ECS due to the ECS_add term. Moreover, before cytosolic IP3 begins to degrade, the Ca2+ flux through the IP3 R decreases as a result of negative feedback from elevated levels of cytosolic Ca2+ . Together, the SERCA pump, PMCA pump, ECS_add, and the deactivation of the IP3 R are able to slow the net change of Ca2+ in the cytosol. As IP3 is degraded and removed from the system, the IP3 R begins to deactivate more rapidly, and the pumps are able to return the system to equilibrium. Even when the Ca2+ response measured in the cytosol has returned to baseline levels, the pumps and channels continue to work at a low level to restore the system back to prestimulus equilibrium. In the model, the process of refilling the ER to pre-stimulus levels is on the order of ∼10 min. Heterogeneity of Model Ca2+ Responses As discussed in section Forms of Ca2+ Response Variability and their Sources, IP3 stochasticity is likely a major source of all three forms of variability in evoked Ca2+ responses considered in this manuscript (see Table 2). To explore this issue, we changed IP3 parameters (A, drise , rrise , and rdec ; linear spacing) over a biologically plausible range of values (see section Materials and Methods and Table 1 for details), while other model parameters were fixed at the default values. We found that this range of IP3 kinetics was sufficient to reproduce our experimentallyobserved Ca2+ responses, compatible with our bifurcation-based classification scheme (for details on how each response type arises from a given IP3 waveform, refer to Handy et al., 2017). Examples of the response types are shown in Figure 4A on the right (solid lines, repeated from Figure 2A), along with the underlying IP3 waveforms (dashed lines). Furthermore, Figure 4A shows a color-coded scatter plot of the four response types generated by this parameter search, plotted vs. response duration and the total area under the simulated Ca2+ response curve (i.e., the total Ca2+ amount). For comparison, in Figure 4B we show a similar scatter plot for the experimental data (N = 71). Model and experimental responses are qualitatively similar, dominated by SP responses for durations <22 s; MP or LL responses for durations >70 s; and a mix of SP, MP, and PL responses for intermediate durations. For both the Model Verification Our single-compartment model includes the Ca2+ handling mechanisms shown in Figure 3B, with IP3 (rather than the GPCR agonist) as the direct input. It consists of three ordinary differential equations describing changes in cytosolic Ca2+ levels, total intracellular Ca2+ levels, and the inactivation variable of the IP3 receptor (IP3 R). See section Materials and Methods for details and Handy et al. (2017) for a general bifurcation analysis. Here, we fit the model to experimental data in order to understand the factors that contribute to response variability in response to brief pulses of ATP. Single-Peak Responses and Resulting Model To verify our model of astrocyte Ca2+ dynamics, we first ensured that our model matches experimentally observed Ca2+ kinetics during a typical SP Ca2+ response, which is the most common astrocyte Ca2+ response type when stimulated with a brief agonist pulse (Figure 2D). We examined experimental shortduration (<25 s) SP Ca2+ responses and found that the average of such a response for the somas (N = 8; Figure 3A) was similar in kinetics to that of the processes (N = 13, data not shown), in agreement with the results discussed in Figure 2C. Given this similarity, and the fact that the somatic responses were less Frontiers in Systems Neuroscience | www.frontiersin.org 10 October 2017 | Volume 11 | Article 79 33 Diversity of Evoked Astrocyte Ca2+ Dynamics Taheri et al. durations are significantly different (Supplementary Figure 1, Kolmogorov-Smirnov test, p = 8e-6), with longer PL and LL responses overrepresented in simulations. Hypothesizing that IP3 kinetics may influence the distributions of model response durations, we explored the effects of restricting the IP3 waveform, with the goal of matching the distribution of experimental response durations without eliminating altogether the longduration responses of Figure 4A. Of the many manipulations we attempted (i.e., limiting IP3 total duration, decay duration, rise duration, and amplitude), we only succeeded in this goal by limiting the IP3 rise duration (drise ) from Equation (4) to <22 s (Kolmogorov-Smirnov test, p = 0.2, max duration = 83 s, Supplementary Figure 1). Thus, the model suggests that shorter IP3 rise durations are more common in the experimental data set than the full range of model IP3 parameters originally considered. Because response durations and types are clearly correlated (Figure 4), we might expect that restricting drise will alter the distributions of response types as well. Figure 5A shows the distribution of modeled response types for the default range of IP3 parameters (left panel) and for restricted values of rise durations (right panel, drise < 22 s). As expected, the short-drise histogram exhibits many more SP responses and substantially fewer longer, more complex responses. This effect is compared with experimental data in section Variability between Astrocyte Subcompartments: Model. Modeled Response Types are Sensitive to the IP3 Waveform FIGURE 4 | Simulated and measured Ca2+ responses. (A) Simulated total Ca2+ amount (the area under the [Ca2+ ]cyt curve) vs. Ca2+ duration as defined in section Materials and Methods. Model responses were generated by choosing IP3 parameter sets as specified in section Materials and Methods and Table 1. Response types are indicated as shown. The right panel repeats example Ca2+ traces from Figure 2A, column 3 with their respective IP3 time courses added. The IP3 parameters are as follows (in order, A, drise , r rise , ddec ): 0.2, 10, 0.2, 90 (SP); 0.375, 34, 0.002, 110 (PL); 0.26, 41, 0.15, 200 (MP); 0.6, 39, 0.002, 220 (LL). Horizontal scale bars, 10 s; vertical scale bars, 0.5 µM. (B) Experimentally measured Ca2+ responses (y-axis shows area under the !F/F trace; N = 19 somas and 52 large processes from a total of 3 mice, 3–6 trials per mouse, 2–5 cells per trial; 16–250 ms stimulus durations) are similar to modeled responses, but more sparsely and non-uniformly distributed. One outlier point was omitted from (B), at a duration and total fluorescence of about 363. Figure 5B shows the total Ca2+ amount vs. the total IP3 amount (i.e., the areas under the Ca2+ and IP3 traces, respectively). Data points are color-coded by response type, and the symbol type denotes the IP3 rise durations (squares: drise < 22 s; circles: drise > 22 s, as described in section Materials and Methods). We draw several conclusions from Figure 5B and associated results. First, the total amounts of IP3 and Ca2+ are strongly positively correlated. In contrast, we found no correlation between other features of the IP3 and Ca2+ waveforms: IP3 rise duration, total duration, decay duration, amplitude, or the ratio of amplitude over duration did not correlate with Ca2+ total amount, duration, or amplitude. Second, although there is some correlation between total IP3 amount and Ca2+ response type, no two features or parameters of the IP3 waveform strictly predict the response type. Third, for intermediate values of total IP3 and Ca2+ amounts, the response type is particularly sensitive to small changes in the IP3 waveform (see the Figure 5B inset and example waveforms in Figure 5C; for more details on the model’s sensitivity to various parameters, refer to Handy et al., 2017). This finding suggests that experimental trial to trial variability discussed earlier could be caused by small trial to trial changes in the IP3 waveform. model and experiments, duration and total area under the curve (i.e., total Ca2+ amount for model, and total fluorescence for experiments) are positively correlated. Additionally, the range of Ca2+ response durations between simulations and experiments are similar. These similarities suggest that the range of our model parameters and selected IP3 kinetics are biologically plausible for evoked, IP3 -dependent astrocyte Ca2+ activity. However, experimental Ca2+ responses are more sparsely and non-uniformly distributed than model Ca2+ responses, which we explore next. Ca2+ Fluxes as a Source of Response Variability In addition to differences in IP3 dynamics, cell to cell and subcompartment to subcompartment response variabilities may also be due to different expression levels or functional properties of channels and pumps (e.g., SOC channels, PMCA pumps, and SERCA pumps) involved in Ca2+ responses (section Forms of IP3 Dynamics as a Source of Response Variability in the Model Although the range of Ca2+ response durations is similar in both model and data (Figure 4), the distributions of response Frontiers in Systems Neuroscience | www.frontiersin.org 11 October 2017 | Volume 11 | Article 79 34 Diversity of Evoked Astrocyte Ca2+ Dynamics Taheri et al. FIGURE 5 | Effect of IP3 kinetics on Ca2+ responses. (A) The left histogram shows the distribution of model response types while scanning IP3 kinetics over the full parameter range from Table 1. The right histogram plots the response type distribution for shorter IP3 rise durations (drise < 22 s). Shorter IP3 rise durations tend to decrease the occurrence of mostly MP and LL responses, with SP being the major response type, while longer IP3 rise durations decrease the percentage of SP responses, but increase other response types. (B) The total IP3 amount correlates with the total Ca2+ amount. However, in most regimes, within a small range of IP3 (and Ca2+ ) amounts, a mixture of Ca2+ response types are generated. An example of such a regime is shown in the zoomed section. Squares indicate responses generated with drise < 22 s (corresponding to data set in the right histogram in (A) and circles indicate responses generated with all other drise values from Table 1. (C) Two example Ca2+ responses are shown with their underlying IP3 dynamics. As seen, a small change in IP3 kinetics is sufficient to change the Ca2+ response type from SP to MP, while the total IP3 amount remains roughly the same (15.28 and 15.17 µM, respectively; Ca2+ amounts are 13.86 and 15.56 µM, respectively). IP3 parameters (A, drise , r rise , ddec ): 0.2, 21, 0.3, 220 (SP), 0.2, 31, 0.3, 179 (MP). Ca2+ Response Variability and their Sources). Therefore, we examined the individual contribution of each of these channels and pumps to Ca2+ responses (Figure 6). We also studied the Frontiers in Systems Neuroscience | www.frontiersin.org effects of modifying Ca2+ leak fluxes between the cytosol and ER or ECS, and the volume ratio of the cytosol to the ER (γ ), and found no major effects (Handy et al., 2017). In unpublished 12 October 2017 | Volume 11 | Article 79 35 Diversity of Evoked Astrocyte Ca2+ Dynamics Taheri et al. FIGURE 6 | Effect of blocking Ca2+ channels and pumps on Ca2+ responses. (A) The effects of blocking SOC channels (green traces), PMCA pumps (red traces), and SERCA pumps (brown traces) on example SP, PL, and MP responses generated with the default model parameters (yellow traces). The underlying IP3 dynamics for each panel are, from left to right (in order, A, drise , r rise , ddec ): 0.2, 21, 0.002, 97 (SP), 0.375, 36, 0.002, 120 (PL), 0.2, 41, 0.15, 179 (MP). The IP3 input was applied 20 s after the start of the simulation. (B) The change in Ca2+ response amplitude and duration after blocking the three channels and pumps. For all 600 IP3 inputs, the mean change in response kinetics from the default Ca2+ response is shown, with its standard deviation. (C) The effects of blocking the channels and pumps on Ca2+ response type distributions. The upper left distribution (corresponding to default parameters) is repeated from the left panel of Figure 5A. (D) The average PL response with default model parameters (IP3 drise < 22 s, as in the right panel of Figure 5A) shown in blue. Using the same underlying IP3 dynamics as the input, we also generated the average Ca2+ response when SOC channels were fully blocked (black trace). As observed, the plateau phase of Ca2+ responses is eliminated when SOC channels are blocked in our model. To generate these average traces, the trace peaks were first aligned, ignoring latency. Error bars indicate standard deviations. work, we have examined the effects of IP3 R parameters including d1 , the IP3 dissocation constant. Increasing d1 shifts to the right the bifurcation that gives rise to the complex transition to more complex and long-lasting responses in Figure 5B (data not shown), but does not lead to qualitative changes in behavior, and can be replicated by scaling IP3 and the parameter d3 . Effects of the more complete set of parameters are considered in the context of bifurcation analysis (Handy et al., 2017). of responses in Figure 6B, left). In fact, for 120 out of the 600 choices for IP3 parameters, blocking SOC channels suppresses the response up to the degree that the response would potentially become undetectable in experiments (amplitude <0.4 µM). In other words, our model predicts that blocking SOC channels decreases the likelihood that the astrocyte will respond to a brief application of agonist. Figure 6C (upper right histogram) illustrates that blocking SOC channels in the model increases the likelihood of observing SP responses, and completely eliminates the incidence of PL and LL responses. These data show that without functional SOC channels, we would expect to only observe SP or MP responses, at least during short-duration agonist applications. The fact that SOC channels have such strong effects on Ca2+ responses is particularly surprising given that the SOC flux rates are low in our model (Figure 3D; examined in more detail in Handy et al., 2017). Blocking SOC Channels Results Mostly in SP Responses The green traces in Figure 6A show how the example Ca2+ responses with default parameters (yellow traces) change when SOC channels are fully blocked. As seen, the durations and amplitudes of the Ca2+ responses to the same IP3 input decrease when SOC channels are blocked. This was true for nearly 99% of the full set of 600 IP3 traces (summarized for complete set Frontiers in Systems Neuroscience | www.frontiersin.org 13 October 2017 | Volume 11 | Article 79 36 Diversity of Evoked Astrocyte Ca2+ Dynamics Taheri et al. In agreement with our model prediction, previous experimental studies have shown that blocking astrocyte SOC channels eliminates the plateau phase of astrocyte Ca2+ responses and transforms these PL responses to SP responses (Malarkey et al., 2008; Pivneva et al., 2008; Wang et al., 2012). Figure 6D (blue trace) shows the average (± standard deviation) PL trace from the same set of model parameters as in the right panel of Figure 5A. Simulating responses using the same IP3 parameters, but with SOC channels fully blocked, eliminates the plateau phase of the responses, transforming them into SP responses (Figure 6D, black trace). the three experimentally-recorded distributions in Figure 2D. A more detailed examination of the response type distributions resulting from each of these subspaces is provided in Handy et al. (2017). The rationale behind this parameter search is that Ca2+ response variability between astrocyte subcompartments stems from variability in both Ca2+ channel/pump properties and the underlying IP3 kinetics (see sections Variability between Astrocyte Subcompartments and Contribution of IP3 to Ca2+ Response Variability). It is noteworthy that the simulations for a given astrocyte subcompartment also include a range of Ca2+ channel/pump parameters and IP3 parameters. Since in the experiments multiple cells were used to generate the data for each astrocyte subcompartment, having a range of parameters in simulations reflects the variability in channel/pump properties and IP3 kinetics inherent to each cell (see section Trial to Trial and Cell to Cell Variability on cell to cell variability). The range of IP3 parameters in simulations for each subcompartment also reflects experimental IP3 stochasticity stemming from trial to trial variability (i.e., the same ROI in the same cell responding differently to identical agonist pulses; see sections Trial to Trial and Cell to Cell Variability and Contribution of IP3 to Ca2+ Response Variability) as well as experimental variability (e.g., pipette distance from the ROIs). Using this parameter search, we found that the only subspace with a distribution that closely resembled the somatic distribution consisted of very short IP3 drise values (≤11 s) and the following parameter ranges: high vSOC (1.83–2.36), medium vPMCA (8.33–11.67), and low vSERCA (0.45–0.75). We did not find any one subspace that matched the distributions of the large and small processes. By looking at combinations of two adjacent subspaces, we found that IP3 drise values ≤21 s and the following parameter ranges provided a distribution similar to that of the large processes: high vSOC (1.83– 2.36), low vPMCA (5.0–8.33), and medium and high vSERCA (0.75–1.35). Using this same parameter subspace of the large processes, but with the full range of IP3 drise values in Table 1 (≤41 s), we obtained a distribution similar to that of the small processes. The three Ca2+ response type distributions from the mathematical model are shown in Figure 7A. The following are the percentages of observed response types in these random simulations (in order, SP, PL, MP, LL) for each subcompartment: 68.42, 17.76, 0.0, 13.82% (soma); 57.44, 16.37, 22.02, 4.17% (large processes); 26.21, 15.46, 52.52, 5.81% (small processes). To confirm that the Ca2+ response kinetics generated from these parameter subspaces are reasonable and comparable to the kinetics in the experimental data, we examined the ranges of Ca2+ kinetics for the soma and large processes of all response types in experiments (from the histogram in Figure 2D) and the random model simulations. We found that, similar to experiments, the model Ca2+ responses consist of a wide range of Ca2+ durations and amplitudes (Figure 7B), which is expected given the trial to trial and cell to cell variability we had discussed previously (Figure 2C). Moreover, while exact amplitude values between the model and experiments are not comparable (due to different measurement units), we see that response durations are very similar. Blocking PMCA Pumps Increases the Occurrence of MP Responses The red traces in Figure 6A show how the example SP, MP, and PL traces transform when PMCA pumps are fully blocked. As seen, the amplitudes, but not durations, of Ca2+ responses to the same IP3 input increase when PMCA pumps are fully blocked. The change in Ca2+ amplitude and duration, for any given IP3 input, arising when PMCA pumps are blocked is shown in Figure 6B, center. The response type distribution (Figure 6C, lower left) illustrates that blocking PMCA pumps almost entirely eliminates PL and LL responses (as was the case when blocking SOC channels). However, in contrast to the effects of blocking SOC channels, blocking PMCA pumps considerably increases the occurrence of MP responses. Partial Block of SERCA Pumps Eliminates MP Responses SERCA pumps are a necessary mechanism to fill the ER with Ca2+ , and completely blocking these pumps experimentally has been shown to lead to apoptosis (Luciani et al., 2009). In our model, completely blocking this term would lead to the ER being entirely emptied, and the system would be unable to evoke a response with IP3 . Therefore, we only investigated a 50% block of this channel. As seen in Figure 6A (brown traces), the durations of Ca2+ responses increase when SERCA pumps are partially blocked, while the amplitudes decrease. This change in Ca2+ kinetics occurred for all 600 IP3 traces and is summarized in Figure 6B, right. Furthermore, the response type distribution (Figure 6C, lower right) shows that MP responses are entirely eliminated when SERCA pumps are 50% blocked, and mostly transform into additional LL responses. Variability between Astrocyte Subcompartments: Model Thus, far, we have examined the separate effects of IP3 dynamics and individual Ca2+ fluxes on resulting Ca2+ dynamics. We now consider the effects of simultaneously changing both IP3 parameters and Ca2+ channel/pump fluxes within a biologically plausible range (Table 1), in order to explore how these differences may shape the variety of response type distributions among the somas, large processes, and small processes (Figure 2D). To do this, we ran simulations with random combinations of the three flux parameters vSOC , vPMCA , and vSERCA , while drawing the IP3 drise parameter from different ranges (for details see section Materials and Methods). We then selected the subspace, or subspaces, that best matched each of Frontiers in Systems Neuroscience | www.frontiersin.org 14 October 2017 | Volume 11 | Article 79 37 Diversity of Evoked Astrocyte Ca2+ Dynamics Taheri et al. FIGURE 7 | Simulation of response type distributions matching experimental distributions of astrocyte subcompartments. (A) Response type distributions in the model with parameters adjusted to the soma, large processes, and small processes. The yellow bars indicate the experimental data from Figure 2D. (B) Ca2+ durations and amplitudes of all response types in the model (from (A); in black) and experiments (from Figure 2D; in yellow) in the soma and large processes. One outlier was excluded from the experimental soma duration plot, with a duration of 363 s (same outlier omitted from Figure 4B). In summary, our results predict that the main differences between the astrocyte somas and large processes that are responsible for different response type distributions are as follows: (1) large processes include IP3 dynamics with slightly longer rise durations, and (2) the Ca2+ flux rates through the PMCA and SERCA pumps are, respectively, lower and higher in the large processes. Response type distributions associated with small processes can be generated by increasing IP3 rise durations even further, but using the same Ca2+ flux parameters as for large processes. This gradient in IP3 rise durations from somas to fine processes could be related to the influx of IP3 through spatial diffusion from other subcompartments, the constricted volume of the smaller subcompartments compared to the larger ones, and different probabilities of the agonist binding to receptors (due to the different subcompartment sizes and GPCR densities or expression levels). plausible kinetics, and found that this range of IP3 parameters was sufficient to reproduce all four Ca2+ response types in the model, without the need for feedback-induced oscillations in the IP3 waveform (Politi et al., 2006). The model was also used to predict how blocking astrocyte Ca2+ channels and pumps would affect the Ca2+ responses. In particular, blocking SOC channels resulted mostly in SP responses, and decreased the response duration and amplitude. Blocking PMCA pumps eliminated most PL and LL responses, and increased response amplitude. Finally, blocking SERCA pumps eliminated MP responses, decreased response amplitude, and increased the duration. Lastly, we propose that the experimentally observed response variability between astrocyte subcompartments can be explained by the differences in their channel/pump flux rates and IP3 kinetics. Namely, our simulations suggest that: (1) astrocyte somas have higher PMCA flux rates and lower SERCA flux rates than the processes, and (2) when moving from the somas to the large and small processes, the underlying IP3 transients tend to include those with longer rise durations. Our modeling results highlight two major advantages of relatively simple, biophysically based, mathematical models. First, they are experimentally falsifiable, and thus can be straightforwardly improved, or if necessary, discarded. Second, verified models of this type can be used as diagnostic tools. For example, because our model makes strong predictions about how specific biophysical mechanisms determine the types of Ca2+ transients evoked in astrocytes, it is potentially very useful in determining which mechanisms are altered in disease states, reactive astrocytosis, or other cases in which Ca2+ events are potentially altered. DISCUSSION We analyzed astrocyte Ca2+ responses to brief focal ATP applications and, based on our results, developed a singlecompartment model of astrocyte Ca2+ activity to investigate mechanisms of Ca2+ response variability. We categorized both experimentally-recorded and model Ca2+ responses into four types: Single-Peak (SP), Multi-Peak (MP), Plateau (PL), and Long-Lasting (LL). We found that, experimentally, SP response kinetics do not differ consistently between the soma, and processes, but we do see variability between cells and trials. However, the distributions of Ca2+ response types vary between different astrocyte subcompartments, with the likelihood of SP responses decreasing, and of MP responses increasing, in the large and small processes relative to the soma. Our model responses were tuned to match the average experimental short-duration SP response kinetics. We then applied IP3 transients to the model with a range of biologically Frontiers in Systems Neuroscience | www.frontiersin.org Contribution of IP3 to Ca2+ Response Variability We have presented three contexts in which astrocyte response variability has been observed (Table 2). Our results suggest that 15 October 2017 | Volume 11 | Article 79 38 Diversity of Evoked Astrocyte Ca2+ Dynamics Taheri et al. et al., 2000, 2003). We explore this issue in more detail in Handy et al. (2017). A similar role may be played by receptor-operated Ca2+ (ROC) channels, which are activated by GPCR agonists rather than by the depleted ER. ROC channels have been found in some astrocytes (Grimaldi et al., 2003; Beskina et al., 2007) and included in earlier models (Croisier et al., 2013). Regardless of the mechanism for plasma membrane Ca2+ fluxes, our model suggests the importance of considering an open-cell model in which total intracellular Ca2+ levels are allowed to fluctuate. IP3 time course variability may be a key contributor to all forms of response variability. For a given agonist application, variability between cells, or subcompartments within a cell, may arise from the profile of agonist reaching the GPCRs (e.g., due to different subcompartment sizes/shapes, or distances from the pipette); the local expression level or properties of GPCRs in a given cell or subcompartment; and differences in the diffusion and degradation of IP3 . Our experimental and computational results suggest that the net effect of such differences between the soma, large processes, and small processes is to increase the effective IP3 rise duration (drise ) in the periphery relative to the soma (Figure 7A). In our model, further differences in PMCA and SERCA flux rates (vPMCA , vSERCA ) are necessary to explain differences between the distributions of responses between somas and processes. For a given ROI, we see much more trial to trial response variability than in IP3 uncaging experiments (Fiacco and McCarthy, 2004). In contrast with Toivari et al. (2011), we thus conclude that the dominant source of trial to trial variability lies in the factors that determine the IP3 waveform in response to repeated agonist applications. IP3 waveform variability and concomitant variability in Ca2+ responses has been observed directly in response to bath application of ATP in other cell types (Tanimura et al., 2009; Nezu et al., 2010). What is the source of this hypothesized trial to trial variability in the IP3 waveform? Because we see no obvious trends in response to repeated stimuli, we speculate that the dominant factor is stochasticity from two sources. First, the number of activated GPCRs may vary stochastically between trials. Second, the mechanisms governing IP3 dynamics downstream of GPCRs are sensitive to varying molecule numbers known to assist with IP3 metabolism (Bartlett et al., 2015), and can lead to robust changes in Ca2+ signals. Another well-described source of stochasticity in Ca2+ responses is the inherent stochasticity of the IP3 R (Falcke, 2003; Dupont and Sneyd, 2009; Dupont et al., 2016) and other Ca2+ channels (Skupin et al., 2008, 2010). However, we believe that stochastic gating of these channels is less dominant for ATP-pulse-evoked Ca2+ transients studied here than for the spontaneous events, due to the larger number of IP3 molecules involved. Our view is supported by Fiacco and McCarthy (2004), who uncaged IP3 , thus presumably reducing variability in the IP3 waveform. They saw much less variability in evoked Ca2+ events compared with our ATP protocol, suggesting that the dominant sources of variability are upstream from the IP3 waveform for evoked transients. Taken together, the body of experimental and modeling results thus suggests that sources of variability are quite different for spontaneous and evoked events. We consider all the potential sources of stochasticity described here ripe for future study. Comparison of Experimental and Model Ca2+ Response Kinetics Although our model accounts very well for the variety of measured Ca2+ transients (Figures 4, 7B), it appears unable to account for a handful of very long (duration >120 s), often oscillatory events. Like nearly any tractable mathematical model, our model was not designed to account for all possibilities that may have been encountered in the experiments. Several factors, not accounted for in the model but possible in the experiments, could contribute positive feedback to extend Ca2+ transients in rare cases. First, applied ATP may cause release of other GPCR agonists from nearby glia or neurons, thus extending the duration of GPCR activation (Anderson et al., 2004). Second, astrocytes produce spontaneous Ca2+ activity via mechanisms that are incompletely understood but distinguishable from those implicated in evoked transients (Aguado et al., 2002; Wang et al., 2006; Haustein et al., 2014; Srinivasan et al., 2015). ATP application or evoked transients may interact with this mechanism to prolong the Ca2+ response in rare cases. Lastly, although single IP3 pulses can drive Ca2+ oscillations in our model, it is not known whether astrocytes generate multiple pulses or oscillations in IP3 levels in response to ATP pulses, as is seen, e.g., in HYS-EA1 cells during agonist bath applications of ATP (Tanimura et al., 2009). Feedback from Ca2+ to IP3 , which has not been demonstrated in astrocytes except in tissue cultures, can in principle generate IP3 oscillations that enhance Ca2+ oscillations and prolong their duration (Höfer et al., 2002; Politi et al., 2006). Similar mechanisms may underlie rare, experimentally observed Ca2+ oscillations with progressively increasing peak heights (e.g., the second MP and PL examples in Figure 2A), which are only reproducible in our model with more than one IP3 pulse (data not shown). Consistent with the suggestion that oscillatory IP3 events underlie growing Ca2+ oscillations, increases in Ca2+ peaks have been observed in HSY-EA1 cells during IP3 oscillations (Tanimura et al., 2009, Figure 5). Moreover, in COS-7 cells where IP3 did not oscillate, Ca2+ oscillations were possible but the initial Ca2+ peak was the largest peak for any given agonist concentration (Tanimura et al., 2009). Several known Ca2+ buffering and exchange mechanisms, including the Na+ /Ca2+ -exchanger and the effects of mitochondria, were not explicitly included in our model. We left these mechanisms out for three reasons. First, we simply lack adequate data from astrocytes to build such a model in good faith. Second, our minimal model was adequate to describe the experimental data sets quantitatively, with the exception of rare, Ca2+ Oscillations and Plasma Membrane Fluxes Our modeling results suggest that the presence of SOC channels allows for sustained Ca2+ oscillations without oscillations in IP3 , despite the low SOC flux rates (Figure 3D). This result is consistent with experimental results from astrocytes (Sergeeva Frontiers in Systems Neuroscience | www.frontiersin.org 16 October 2017 | Volume 11 | Article 79 39 Diversity of Evoked Astrocyte Ca2+ Dynamics Taheri et al. long-lasting events. Third, our model is simple enough to allow formal mathematical analysis (Handy et al., 2017) that gives a significantly deeper understanding of the model’s behavior. Development of a more mechanistically complex and detailed model awaits more detailed data, collected from astrocytes in the presence of appropriate blockers, so that the effects of each buffering component can be studied in isolation. smoothly with small differences in the upstream triggering events (Figure 5B). AUTHOR CONTRIBUTIONS MT collected and analyzed the experimental data under the guidance of JW. MT and GH built the mathematical model, ran the simulations, and analyzed the results under the guidance of AB. AB and JW oversaw the design and execution of project. MT, GH, AB, and JW wrote and reviewed the manuscript. Role of Astrocyte Ca2+ Responses Ca2+ transients have been hypothesized to have diverse downstream effects, including multiple forms of gliotransmission, modulation of transporters, and gene expression (reviewed by Bazargani and Attwell, 2016). It is likely that transients generated in different subcompartments generate different outcomes. For example, due to peripheral processes’ proximity to neuronal synapses and blood vessels, small processes may be more involved in regulation of synaptic function and blood flow. The different response type distributions of each astrocyte subcompartment (Figure 2D), may be a reflection of their different roles. Much of our analysis has focused on dividing Ca2+ responses into one of four types. This approach can be a useful tool for researchers, because the distribution of response types is clearly related to the underlying biophysics. However, it is unlikely that astrocytes encode information based solely on Ca2+ response type, since small changes in IP3 time course may change response types (Figure 5C). Instead, our results suggest that the most controllable way to reliably “encode” for different outcomes (Bazargani and Attwell, 2016) is via total Ca2+ amount (related to response type, duration, and amplitude), which varies more FUNDING This work was supported by the National Science Foundation (DMS-1022945 to AB; DMS-1148230, to AB and GH) and the National Institutes of Health (R01 NS078331, to JW and K. S. Wilcox). ACKNOWLEDGMENTS We thank Drs. Fernando R. Fernandez, Nathan A. Smith, and Karen A. Wilcox for helpful comments on this project and manuscript. SUPPLEMENTARY MATERIAL The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fnsys. 2017.00079/full#supplementary-material REFERENCES Cahoy, J. D., Emery, B., Kaushal, A., Foo, L. C., Zamanian, J. 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The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms. 19 October 2017 | Volume 11 | Article 79 42 Supplementary Material Diversity of Evoked Astrocyte Ca2+ Dynamics Quantified Through Experimental Measurements and Mathematical Modeling Marsa Taheri+, Gregory Handy+, Alla Borisyuk*++, John A. White*++ +: Co-first authors. ++: Co-corresponding authors * Correspondence: Alla Borisyuk: [e-mail], John A. White: [e-mail] 1 Supplementary Figure Supplementary Figure 1. Distribution of experimental and model Ca2+ transient durations. The distribution of Ca2+ response durations from experimental data (top left; same data as in Fig. 4B, with the same outlier omitted) was significantly different from the distribution of model Ca2+ transients (top right; same data as in Fig. 4A). Limiting the IP3 parameters in the model (total IP3 duration, IP3 decay duration, IP3 rise duration, and IP3 amplitude) changes the distribution of Ca2+ transient durations as shown. The only distribution that is not significantly different from the distribution of experimental Ca2+ durations is the one where IP3 rise durations are limited to less than 22 s (bottom left). CHAPTER 3 MATHEMATICAL INVESTIGATION OF IP3-DEPENDENT CALCIUM DYNAMICS IN ASTROCYTES This chapter was reprinted by permission from Springer Nature. The original publication appeared in the Journal of Computational Neuroscience on June 2017 (Volume 42, Issue 3, pages 257- 273). The contributing authors to this work are myself, Gregory Handy, John A. White, and Alla Borisyuk. GH and I built the mathematical model together, with equal contribution. Together with GH, I ran the model simulations, analyzed the model, and wrote the initial draft of the manuscript (which was then edited by all co-authors). 44 HHS Public Access Author manuscript Author Manuscript J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. Published in final edited form as: J Comput Neurosci. 2017 June ; 42(3): 257–273. doi:10.1007/s10827-017-0640-1. Mathematical Investigation of IP3-Dependent Calcium Dynamics in Astrocytes Gregory Handy, Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA Author Manuscript Marsa Taheri, Department of Bioengineering, University of Utah, Salt Lake City, UT 84112, USA John A. White, and Department of Biomedical Engineering, Boston University, Boston, MA 02215, USA Alla Borisyuk Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA, Phone: [phone], Fax: [phone] (Attn: Alla) Abstract Author Manuscript We study evoked calcium dynamics in astrocytes, a major cell type in the mammalian brain. Experimental evidence has shown that such dynamics are highly variable between different trials, cells, and cell subcompartments. Here we present a qualitative analysis of a recent mathematical model of astrocyte calcium responses. We show how the major response types are generated in the model as a result of the underlying bifurcation structure. By varying key channel parameters, mimicking blockers used by experimentalists, we manipulate this underlying bifurcation structure and predict how the distributions of responses can change. We find that store-operated calcium channels, plasma membrane bound channels with little activity during calcium transients, have a surprisingly strong effect, underscoring the importance of considering these channels in both experiments and mathematical settings. Variation in the maximum flow in different calcium channels is also shown to determine the range of stable oscillations, as well as set the range of frequencies of the oscillations. Further, by conducting a randomized search through the parameter space and recording the resulting calcium responses, we create a database that can be used by experimentalists to help estimate the underlying channel distribution of their cells. Author Manuscript Keywords Astrocytes; Calcium response; Hopf Bifurcation; Store operated calcium channels 1 Introduction Astrocytes make up approximately 50% of human brain volume (Tower and Young, 1973) and are roughly as numerous as neurons in the mammalian brain (Nedergaard et al., 2003). G. Handy and M. Taheri are co-first authors. 45 Handy et al. Page 2 Author Manuscript Filling most of the space between neurons, astrocytes are known for their passive roles in the brain, such as regulating neurotransmitter concentrations (Anderson and Swanson, 2000; Zhou and Danbolt, 2013) and buffering ions released during synaptic activity (Wallraf et al., 2006; Larsen et al., 2014). Recently, astrocytes have also been shown to serve a number of active roles in the brain, including the release of neuroactive compounds (e.g. glutamate, DSerine, ATP), which is commonly known as gliotransmission (Anderson and Swanson, 2000; Wang et al., 2013; Bezzi et al., 1998; Haydon, 2001; Liu et al., 2004), mediation of the blood-brain barrier, and control of the flow of blood to brain regions based on metabolic demand (Verkhratsky et al., 2012b). Calcium is believed to be a crucial second messenger in many of these pathways. Calcium transients are also observed experimentally after astrocytes are activated by the application of neurotransmitters, such as glutamate and ATP (Wang et al., 2012), even though there is little consensus on the exact role of astrocyte calcium signaling in the brain. Author Manuscript Existing mathematical models of astrocytes helped explore some of the possible roles astrocyte calcium signals play in the brain. Specifically, models have shown that through gliotransmission, astrocytes may be able to significantly change the firing patterns of nearby neurons (Di Garbo et al., 2007; Wade et al., 2011), alter the induction of long-term potentiation and long-term depression (De Pittà and Brunel, 2016), and have consequences for the overall network behavior (Amiri et al., 2012; Reato et al., 2012). However, gliotransmission and other communication pathways are under active investigation and debate in the experimental community (Nedergaard and Verkhratsky, 2012; Fujita et al., 2014; Haydon and Nedergaard, 2015). In the present work, we choose to focus on the detailed exploration of the calcium dynamics itself, and leave the investigation of its role in astrocyte-neuronal communication for future work. Author Manuscript Evoked calcium responses in astrocytes have also been the subject of some modeling studies. In most of these, as well as the model we present in this paper, it is assumed that the calcium release is triggered through the signaling molecule inositol (1,4,5)-trisphosphate (IP3). The modeling has shown that astrocytes may be able to convey signal information via frequency or amplitude of calcium oscillations (De Pittà et al., 2009), that IP3 diffusion may mediate the synchronization of calcium oscillations in neighboring astrocytes (Ullah et al., 2006), and that receptor stochasticity may play an important role in calcium response variability (Toivari et al., 2011). It is important to note, however, that the models found in Ullah et al. (2006) and Toivari et al. (2011) were created in reference to experiments with bath application of agonists to cultured astrocytes (a highly idealized experimental condition), while De Pittà et al. (2009) model includes a closed-cell assumption, i.e. assumes that the total calcium level inside an astrocyte is constant. Author Manuscript Recently, we investigated calcium transients generated in response to a brief pulse of ATP applied near the soma and processes of an astrocyte (Taheri et al., submitted). Based on our results and the astrocyte literature, we developed an open-cell model of astrocyte Ca responses and used the model to show that the observed response variability can be explained by differences in calcium channels and IP3 dynamics (Taheri et al., submitted). J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 46 Handy et al. Page 3 Author Manuscript Author Manuscript In this manuscript we present a more detailed mathematical analysis of this model to reveal key mathematical features underlying the diversity of responses. First, we show the bifurcation structure that allows the model to generate four main experimentally-observed response types (Single-Peak, Multi-Peak, Plateau, and Long-Lasting). We then manipulate the maximal flow through three of our included calcium channels/pumps, specifically storeoperated calcium (SOC) channels, sarco/endoplasmic reticulum calcium ATPase (SERCA) pumps, and plasma membrane calcium-ATPase (PMCA) pumps. This manipulation mimics application of non-competitive blockers for these channels in experiments. The results reveal, surprisingly, that blocking store-operated calcium channels, which remain relatively inactive during a calcium transient, can have a large effect on the shapes of calcium transients, eliminating the occurrence of Plateau and Long-Lasting response types entirely. This analysis also explains previous results in Taheri et al. (submitted), where we found that blocking each of the channels changes the observed distribution of calcium response types. This observation results from shortening or lengthening of the oscillatory regime in the bifurcation structure, as well as changing the frequency range of these oscillations. Further, by randomly varying channel parameters and recording the resulting calcium responses, we create a database that can be used by experimentalists to help determine the structure of their cells. 2 The Mathematical Model Author Manuscript Author Manuscript Fig. 1 shows a simplified schematic of the calcium activity components in our model. The corresponding mathematical model is first presented in Taheri et al. (submitted), and the model equations can be found in the Appendix. Briefly, when membrane receptors are activated by their agonist (e.g. ATP, glutamate), it triggers a cascade of reactions, resulting in the production of IP3 and its increased concentration in the cytosol (Agulhon et al., 2012). After peaking within seconds, the IP3 concentration then decays as IP3 diffuses away or degrades (Höfer et al., 2002). In our model we explicitly postulate the time evolution of the amount of IP3 as growing to a maximal level, then exponentially decaying. IP3 is then responsible for activating the calcium pathway via the IP3 receptor (IP3R), which releases calcium from the endoplasmic reticulum (ER), a calcium store in astrocytes, in the cytosol. The other fluxes, JSERCA (pumping calcium back into the ER), JPMCA (pumping calcium out of the cell), JSOC (activated by ER calcium depletion), the leak flux between the ER and cytosol JER_leak, and the additional fluxes on the plasma membrane JECS_add, all play an active role in shaping the calcium transient and returning the system to steady state postevoked response. Investigating the exact role of these fluxes is a major focus of this paper. Throughout this paper, we denote cytosolic calcium as c, total free calcium in the cell as ctot, calcium in the ER as cER, and IP3 as p. 2.1 Calcium Response Types In our recent work, we experimentally observed four distinct types of calcium responses in astrocytes after focally applying a short (<250 ms) pulse of ATP: a Single-Peak (SP) response, a Plateau (PL) response, a Multi-Peak (MP) response, and a Long-Lasting (LL) response (Taheri et al., submitted). By only varying key IP3 parameters (Table 1), the mathematical model is capable of reproducing these four response types (Fig. 2). A detailed J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 47 Handy et al. Page 4 Author Manuscript classification method is given in Taheri et al. (submitted). Furthermore, the general shape of the IP3 time course and a reasonable range of IP3 parameters (A, drise, rrise, and ddecay; see Table 1) for use in our simulations is based on previous experimental (Pasti et al., 1995; Tanimura et al., 2009; Nezu et al., 2010) and modeling (De Pittà et al., 2009) studies, as described in more detail in Taheri et al. (submitted). For example, the rise duration estimates vary from about 1 to 40 seconds and could vary between different subcompartments (soma vs. processes) of an astrocyte. The full model, as well as this classification algorithm can be found at ModelDB, (Hines et al. (2004); http://senselab.med.yale.edu/modeldb/default.asp; Model no. 189344). 3 Results Author Manuscript Building off our previous work focused on reproducing all four categories of experimentally-observed calcium response types (Taheri et al., submitted), we now investigate in more detail the underlying mathematical dynamics that allow for this diversity of calcium responses. We start with a bifurcation analysis of the system using the default parameters. Then, by systematically weakening or blocking specific calcium-handling mechanisms (i.e. calcium channels and pumps), we examine how this underlying dynamical landscape changes. 3.1 The Default Dynamical Landscape: Role of IP3 Author Manuscript 3.1.1 Mechanisms Underlying the Four Types of Calcium Responses—We first consider the SP response and the individual calcium fluxes governing it (Fig. 3A). Before the IP3 (black, dashed curve) input begins, the system is at rest with most of the fluxes being completely inactivated. The steady state of the system is created by a balance between the SERCA pump and calcium leak from the ER (black, dashed and light gray, dashed curves respectively). If this balance is upset, for example by blocking of SERCA pumps, it would result in a prompt emptying of ER stores, as observed experimentally (Jousset et al., 2007). Author Manuscript As IP3 is added into the system, the IP3 receptor (IP3R) is activated, causing this channel to open and release calcium into the cytosol from the ER (JIP3R, black curve). This initial calcium release has a positive feedback onto the IP3R, and the channel opens very quickly. Consequently, the SERCA pump responds by pumping some of the newly released calcium back into the ER (JSERCA, black, dashed curve). Initially, cytosolic calcium concentration continues to rise, because the SERCA pump is unable to completely counteract the flux out of the ER through the IP3R. Eventually, the PMCA (gray, dashed curve) and the additional fluxes on the plasma membrane (light gray curve) begin to release calcium from the cytosol into the extracellular space (JPMCA and JECS_add), and the cytosolic calcium concentration begins to decrease. Meanwhile, the flux through the IP3R is eliminated, as IP3 concentration decreases, along with the negative feedback the receptor feels from the elevated cytosolic calcium levels. The SOC channel flux (JSOC, gray) remains largely inactive during this single calcium response, since the calcium levels in the ER do not approach the halfsaturation constant of these channels (kSOC = 90). Nonetheless, this channel plays a significant role in IP3-induced calcium dynamics as described later in Section 3.2. J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 48 Handy et al. Page 5 Author Manuscript Overall, this SP calcium response causes a ∼12% decrease in ER calcium concentration (inset of Fig. 3A top). Although the remaining pumps and channels appear to return to steady state levels immediately after the calcium transient, the calcium concentration in the ER remains below steady state well after the calcium transient in the cytosol is completed, and is restored to > 95% after approximately 10 minutes (not shown). During this time, the pumps and channels continue to work at low levels to completely restore the system. However, even during this refilling period, there is a sufficient amount of calcium in the ER to produce another calcium transient if the astrocyte is stimulated again. Author Manuscript Author Manuscript Extending our analysis now to all response types, we found that it was only necessary to vary the amplitude (A) of the IP3 trace in order for the model to generate SP, MP and PL responses, and increasing the duration (ddecay) of the IP3 input, results in a LL response (figure not shown). Further, our previous study found that varying all four of the IP3 transient kinetics (amplitude (A), rise duration (drise), decay duration (ddecay), and the rise rate (rrise)) within a biologically plausible range (see Section 2.1) results in similar calcium kinetics and response types (Taheri et al., submitted). Using these results, we now propose specific mechanisms driving the MP, PL and LL calcium responses before taking a closer look at the underlying mathematical structure. The MP example trace in Fig. 3B has an IP3 transient with the same amplitude as the IP3 transient underlying the SP response in Figure 3A; however, the IP3 transient responsible for the MP response remains elevated for a longer period of time (greater drise and ddecay, as well as an increased rrise). With this IP3 transient, after the initial calcium transient is completed and the negative feedback on the IP3R wears off, another response, although a smaller one, is generated due to the ongoing elevation of IP3. The example PL response (Fig. 3C) has an IP3 trace that has a higher amplitude than the SP and MP examples (with the same rrise as the SP response and drise and ddecay values lower than the MP response). The IP3 levels underlying this PL response are high enough to be able to counteract the negative feedback onto the IP3 receptor from cytosolic calcium; therefore, the channel associated with the IP3R remains open for longer, leading to an elevated calcium response. The final experimentally observed response (LL) remained sufficiently elevated for more than 70 s (figure not shown). We found that LL responses are fundamentally similar to those underlying PL responses, but have larger IP3 amplitude and ddecay, allowing for the calcium response, or elevated oscillations, to linger for a longer time. For this reason, the rest of the paper focuses primarily on the SP, MP, and PL response types. Author Manuscript To elucidate the mathematical structures underlying the above-described dynamics, we note that although the concentration of IP3 is a dynamic variable, it is uncoupled from the full system and its time course is set. Therefore, if we consider IP3 transients with a much slower timescale than the other model variables, then key features of the full model can be explained, by examining the three dimensional system consisting of only c, ctot, and h (Appendix: Equations (1)-(3)), while treating IP3 as a bifurcation parameter. The bifurcation diagram of this subsystem can be seen in Fig. 3D. As illustrated in this figure, a supercritical Hopf bifurcation occurs at [IP3]≈0.18 (black asterisk 1) that causes the single stable steady state to transition into a stable limit cycle, shown as the gray, dot-dashed curve in the figure. This stable limit cycle is eliminated by an unstable limit cycle (light gray open circles) originating at a subcritical Hopf bifurcation at [IP3]≈0.35 (black asterisk 2). J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 49 Handy et al. Page 6 Author Manuscript With this diagram in mind, one can visualize how a stereotypical SP, MP, and PL response can be generated if the underlying IP3 transient was on a much slower timescale compared to the rest of the system. Due to the excitability of the IP3R, a large enough IP3 transient produces an initial calcium response. If the amplitude of this IP3 transient is small enough (below the supercritical Hopf bifurcation point), the calcium levels simply return to steady state, generating a SP response. On the other hand, if the IP3 transient amplitude goes beyond this first bifurcation point (but under the second bifurcation point), we are placed in the oscillatory regime, and if IP3 remains elevated long enough, a MP response is generated. Lastly, if the IP3 transient amplitude goes past the second bifurcation point, the calcium levels land on the higher steady state before decaying back to baseline; thus, a PL response is generated. Author Manuscript Note that the above analysis is simplified, since the IP3 timescale in our simulations is not always slower than the rest of the system. However, this general framework proved useful in the classification of calcium responses types, and changes in the underlying bifurcation diagram will be used to explain differences between different parameter regimes. Author Manuscript 3.1.2 Sustained Calcium Oscillations May Be Attained without IP3 Oscillations —The bifurcation diagram of the system suggests that sustained calcium oscillations can be generated without IP3 oscillations, as long as IP3 is held constant at certain levels (between the supercritical and subcritical Hopf bifurcations at 0.18 - 0.35 µM). This is in agreement with the work done by Li and Rinzel (1996) and Sneyd et al. (2006), which noted that the positive and negative feedback mechanisms in the Li-Rinzel IP3 receptor model are capable of supporting calcium oscillations. Viewing IP3 as a slow changing parameter is realized in experiments where intracellular IP3 concentration is fixed (e.g. uncaging experiments or, possibly, agonist bath applications), further motivating our pursuit of this analysis. Fig. 3E illustrates the calcium dynamics in response to step-wise increases in IP3 concentration. In agreement with the bifurcation diagram, sustained oscillations exists for IP3 concentrations in the range of 0.18 - 0.35 µM. In addition, this figure also illustrates that steps from a lower to higher IP3 concentration are accompanied by a brief, high amplitude transient, either in addition to the sustained oscillations or as a stand alone. We predict that such a transient would be observable experimentally. However, this transient does not occur if we step from a higher IP3 concentration to a lower one. This result can be explained by the dynamics of the IP3R. When rapidly increasing the concentration of IP3, the IP3Rs quickly open, resulting in a sudden initial increase and transient as the system settles into its new steady state. But when the concentration of IP3 is lowered, IP3Rs remain closed and do not contribute to a transient oscillation. Author Manuscript 3.2 Contribution of SOC channels to the Dynamical Landscape 3.2.1 Calcium Flux Levels through SOC Channels are Low, but Play a Major Role in Shaping Calcium Responses—In Taheri et al. (submitted), we showed that completely blocking SOC channels significantly changed the distribution of possible response types across the range of IP3 parameters, entirely eliminating the occurrence of PL and LL response types. We investigate this result more closely by examining the effect of a partial and complete block of SOC channels on the response types. Specifically, Fig. 4A-C J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 50 Handy et al. Page 7 Author Manuscript show that with a partial SOC block (0.20 * υSOC, gray curves), the SP response becomes shorter, the second peak in the MP response is eliminated, turning it into an SP response, and the PL response develops a second peak to become an MP response. Further, when SOC is completely blocked in these examples (light gray curves), the SP and MP responses disappear, while the PL response is reduced to a SP response. Thus, SOC channels are instrumental in shaping calcium response dynamics, however, in the previous flux traces (Fig. 3A-C), these channels appeared as relatively inactive during all of the calcium transients. However, the influence of SOC channels is misrepresented in these figures. Since the other fluxes work to counteract each other (i.e. JIP3R vs. JSERCA), the following partial sum of fluxes, Author Manuscript has, in fact, a similar magnitude as δ · Jsoc (figure not shown). As a result, eliminating the flux through SOC channels will have a larger impact on calcium transients than one might initially expect. Author Manuscript We can also gain insight into this rather counterintuitive result by investigating the projection of our system onto the (c, ctot) phase space (Fig. 4D). As the figure illustrates, JSOC plays a crucial role in shaping the underlying dynamics, particularly the shape of the ctot-nullcline. The progression of the ctot-nullcline (gray curve) in Fig. 4D from left to right, corresponding to decreasing vSOC from its default value of 1.57, shows that as υSOC is blocked, this nullcline flattens out and becomes completely vertical when υSOC = 0. Examining the equation for this ctot-nullcline, we have The JSOC term makes this a function of c and cER, as opposed to just c. When the SOC component is eliminated, one obtains a quartic function of c that has an unique solution, and is therefore constant for all levels of IP3 (Fig. 4E, inset). The background color of these figures illustrate that while υSOC has the ability to shift nullclines and reshape the direction field, calcium transients exists in the region where JSOC remains small (white background). Author Manuscript Additional evidence supporting the surprisingly large role SOC channels have in the model can be found by investigating the two-parameter bifurcation diagram presented in Fig. 4F, which uses IP3 and υSOC as parameters (note log scale on y-axis). The shaded region represents the parameter regime that supports stable oscillations, and the dotted line indicates the default vSOC parameter value (vSOC = 1.57), with the gray dots representing the original Hopf bifurcation points (from Fig. 3D). It is clear from this figure that as υSOC goes to zero, the leftmost Hopf bifurcation point increases beyond a realistic value of IP3, and so with zero SOC flux the sustained oscillations are not supported. However, the oscillatory region lies close to the IP3-axis, and, as a result, even small levels of SOC flux can move the J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 51 Handy et al. Page 8 Author Manuscript solution across the boundary into the oscillatory region, changing the dynamics of the system. However, the oscillatory regime remain close to the IP3-axis, and as a result, even small values of JSOC can result in large changes in the underlying dynamics of the system. Author Manuscript 3.2.2 Sustained Calcium Oscillations are Not Possible without Functional SOC Channels—Even though the system can only sustain stable calcium oscillations for positive values of υSOC, MP responses are still observable when υSOC=0, though they occur through a different mechanism. In fact, Fig. 4E shows that elevating IP3 and holding it elevated can result in such a response. These MP responses with υSOC=0 are created by the system spiraling back to the steady state level for cytosolic calcium. These are not sustainable oscillations in a sense that their amplitude decays to zero even if the IP3 level stays constant. As a result, this is an inherently different type of MP response than those with υSOC at the default value (Fig. 3B and Fig. 3E), where the MP response was created due to the underlying oscillatory regime in the bifurcation diagram. In addition, the bifurcation diagram found in Fig. 4E (inset) demonstrates the fact that PL and LL responses are no longer possible when υSOC = 0, since there exists no elevated steady state. Consistent with this analysis, we have previously shown that blocking SOC channels allows for only SP and MP responses (Taheri et al., submitted). Author Manuscript 3.2.3 Partially blocking SOC Channels Increases the Incidence of Calcium Oscillations—The two-parameter bifurcation diagram in Fig. 4F also indicates that there is a drastic difference between completely blocking and partially blocking SOC channels. A complete block will eliminate stable oscillations, whereas a partial block, such as decreasing υSOC by over 85% of its default value (υSOC = 0.2), will result in a significantly larger region of stable oscillations. This is a testable prediction that can be verified experimentally using partial blocking techniques, such as those used by Jousset et al. (2007). This also provides a metric into the effectiveness of an SOC blocker: a more effective, though partial, blocker will have a larger regime of stable calcium oscillations, which could be potentially measured using an IP3 uncaging technique (Kantevari et al., 2011). Author Manuscript 3.2.4 Verification of SOC Channel Activation—Despite remaining relatively inactive during calcium transients, the flux through SOC channels in our model can be activated when the ER is sufficiently depleted. In fact, a common method to test for the presence of SOC channels experimentally involves completely depleting the calcium in the ER by blocking the re-uptake of calcium through SERCA blockers (Jousset et al., 2007; Roos et al., 2005; Malarkey et al., 2008). In one such experiment, extracellular calcium is initially removed, driving some amount of calcium out of the cytosol and into the extracellular space. Then, by applying a SERCA pump blocker such as Thapsigargin (a permanent blocker), calcium is prevented from returning to the ER and therefore continues to leave the cell. Once the ER is empty (thus, activating SOC channels), calcium is introduced back into the extracellular space. This results in calcium entry to the cytosol through the SOC channel. Our model can be used to simulate similar experimental conditions (Fig. 5). Beginning at steady state (Phase I), we simulate the removal of calcium from the extracellular space by J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 52 Handy et al. Page 9 Author Manuscript setting υin = υSOC = 0 at 50 s. As seen, a slight amount of calcium is observed to leave the cytosol and enter the extracellular space (Phase II). We then block SERCA pumps by setting υSERCA = 0 at 100 s, causing a calcium spike in the cytosol via the leak out of the ER (Phase III). This causes fluxes in both the PMCA pump and ECS leak, and would eventually lead to the majority of intracellular calcium to be removed. Before this occurs, calcium is reintroduced to the extracellular space (υin = 0.05, υSOC = 1.57) and calcium rapidly reenters the cell (Phase IV). In experiments, this increase is believed to be caused primarily by an influx via SOC channels since the response is largely eliminated by knocking down the gene responsible for the creation of STIM1 proteins (Jousset et al., 2007). Indeed, this is true in our model as well. 3.3 Role of SERCA Pumps, PMCA Pumps, and leak terms Author Manuscript 3.3.1 Partially Blocked SERCA Pumps Prevents Oscillations—We have previously reported that blocking SERCA pumps by 50% entirely eliminates MP responses, while increasing the number of LL responses, for our given range of IP3 transients (Taheri et al., submitted). Here, Fig. 6A-C demonstrates that blocking SERCA pumps by 50% affects the default response types in the following way: it shortens the amplitude, but increases the duration of the SP response, transforms the MP response into a PL, and flattens out the PL response. Author Manuscript By investigating the two parameter bifurcation diagram (Fig. 6D), now using υSERCA as the second parameter, we see in fact that oscillations are eliminated by decreasing υSERCA. In fact, 0.5 υSERCA rests a sufficient distance away from this oscillatory regime, unlike in the SOC channel case, and the steady state is not a spiral. Furthermore, unlike the bifurcation diagram found in Fig. 4E (inset), the single parameter bifurcation diagram for 0.50 * υSERCA (i.e. when SERCA is 50% blocked) illustrates that decreasing υSERCA does not lead to a constant value for the steady state of cytosolic calcium. Instead, it remains a monotonically increasing function of IP3 concentration, without an oscillatory regime. Without this oscillatory regime and the steady state not being a spiral, MP response types are no longer possible, which explains why we observed a higher distribution of LL responses when SERCA channels are 50% blocked in Taheri et al. (submitted). Author Manuscript 3.3.2 Blocking PMCA Pumps Creates a Larger Oscillatory Regime—We also reported in Taheri et al. (submitted) that blocking PMCA pumps fully almost entirely eliminates PL and LL responses, while the incidence of MP responses increase. Examples in Figure 7A-C also illustrate that while blocking PMCA pumps increases the max amplitude in all three cases, the response type of the first two cases (SP and MP) remain intact. On the other hand, the PL response changes noticeably and becomes a MP response. This can be explained by the two-parameter bifurcation diagram in Figure 7D, using IP3 and υPMCA as parameters. Figure 7D shows that decreasing υPMCA increases the width of the oscillatory regime caused by the Hopf bifurcations. This increased oscillatory region increases the occurrences of MP responses. Figure 7E illustrates this fact by overlaying the MP calcium trace found in Fig. 7C (gray curve) with the elongated bifurcation diagram caused by blocking PMCA pumps (the gray dots on the x-axis mark the location of the Hopf bifurcation points with the default parameters). J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 53 Handy et al. Page 10 Author Manuscript 3.3.3 υin and γ Parameters Have Minor Influences on Calcium Responses— Unlike the previous parameters, significantly changing υin (increasing by 300% and decreasing the value to 0), showed very little change in the underlying bifurcation diagram (figure not shown). However, this parameter does play a key role in setting the steady state calcium concentrations in the cytosol and the ER. Specifically, decreasing υin to 0 decreases the cytosolic calcium steady state by approximately 11%. This inward leak term implicitly represents many of the other channels not explicitly included in the model, such as TRPA channels. This channel has been shown experimentally to play a large role in setting the basal concentration of calcium (Shigetomi et al., 2012), which is consistent with our simulations. Author Manuscript We also investigated the effects of changing the γ, the parameter that represents the ratio of cytosolic volume to ER. This parameter helps set the total amount of free calcium in the cell, but does not change the steady state levels of calcium in the ER and cytosol. Decreasing γ from 5.4054 to 1, representing a cell that has an ER equal to volume as the cytosol, marginally increased the range of stable calcium oscillations from 0.1711-0.3569 µM to 0.1693-0.3722 µM. Further, increasing γ to 20 marginally decreased the rang of stable calcium oscillations to 0.1796-0.3041 µM (figure not shown). 3.4 Frequency of Calcium Oscillations Author Manuscript The intrinsic frequency of calcium oscillations are viewed as potentially crucial in regulating mechanisms such as gliotransmission (De Pittà et al., 2009; Croft et al., 2016). As a result, in this section we investigate how the parameters υSOC, υSERCA, and υPMCA affect the range of supported frequency, known as the dynamic range, of these calcium oscillations. Building off of Sections 3.2 and 3.3, where we saw that decreasing υSOC and υPMCA, and increasing υSERCA lead to an increase in the range of stable oscillations, Fig. 8 shows how the frequency of these oscillations change. In the first panel, decreasing υSOC sharply drops the frequency curve for a wide range of IP3 values, shifting the oscillation to lower frequencies, while having a minor effect on the dynamical range. This further emphasizes that small changes in υSOC will result in large changes in the dynamical system. Increasing υSERCA shows a sharp initial drop in frequency, with quick rise, hence increasing the dynamic range, but leaving it centered in a similar frequency range as the default case. These frequency curves do not show the same separation as was seen with changes in υSOC. Lastly, unlike the other two parameters, decreasing υPMCA has very little effect on the underlying frequency curve. Author Manuscript Overall, this figure suggests that these currents can play very different roles in controlling the frequency of calcium oscillations, with PMCA pumps having no effect, SERCA pumps adjusting the width of the dynamic range, and SOC channels shifting the overall location of the dynamic range. This effect is in addition to determining the type of calcium responses the cell can support, as discussed in previous sections. 3.5 Tuning Model Parameters to Reproduce Specific Experimental Conditions Experimentally, the expression levels or functional properties of the model components discussed above may be different in different astrocytes or different subcompartments (i.e. J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 54 Handy et al. Page 11 Author Manuscript soma, large processes, small processes) of one astrocyte. For example, there may be a difference between astrocytes in the densities of their agonist receptors (which would change the resulting IP3 kinetics) or calcium channels/pumps. Moreover, these channels, pumps, and receptors may be up- or down-regulated (or may change in their functional properties) under different conditions, such as when astrocytes become reactive in disease (de Lanerolle et al., 2010; Takahashi et al., 2010). Author Manuscript To examine such changes between different astrocytes, we consider how simultaneous changes in relevant model parameters affect the distribution of generated calcium response types. From a theoretical point of view, this addresses the sensitivity of the calcium response distribution to major parameters in the model. For each IP3 transient (with parameter ranges as in Table 1, the total of N = 600 IP3 transients), we randomly chose the parameters υPMCA, υSERCA, υSOC within ±50% interval around the default values, and repeated this process 30 times for each IP3 transient. We then binned the υPMCA, υSERCA, vSOC parameter space into 27 boxes, and reported the distribution of calcium response types within each box, as seen in Fig. 9. For example, the distribution in the upper left-hand corner represents the parameter box (υPMCA, υSERCA, υSOC) ∈ [5.0, 8.33] × [1.05, 1.35] × [0.785, 1.31]. As the figure illustrates, different boxes in the parameter space have different distributions of calcium responses. This allows us to visualize how changes in parameter values effect these response type distributions. For instance, we can quickly investigate the situation where we simultaneously increase the calcium flux through SERCA pumps and SOC channels, while keeping PMCA constant (i.e. moving diagonally within the same panel from bottom left to top right). In all three panels, we see that the number of observed MP responses increases, while PL responses decrease. Author Manuscript We can also examine how the distributions in each of these subspaces change with different IP3 kinetics. Figure 10 illustrates the same 27 subspaces for low amplitude (A ≤ 0.3) IP3 transients. For the subspaces where calcium flux through PMCA and SERCA pumps is high and SOC is constant (i.e. distributions in the top row in the third panel), decreasing the IP3 amplitude decreases the incidence of MP responses and increases the incidence of SP responses. Author Manuscript Such a parameter map can be used by experimentalists to elucidate differences in calcium handling mechanisms between various astrocyte populations. For instance, the response type distribution profile of reactive astrocytes in disease, such as in epilepsy, can be compared against that of healthy astrocytes and be reproduced in the model. The model parameters that need to be modified to reproduce such calcium distributions could indicate how the expression levels and functional properties of channels, pumps, and membrane bound receptors may change in reactive astrocytes. This approach was used to explain the variability in responses of different astrocyte subcompartments (Taheri et al., submitted). 4 Discussion Through bifurcation analysis and simulations, we investigated the role SOC channels, SERCA pumps, and PMCA pumps have in shaping the underlying dynamical landscape of J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 55 Handy et al. Page 12 Author Manuscript our model of evoked calcium responses in astrocytes. Surprisingly, despite the low amount of calcium that flows through SOC channels during a calcium transient, we showed that these channels are necessary for PL and LL response types, as well as for stable oscillations. Partially blocking SERCA also eliminated stable oscillations, but PL and LL response types were observed in the system. Further, blocking PMCA pumps significantly widened the IP3 range that supported oscillations, and thus, increased the occurrences of MP response types. We have also shown that the strength of SOC channels is a critical parameter in determining the location of the dynamical range of calcium oscillations, while the parameter corresponding to the strength of SERCA pumps has the ability to tune the width of supported frequencies. Lastly, we presented a novel method for fitting parameters of the model given experimental data of calcium response type alone. In this section, we discuss the potential physiological role of SOC channels, as well as discuss specific limitations of our model. Author Manuscript 4.1 Analysis of Calcium Dynamics Author Manuscript Open-cell calcium dynamics have been the focus of numerous research publications (Sneyd et al., 2004; Ullah et al., 2006; De Pittà et al., 2009; Croisier et al., 2013), and have been reviewed in Falcke (2004), Dupont (2014), and Dupont et al. (2016). Building off this previous work, this paper analyzes the calcium dynamics stemming from a unique and astrocyte-specific combination of calcium fluxes and external stimulation. In addition, the models parameters were chosen in order to capture qualitative and quantitative behavior from astrocyte-specific experiments. Further, while there has been a recent focus of some mathematical models to investigate the role of SOC channels in calcium dynamics, the fields understanding of this channel is still incomplete (Liu et al., 2010; Croisier et al., 2013). In this work, we provide a detailed analysis of this flux and how the underlying dynamical structure is dependent on this quiet, yet influential channel. Our systematic approach to blocking and adjusting the relative strengths of individual fluxes (JPMCA, JSERCA, and JSOC) allowed us to provide an in-depth analysis of the role each of these channels have on the underlying bifurcation structure, the diversity of calcium transients, and the frequency of calcium oscillations. 4.2 Mechanism and Role of SOC Channels Author Manuscript While the expression of SOC channels has been identified in astrocytes, specifically the transient receptor potential channel, the exact mechanism allowing these channels to communicate with the ER and allow for the flow of calcium is still under investigation (Verkhratsky et al., 2012b). Instead of modeling this complex mechanism, we focused on qualitatively capturing the characteristic that SOC channels open after calcium is depleted from the ER. Therefore, as a simplified first step, we used a reverse Hill equation to model the activation of these channels, as in Cao et al. (2014). Store-operated calcium entry is a subject of intense ongoing research, and the model will need to be re-evaluated and updated as more data becomes available. For example, there is evidence presented in Courjaret and Machaca (2014) that in oocytes at least some fraction of calcium entering through SOC channels is transported directly into the ER, and would thus need to be included as part of ER calcium in the model. Similar mechanisms may be present in astrocytes as well, and their experimental and theoretical consequences will need to be explored in future work. J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 56 Handy et al. Page 13 Author Manuscript An additional delay (capturing the potential time it takes the ER to communicate with the plasma membrane) was also considered, but did not significantly change the results, primarily because JSOC remained silent during the calcium transients. Experimentally, the presence of SOC channels is confirmed via SERCA blockers that completely empty the ER (Malarkey et al., 2008). The model suggests that this type of confirmation may be necessary, because SOC channels remain mostly silent during calcium transients. Nonetheless, despite little activity during transients, SOC channels shape the background dynamics necessary for prolonged responses. 4.3 IP3 Dynamics Author Manuscript Detailed, mechanistic models of IP3 dynamics exists in the literature (De Pittà et al., 2009), but the necessary kinetic details of such models are unknown in astrocytes. New information is even being revealed in non-astrocytic cells, e.g. Bartlett et al. (2015) recently showed how differential regulation of intermediate steps in IP3 creation can have a significant impact on calcium transients. The range of IP3 transients considered here attempt to indirectly account for the uncertainty caused by the number of membrane-bound receptors that are bound and activated by ATP molecules, the duration of this activation, calcium feedback, and PKC regulation to name a few. However, our explicit function for IP3 (Equation (4)) is potentially missing details that would be found in experimental measurements of IP3 concentration, and can be extended in future work. 4.4 Modeling of Non-competitive Blockers To mathematically model the presence of channel blockers, we simply reduce the corresponding maximum flux term associated the channel of interest. For example, a 50% block of an unspecified channel is simply Author Manuscript This approach includes the implicit assumption that the blockers are assumed to decrease the overall activity of the channel, while not competing directly with calcium for binding sites. Examples of such non-competitive blockers include thapsigargin and Cyclopiazonic Acid (CPA) for the SERCA pump (Plenge-Tellechea et al., 1997; Singh et al., 2005), and eosin for the PMCA pump (Gatto and Milanick, 1993). Due to the relative novelty of SOC channels, specific non-competitive blockers have not been developed. However, genetic knockdown of the STIM1 protein, has been shown to down-regulate these channels (Jousset et al., 2007), which we assume to have a similar effect as a non-competitive blocker. Author Manuscript 4.5 Higher Dimensional Parameter Spaces This work focused primarily on investigating how the parameters υSOC, υPMCA, and υSERCA influence the underlying dynamics of the model proposed in (Taheri et al., submitted). Further, even though we did not vary υIP3R directly, we considered a range of IP3 amplitudes, a parameter which has a similar effect as varying the maximum flux through the IP3R. However, other parameters of the model also influence the calcium response types. J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 57 Handy et al. Page 14 Author Manuscript For example, varying the parameter a2, which controls the time constant of the IP3R inactivation variable h(t), can significantly change the type of calcium response types observed. Specifically, decreasing a2 (effectively increasing the time constant of h(t)), prevents the inactivation of IP3Rs, leading most calcium responses to become SPs, while increasing a2 leads to a faster and stronger inactivation of IP3Rs, allowing for MPs to become more prevalent (figure not shown). Though not completed here, the above analysis, specifically the tuning method outlined in Section 3.5, can be extended to account for variations in additional parameters. To account for a higher dimensional space, Latin hypercube sampling may be used as a more effective way to sample the multidimensional parameter distribution (McKay et al., 1979), and an automated least squares method can be used to find the region most closely related to one's experimental calcium response type distributions. Author Manuscript 4.6 Healthy and Reactive Astrocytes Author Manuscript While we found that the value of γ played a minor role in determining the dynamics of the model, the ratio of ER volume to cytosol volume is known to vary between healthy and reactive astrocytes. Reactive astrocytes are characterized in part by hypertrophy and upregulation of glial fibrillary acidic protein (GFAP) (Agulhon et al., 2012). Specifically when the central nervous system is injured, astrocytes extend their processes and increase in volume by producing additional GFAP (Ridet et al., 1997). While experimentally observed calcium transients vary in these two states (Aguado et al., 2002), the model suggests that this volume ratio change is not sufficient to produce the change from one state to the other. Thus, a model of reactive astrocytes would need to account for other parameters that may be affected by this ratio change. These could include the change in density of different channels/pumps, the distance of the plasma membrane from the ER and its effects on SOC channel function, and how quickly the SERCA pump can uptake cytosolic calcium, to name a few. Acknowledgments This work was supported by the National Science Foundation (DMS-1022945 to A. Borisyuk; DMS-1148230, to A. Borisyuk and G. Handy) and the National Institutes of Health (R01 NS078331, to J.A. White and K.S. Wilcox). Appendix: Mathematical Model The differential equations driving the model are Author Manuscript (1) J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 58 Handy et al. Page 15 Author Manuscript (2) (3) where we denote the calcium concentration in the ER as cER = (ctot − c)γ, and IP3 concentration as p. The Ji's are the fluxes found in Fig. 1. Specifically, we use the Li-Rinzel IP3 receptor model to capture the calcium dynamics through the IP3R channel (Li and Rinzel, 1996), which is governed by the following equations Author Manuscript where and Author Manuscript The SERCA and PMCA pumps are both model as Hill functions, the forms found in Cao et al. (2014) and Croisier et al. (2013) respectively, and are given by the equations, Author Manuscript and J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 59 Handy et al. Page 16 Author Manuscript Similar to the work in Cao et al. (2014), we model SOC channels as the following reverse Hill function since it has been shown that they open when calcium is depleted in the ER (Verkhratsky et al., 2012b). The model also includes an IP3R-independent leak between the cytosol and the ER with the following equation Author Manuscript Further, we account for additional fluxes across the plasma membrane with the equation where υin captures the constant leak from the extracellular space, and −koutc accounts for additional calcium extrusion not explicitly model, such as the sodium-calcium exchanger (Höfer et al., 2002; Ullah et al., 2006; Keener and Sneyd, 2009; Verkhratsky et al., 2012a). Lastly, the explicit equation for IP3 is Author Manuscript (4) where Author Manuscript t* is the time of stimulus, A is the max amplitude, rrise and rdecay are the rate of rise and decay respectively, and drise and ddecay are the duration of the rising and decaying phase. These parameters allow us the flexibility to explore a large distribution of IP3 responses easily and effectively. The complete range of IP3 parameters, as well as the other parameters mentioned in this section, are included in Table 1. 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J Neuroscience. 2013; 33(44):17404–17412. Zhou Y, Danbolt NC. Gaba and glutamate transporters in brain. Front Endocrinol. 2013; 4(165) Author Manuscript J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 63 Handy et al. Page 20 Author Manuscript Author Manuscript Author Manuscript Fig. 1. Simplified schematic of an astrocyte and its major calcium components. Arrows show the direction of calcium flux. Author Manuscript J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 64 Handy et al. Page 21 Author Manuscript Fig. 2. Author Manuscript Example traces of the four calcium response types produced from the model. From left to right these response types and IP3 parameters are: SP (A=0.2, drise = 10, rrise = 0. 2, and ddecay = 97), PL (A=0.375, drise =34, rrise=0.002, and ddecay= 138), MP (A=0.26, drise=41, rrise=0.12, and ddecay= 200), and LL (A=0.55, drise=39, rrise=0.002, and ddecay=179). Default channel parameters, found in Table 1, were used in all response types. The x- and y-scale bars are 10 seconds and 0.5 µM respectively. Author Manuscript Author Manuscript J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 65 Handy et al. Page 22 Author Manuscript Author Manuscript Fig. 3. Author Manuscript Variability of calcium response types and underlying bifurcation structure. A-C: (top panel) calcium time course (black) and IP3 stimulus (black, dashed) with default channel parameters. (lower panel) Calcium fluxes through channels. Positive value represents flow into the cytosol. A: SP, with A = 0.2 drise = 21, and rrise = 0.002, and ddecay = 97, (inset shows calcium dynamics in the ER). B: MP with A = 0.2, drise = 41, rrise = 0.15, and ddecay = 179. C: PL with A = 0.375, drise = 36, rrise = 0.002, and ddecay = 120. D: Bifurcation diagram using IP3 as the parameter. Shows stable steady states (black, curve), unstable steady states (black, dashed curve), and projection of maximum and minimum values of stable (gray, dotdashed curve) and unstable (light gray circles) oscillations. Hopf bifurcations occur at labeled points (1 and 2). E: Calcium times course (black curve) in the presence of step increases in IP3 (black, dashed curve). Author Manuscript J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 66 Handy et al. Page 23 Author Manuscript Author Manuscript Fig. 4. Author Manuscript Effect of blocking SOC channels on response times and phase space. A-C: Calcium response types with default parameters (black curve), υSOC = 0.3140 (gray curve) and υSOC = 0 (light gray curve). A: SP, with A = 0.2 drise = 21, and rrise = 0.002, and ddecay = 97. B: MP with A = 0.2, drise = 41, rrise = 0.15, and ddecay = 179. C: PL with A = 0.375, drise = 36, rrise = 0.002, and ddecay = 120. D: The(c, ctot)-phase space with c-nullcline (light gray, dashed) and ctotnullcline (gray, dashed curve), along with the calcium transients (in black, gray, and light gray curves) corresponding to those in A (left panel). The color scale corresponds to the value of δ · JSOC(cER). E: Calcium response (black curve) to a prolonged IP3 stimulus (black, dashed curve) with υSOC = 0. Inset shows the corresponding bifurcation diagram of the system. F: Two parameter bifurcation diagram tracking the Hopf bifurcation points (black curve). Shaded region corresponds to regions that support stable oscillations. Gray dots represent the Hopf bifurcation points with default parameters. Notice log-scale on the y-axis. Author Manuscript J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 67 Handy et al. Page 24 Author Manuscript Fig. 5. Author Manuscript Simulation of dual blocking (SERCA and SOC) experiment. A: The system rests at steady (Phase I), and at t = 50s, calcium is removed from the system by setting υin = υSOC = 0 (Phase II). At t = 100s, υSERCA = 0 in order to simulate the application of the drug thapsigargin, and an increase in cytosolic calcium in observed (Phase III). Finally, at t = 350s, calcium is reintroduced to the system by restoring υin = 0.05, and υSOC to various amounts (Phase IV). υSOC = 1.57 is shown in black, υSOC = 0.4239 in gray, and υSOC = 0 in light gray. B: Figure show the corresponding fluxes for the case υSOC= 1.57. JIP3R is 0 throughout this simulation (not shown). Author Manuscript Author Manuscript J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 68 Handy et al. Page 25 Author Manuscript Author Manuscript Fig. 6. Effect of blocking SERCA channels on response times and bifurcation structure. A-C: Calcium response types with default parameters (black), υSERCA = 0.45 (gray). A: SP, with A = 0.2 drise = 21, and rrise = 0.002, and ddecay = 97. B: MP with A = 0.2, drise = 41, rrise = 0.15, and ddecay = 179. C: PL with A = 0.375, drise = 36, rrise = 0.002, and ddecay = 120. D: Two parameter bifurcation diagram tracking the Hopf bifurcation points (black curve). Shaded region corresponds to regions that support stable oscillations. Gray dots represent the Hopf bifurcation points with default parameters. E: Bifurcation diagram with υSERCA = 0.45 (stable steady state shown in black), overlaid with the calcium transient from panel B (gray). Author Manuscript Author Manuscript J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 69 Handy et al. Page 26 Author Manuscript Author Manuscript Fig. 7. Author Manuscript Effect of blocking PMCA channels on response times and bifurcation structure. A-C: Calcium response types with default parameters (black curve), υPMCA = 0 (gray curve). A: SP, with A = 0.2 drise = 21, and rrise = 0.002, and ddecay = 97. B: MP with A = 0.2, drise = 41, rrise = 0.15, and ddecay = 179. C: PL with A = 0.375, drise = 36, rrise = 0.002, and ddecay = 120. D: Two parameter bifurcation diagram tracking the Hopf bifurcation points (black curve). Shaded region corresponds to regions that support stable oscillations. Gray dots represent the Hopf bifurcation points with default parameters. E: Bifurcation diagram with υPMCA = 0, overlaid with the calcium transient from panel C (gray curve). Shows stable steady states (black, curve), unstable steady states (black, dashed curve), and projection of maximum and minimum values of stable (gray, dot-dashed curve) and unstable (light gray circles) oscillations. Author Manuscript J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 70 Handy et al. Page 27 Author Manuscript Author Manuscript Fig. 8. Frequency curves for the oscillations under different parameters. Left: decreasing υSOC Middle: increasing υSERCA Right: decreasing υPMCA. Author Manuscript Author Manuscript J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 71 Handy et al. Page 28 Author Manuscript Author Manuscript Fig. 9. Three-dimensional space of parameter variation over entire IP3 distribution showing differences in calcium response type distributions. From left to right, the three panels consider the followings bins for υPMCA: [5.0, 8.33], [8.33, 11.67], and [11.67, 15.0]. The three rows bin υSERCA according to [0.45, 0.75] (bottom), [0.75, 1.05] (middle), and [1.05, 1.35] (top). Within each panel, the columns show variations in υSOC according to [0.785, 1.31] (left), [1.31, 1.83] (center), and [1.83, 2.36] (right). The distributions represent that calcium response types recorded in each subspace (SPs are in black, PLs are in gray, MPs are in light gray, and LLs are in white). Author Manuscript Author Manuscript J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 72 Handy et al. Page 29 Author Manuscript Fig. 10. Author Manuscript Three-dimensional space of parameter variation over low amplitude IP3 transients. Organization follows the same as Fig. 9. Author Manuscript Author Manuscript J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. 73 Handy et al. Page 30 Table 1 Author Manuscript Model parameters. γ is from Ullah et al. (2006), kSERCA and υSERCA are from De Pittà et al. (2009), υin is from Lavrentovich and Hemkin (2008), υSOC is found in Croisier et al. (2013), and d1, d2, d3, and d5 are from the model developed by De Young and Keizer (1992). Author Manuscript Author Manuscript Parameter Description Value Units γ (Cyt vol) / (ER vol) 5.4054 – υIP3R Max IP3 Receptor Flux 0.222 s-1 υER_leak Cytosol to ER leak 0.002 s-1 υin Rate of leak into Cytosol from Plasma Membrane 0.05 µM s-1 kout Rate of leak out of Cytosol from Plasma Membrane 1.2 s-1 υSERCA Max SERCA Flux 0.9 µM s-1 kSERCA Half-Saturation for SERCA 0.1 µM υPMCA Max PMCA Flux 10 µM s-1 kPMCA Half-Saturation for PMCA 2.5 µM υSOC Max SOC channels Flux 1.57 µM s-1 kSOC Half-Saturation for SOC channels 90 µM δ Ratio of membrane transport to ER transport 0.2 – d1 Dissociation constant for IP3 0.13 µM d2 Dissociation constant for Ca2+ inhibition 1.049 µM d3 Receptor dissociation constant for IP3 0.9434 µM d5 Ca2+ activation constant 0.08234 µM a2 Ca2+ inhibition constant 0.04 µM s-1 drise Rate of Exponential Growth 0.002-12 s-1 ddecay Duration of IP3 decline 15-220 s drise Duration of IP3 increase 1-41 s A Max amplitude of IP3 transient 0.2-0.9 µM Author Manuscript J Comput Neurosci. Author manuscript; available in PMC 2018 June 01. CHAPTER 4 CA2+ MEASUREMENTS AND PROBABILISTIC MODELING REVEAL STIMULATION FREQUENCY-DEPENDENT RESPONSE PATTERNS IN ASTROCYTES 4.1 Abstract Astrocytes communicate bidirectionally with neurons and perform important functions in the brain, often through their intracellular Ca2+ signaling. We investigated how astrocyte Ca2+ activity, despite its heterogeneity, reflects the signals astrocytes receive and changes with repeated stimulation. We examined astrocyte spontaneous activity and responses to focally-applied, periodic ATP stimulation with varying frequencies in the mouse cortex. We developed a data-driven, time-varying Hidden Markov Model that describes the stochasticity of astrocyte Ca2+ activity and used it to interpret our experimental data. We found that astrocytes encode the presence of stimuli in a probabilistic manner and that, depending on the stimulation frequency, their response probability increases or decreases with successive stimulations, relative to spontaneous activity levels. We characterized potential refractory and negative feedback mechanisms responsible for this dual behavior. Our results provide insight into how astrocytes may respond to ongoing neuronal activity and our model provides a novel tool for future analyses. 75 4.2 Introduction Growing evidence indicates that astrocytes, through their Ca2+ activity, perform a wide variety of essential brain functions, including synaptic modulation (Covelo and Araque, 2018; Di Castro et al., 2011; Kang et al., 1998; Khakh and Sofroniew, 2015; Liu et al., 2005; Wang et al., 2012). Similarly, neuronal activity is able to modulate astrocyte Ca2+ activity, particularly through G-Protein Coupled Receptor (GPCR) activation (Covelo and Araque, 2018; Haydon, 2001; Porter and McCarthy, 1996). Astrocyte Ca2+ activity is also altered during different physiological (Ding et al., 2013; Nimmerjahn et al., 2009; Paukert et al., 2014; Srinivasan et al., 2015) and disease states (ÁlvarezFerradas et al., 2015; Delekate et al., 2014; Jiang et al., 2016; Kuchibhotla et al., 2009; Plata et al., 2018). However, it is unclear what these changes in Ca2+ activity reveal about the signals astrocytes receive and which of its features are most informative and allow for the encoding of information. It is important to investigate these issues, given the numerous Ca2+-mediated functions of astrocytes, their bidirectional communication with neurons, and their sensitivity to changes within the brain. Studying astrocyte Ca2+ signals is challenging, for a number of reasons. First, we require a convenient way to stimulate the astrocytes, but available tools are very limited (Khakh and McCarthy, 2015). Here, we used focal, brief ATP pulses to induce Ca2+ transients via purinergic receptors (Thrane et al., 2011, 2012), as a surrogate for neural input, in order to have better control over the type and timing of inputs to astrocytes (see Discussion). Second, evoked astrocyte Ca2+ activity is highly variable among different astrocytes, among different regions within an astrocyte, and within one astrocyte region (James et al., 2011; Jiang et al., 2016; Skupin and Falcke, 2007; Taheri et al., 2017; Tang 76 et al., 2015; Xie et al., 2012). These evoked events occur primarily through GPCR activation and subsequent inositol (1, 4, 5)-trisphosphate (IP3) production, leading to the release of Ca2+ from intracellular stores through IP3 receptor (IP3R) channel activation (Haydon, 2001). Third, astrocytes also exhibit spontaneous Ca2+ events (Agarwal et al., 2017; Aguado et al., 2002; Gee et al., 2015; Haustein et al., 2014; Skupin et al., 2008; Srinivasan et al., 2015; Wang et al., 2006), which occur stochastically (Croft et al., 2016; Skupin et al., 2008, 2010) and are often difficult to distinguish from evoked events. Spontaneous events likely arise from multiple pathways: single IP3R channel fluctuations that trigger collective channel openings within a cluster (Falcke, 2003; Skupin and Falcke, 2010), transient openings of mitochondrial permeability transition pores (Agarwal et al., 2017), and extracellular Ca2+ influx (Rungta et al., 2016; Srinivasan et al., 2015). Both spontaneous and evoked astrocyte Ca2+ events occur in a probabilistic manner (Croft et al., 2016; Skupin et al., 2008) and have diverse spatial and temporal patterns which represent a continuum, despite often being divided into discrete categories for data analysis (Croft et al., 2016; Handy et al., 2017; Jiang et al., 2016; Taheri et al., 2017; Wu et al., 2014). Due to their complexity, it is necessary to develop data analysis and computational tools to study astrocyte Ca2+ signals. Most computational models of astrocyte Ca2+ activity are mechanistic models based on differential equations, thus having numerous free variables. While such models can be extremely useful, they necessitate several assumptions and simplifications, e.g. ignoring spontaneous Ca2+ activity or the heterogeneity of Ca2+ events (see (Taheri et al., 2017; Toivari et al., 2011) for a discussion and (Manninen et al., 2018) for a recent review of models). Few models of astrocytes and other cell types have taken a top-down, 77 statistical modeling approach to describe Ca2+ events directly (e.g. (Skupin and Falcke, 2007, 2010; Tilūnaitė et al., 2017)). These models often focus on activity in astrocyte somas, simplify Ca2+ activity to a point process, and do not consider the dynamic time course of stimulation that cells are exposed to in vivo. Nevertheless, statistical models are advantageous for several reasons: They require minimal assumptions about the biological system’s mechanisms, and thus are especially useful when systems are as complex as astrocyte Ca2+ signaling; they are typically computationally cheap; they are valuable tools for interpreting experiments from a population of cells, such as Ca2+ imaging data; and they can have biological significance since they often implicitly account for certain mechanisms. In this work, we investigate how astrocytes translate the chemical signals they receive into Ca2+ signals and whether their responses to successive stimulations change depending on the stimulation frequency. Focusing especially on astrocyte fine processes, we examine both spontaneous and evoked Ca2+ activity; for the latter, we focally apply brief (60 ms) pulses of ATP with varying frequencies (ranging from 0.25 to 4 stimulations/min). Using a subset of our data and considering both Ca2+ event durations and inter-event intervals, we develop a Hidden Markov Model (HMM) with time-varying transition probabilities that describes astrocyte Ca2+ activity. Comparing results from our model and experiments, we find that, depending on the stimulation frequency, astrocyte response probability can either increase or decrease relative to spontaneous activity levels. This dual behavior is observed at the population level, rather than the single-cell level. We then use the model to identify potential biological mechanisms resulting in this complex behavior. Our study provides insight into how astrocytes may encode 78 information and what their Ca2+ dynamics reveal about the signals they receive from their environment and nearby cells. Moreover, our modeling approach provides a tool for studying astrocyte Ca2+ signaling in a variety of physiological and pathophysiological contexts. 4.3 Results 4.3.1 Spontaneous and periodically-evoked astrocyte Ca2+ events are highly variable We examined both spontaneous astrocyte Ca2+ activity, as well as responses to periodic focal ATP pulses (60 ms durations, with varying application frequencies). Since we mainly used tdTomato to select regions of interest (ROIs) and we selected a specific number of ROIs from each astrocyte (see Methods), we were able to analyze astrocyte Ca2+ signals with minimal selection bias. Time-lapse images of astrocyte activity before and after one ATP stimulus are shown in Figure 4.1A. To show the variability observed during spontaneous and evoked activity, we present, in Figure 4.1B, example Ca2+ traces from five astrocyte somas (same ones as in Fig. 4.1A) and one process from each. Consistent with other studies (Agarwal et al., 2017; Gee et al., 2015; Haustein et al., 2014; Skupin et al., 2008, 2010), we saw irregular, variable spontaneous Ca2+ events in astrocytes. Also, within a given astrocyte, spontaneous events of somas and processes were often dissimilar. When multiple pulses of ATP were introduced, similar variability was seen among different cells and between the soma and processes of the same cell. Moreover, a given astrocyte ROI often responded differently to multiple experimental trials that used the same stimulus train (Fig. 4.1B), despite long delays between trials (10 mins in all our experiments). Lastly, the same astrocyte region responds heterogeneously 79 A -5s -5s tdT, Alexa 594 GCaMP5 Merge 1 2 3 5s 5 4 + ATP 5s 20 s 20 s 40 s B Spontaneous Stim. Period: 1 min (Trial 1) Stim. Period: 1 min (Trial 2) Somas 100 0 -100 -200 -300 250 1 2 -400 150 Processes 0 -300 -400 -100 50 3 4 5 400 0 -200 -50 -300 -100 0 20 40 60 -150 80 -400 100 -200 300 0 50 100 150 200 250 300 1 2 40 60 3 4 -200 -300 5 5 300 300 -400 400 0 50 100 150 200 250 300 350200 -500 100 0 -100 -200 -200 -200 3 4 -300 5 20 -100 3 4 -100 -400 -500 80 0 -400 -500 100 -600 -300 5 -400 0 0 -100 3 -3004 4 5 1 2 100 1 20 0 3 1 2 100 100 2 350 50 100 150 200 250 300 ATP 200 200 200 1 -100 -200 0 100 200 100 100 1 2 200 3 4 5 300 200 300 200 Stim. Period: 30 s ATP ATP -600 350 0 400 50 100 150 200 250 300 350 1 400 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 2 3 4 5 400 Figure 4.1. Variability of spontaneous and periodically-evoked astrocyte Ca2+ events. (A) The first image shows the tdTomato signal, indicating astrocyte locations, as well as the pipette, filled with ATP and red dye (Alexa 594) for visualization. The other images are time-lapse images of the Ca2+ activity during one ATP application (corresponding to the first ATP stimulus in the 1 min stimulation period traces in B); the times in the upper right corners corresponds to the time relative to the ATP application. The cell numbers indicated in the image from t = -5 s correspond to the cells numbered in B. Zoomed in time-lapse images of GCamP5 signals from Cell 3 (boxed region in t = - 5 s) are shown to the right, with an example of a selected ROI in its process shown (scale bars, 20 µm). (B) Example Ca2+ traces from the same cells in A, including spontaneous activity and periodic ATP stimulation with 1 minute (two trials) and 30 s inter-stimulus intervals. Dashed lines indicate ATP stimulation times (Scale bars: 50% ΔF/F0 and 25 s). 80 to consecutive ATP applications within one experimental trial, regardless of the time interval between applications. In response to periodic stimuli, multiple factors result in even greater variability than seen during single ATP pulses (examined in our previous work (Taheri et al., 2017)). These factors include spontaneous Ca2+ events (which are seen more frequently in longer recordings than in short recordings of merely one ATP pulse), temporally heterogeneous Ca2+ responses to each stimulus, variable response latencies, and the interaction of these factors with the timing of periodic stimuli. Given this level of variability, it is often difficult to know whether a Ca2+ event is a spontaneous event (independent of the ATP stimulation), evoked directly from the ATP stimulation, or a result of subsequent inter- or intra-cellular mechanisms triggered by the ATP stimulation (e.g. possible feedback from Ca2+ to IP3 (Höfer et al., 2002; Politi et al., 2006), or intercellular Ca2+ waves traveling through gap junctions (Fujii et al., 2017; Höfer et al., 2002; Newman, 2001)). For example, the soma of Cell 3 (in Fig. 4.1B) responded clearly to the first three stimuli during the first 1 minute stimulation period trial; however, in subsequent trials (rightmost two columns), spontaneous events occurred before the first stimulus, making it difficult to interpret how well the cell responded to subsequent stimuli. This high degree of variability suggests that a probabilistic model incorporating the stochasticity of astrocyte Ca2+ transients can be helpful for interpreting the data. Prior to modeling, in the next section, we characterize whether these evoked transients differ from spontaneous transients and depend on stimulus frequency. 81 4.3.2 Evoked Ca2+ events differ from spontaneous events, but their kinetics do not depend systematically on the stimulation frequency From comparing spontaneous and evoked activity, such as those in Figure 4.1B, it appears that when ATP is present, there are more Ca2+ events with potentially different kinetics (e.g. prolonged Ca2+ responses in Cell 5’s soma and Cell 3’s process during 30 s interval pulses). We assessed such potential differences for astrocyte somas and fine processes, as well as potential differences during different stimulation frequencies, in Figure 4.2. First, spontaneous Ca2+ events were more likely to be observed in astrocyte processes than in somas (e.g. Fig. 4.1B); however, this result is confounded by the much larger number of ROIs in processes than in somas. Correcting for this confound, and recording for 2 minutes, we found no major difference in the proportion of active somas versus processes (44.7% and 41.5%, respectively; Fig. 4.2A in blue). However, active somas had fewer Ca2+ events per minute than active processes (mean frequency ± standard deviation of active regions: 0.88 ± 0.36 events/min in somas vs. 1.07 ± 0.54 events/min in processes; histograms and p-values in Fig. 4.2B), implying that active somas were more likely to be missed in shorter recordings. Moreover, spontaneous somatic Ca2+ events had smaller amplitudes than spontaneous events in processes (82.71 ± 57.96% vs. 113.76 ± 51.25%; histograms and p-values in Fig. 4.2B). In contrast, no difference in the event durations was observed (Fig. 4.2B; 23.42 ± 17.23 s for somas vs. 18.32 ± 13.63 s for processes). Next, we compared spontaneous and evoked activity (for somas and processes, separately). We estimated the proportion of spontaneously active ROIs in recordings of 82 Figure 4.2. Astrocyte responsiveness and Ca2+ kinetics are highly variable but differ between spontaneous and evoked activity. (A) The bar graph shows the percentage of ROIs in somas and processes that had at least one Ca2+ event at any time during recordings of spontaneous (restricted to 2 min) and evoked activity. Based on the percentage of spontaneously active processes, we estimated the expected percentages of responding ROIs for different recording durations, corresponding to each of the stimulation protocols (overlaid black boxes). (Here, recordings from the 15 s interstimulus interval protocol were restricted to four, rather than eight, stimuli.) (B) Histograms for the frequency, peak amplitude, and duration of Ca2+ events in astrocyte somas and processes during spontaneous and evoked (1 min inter-stimulus interval) activity. Non-responding ROIs were excluded. P-values were calculated using the MannWhitney U (for frequency) or Kruskal-Wallis (for amplitude and duration) tests. (C) The return maps in the two left columns show the dependence of each Ca2+ event amplitude or duration on the preceding event amplitude or duration (hence, only Ca2+ responses from traces with 2 or more events were included; data are from astrocyte processes). The return maps in the right column show the dependence of the total Ca2+ activity duration in each stimulation interval on the total duration in the preceding interval; thus, the maximum possible Ca2+ duration is equivalent to the stimulation period (zero durations indicate that no Ca2+ activity was present within the stimulation interval). The schematic illustrates the difference between the middle and right columns. These return maps are summarized in the bottom bar graphs, representing the mean (and s.e.m.) change in amplitude or duration in consecutive responses, or consecutive stimulation intervals (see text for details on statistics). For all graphs, the number of ROIs are (in order for processes and somas): Nspont=649 and 103, N2min=125 and 27, N1min=320 and 63, N30s=144 and 26, N15s=167 and 34; for each stimulation protocol, 4 to 10 experimental trials from 4 to 6 mice were used; for spontaneous data, 17 experimental trials from 9 mice were used. 83 84 different lengths (corresponding to recording times for each stimulation protocol) based on the proportion of spontaneously active processes in the first minute (37.8%) and first 2 minutes (41.5%) of recordings, and assuming that whether or not a ROI responds is random (with the same probability of responding among all ROIs). We found that the proportions of active ROIs (somas and processes) for all stimulation protocols were greater than the proportions of spontaneously active ROIs, implying that the presence of ATP recruits Ca2+ events in additional ROIs (Fig. 4.2A). Moreover, for somas and processes alike, Ca2+ event amplitudes were larger during stimulation than during spontaneous activity (mean ± standard deviation during 1 min period stimulation vs. spontaneous activity: 118.25 ± 77.53% vs. 82.71 ± 57.96% for somas, 153.02 ± 96.40% vs. 113.76 ± 51.25% for processes; histograms and p-values in Fig. 4.2B). On the other hand, event durations only increased significantly for processes during stimulation, even though stimulated somas exhibited a few rare events with prolonged durations (mean ± standard deviation during stimulation vs. spontaneous activity: 24.73 ± 34.95 s vs. 23.42 ± 17.23 s for somas, 23.69 ± 26.03 s vs. 18.32 ± 13.63 s for processes; histograms and pvalues in Fig. 4.2B). We then examined how different stimulation frequencies might affect astrocyte Ca2+ activity differently (e.g. possible facilitation or depression in Ca2+ events), and at what point, if any, refractoriness in Ca2+ activity inhibits astrocyte responses. First, from Figure 4.2A alone and given the variability of Ca2+ activity in different cells and trials, it is not clear whether the different proportions of active ROIs during different stimulation frequencies are meaningful. When comparing the Ca2+ event amplitudes and durations during different stimulation frequencies (data not shown), we also found no meaningful 85 trend or evidence of refractoriness. To delve deeper, we assessed how Ca2+ kinetics might change over repeated stimulations during different stimulation frequencies. We examined the change in amplitude and duration between consecutive events in each Ca2+ trace (Fig. 4.2C, left and middle columns, for spontaneous activity and 1 min and 15 s inter-stimulus intervals, for astrocyte processes). In these return maps, each point represents how the amplitude or duration changed from one Ca2+ event (the nth event, with Amp(n) and Dur(n)) to the next (the (n+1)th event, with Amp(n+1) and Dur(n+1)), within one recording of a given ROI. One would expect that if facilitation in Ca2+ responses occurs within one ROI, the amplitude or duration would increase with successive events (i.e. points would be above the dashed line), while the opposite would be observed if depression in Ca2+ responses within one ROI occurred. While some Ca2+ traces exhibited an apparent increase (facilitation) or decrease (depression) in consecutive event kinetics at some point, both phenomena were present in many ROIs and in the overall population. Interestingly, similar apparent facilitation and depression were also observed in spontaneous activity, indicating that ATP stimulation did not have a clear effect on changing Ca2+ kinetics in consecutive events. We tested this by examining the change in kinetics (mean ± s.e.m. in Fig. 4.2C bar graphs) for the spontaneous case and different stimulation protocols. We did not find any significant difference between these cases (Kruskal-Wallis, p>0.3 for amplitude, and p>0.71 for duration, for all pairs) or any evidence that the overall change in kinetics is nonzero (Wilcoxon signed rank test, p>0.087 for amplitude and p>0.078 for duration, for all cases). For the return maps in the rightmost column of Figure 4.2C, we separated each 86 Ca2+ trace into the four intervals of stimulation (or eight intervals, for 15 s stimulation periods), and measured the total duration of Ca2+ activity during each consecutive stimulation interval (illustrated in the schematic in Fig. 4.2C). Here, points on the x or y axes (with zero durations) indicate that no Ca2+ activity was observed during the stimulation interval. One would expect that facilitation or depression within a Ca2+ trace would result in more points on the y-axis or x-axis, respectively, or in significantly more points above or below the dashed line. Once again, we found little evidence of facilitation or depression: The change in duration was not significantly different between different stimulation protocols (Kruskal-Wallis, p>0.27 for all pairs) and, in all except the 1 minute inter-stimulus interval stimulation protocol (Wilcoxon signed rank test, p=0.033), the overall change was zero (p>0.19 for the other three protocols). Even for the 1 minute inter-stimulus interval protocol, the mean decrease in Ca2+ duration was merely 1.71 s. Taken together, these results suggest that, despite high variability between different astrocyte regions and between trials, the astrocyte population as a whole responds to stimulation with an increase in the number of active ROIs and increased Ca2+ event amplitudes (for somas and processes) and durations (for processes). However, there are no overarching trends in how the kinetics of Ca2+ events change with repeated stimulation and during different stimulation frequencies. While some Ca2+ activity traces may exhibit an increase or decrease in kinetics with consecutive stimuli, these changes in kinetics vary between regions, vary over time for a given region, and do not appear to be systematic or different from those that occur randomly during spontaneous activity. 87 4.3.3 Astrocyte spontaneous and evoked activity can be described by Hidden Markov Models In response to periodic stimulation, we saw an increase in astrocyte Ca2+ activity compared to spontaneous levels; but a simple analysis suggests no clear differences in response kinetics, or the tendency for facilitation or depression, during different stimulation frequencies. To analyze these finite-length data with more rigor and in more detail, we needed to account for the stochasticity of Ca2+ activity. Therefore, we chose to represent the data using a probabilistic framework with Hidden Markov Models (HMMs). A HMM in this context implies that a given astrocyte region has a certain probability of beginning or ending a Ca2+ event at any moment in time. This probability is generally independent of the astrocyte’s Ca2+ activity history but depends on the cell’s current state. However, whether or not the probability changes over time and under different experimental conditions depends on how the model is constructed. To develop and validate the model, we focus first on data from astrocyte processes due to the larger body of data available for these regions compared to the somas (necessary for statistical modeling). Further, because the processes are in contact with neurons and blood vessels and likely play a role in regulating their activity, Ca2+ events in the processes are possibly more physiologically relevant. HMMs consist of “hidden” states, i.e. not experimentally discernible, with specific probabilities of transitioning between connected states. Each hidden state corresponds to an “observable” state, which may or may not be shared between the hidden states. We first binarized the experimental data to include only two observable states: On and Off (corresponding to the presence or absence of a Ca2+ event, 88 respectively). A few example traces, with events detected automatically and transformed into a binary signal, are shown in Figure 4.3A. This approach of ignoring response amplitudes allowed us to reduce the number of free parameters substantially and thus analyze these highly variable data with confidence. We had also considered modeling multiple On states to distinguish between small and large events, by categorizing events based on their amplitudes or based on both their amplitudes and durations. However, our data showed a clear monomodal distribution of amplitudes (Fig. 4.2B) and we found that amplitude and duration were not correlated (Fig. 4.3B; correlation coefficient < 0.5 for spontaneous and < 0.25 for evoked events with all stimulation frequencies). Moreover, since the observation of very large amplitude or prolonged Ca2+ events is limited to evoked conditions (Fig. 4.3B), such categorization would be unreasonable for a comprehensive model of astrocyte Ca2+ activity. Hence, we found it reasonable to place all Ca2+ events into one observable category (an On state) and use a HMM with two observable states. We developed and considered multiple HMMs (Fig. 4.3C), with different numbers of and connectivity between hidden states, based on our knowledge of the biological mechanisms underlying Ca2+ responses (including IP3 dependence, recruitment of various Ca2+ channels and pumps during a Ca2+ event, and depletion of intracellular Ca2+ stores). For each model, we used a subset of experimental, binarized Ca2+ traces to estimate two sets of transition rates between the hidden states: one for spontaneous activity and another for evoked activity (specifically, the first stimulation interval in the 1 stim/min experiments; see Methods). Our goal was to find the simplest model – one with the fewest hidden states – that 89 Figure 4.3. Data simplification and development of a Hidden Markov Model to describe astrocyte activity occurring spontaneously and in response to a single stimulus. (A) Example Ca2+ traces (blue), with automatically detected events (peaks indicated with open circles, start and end points indicated with filled circles), and the binarized trace (black) consisting of On and Off states. Traces are from 2 minute stimulation periods (dashed lines indicate stimulation times). (B) Amplitude versus duration of individual Ca2+ events during spontaneous and evoked (1 min stimulation periods) activity. Nspont=327 events in 649 trace from 17 experimental trials, N1min=839 events in 320 traces from 10 experimental trials. Inset is zoomed in lower amplitudes and durations for better visibility. (C) Hidden Markov Models examined to fit to spontaneous Ca2+ activity and a subset of evoked data (only the first stimulation interval of 1 min stimulation period experiments). The hidden state abbreviations, and corresponding observable states, are as follows: C = Closed (Off); A = Activated (Off); R = Refractory (Off); O = Open (On). (D) Experimental On and Off dwell times (gray histograms and corresponding black dashed curves (median filtered from the histograms) during spontaneous (top) and evoked (bottom) activity, overlaid with dwell times generated from four models from B (numbers 1, 2, 4, and 8; legend names correspond to model hidden states; 5,000 simulations for each). Dwell times from experimental data are restricted to the first minute of spontaneous and ATP-evoked activity. (E) Illustration explaining the potential biological meaning of the COO model’s hidden states. 90 91 describes the spontaneous and evoked (1 min after one ATP pulse) data. We evaluated this by examining the estimated transition rates (see Methods) and qualitatively comparing the On and Off dwell times generated from the models with those from the experimental data (Fig. 4.3D). We found that two Open states (O1 and O2) were required in order for the model’s On dwell times to match experimental ones; otherwise, very low On dwell times were overrepresented (e.g. CAO and CO models in Fig. 4.3D). The CAROO and COO models had distributions of On and Off dwell times closest to experimental distributions and similar to one another (Fig. 4.3D). We decided on the COO model (#8 in Fig. 4.3C) to describe astrocyte Ca2+ activity because it was the simpler of the two best-fit models, thus reducing the chances of overfitting. The estimated transition rates of the COO model are listed in Table 4.1. The prior distribution (fixed for the spontaneous and evoked cases; see Methods) was 0.86 (for C), 0.03 (for O1), and 0.11 (for O2). Interestingly, the transition rate from O1 to C states (𝒓𝑶𝟏→𝑪 ) was estimated to be approximately zero for both spontaneous and evoked cases, meaning once the cell goes into an Open (or On) state, it can only return to the Closed (or Off) state by first going to the second Open state (O2), and cannot return immediately from O1. This allows for a lower occurrence of very short duration Ca2+ events in the model, consistent with Table 4.1. Estimated model transition rates. Transition rates, 𝒓 (s-1), between the three hidden states of the COO model during spontaneous and evoked conditions. 𝒓𝑪→𝑶𝟏 𝒓𝑶𝟏→𝑶𝟐 𝒓𝑶𝟐→𝑶𝟏 𝒓𝑶𝟐→𝑪 Spontaneous 0.0070 0.2750 0.0040 0.0500 Evoked 0.0122 0.1468 0.0602 0.0646 92 experimental data (histograms in Fig. 4.2B and 4.3D). This is also biologically plausible, as it indicates that the probability of the cell going from On to Off changes over time. The On to Off transition probability is initially very low, e.g. due to positive feedback from cytosolic Ca2+ onto IP3Rs (Patel et al., 1999; Taylor, 1998) after these receptors initially open and allow for an influx of endoplasmic reticulum (ER) Ca2+ into the cytosol. After this initial stage, cytosolic Ca2+ levels may be high enough to deactivate IP3Rs (delayed negative feedback (Patel et al., 1999; Taylor, 1998)) and cytosolic IP3 may have decreased to levels too low to keep the IP3Rs open. At this second stage of a Ca2+ event, the likelihood of ending the event and transitioning to an Off state is much more likely. If at this second stage, before the Ca2+ event has fully subsided (i.e. before the cell returns to the Off/Closed state), IP3Rs become reactivated (e.g. due to IP3 diffusion from other astrocyte regions or stochastic opening of IP3Rs (Dupont and Croisier, 2010; Falcke, 2003)), the cell may re-enter the first stage of Ca2+ activity for some additional time, during which returning to the Off state is, again, highly unlikely. The illustration in Figure 4.3E portrays, based on this description, the hidden states that conceivably underlie two example astrocyte Ca2+ events. Given this interpretation of the biological mechanisms behind the COO model’s hidden states, it makes sense that in the evoked case, the probability of transitioning from O1 to O2 decreases (i.e. the cell stays in the first stage of Ca2+ activity for a longer time) and the probability of O2 to O1 increases (i.e. the cell goes back to the first stage of Ca2+ activity) due to the presence of ATP and subsequent IP3 elevation. Similarly, the higher transition probability from C to O1 in the evoked case, compared to the spontaneous case, is expected. Interestingly, the probability of transitioning from O2 to C increases in the 93 evoked case, which may be related to the delayed negative feedback of Ca2+ on IP3Rs: additional increase in Ca2+ concentration resulting from stimulation deactivates IP3Rs sooner. To better understand the effect of transition probability changes from spontaneous to evoked cases, we generated a set of simulations for each case (N=5,000, each 4 min long). Using the generated sequences of hidden states, we find the percentage of the time that the cell spends in each hidden state, and the percentage of the cell’s transitions into each state over the course of the simulation (Table 4.2). As expected, when the cell is stimulated, it is less likely to transition into and stay in the C state, but more likely to transition into and stay in the O1 and O2 states; in particular, the amount of time spent in O1 increases. We conclude that the three-state COO model is able to describe both spontaneous and evoked astrocyte Ca2+ activity, with differences in the two cases arising from differences in transition probabilities between the model’s hidden states. Also, even though the COO model is a HMM, following the Markov property of being memoryless and having constant transition probabilities, this model, in effect, allows a single Ca2+ event to have different probabilities of turning off throughout its course, consistent with Table 4.2. Effects of transition rates corresponding to spontaneous or evoked activity. Proportion of time spent in, or of transitions into, each of the three hidden states of the COO model, given the estimated transition rates during spontaneous and evoked activity. Hidden States C O1 O2 % of Time Spent in Each State % of Transitions into Each State Spontaneous Evoked Spontaneous Evoked 85.62 2.37 12.01 74.89 11.59 13.52 51.48 23.21 25.31 32.42 33.92 33.66 94 its underlying biological mechanisms. In the next section, we extend the model to one with transition probabilities that vary over time, similar to what is biologically expected during multiple ATP stimulations of astrocytes. 4.3.4 Astrocyte responses to multiple stimulations can be described by a time-varying Hidden Markov Model When an astrocyte is stimulated with an ATP pulse, we expect that the effect of the agonist lasts for some time, but is also not constant over this time. In other words, the agonist effect is greater when it is first introduced, but then gradually decreases. To account for this, we extend the COO model to one with transition probabilities that vary over time. Specifically, we expect that the transition probabilities are initially set at spontaneous levels, but reach evoked levels when an ATP pulse is applied. Eventually, after some unknown time, the transition probabilities return to spontaneous levels and remain at those levels until the next ATP pulse is applied. We assume that the switch from spontaneous to evoked rates with an ATP pulse is immediate, while the return to spontaneous rates is slow and linear (discussed further below), occurring over a total effect time of Teff (the only free parameter in our model). Figure 4.4 (A-B) illustrates this time-varying Hidden Markov Model (TV-HMM). Using this TV-HMM, with Teff = 80 s (see Fig. 4.4—figure supplement 1 for a description of how this value was chosen), we generated sequences comparable to experimental On-Off traces (Fig. 4.4C, both hidden and observable states are shown for the 1 stim/min protocol). As seen, with this model we are able to achieve On-Off traces with variability closely resembling the variability of experimental data: A cellular region 95 Figure 4.4. A Hidden Markov Model with time-varying transition rates describes astrocyte responses to multiple stimuli. (A-B) Schematic of the proposed time-varying HMM. During spontaneous activity, the cell’s Ca2+ activity follows the COO model with transition rates at the spontaneous level. When the ATP pulse is introduced, the transition rates of the Hidden Markov Model immediately switch to evoked levels (plus and minus signs indicate the increase or decrease of each rate, based on estimations in Section 1.3), then return to spontaneous levels linearly, with a total return (or effect) time of Teff. The cycle in A repeats for each stimulus presented. Depending on how long the stimulation period is relative to Teff, the transition rates between pulses may or may not return to spontaneous levels, as shown in the schematic in B (the transition rates are denoted by α; arrows indicate stimulation times). (C) Using the proposed model, with Teff = 80 s, we generated sequences of the hidden (blue; C, O1, or O2) and observable states (gray; Off or On) during the 1 minute stimulation protocol (examples shown; dashed lines indicate stimulation times). By averaging over all simulations (N=300), we obtain the probability of a cellular region being active, or On, at any time (green and cyan traces, each for one set of simulations). The corresponding histograms of On probabilities (Pr(On)) are plotted vertically. (D) Pr(On) histograms for N=300 simulations during spontaneous activity (blue) and evoked activity (red), compared to similar histograms for experimental data (same N values as reported in Fig. 4.2 for processes). Top two histograms are from 4 minutes of spontaneous or evoked (4 ATP stimulations, 1 min period, starting from the first stimulation) activity, and the bottom two are from 1 minute of activity (only the first stimulation interval of the 1 min period protocol). See text for p-values. (E) Model prediction of On probabilities over time for different stimulation protocols, given a very large number of simulations (i.e. large population of astrocyte ROIs) (N=5,000). When averaging over more sequences, a clearer pattern emerges in the Pr(On) traces (e.g. teal trace in comparison to the traces in C for the same 1 min period protocol; dashed lines correspond to the 1 min period stimulation times). See also Figure 4.4—figure supplement 1. 96 97 may respond to some stimuli, but not all, and each response may have varying durations and latencies (e.g. traces 4 and 5 in Fig. 4.4C). Also, these responses are confounded by spontaneous activity (e.g. traces 2 and 3 in Fig. 4.4C). Similar to experimental Ca2+ traces (Fig. 4.1B and 4.3A), one cannot infer the timing of stimuli from examining a handful of the simulated sequences. By averaging over many sequences, we can obtain the probability of having a Ca2+ event (i.e. being in the On state, Pr(On)) at any moment in time. With a fairly high number of sequences (e.g. N=300 simulations for each of the two Pr(On) traces in Fig. 4.4C), we begin to notice that stimulations increase the Pr(On) for the overall population. However, there is a high degree of variability between different simulation sets (green vs. cyan Pr(On) traces in Fig. 4.4C) and, still, one cannot accurately infer the timing of stimuli from such Pr(On) traces. Nonetheless, with a much higher number of sequences (e.g. N=5,000 in Fig. 4.4E), a pattern emerges in the Pr(On) trace and the timing of stimuli becomes clearer (the teal trace corresponds to the same 1 stim/min protocol in Fig. 4.4C). Notably, the immediate switch to evoked transition rates and the linear return to spontaneous levels in the model (Fig. 4.4B) is not reflected instantaneously in Ca2+ (OnOff) activity (Fig. 4.4E): the maximum Pr(On) is not achieved immediately and the Pr(On) does not reach spontaneous levels linearly or within the time Teff. This noninstantaneous manifestation of the model’s time-varying transition rates into observable On-Off states occurs because it takes time for a new set of transition rates to fully affect the system (i.e. the cell’s activity). Based on this model prediction, astrocyte Ca2+ activity in response to an ATP stimulation reaches its peak level ~28 s after the stimulus onset. At this time point, the largest number of ROIs are in an active/On state, as a result of two 98 phenomena: (1) most ROIs that became active early on remain active until this time (the average duration of evoked Ca2+ events in astrocyte processes is roughly 23 s (Fig. 4.2B), with some events lasting much longer); (2) newer ROIs continue being recruited until this time since different ROIs become active with different latencies (possibly, those directly stimulated by ATP become activated first, followed by those activated indirectly, e.g. through IP3 diffusion from neighboring astrocytes; see (Fujii et al., 2017)). Assuming we had an extremely large number of Ca2+ traces (N≈5,000), our TVHMM predicts that the Pr(On) trace for different stimulation frequencies would be as shown in Figure 4.4E. As indicated, for faster stimulations, each stimulus adds to the previous one and increases the Pr(On) until a maximum value is reached (which is equivalent to the Pr(On) levels reached if transition rates were kept constant at the estimated evoked rates). On the other hand, during slower stimulations, the Pr(On) during each stimulation interval is roughly the same; and if the stimulation rate is slow enough, the cellular activity is able to reach spontaneous Pr(On) levels between stimuli. To evaluate the model’s ability in predicting experimental results from different stimulation rates, we need to account for the variability between smaller sets of data, as we do next, because we do not have such an extremely large number of data points on the order of several thousands. In Figure 4.4D, we examined whether the predicted change in the Pr(On) levels during stimulation occurs in experimental data. From the model, we generated sequences (4 min each; N=300, comparable to the number of ROIs in experimental data) for spontaneous activity (in blue; transition rates kept constant at spontaneous levels) or evoked activity (in red; 4 stimulations, each 1 min apart; the first stimulus occurs at the 99 beginning of each simulation, i.e. no initial spontaneous activity is included). As seen from the Pr(On) histograms (top histograms in Fig. 4.4D), the model predicts a clear increase in the Pr(On) during stimulation relative to the spontaneous Pr(On) (mean ± standard dev., in order: 0.2280 ± 0.0253 vs. 0.1392 ± 0.0184; p<<1e-8, bootstrap test). We also examined Pr(On) histograms for 1 minute after the first stimulation (i.e. the first stimulus interval) and 1 minute of spontaneous activity, and found a similar increase in the Pr(On) (mean ± standard dev.: 0.2149 ± 0.0247 vs. 0.1419 ± 0.0170; p<<1e-8, bootstrap test). Interestingly, we found very similar Pr(On) values in the experimental data for both evoked and spontaneous activity, with a similarly clear increase in Pr(On) in the presence of ATP stimulation. Mean ± standard deviation for all four stimuli and for the duration of spontaneous recordings, in order: 0.2142 ± 0.0367 versus 0.1307 ± 0.0285 (second row of histograms in Fig. 4.4D; p<<1e-8, bootstrap test). Mean ± standard deviation for the first stimulation interval and for the first minute of spontaneous recordings: 0.2383 ± 0.0276 versus 0.1348 ± 0.0172 (last row of histograms in Fig. 4.4D; p<<1e-8, bootstrap test). It is noteworthy that, given the variability in different sets of simulations of ~N=300 each (Fig. 4.4C-D), the Pr(On) histogram shapes will be somewhat different in each simulation set, but will have consistent means and standard deviations, similar to experimental results. Our results from the model, also validated thus far in the 1 stim/min experiments, indicate that astrocytes encode the presence of stimuli in a probabilistic manner, with increased likelihood of exhibiting Ca2+ events, despite high variability in the response pattern of individual astrocyte regions. Moreover, our time-varying model correctly predicts the probability of an astrocyte process being active during spontaneous activity, 100 in response to a single stimulus, and during periodic stimulation at a rate of 1 stim/min. It is also encouraging that only one free parameter – the duration of the linear switch between transition rates after stimulation (Teff) – is required to fit these data well, with the other parameters (the transition probabilities) estimated by a machine-learning algorithm using a subset of data. 4.3.5 Astrocytes respond to repeated stimulation in two opposing manners depending on the stimulation frequency We next examined how Pr(On) depends on the stimulation rate, in order to extract information about adaptation or refractoriness in astrocyte Ca2+ activity. To contend with the high variability of ensemble averages in experimental data (Fig. 4.4C), we averaged Pr(On) values in each stimulation interval for a given experimental trial, then calculated the mean and standard deviation among different experimental trials (Fig. 4.5, in red). Similarly, in the model, we ran 6 sets of simulations (representing 6 experimental trials) with 50 sequences generated for each (representing 50 ROIs); we then averaged the Pr(On) values for each interval (Fig. 4.5, in black) but also show the individual sets of simulations (Fig. 4.5, in gray) to illustrate the level of variability expected in individual trials resulting from intrinsic data variability. For spontaneous activity, we calculated mean Pr(On) values separately for 30 s intervals throughout the Ca2+ recordings (and model simulations) to show the stimulus-independent variability between Pr(On) values in each time interval. For all stimulation rates except the highest (4 stim/min), we saw consistent results: (1) Pr(On) values were consistently above spontaneous Pr(On) values, further 101 Figure 4.5. Responses of astrocyte processes to consecutive stimuli depend on stimulation frequency. The probability of having a Ca2+ response, Pr(On), at any second for a given astrocyte process during different stimulation protocols (in red, except for the 4 min period protocol which is shown in maroon and overlaid with the 2 min protocol, due to different recording protocol; see text and Methods). Symbols and error bars show mean +/- standard deviation. Pr(On) values from the COO model for the same stimulation protocols are overlaid (individual trials in gray, mean +/- standard deviation in black). For spontaneous activity (first graph), recordings were divided into 30 s intervals to show the variability in Pr(On) levels in the absence of stimulation. Average Pr(On) during spontaneous activity shown in dashed blue lines. Only experimental results for the 15 s protocol deviated from the model (* and ** indicate p<0.05 and p<0.01; bootstrap test). The lower right graph shows, for each protocol, the percentage of responding ROIs during each stimulation (mean ± s.e.m.; excluding ROIs that never responded during a recording). Same N values as those reported for astrocyte processes in Figure 4.2. See also Figure 4.5—figure supplement 1. 102 validating that astrocytes encode the presence of stimuli probabilistically with an increased Pr(On) at any given moment in time; (2) our model estimated correct Pr(On) levels (p>0.05, bootstrap test); (3) no change in Pr(On) levels between consecutive stimulations was detectable (given the expected level of data variability), implying no adaptation (depression or facilitation) in Ca2+ responses during low frequency stimulation. Conversely, during high frequency stimulation (4 stim/min), astrocyte processes initially responded as expected, but from the second stimulus onward displayed decreased Pr(On) values. In fact, the Pr(On) dropped to lower than spontaneous levels after the third stimulus, and remained at these depressed levels. This experimental finding is in stark contrast to the model prediction of Pr(On) levels (p-values in Fig. 4.5), suggesting that some form of refractoriness in astrocyte processes exists, resulting in depression of Ca2+ activity, during repeated stimulation at high frequency. To investigate the reason for this decreased Pr(On) level, we examined the mean Ca2+ duration (time spent in the On state) per stimulation interval (Fig. 4.5—figure supplement 1A) and the fraction of active ROIs responding to each stimulus (Fig. 4.5, lower right graph). While the mean Ca2+ duration was fairly constant throughout the stimulation, the fraction of responding ROIs decreased with consecutive stimulation and was responsible for the lower Pr(On) levels observed during high frequency stimulation. 103 4.3.6 Refractory and negative feedback mechanisms underlie astrocyte response patterns to high frequency stimulation To characterize the mechanisms underlying decreased Pr(On) levels during high frequency stimulation, we started by accounting for refractoriness in Ca2+ responses in the model in two ways (Fig. 4.6). First, we considered an extreme form of refractoriness in the response of astrocytes to stimuli by assuming that the astrocyte is only affected by the first stimulus and ignores the rest completely (top graphs in Fig. 4.6A-B). This refractoriness may be due to receptor desensitization or decreased IP3 formation from other pathways downstream of receptor activation. This modification somewhat lowered the expected Pr(On) levels (left black and gray curves in Fig. 4.5B), but not to the levels observed experimentally. Second, we considered that perhaps internal Ca2+ stores are depleted for a longer time during intense stimulation. The COO model already accounts for some store depletion since, after a Ca2+ event, the astrocyte enters and remains in the C state for some time. To include additional store depletion, we added a Refractory (R) state in the COO model, causing the astrocyte to linger in the Off state for longer after it leaves the On state (middle graphs in Fig. 4.6A-B; see Methods for details). We found that even with low R to C transition rates (µ = 30 s in Fig. 4.6A), the Pr(On) values do not decrease to lower than spontaneous values as quickly as the experimental data. This discrepancy arises because, in the model, a sufficient number of astrocytes must first become active (reach the O2 state) before they can enter the R state; this requires a higher number of stimulations. The model’s inability to reach low enough Pr(On) values during high frequency 104 Figure 4.6. Refractory and negative feedback mechanisms underlie the response pattern of astrocyte processes during high frequency stimulation. (A) Pr(On) values during high frequency stimulation (4 stim/min) for experiments and modified versions of the model (details in B) that take into account various forms of refractory and negative feedback mechanisms. In the lower plot, the model with all three mechanisms (two forms of refractoriness and negative feedback) reproduces experimental results (* and ** indicate p<0.05 and p<0.01; bootstrap test). (B) To incorporate the first form of refractoriness (e.g. extreme form of receptor desensitization) in the model, transition rates were affected only by the first stimulus (top schematic). To account for the second form of refractoriness (e.g. additional store depletion and slow recovery from Off state during intense stimulation), a fourth hidden state was added to the COO model (middle schematic). To include negative feedback (lower schematic), the transition rate from the C to O1 state was decreased as specified in Methods and in Figure 4.6—figure supplement 1A. (C) The decrease in Pr(On) levels in the model stems from a lower percent of active ROIs per stimulation interval, similar to experimental data (red trace is the same as purple trace in Fig. 4.5, lower right plot). Note the apparent partial recovery of Pr(On) and percent of active ROIs in B and C in both the model and experiments. In the model, this occurs because ROIs in the R state start recovering. See also Figure 4.6— figure supplement 1. 105 stimulation, despite accounting for multiple sources of refractoriness, suggests that the depressed Ca2+ activity results from not only refractoriness, but also negative feedback onto ROIs. More specifically, the Ca2+ activity of even those ROIs that never responded to the stimulus must be suppressed. (The biological mechanisms are unknown and need to be investigated experimentally; see Discussion for more.) Indeed, Pr(On) levels similar to experimental data can be achieved in the model (Fig. 4.6A, bottom graph) by decreasing the C to O1 transition rate to include negative feedback (for details, see Methods, bottom schematic in Fig. 4.6B, and Fig. 4.6—figure supplement 1A). Furthermore, similar to experimental findings, the reason for the model’s lower Pr(On) values stems from a lower fraction of responding ROIs (Fig. 4.6C). Consistent with our results, inspection of the experimental raster plots reveals that, while some astrocyte processes are able to continue responding to successive high frequency ATP stimulations, most astrocyte processes, including those that did not respond to the stimuli, become silent and decrease their spontaneous activity as well (Fig. 4.5—figure supplement 1B). These findings suggest that the depression in Ca2+ activity is not so much a phenomenon observed in individual processes as it is a phenomenon of the population activity. Hence, we were not able to detect this effect of high frequency stimulation when examining individual ROIs in Figure 4.2C. We also examined Pr(On) levels for astrocyte somas. Given that there are far fewer somas than processes (N values reported in Fig. 4.2), we expect a much higher variability in the Pr(On) levels. Still, in all stimulation protocols, Pr(On) values were higher than during spontaneous activity (data not shown), with spontaneous Pr(On) levels almost identical to that of processes (mean ± standard dev.: 0.1325 ± 0.0373). In contrast 106 to astrocyte processes, somatic Pr(On) levels do not drop below spontaneous levels during high frequency stimulation (Fig. 4.6—figure supplement 1B). By the fourth stimulus, the Pr(On) value decreases to spontaneous levels, but increases again before another decrease, in a seemingly cyclic manner. A similar behavior is reproducible in the model by including the mechanisms discussed above, with the following modifications in their implementation for somas compared to processes: First, both the first and sixth stimuli are effective in the sense that they result in the switch of transition rates from spontaneous to evoked levels; second, µ is kept at 20 s (using 30 s, as in the processes, prevents the somatic Pr(On) from recovering after the fourth stimulus); third, no negative feedback mechanism is included. By comparing our experimental Pr(On) values for astrocyte somas and processes with our model predictions, we conclude that astrocytes overall do not exhibit any noticeable adaptation in their Ca2+ activity during low frequency stimulation, and continue responding to stimulation with an increased probability of exhibiting a Ca2+ event at any given second (~1.7 times the spontaneous levels). However, during high frequency stimulation (~4 stim/min), astrocyte Ca2+ activity in both somas and processes exhibits depression resulting, at least partially, from multiple sources of refractoriness. Additional negative feedback mechanisms suppress Ca2+ activity in astrocyte processes and contribute to further depression of Pr(On) levels to below spontaneous levels. 4.3.7 Individual astrocyte processes only partially reflect the timing of stimuli We investigated whether during different stimulation frequencies, the timing of astrocyte Ca2+ events changes such that it can be used for determining the timing of 107 stimuli. First, we examined whether or not the frequency of Ca2+ events reflects the stimulation frequency in any way. When considering Ca2+ traces from all ROIs, including those that did not necessarily display Ca2+ activity, we found no difference in the Ca2+ event frequency during different stimulation rates, though there was some significant difference during stimulation compared to spontaneous activity (Fig. 4.7—figure supplement 1). However, when we considered only those Ca2+ traces that had at least one event, we found that a trend in Ca2+ frequency with various stimulation rates emerges (Fig. 4.7A). Specifically, during spontaneous activity, the frequency of events is highly variable, ranging from ~0.525 to 3.6 events/min. When stimulated at a slow rate of 0.5 stim/min, the Ca2+ frequency decreases and appears to become more regulated (with a range of ~0.125 to 1.375 events/min). As the stimulation rate increases, the Ca2+ frequency, including its range, increases. The increase in Ca2+ frequency range implies that some astrocyte processes are better able to keep up with the stimulations, while others are not, and this discrepancy increases with higher stimulation rates. Note that the minimum Ca2+ frequency increases as well with increasing stimulation rates, suggesting that the activity of more or less all responsive astrocytes changes in accordance with the stimulation rate, even if this change is not sufficient to keep up with the stimuli. When examining Ca2+ frequencies during fast stimulation (4 stim/min), the frequency decreases significantly when all 8 stimulations were considered, which is expected given the refractoriness observed in the previous section. Intriguingly, all stimulation rates resulted in Ca2+ frequencies that were significantly different from spontaneous Ca2+ frequencies, except the rate of 2 stim/min (inter-stimulus interval of 30 s). This was true whether 108 A * ** ** ** ** ** ** ** 3 2 1 Experiments ** 0.6 0.4 0.2 0 0.8 Model 0.6 0.4 0.2 0 3 0 0.5 Experiments Model 2 C 1 0 1 Spont. 0.5 1 2 4 Stimulation Frequency (Stim/min) 2 Four Stimuli 4 4 ht Eig muli Sti Stimulation Frequency (Stim/min) Percent of Responsive ROIs Ca2+ Frequency (Events/min) 1 0.8 Reliability Ratio 4 B ** Responding to: 1 stimulus 2 stimuli 40 3 stimuli 4 stimuli 20 0 0.5 1 2 4 Stimulation Frequency (Stim/min) Figure 4.7. Astrocyte Ca2+ event frequency and ability to respond to each stimulus when stimulated at different rates. (A) Frequency of Ca2+ events during spontaneous activity and as a function of stimulation frequency, in experiments and the model. Only those ROIs were included that had Ca2+ activity at some point (i.e. zero frequency traces were eliminated). The model during high frequency stimulation includes the different refractory and negative feedback mechanisms described earlier. (Here, * and ** indicate p=0.047 and p<0.008, respectively, using the Kruskal-Wallis test; Ca2+ frequency for 8 stimuli during high frequency stimulation was only significantly different from spontanoues activity and stimulation at 2 stim/min). (B) The Reliability Ratio (fraction of stimulation intervals in which astrocytes display Ca2+ activity), indicative of the ability of an astrocyte to keep up with stimuli, as a function of stimulation frequency, in experiments and the model (for the 0.5 stim/min rate, the stimulation interval was restricted to 60 s to minimize the inclusion of solely spontaneous Ca2+ events). (C) Bar graph showing the percentage of astrocyte processes (of those that responded at some point) which responded to 1, 2, 3, or 4 of presented stimuli (out of 4 total), during different stimulation frequencies. The data presented here are the same as those in B for experimental data, except the number of stimulations for the 4 stim/min rate was restricted to 4 total for consistency. See also Figure 4.7—figure supplement 1. 109 considering all ROIs (Fig. 4.7—figure supplement 1) or only those that exhibited Ca2+ events (Fig. 4.7A). We also examined our model and found that it reproduces Ca2+ frequencies comparable to experimental data (Fig. 4.7A and Fig. 4.7—figure supplement 1). The same trend in Ca2+ frequency during spontaneous activity and different stimulation rates was observed, while the ranges were often less than in experiments. These similarities suggest that our model is robust and able to reproduce multiple aspects of experimentally observed Ca2+ events, though it tends to represent the average ROIs rather than extremes. This is reasonable given that our model treats each ROI as completely independent from others and assumes they all have the same transition probabilities (see Discussion). While Ca2+ event frequency increased with higher stimulation frequencies, it is not clear whether the Ca2+ events occurred in synchrony with each stimulus. To evaluate how well astrocytes were able to respond to each stimulus during different stimulation frequencies, we examined their Reliability Ratio, also known as Phase Locking Ratio (PLR) in (Jovic et al., 2010). The Reliability Ratio is an indicator of how reliable the cell’s Ca2+ responses are in encoding the timing of stimuli. For a given ROI, it is calculated as the number of stimuli that result in a Ca2+ response divided by the total number of presented stimuli. Figure 4.7B shows the mean Reliability Ratio (and the 25th and 75th percentiles) for astrocyte processes during different stimulation frequencies. As expected, the Reliability Ratio decreased with increasing stimulation frequency, indicating that astrocyte Ca2+ activity is unable to keep up with fast stimulation (similar results in the model, shown in gray). This finding is also illustrated in Figure 4.7C: from the ROIs that responded at some point during the recording, the percentage of those that 110 responded to 1, 2, 3, or all 4 stimuli during different stimulation frequencies is shown. As seen, during slower stimulation rates, the majority of active ROIs responded to 3 out of 4 (~33%) or all 4 stimuli (~34%). During stimulation rates of 2 stim/min, the proportion of ROIs responding to all 4 stimuli decreases to 23%. During faster stimulations (4 stim/min), only 5% of ROIs are able to respond to all 4 stimuli, and most astrocyte processes are able to respond to only 1 (~34%) or 2 (~38%) stimuli. This occurs even before refractoriness and negative feedback for high frequency stimulation take full effect, since it is restricted to the first 4 stimulations. Previous biophysical modeling studies of GPCR-dependent Ca2+ activity (in cell types other than astrocytes; (Giraldo et al., 2011; Jovic et al., 2010)) suggest that a cell’s Reliability Ratio (or, PLR) during periodic stimulation with varying frequencies provides insight into the biological mechanisms underlying its Ca2+ responses (see Discussion). Overall, we find that astrocyte population activity depends reliably on stimulation frequency, which may allow for encoding the rate of stimuli, particularly during lower frequency stimulation. Moreover, astrocyte processes are heterogenous in their ability to respond to different stimulation frequencies and this heterogeneity becomes more evident as the stimulation frequency increases. Therefore, even during higher frequency stimulation, some astrocyte processes may be able to encode the rate of stimuli. For the most part, this response heterogeneity can be replicated merely through the stochasticity of astrocyte activity, using Hidden Markov Modeling, without the need to incorporate explicit heterogeneity between individual astrocytes. 111 4.4 Discussion In this work, we sought to gain insight into several questions concerning astrocyte Ca2+ signaling: How does astrocyte Ca2+ activity, despite its heterogeneity, reflect the signals astrocytes receive (e.g. during ongoing neuronal activity or under pathophysiological conditions)? How can we decode and interpret highly variable astrocyte Ca2+ activity? Do Ca2+ responses to successive stimuli exhibit any higher order response properties (e.g. facilitation or depression) that depend on the stimulation frequency; if so, what are their potential mechanistic bases? To address these questions, we examined both astrocyte spontaneous Ca2+ activity as well as responses to periodic, brief ATP stimulation. Due to the high cell-to-cell and trial-to-trial variability of these Ca2+ dynamics, our initial analyses revealed little difference in Ca2+ activity during stimulation with different frequencies. To overcome the issue of high variability in our data analyses, we utilized a probabilistic modeling approach. We show that a Hidden Markov Model (HMM) with three hidden states (Closed, Open 1, and Open 2) and timevarying transition probabilities reproduces the experimentally-observed variability of astrocyte Ca2+ signals and correctly predicts several experimental results from novel data sets not used to determine model parameters. In both the model and experiments, we found that: (1) During low frequency ATP stimulation, astrocyte response probability at each second increased to ~1.7 times the spontaneous levels; (2) Astrocyte Ca2+ activity showed no signs of facilitation or depression with repeated stimulation when the frequency was ≤ 2 stim/min; (3) Ca2+ event frequency varied widely among astrocyte processes during spontaneous activity, became less variable during low frequency stimulation, and increased (in terms of both its 112 average and range) with increased stimulation frequency; (4) As the stimulation frequency increased, astrocytes responded less reliably to each stimulus (i.e. decreased Reliability Ratio). Importantly, and in contrast to our three-state model, we found that astrocytes responded in two opposing manners to ATP stimulation depending on the stimulation frequency. This dual behavior was only observed at the population level, not at the level of single cells. Specifically, when the stimulation rate was 4 stim/min, astrocyte processes exhibited an initial increase in Pr(On) levels, quickly followed by a decrease below spontaneous levels. To explain the mechanisms underlying this behavior, we examined modified versions of our model, which suggest that this dual behavior of astrocyte processes arises from two sets of mechanisms: first, their inability to respond to fast stimulation (refractoriness from receptor desensitization, Ca2+ store depletion, etc.); second, negative feedback during fast stimulation that suppresses regular Ca2+ activity in astrocyte processes (associated biological mechanisms need further experimental investigation). Interestingly, astrocyte somas also exhibited decreased Pr(On) levels during high frequency stimulation, but not to levels lower than spontaneous levels, suggesting that negative feedback mechanisms do not apply to somatic responses. These findings and potential biological mechanisms underlying responses of astrocyte somas and processes during high frequency stimulation are summarized in Figure 4.8. 4.4.1 Role of astrocyte spontaneous activity and dual response pattern during high versus low frequency stimulation Astrocytes have been thought to respond to both excitatory and inhibitory neurotransmitters through increased Ca2+ activity (Guerra-Gomes et al., 2018). In this 113 Figure 4.8. Astrocyte Ca2+ activity during low and high frequency stimulation and its potential underlying biological mechanisms. (A) Schematics showing that during low frequency stimulation, the probability of Ca2+ activity at each moment in time increases compared to spontaneous activity levels, in both somas and processes. This occurs due to mechanisms including GPCR activation, IP3 production, IP3 diffusion from other cellular regions, Ca2+ released from internal stores (e.g. ER), and Ca2+ entry from the extracellular space. (B) During high frequency stimulation, Pr(On) levels vary between the astrocyte somas and processes, but decrease in both. The biological mechanisms responsible for decreased Pr(On) levels likely include (based on our model and experimental results, Fig. 4.5 and 4.6) various forms of refractoriness (e.g. receptor desensitization and internal store depletion in somas and some processes) as well as negative feedback (in processes, including those that do not respond to the stimulus). In astrocyte processes, this negative feedback results in Pr(On) levels that are lower than spontaneous levels. study, we showed that astrocytes, in terms of their population activity, have the ability to respond to the same agonist (here, ATP) in two distinct manners—both increased and decreased Ca2+ activity relative to spontaneous activity levels—depending on the rate of stimuli. This dual behavior may play an important role in how astrocytes encode signals from nearby neurons and other cells (e.g. other astrocytes and microglia). Perea and Araque (2005) previously showed that astrocyte Ca2+ responses elicited through 114 simultaneous stimulation of two different neuronal pathways (containing glutamatergic and cholinergic axons) had an inverse relation to the neuronal stimulation frequency. More specifically, during low (1-10 Hz) and high (30-50 Hz) frequency stimulation, the resulting astrocyte Ca2+ responses were, respectively, greater and less than the linear summation of responses elicited from separately stimulating the two neuronal pathways. While the measurements and experimental setup in this study were different from ours, both studies indicate that astrocyte Ca2+ activity can be bidirectionally modulated depending on the input frequency. This suggests that astrocyte Ca2+ signaling depends systematically on nearby synaptic activity. These differentially modulated astrocyte Ca2+ responses are likely to have distinct downstream effects on brain circuit activity and, ultimately, on behavior. Indeed, previous studies suggest that depending on the timing of neuronal inputs, astrocytes can regulate synaptic activity in opposing manners (Covelo and Araque, 2018) and regulate long-distance modulatory effects of astrocytes through either enhancing or inhibiting intracellular Ca2+ propagation (Perea and Araque, 2005). It has also been shown that both enhanced and reduced astrocyte Ca2+ activity are associated with distinct behavioral outcomes (Chen et al., 2016; Ding et al., 2013; Nimmerjahn et al., 2009; Paukert et al., 2014; Srinivasan et al., 2015; Yu et al., 2018). However, these studies have not specifically examined how evoked astrocyte Ca2+ responses are modulated relative to spontaneous activity levels. This is a particularly interesting question since our results indicate a potential role for astrocyte spontaneous Ca2+ activity: It may serve as a baseline against which evoked Ca2+ activity is measured, resulting in distinct roles for suppressed or enhance Ca2+ activity. Consistent with this 115 proposed role of astrocyte spontaneous Ca2+ activity, from an information theory perspective, randomly generated Ca2+ events during spontaneous activity allow for efficient signal detection (Skupin and Falcke, 2007, 2010). Due to the sparseness and variability of spontaneous Ca2+ events for a single astrocyte region, comparisons of evoked versus spontaneous activity must be done over a large population of cellular regions or a large number of trials. Hence, quantifying astrocyte Ca2+ dynamics under different experimental conditions from a probabilistic standpoint and using computational tools such as HMMs can provide further insight into the roles of spontaneous and differentially modulated astrocyte activity. 4.4.2 Time-varying Hidden Markov Modeling as a tool for experimental analyses We showed that the stochasticity of astrocyte Ca2+ activity is well-described by a three-state HMM, with differences between spontaneous and evoked activity arising from differences in transition probabilities between these hidden states. By comparing our experimental data against this model, we identified meaningful changes in Ca2+ activity during repeated high frequency stimulation as well as its potential underlying biological mechanisms. Similarly, our model can be used to assess how the transition probabilities (or even the necessary number of hidden states) that best describe the data change under different experimental conditions (e.g. spontaneous activity in the epileptic brain or during specific physiological behavior). Such analyses could allow for detecting changes in astrocyte Ca2+ dynamics that would otherwise be obscured by its stochasticity and heterogeneity. Additionally, these analyses could point to underlying biological 116 mechanisms that are altered under these conditions and could be used to guide further experiments. To address our experimental questions and examine the effects of stimulation frequency on Ca2+ responses, it was sufficient to treat each individual astrocyte region in the model as identical (i.e. with the same transition probabilities) and independent of one another. Under these assumptions, heterogeneity in the population activity arises from stochasticity of each individual astrocyte region. However, even greater heterogeneity (as seen experimentally, e.g. Fig. 4.5 and 4.7A) can be achieved by: (1) introducing inherent variability between different astrocyte subpopulations (through different sets of transition probabilities between the model’s states for different astrocyte subpopulations, both above and below the current probabilities); (2) including connections between certain astrocyte regions by introducing an additional time-varying component in the transition probabilities (e.g. when one region enters the O1 state, a connected region has a higher likelihood of entering O1 as well). These model modifications would add complexities that may be useful depending on the experimental questions being asked. 4.4.3 Time-varying Hidden Markov Modeling as a tool for improving biophysical models Phenomenological models, such as the HMM in this work, can also be used to improve biophysical models of astrocyte activity. For example, our model can be used to include noise and stochasticity relating to the timing and durations of Ca2+ events, both spontaneous and evoked. If spontaneous Ca2+ activity is included at all in biophysical models of astrocyte Ca2+ dynamics, it is often incorporated as a regular (i.e. 117 deterministic) oscillation (Lavrentovich and Hemkin, 2008; Li et al., 2012; Zeng et al., 2009). Given the stochastic nature of these spontaneous Ca2+ events and its likely importance in information processing (Falcke, 2003; Skupin and Falcke, 2007, 2010; Thurley et al., 2014), it seems necessary to include the randomness of spontaneous activity in astrocyte models. A major difficulty in doing so is that the sources underlying this stochasticity and spontaneous Ca2+ activity are incompletely understood and are likely complex, stemming from multiple sources (e.g. IP3Rs, mitochondria, and different extracellar fluxes) (Agarwal et al., 2017; Falcke, 2003; Nett et al., 2002; Parri et al., 2001; Rungta et al., 2016; Skupin and Falcke, 2010; Srinivasan et al., 2015). Hence, it would be unreasonable to include each contributing pathway explicitly in biophysical models. One could, instead, implicitly account for the variability of Ca2+ activity by including appropriate levels of noise in relevant model equations, deriving their statistics from models such as the HMM here. Additionally, the time scales incorporated in a phenomenological model like ours can help identify biological mechanisms and estimate their time scales. For instance, the value Teff in our model represents the time scale of the ATP effect on astrocytes. In terms of biophysical models, this value relates to the time scales of GPCR activation and elevated IP3 levels—either directly from GPCR activation or indirectly through intra- and inter-cellular IP3 diffusion. Likewise, the time scales relating to the different biophysical mechanisms that lead to Ca2+ depression during stimulation with a 15 s period can be examined in biophysical models. One of these mechanisms may be related to IP3 recovery dynamics (described next). 118 4.4.4 Suppressed Ca2+ responses during high frequency stimulation and IP3 recovery dynamics Biophysical modeling studies of GPCR-dependent Ca2+ activity (Giraldo et al., 2011; Jovic et al., 2010) suggest that cells in which the Reliability Ratio (a.k.a. PLR) decreases with increased stimulation frequency (such as astrocytes, Fig. 4.7B) likely have specific IP3 recovery mechanisms that are responsible for such Ca2+ dynamics. In such cells, during higher frequency stimulation, the rest time between stimuli is insufficient for the recovery of IP3 to basal levels. Hence, the likelihood of IP3 crossing the threshold required for a Ca2+ event decreases. This mechanism for decreased likelihood of Ca2+ events during higher frequency stimulation may, at least partially, contribute to the astrocyte refractoriness as well as negative feedback mechanisms proposed by our results. The latter is achieved if basal IP3 levels are needed to maintain regular spontaneous activity, which appears to be the case since a proportion of astrocyte spontaneous activity is IP3R-dependent (Agarwal et al., 2017; Jiang et al., 2016). With such IP3 recovery dynamics, the likelihood of a Ca2+ event (spontaneous or evoked) decreases regardless of how large the IP3 peak was and whether or not it resulted in a detectable Ca2+ event. These refractory and negative feedback mechanisms involving IP3 dynamics can be considered in future biophysical modeling studies of astrocyte Ca2+ activity to reproduce the two distinct behaviors seen in response to low versus high frequency stimulation. Also, other potential biological mechanisms responsible for the negative feedback (which suppresses spontaneous activity, even in non-responsive processes) should be examined in future experimental work. 119 4.4.5 Interpreting in vivo astrocyte Ca2+ activity based on our results By directly applying focal, brief pulses of ATP, we controlled the type and timing of inputs to astrocytes to elucidate how astrocyte Ca2+ activity reflects input signals. While indirect astrocyte stimulation through driving neuronal activity would have been more physiological, it creates several complexities: First, it would be unclear which astrocyte process received an input and which did not since the neuronal connections to individual processes that are also functional are unclear; second, the timing and identity of inputs to astrocytes would be less controllable; third, neuronal activity affects astrocyte Ca2+ signaling through multiple complex pathways involving GPCR activation as well as K+ and neurotransmitter uptake (Anderson and Swanson, 2000; Sibille et al., 2014; Wang et al., 2012). Therefore, applying ATP pulses was a more controllable way to stimulate astrocytes. Furthermore, we also performed our experiments (spontaneous and periodically-evoked activity, for all frequencies except 0.5 stim/min) while bath applying TTX to block neuronal action potentials (data not shown). We found no difference in our results when analyzing astrocyte activity in the presence or absence of TTX, suggesting that astrocyte Ca2+ responses to ATP and during spontaneous activity were independent of neuronal action potentials. We focused on presenting results from experiments without TTX to allow our results and model to be applicable to in vivo astrocyte activity. Based on our results, one feature of astrocyte Ca2+ signaling that can be informative during different experimental conditions is changes in the statistics of its population activity over time. A decrease or increase in activity (Pr(On) levels) over time may reveal different patterns of input signals to astrocytes throughout an experiment. Related to this, the input frequency range that switches astrocyte activity from enhanced 120 to suppressed is likely to vary under various contexts. With our experimental setup where ATP was dispersed over an area containing multiple astrocytes, Ca2+ signaling switched to a suppressed state with an inter-stimulus interval of ~15 s. The time-scale of this switch may be similar in experiments where stimulation occurs through strong synchronous neuronal activity over a larger area (e.g. rhythmic activity in the brain or epileptiform activity). Conversely, if the inputs are smaller synaptic inputs to individual astrocyte processes, the time-scale of switching to suppressed activity is likely to be much smaller due to the different neurotransmitter concentrations and diffusion properties within a single synapse. Another informative Ca2+ activity pattern, based on our results, is the similarity or difference between activity in somas and fine processes. During high frequency stimulation, we observed a discrepancy in the Pr(On) of the two regions compared to baseline spontaneous Pr(On) levels. Moreover, our results indicate the importance of examining the level of Ca2+ activity heterogeneity within the astrocyte population, in addition to comparing average changes. For example, we had found that the heterogeneity of Ca2+ frequency among astrocyte regions is large during spontaneous activity but decreases during low frequency stimulation (Fig. 4.7A). Finally, while we did not explicitly examine spatial properties of the Ca2+ events—e.g. whether they were highly localized microdomain activity or global events that spread throughout the astrocyte (Khakh and Sofroniew, 2015)—the increased Pr(On) levels we found correlate with greater likelihood of global events. On one hand, the synchronous activity of multiple regions (manifested in the higher Pr(On) levels) increases the chances of a global Ca2+ event (Croft et al., 2016). On the other hand, 121 during a global event, more regions are active, hence Pr(On) increases. Still, further examination is needed to determine the relationship between various spatial patterns of astrocyte Ca2+ activity and changes in Pr(On) levels for individual astrocyte regions under different experimental conditions. 4.5 Materials and Methods 4.5.1 Transgenic mouse line All animal experiments were carried out in accordance with the NIH Guide for the Care and Use of Laboratory Animals and approved by the University of Utah Institutional Animal Care and Use Committee. To express the GCaMP5G geneticallyencoded Ca2+ indicator in astrocytes, targeted reporter mice (PC::G5-tdT, (Gee et al., 2014); JAX labs, stock no. 024477) were crossed with the GFAP-Cre+ mouse line ((Gregorian et al., 2009); JAX labs, stock no. 024098). Both male and female mice, aged 5–8 weeks old, were used. 4.5.2 Acute brain slice preparation To extract the brain, the mice were anesthetized in a closed chamber with isoflurane (1.5%) and decapitated. The brains were then rapidly removed and immersed in ice-cold cutting solution, containing 230 mM sucrose, 1 mM KCl, 0.5 mM CaCl2, 10 mM MgSO4, 26 mM NaHCO3, 1.25 mM NaH2PO4, 0.04 mM Na-Ascorbate, and 10 mM glucose (pH = 7.2–7.4). 400 µm-thick coronal slices were cut using a VT1200 Vibratome (Leica Microsystems, Wetzlar, Germany) and transferred to oxygenated artificial cerebrospinal fluid (aCSF) that contained 124 mM NaCl, 2.5 mM KCl, 2 mM CaCl2, 2 mM MgSO4, 26 mM NaHCO3, 1.25 mM NaH2PO4, 0.004 mM Na-Ascorbate, and 10 122 mM glucose (pH = 7.27.4; osmolarity = 310 mOsm). Slices were allowed to recover in oxygenated aCSF at room temperature for 1 h before experiments. During the recordings, the slices were placed in a perfusion chamber and superfused with aCSF gassed with 95% O2 and 5% CO2 at room temperature. For a subset of experiments where tetrodotoxin (TTX) was applied to suppress neuronal action potentials (see Discussion), we included 1 µM TTX in the aCSF solution. 4.5.3 Ca2+ imaging Astrocyte Ca2+ imaging was performed using a Prairie two-photon microscope with a mode-locked Ti:Sapphire laser source emitting 140 fs pulses at an 80 MHz repetition rate with a wavelength adjustable from 690 to 1,040 nm (Chameleon Ultra I; Coherent, Santa Clara, CA). We used laser emission wavelengths of 920 nm to excite GCaMP5G or 1,040 nm to excite tdTomato. The laser intensity was measured above the sample under the objective with a PM100A power meter (Thorlabs, Newton, New Jersey, USA) and held between 6-9 mW to prevent damage to astrocytes from the laser light (Kuga et al., 2011). Two-photon imaging was performed using a 20 × 0.95 NA waterimmersion objective (Olympus, Tokyo, Japan) and at a frame rate of 1 Hz. All imaging was done on astrocytes in the primary somatosensory cortex. Experimental recordings included both spontaneous astrocyte Ca2+ activity (for a duration of 1 to 4 mins) and chemically-evoked Ca2+ activity. To chemically evoke astrocyte Ca2+ responses, 500 µM ATP (Tocris Bioscience, Bristol, UK, catalog no. 3245; same concentration used by (Kim et al., 2016; Otsu et al., 2015; Taheri et al., 2017; Thrane et al., 2012)) dissolved in aCSF was delivered locally and briefly via a glass 123 pipette (10 psi, 60 ms) using a Picospritzer III (Parker Instrumentation, Chicago, IL). The pipette also contained 5 µM Alexa Fluor 594 in order to visualize its location and the distance of agonist dispersion. ATP pulses were delivered periodically (beginning at ~40 s after the start of imaging) with varying time intervals between pulses (stimulation periods: 15 s, 30 s, 1 min, 2 min, and 4 min). A total of 4 ATP pulses for all stimulation protocols were delivered, unless otherwise noted and except for the fastest protocol (15 s period) where 8 pulses were applied. Recording the 4 minute stimulation period protocol with 4 ATP pulses would take >16 minutes; however, since we wanted to limit the duration of each experiment in order to avoid damaging or activating astrocytes from exposure to prolonged laser light, for these experiments, Ca2+ activity was recorded separately after each stimulus for ~140 s (to allow the tissue ~1 min of rest from the laser light). Due to this, some analyses for other stimulation protocols needed to be modified for (e.g. Pr(On)), or were not applicable to (e.g. frequency of Ca2+ events), the 4 minute stimulation protocols. For each brain slice, spontaneous Ca2+ activity without any ATP application was recorded at least once at the beginning and up to two more times throughout the experiment. These spontaneous recordings as well as recordings with 3 to 4 stimulation protocols (sometimes repeated protocols) were performed in a random order, with a resting time of at least 10 minutes between each protocol (including before and after recordings of spontaneous activity) to allow the cells to recover from ATP stimulation and/or exposure to the laser light. Given the resting time between protocols and the long stimulation periods for some, the number of stimulation protocols performed for each animal was limited to 3-4 in order to limit the duration of the experiment. 124 4.5.4 Analysis of Ca2+ recordings The two-photon images were processed and analyzed using custom-written MATLAB (2015b; MathWorks, Natick, MA) scripts. Each time-lapse image was first processed with a 3 × 3 median filter. For each pixel, the median fluorescence of the first 40 frames (which, in ATP-evoked Ca2+ recordings, was prior to the agonist application) was used to calculate the baseline fluorescence (F0). Next, the percent change in fluorescence (100 × ΔF/F0) was calculated for each pixel throughout the time-lapse image. To select regions of interest (ROIs), the maximum ΔF/F0 projections were overlaid with an amplified tdTomato fluorescence signal. Because the tdTomato signal indicates astrocyte locations and remains unchanged over time (Gee et al., 2014), using it to select ROIs reduces selection bias (as opposed to using only the GCaMP5G signal to select ROIs). ROIs were manually selected for astrocyte somas and processes from the agonist puff area (detected from the dispersion of the dye from the pipette). Astrocytes that were exposed to the dye and analyzed were approximately within a 110 µm distance from the pipette (comparable to the maximum distance used in analyses by (Thrane et al., 2012)). Analyzed astrocyte processes included mostly branchlets, but also some branches, as defined by (Khakh and Sofroniew, 2015). For each astrocyte that had somas visible in the field of view, 3-5 ROIs (constrained to 2 × 2 µm boxes) from their processes were selected (one ROI per major branch) at a distance of ~4-18 µm from the soma perimeters. Roughly 5 additional ROIs were selected from other visible processes of astrocytes whose somas were not clearly in the field of view. The ΔF/F0 trace for each ROI was calculated, then filtered using an order-3 one- 125 dimensional median filter. Some traces that showed a drift in their baseline levels were detrended by fitting either a line or a third degree polynomial curve to the trace. Ca2+ event peaks were automatically detected when the ΔF/F0 exceeded 8 (for somas) or 6 (for processes) standard deviations above the baseline value and had a minimum value of 15% (for somas) or 35% (for processes). The start and end points of the Ca2+ events were also automatically detected, and defined as, respectively, the last or first data point before or after the Ca2+ peak that had a ΔF/F0 value less than 5 (for somas) or 3.3 (for processes) standard deviations above the baseline. The time between the two troughs determined the duration of an On event when binarizing the Ca2+ traces into Off and On states (used for the hidden Markov model, or HMM). All Ca2+ traces with automatically detected event peaks and troughs were manually examined and those with detected Ca2+ events that were not visibly distinguishable from potential signal noise were eliminated. 4.5.5 Hidden Markov Model (HMM) To estimate the transition rates (separately for evoked and spontaneous conditions) of each HMM examined in Section 1.3, we used the On-Off Ca2+ traces from a small subset of ATP-evoked Ca2+ recordings (specifically, only 1 min after the first stimulus in 1 stim/min recordings) and from spontaneous recordings (ranging from 1 min to 4 min). First, transition rates (equivalent to transition probabilities here, since all our data were recorded at 1 Hz) for spontaneous activity were estimated with the hmmtrain function in MATLAB (Statistics and Machine Learning Toolbox) which uses the BaumWelch algorithm (a special case of the Expectation-Maximization (EM) algorithm; (Baum, 1972)) for the estimation (given the experimental On-Off sequences and the 126 model as determined by the user in the initially-guessed transition and emission matrices). It also estimates the prior probabilities (i.e. the probability of being in each hidden state at the start time). Using MATLAB’s hmmgenerate function, we then generated 5,000 sequences (each with a length of 60 time steps, or seconds) for the model with the estimated transition rates and found the overall probability of being in each state for the entire duration of all sequences. Since during the entire duration of spontaneous activity, the probability of a cellular region (ROI) being in any given state should remain constant, we choose this calculated probability distribution as the fixed prior and used this prior to recalculate the transition rates for spontaneous activity (𝑟)*+,- ). We used the same fixed prior to also estimate the rates for evoked activity (𝑟./+01 ) since at the initial time when the first stimulus is applied, the cellular region still has the same probabilities of being in any given state as during spontaneous activity. To fix the prior in these final transition rate calculations, a different HMM toolbox was used, which uses the same EM algorithm as hmmtrain to estimate transition rates, but allows for fixing the prior probability distribution (toolbox accessed on January 14, 2018 from: https://www.cs.ubc.ca/~murphyk/Software/HMM/hmm.html). To evaluate the HMM that best fit our experimental data, we first examined the estimated transition rates (for both the spontaneous and evoked cases) to ensure that any rates found to be zero do not fundamentally change the model or make the model biologically unreasonable (e.g. for model 5 in Fig. 4.3C, the transition rate from R to C states was zero, meaning that once the system/cell goes to the R state, it stays there indefinitely). From those models with reasonable transition rates, we generated two sets of sequences (5,000 each) using hmmgenerate with the two sets of transition rates 127 (spontaneous or evoked). We qualitatively compared the On and Off dwell times from these sequences with those from the experimental data (restricted to the first minute of spontaneous and ATP-evoked activity). In subsequent sections, to generate simulations from the time-varying HMM, we used a custom-written modified version of hmmgenerate which allowed for incorporating time-varying transition rates. When modifying the three-state COO model to incorporate additional store depletion during high frequency stimulation (Fig. 4.6), we included a Refractory (R) hidden state, which the cell enters after leaving the O2 sate and before reaching the C state. All transition rates were kept at the same levels of the original COO model, with the O2 to C transition rate now being the O2 to R rate. The transition rate from states R to C (𝑟 2→3 ) was kept constant throughout the simulation, regardless of the time since the first stimulus. To compare the model’s results with experimental data, we examined different R to C transition rates, defined as 𝑟 2→3 = 1/µ, where is µ is the average time spent in the R state once the astrocyte enters this state. To incorporate negative feedback into the model during high frequency stimulation (Fig. 4.6), only the transition rate from C to O1 was modified. Within one 3→67 time step (1 s) after the second stimulus, this transition rate was decreased to rate 𝑟415 (hence, its effects are visible in the Pr(On) levels of the second stimulation interval). The reduced transition rate was kept constant at that same level during all stimulations thereafter (see Fig. 4.6—figure supplement 1, A). The amount of transition rate reduction 3→67 from the spontaneous rate (𝑟)*+,) was half of the rate increase after one stimulation 3→67 (𝑟./+018 ): 3→67 3→67 8.→67 3→67 𝑟415 – 𝑟)*+,= – (𝑟./+01 – 𝑟)*+,)/2 = – 0.0026 128 4.5.6 Statistics All analyses were performed using MATLAB. The statistical tests used for each result are specified in the corresponding text and/or figure legends. Significance was achieved at α = 0.05 level and was indicated with *, or with ** when α < 0.01. 4.6 Supplementary Figures Figure 4.4—figure supplement 1. Determining the Teff parameter in the timevarying HMM model. (A) Average Ca2+ responses from two different experimental recordings during stimulation at a rate of 1 stim/min. (Specifically, the two-photon imaging area was divided into contiguous boxes, each 8x8 pixels in size. The ΔF/F per pixel was averaged for each box, then the ΔF/F of all boxes were averaged and normalized.) The average responses of different experiments were different from one another, but based on examples such as the ones shown here, we estimated that the effect of ATP on astrocyte Ca2+ signaling ranged between approximately 50 s and 100 s. Also, in our previous study (Taheri et al., 2017), we had estimated the effect of one ATP pulse on intracellular IP3 duration based on multiple experimental and modeling results, which was on the higher end of this range. (B) In the TV-HMM model, since the transition probabilities take some time to fully effect the system, small differences in the Teff have virtually no effect on the Pr(On) levels, especially when the number of simulations are low and the variability is high. Here, the Pr(On) values from the COO model are shown during stimulation at a rate of 1 stim/min (top) and 0.5 stim/min (bottom) for different Teff values (N=2,000). As seen, even with such a high number of simulations, the effect of this range of Teff values on Pr(On) is minimal and will not affect our conclusions. Hence, we chose Teff = 80 s in this study. 129 Figure 4.5—figure supplement 1. The proportion of active astrocytes decreases with repeated high frequency stimulation. (A) (Left) Pr(On) levels for all stimulation protocols (mean ± s.e.m.) overlaid. (Right) For each protocol, the total time in the On state (i.e. total Ca2+ duration; mean ± s.e.m.) during each stimulation interval is plotted against stimulus number. The maximum duration for each protocol is equivalent to that protocol’s stimulation period or 60 s (to limit spontaneous activity that occurs after stimulations), whichever is less. (B) Raster plots and the population activity (i.e. the average ROIs active, equivalent to Pr(On)) during low (2 stim/min) and high (4 stim/min) stimulation frequency. The black horizontal lines in the raster plots show the times at which each astrocyte process had Ca2+ activity. 130 Figure 4.6—figure supplement 1. Modeling the negative feedback mechanism and the somatic responses to high frequency stimulation. (A) To implement a negative feedback mechanism in the TV-HMM model for astrocyte processes during high frequency stimulation (4 stim/min), the transition rate from C to O1 was modified as shown in the schematic. In the time step (1 s) following the second stimulus, the C to O1 transition rate immediately decreases and stays at the lower level continuously. The amount of rate suppression (h) is half the rate increase during stimulation (2h). (B) Experimental Pr(On) values for astrocyte somas during high frequency stimulation (red) and the average Pr(On) for somas during spontaneous activity (blue; N values same as in Fig. 4.2). The black trace is the Pr(On) from the model that is modified for somatic activity as described in the text. 131 *p=0.048 **p=0.0037 4 **p=0.0034 * Ca2+ Frequency (Events/min) 3 2 1 0 3 Experiments Model 2 1 0 Spont. 0.5 1 2 Four Stimuli 4 4 ht Eig muli Sti Stimulation Frequency (Stim./min) Figure 4.7—figure supplement 1. Astrocyte Ca2+ event frequency from all processes. Ca2+ frequency in the model and experiments from all ROIs, including those that did not respond (i.e. had zero frequency). 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CHAPTER 5 CONCLUSION 5.1 Major Findings and Implications Investigating the role of astrocyte Ca2+ signaling in the brain is an area of active research. In our studies, we used coordinated Ca2+ imaging experiments and computational modeling to examine the heterogeneity and response properties of astrocyte Ca2+ activity and the contributing biological mechanisms. Our experimental findings show that astrocytes exhibit variable Ca2+ patterns, both spontaneously and when chemically stimulated, and encode the presence of stimuli in a probabilistic manner. The two models of astrocyte Ca2+ activity that we developed based on our experimental data—a biophysical, mechanistic model and a probabilistic, phenomenological model— provided significant insight into astrocyte signaling patterns and underlying mechanisms. Our results imply that specific features of astrocyte Ca2+ activity are informative and should be considered under various experimental conditions. For instance, the likelihood of observing various astrocyte response types (e.g. Single-Peak vs. Multi-Peak) reveal details about the cell’s Ca2+ machinery, which may change under certain experimental settings, such as in disease. Also, differences in astrocyte population activity patterns may indicate different input signal patterns to astrocytes. Below, I describe some major findings from our studies and give specific examples of how each finding can be used to 139 improve future studies of astrocyte signaling. In the first study presented here, we characterized the variety of astrocyte response patterns to a single ATP pulse by examining the Ca2+ response kinetics (amplitude, duration, rise time, and decay time) and the distribution of response types (Single-Peak, Multi-Peak, Plateau, and Long-Lasting). One finding was that the response type distributions differ between astrocyte subcompartments: Moving from the astrocyte somas to the large processes and then the small processes, the probability of Single-Peak and Multi-Peak responses respectively decrease and increase. Conversely, when comparing the kinetics of one response type between astrocyte subcompartments, we found little difference. Moreover, our model suggested that the different response type distributions arise from systematic differences in the underlying Ca2+ machinery and IP3 dynamics of different astrocyte subcompartments. Many studies that compare Ca2+ activity under different experimental conditions (e.g. health vs. disease) look for changes in Ca2+ kinetics. However, our results suggest that examining evoked Ca2+ response types (which is based on the response shape and duration) may reveal more about the differences between the experimental conditions. In addition, using our modeling studies (from Chapters 2 and 3), these differences can be used to identify potential differences in cellular mechanisms. As an example, one might find that in a certain disease, the incidence of evoked Multi-Peak responses decreases dramatically while the incidence of Plateau responses increases. Our biophysical model suggests that this change may be due to decreased SERCA pump activity. This provides a new testable hypothesis that can guide future experiments and point to possible treatments for recovering normal astrocyte Ca2+ activity and, ultimately, downstream 140 functions. In the second body of work, we examined in detail the biophysical model that we developed using experimental data from the first study. We identified the roles of individual Ca2+ channels and pumps in generating a variety of response patterns. One notable finding was the significant role of Store-Operated Ca2+ (SOC) channels. We found that despite the very small Ca2+ flux through these channels (drastically less than any other flux in our model), they define the dynamical landscape of the model. On one hand, completely blocking SOC channels eliminates sustained Ca2+ oscillations (even though Multi-Peak responses can be observed, as explored in this study and the previous study). On the other hand, partially blocking SOC channels increases the incidence of sustained oscillations. These are surprising results that change the way one can view experimental data. First, this implies that a channel or pump with a small Ca2+ flux is not necessarily negligible and may prove more significant than one with a greater flux. Indeed, the role of SOC channels in astrocyte oscillations has been suggested by previous studies (Sergeeva et al., 2000, 2003). Second, the fact that partial and complete SOC channel blocking have opposite effects implies that pharmacological blocking experiments should be interpreted with caution. If a pharmacological blocker is not fully effective, the experimental conclusions made may be incorrect, and may contradict conclusions from a more effective blocker. Third, the characteristics of Ca2+ oscillatory activity in astrocytes during prolonged IP3 elevations, especially when sustained at different levels, can reveal its cellular mechanisms and potentially be used to examine astrocyte heterogeneity, within one brain region or between regions. One method for maintaining higher IP3 levels 141 in vivo, though not necessarily at a constant level, would be to use mice expressing GqDREADDs (Designer Receptor Exclusively Activated by Designer Drugs) in astrocytes and intraperitoneally administer CNO (clozapine N-oxide) at different concentrations (Roth, 2016; Xie et al., 2015b). The final work presented in this dissertation addressed both spontaneous and periodically-evoked astrocyte Ca2+ activity. Experimental data were used to characterize these two forms of activity and to develop a Hidden Markov Model (HMM) that describes them. The model was then extended to a HMM with time-varying transition probabilities, in order to account for the dynamics of having multiple stimulations. Using this model, we were able to account for the high variability of astrocyte Ca2+ activity and, thus, interpret our experimental data. To our knowledge, this is the first study showing that astrocytes have the ability to respond in two opposing manners to the same agonist— either increasing or decreasing their probability of being active relative to spontaneous activity levels—depending only on the stimulation frequency. This finding has important implications. It suggests that astrocytes do not act as simple, linear input-output systems; rather, they can integrate signals in a complex manner and respond distinctly to different input patterns. It also suggests that astrocyte spontaneous activity may provide a baseline against which evoked activity should be measured, and a relative increase or decrease in activity can have distinct downstream functions. Hence, examining the statistics of astrocyte Ca2+ activity over the course of an experiment may provide essential information. Comparing our experimental data and probabilistic model, we also found evidence for a unique form of depression in Ca2+ activity: During high frequency 142 stimulation, in addition to typical refractory mechanisms (e.g. receptor desensitization), negative feedback suppresses astrocyte spontaneous activity, even in astrocyte processes that have not responded to the agonist. This unique form of astrocyte “silencing” has two major implications for experimental studies. First, this silencing is visible at a population level, rather than at the level of individual astrocytes. This occurs because, while the response probability decreases with repeated stimulation, some astrocyte processes are still able to respond to the stimuli or exhibit spontaneous activity. Due to this variability between astrocyte processes, we were not able to detect any overarching form of Ca2+ depression in individual processes during high frequency stimulation. Thus, examining both individual astrocytes and the population activity can provide different insights into astrocyte function under various experimental conditions. Second, to our knowledge, this astrocyte silencing in response to stimulation has never been shown before. In fact, astrocytes have been known to respond with increased Ca2+ activity to different types of input neurotransmitters (excitatory or inhibitory), even when different GPCR types become activated (Gq vs. Gi-GPCRs), suggesting that astrocytes act as a redundant layer of integration in brain circuits (Guerra-Gomes et al., 2018). Our findings, however, challenge this notion by suggesting that astrocytes can decrease their activity in response to the same type of agonist, depending on the input frequency. This result still needs to be further examined for different neurotransmitter types and with more physiological stimulation (e.g. in vivo optogenetic activation of excitatory or inhibitory neurons at different rates, with simultaneous recordings of astrocyte Ca2+ activity); however, it suggests a new way to think about astrocyte activity when analyzing experimental data. 143 The two computational models we introduced in this dissertation can also move the astrocyte mathematical modeling field forward. The biophysical, mechanistic model is novel because it is one of the few astrocyte models that is based on experimental measurements from non-cultured astrocytes (astrocytes in slice and in vivo are considered to be significantly different from cultured astrocytes (Cahoy et al., 2008b)). Additionally, it is based on experimental data consisting of stimulation with focal, brief agonist pulses, which mimic physiological stimulation better than agonist bath application (Pasti et al., 1995). Importantly, it is one of the few models that considers the heterogeneity of Ca2+ responses, rather than treating them all as one stereotyped response. Finally, our model is an open-cell model (i.e. total intracellular Ca2+ levels are able to fluctuate due to plasma membrane channel/pump activity), which is physiologically more accurate than a closed-cell model and, as we found, has important implications in the model results. The probabilistic HMM model we developed is also novel for several reasons. First, it is one of the few modeling studies that describes Ca2+ dynamics (in astrocytes or other cell types) directly, using a top-down, statistical approach (Skupin and Falcke, 2007, 2010; Tilūnaitė et al., 2017). Additionally, it is the only such astrocyte model that considers the dynamic time course of stimulation that astrocytes are exposed to in vivo, rather than considering constant stimulation (Tilūnaitė et al. (2017) did so as well, but they studied Ca2+ activity in HEK293T cells, not astrocytes). Third, while Ca2+ traces were simplified to binary (On-Off) traces, we took into account the duration of Ca2+ events rather than considering all Ca2+ events equal by reducing them to Ca2+ spikes, as done in other statistical models of Ca2+ activity (Skupin and Falcke, 2007, 2010; 144 Tilūnaitė et al., 2017). Fourth, our model has merely one free parameter (Teff), with the other parameters (only 4-5 transition probabilities) determined by a subset of experimental data using machine learning training algorithms. In addition, the data used to train and evaluate the HMM was from astrocyte processes (they are in close proximity to neuronal synapses and blood vessels, thus are likely more relevant than somas when considering astrocyte-neuron interactions), unlike other statistical models that focused mainly on activity in astrocyte somas (Skupin and Falcke, 2007, 2010). 5.2 Future Directions and Preliminary Results Our studies have shed light on how astrocyte Ca2+ signals reflect the inputs that astrocytes receive, and on different features of these Ca2+ dynamics which are informative when interpreting experimental data. Further studies would need to examine these results in neuronal-induced astrocyte activity and under more physiological conditions. Additionally, future work should examine the potential effects of the different Ca2+ activity patterns that we described, on neuronal activity. The computational models we introduced here should also be extended to become more comprehensive and to include astrocyte-neuron interactions. Here, we describe, in greater detail, a few example experimental and modeling studies and show some preliminary experimental results. Since astrocytes are able to respond differently to different input frequencies (shown in Chapter 4), we expect that in response to various patterns of neuronal firing, they would also exhibit distinct forms of Ca2+ activity (with each having distinct functional outcomes). Consistent with this, previous studies showed that, depending on the neuronal firing frequency, astrocytes can behave differently (Covelo and Araque, 145 2018; Perea and Araque, 2005). However, it is unclear how astrocytes respond to different neuronal firing patterns through their Ca2+ activity, and how this activity translates into different effects of astrocytes onto neurons. One experimental limitation in studying the effects of neuronal firing on astrocytes is that the functional connectivity of individual neurons and astrocytes are often unclear: When stimulating a neuron, it is not obvious which astrocyte processes are functionally connected to that particular synapse and should be responsive. This can confound experimental interpretations. Using an experimental setup where the activity of a population of neurons is synchronized increases the likelihood that individual astrocyte processes in the region are receiving similar inputs. The isolated hippocampal preparation can be used as such an experimental setup, since it intrinsically (without pharmacological manipulation) generates theta rhythms (Goutagny et al., 2009). This preparation has been established in our lab (by Feliks Royzen, a member of Dr. John White’s lab). Neuronal local field potentials (LFP) have been recorded simultaneously with whole-cell and on-cell recordings from individual neurons, confirming a tight correlation between pyramidal cell spiking and LFP peaks. An additional benefit of using this preparation for studying astrocyte responses to neuronal activity is that is allows for pharmacological manipulation, unlike in vivo experimental settings. Studying astrocyte activity in this theta-generating hippocampal preparation is also intriguing for other reasons: It has been proposed in several bodies of work that astrocytes play a key role in brain rhythms, such as hippocampal theta rhythms, and, consequently, are involved in learning, memory, and cognition (Hassanpoor et al., 2014; Parpura and Haydon, 2009; Sibille et al., 2015b). Also, earlier studies have suggested that 146 astrocytes in the hippocampus respond to neuronal activity (Haustein et al., 2014; Khakh and Sofroniew, 2015; Panatier et al., 2011), control neuronal network synchronization through their Ca2+ transients (Sasaki et al., 2014), and exhibit ~4 Hz oscillations in their membrane potentials during hippocampal theta rhythms (Mishima and Hirase, 2010). Using this isolated hippocampal preparation in GCaMP5G x GFAP-Cre mice (same mice used in Chapter 4; N=5, aged p13-16), we simultaneously recorded astrocyte Ca2+ activity and neuronal LFPs (Fig. 5.1) while performing two sets of manipulations: (1) disrupting theta rhythms by applying DNQX (6,7-dinitroquinoxaline-2, 3-dione), an ionotropic glutamate receptor antagonist which blocks synaptic transmission (Honoré et al., 1988; Watkins et al., 1990); (2) decreasing and increasing extracellular K+ concentrations (3 and 6 mM, respectively, with the normal concentration being 4.5 mM), which can affect theta power and frequency (Goutagny et al., 2009) as well as astrocyte function (e.g. through modifying the activity of Kir4.1 channels or NKA pumps; see Introduction). Using a combination of these manipulations, we have a total of six experimental conditions. We also ran another set of control experiments to determine the difference between spontaneous astrocyte Ca2+ activity in acute hippocampal slices and the isolated hippocampal preparation (data not shown; N=2, aged p12-13). This was necessary because the intrinsic theta rhythms in this hippocampal preparation are only observed in young mice (~p12-16), and our previous experimental data of spontaneous activity (in Chapter 4) were from adult mice (>4 weeks old) and in the somatosensory cortex. Here, I present a brief summary of our preliminary results; however, more indepth analysis will need to be done later. 100 % ∆F/F 147 Frequency (Hz) Power (μV2/Hz) 1200 12 1000 800 8 600 400 4 200 0 0.5 1 1.5 2 2.5 3 Time (min) A Mean Ca2+ Freq. (mHz) Figure 5.1. Simultaneous recordings of astrocyte Ca2+ activity and hippocampal theta rhythms. (Top) Example Ca2+ traces from astrocyte processes (average of all processes for each of the two astrocytes) during hippocampal theta rhythms ([K+] = 4.5 mM). (Bottom) Spectrogram of the LFP recording performed simultaneously with the astrocyte activity above. B Soma In our preliminary analyses, astrocyte Ca2+ events (from the subiculum and CA1 regions) were automatically detected from Ca2+ traces (similar method as in Chapter 4) Mean Ca2+ Freq. (Hz) from somas and processes; here, responses of processes were averaged over all processes of individual astrocytes. Ca2+ Peak eventFreq.(Hz) kinetics, event frequencies, and the percentage of Mean Ca2+ Duration (s) active ROIs were measured. Also, the LFP recordings (from the subiculum) were Proc. analyzed to detect the theta band (~2-8 Hz) power, the peak frequency, and the power at the peak frequency. Surprisingly, we found no correlation between these different measurements (Fig. 5.2A). In the future, these measurements should be repeated for Ca2+ Duration (s) Active POI Norm. Power at Peak Freq. than the averaged fluorescence of all the individual ROIs from astrocyte processes (rather Proc. Time (min) 148 B Mean Ca2+ Freq. (mHz) Soma 30 20 10 Mean Ca2+ Duration (s) Peak Freq.(Hz) 0 Ca2+ Freq. (mHz) Proc. 30 20 10 0 Norm. Power at Peak Freq. 20 15 10 5 Norm. Theta Band Power [K+] = 3 mM QX DN + + DN QX 0 DN QX Ca2+ Duration (s) % of Active ROI Proc. 25 + A 4.5 mM 6 mM Figure 5.2. Astrocyte Ca2+ activity does not correlate with LFP power or frequency, but changes under different experimental conditions. (A) Different features of astrocyte Ca2+ activity in CA1 plotted against LFP features. Each data point represents the average from all astrocytes in the particular recording. Blue and red circles indicate somas and averaged processes, respectively. (B) Box plots of CA1 astrocyte features plotted for each of the six experimental conditions, for the somas and processes, as indicated on the right. 149 processes of each astrocyte, which is presented here). However, the fact that we saw no correlations here suggests that the range of neuronal firing rates that we examined was too small to alter astrocyte Ca2+ activity in significantly different manners. A broader range of neuronal firing rates need to be examined in future studies, perhaps through the use of optogenetic neuronal activation at different rates. Next, we examined a variety of astrocyte Ca2+ features (a few examples shown in Fig. 5.2B) during each of the six experimental conditions, for astrocyte somas and processes in each brain region (subiculum vs. CA1). We found that many of these features changed significantly under different experimental conditions. For instance, in CA1, astrocyte somas under normal conditions ([K+] = 4.5 mM, no DNQX) had a significantly different Ca2+ event frequency distribution compared to high [K+] conditions (with or without DNQX; Kruskal-Wallis test, p<0.01; Fig. 5.2B, top plot). However, no differences in somatic Ca2+ durations, rise times, or decay times were detected (data not shown). Astrocyte processes, on the other hand, exhibited different Ca2+ event frequencies during the presence or absence of theta rhythms only with normal [K+] (Kruskal-Wallis test, p<0.01; Fig. 5.2B, middle plot). Astrocyte processes also had different Ca2+ event durations (Fig. 5.2B, bottom plot), decay times, and peak amplitudes (not shown) under different conditions. Despite having found some significant differences, no meaningful trend was found in how these Ca2+ features relate to different extracellular [K+] levels or to synchronized neuronal activity (i.e. theta rhythms). Further analyses regarding these differences are yet to be done. We are also examining the correlation in activity among different astrocyte processes and comparing to control experiments. A preliminary 150 examination of these indicates that such correlations may be affected drastically in the presence and absence of synchronized neuronal activity. Other future studies include developing more comprehensive mathematical models of astrocyte Ca2+ activity. To achieve this, one possible direction to take is integrating the two models introduced in this dissertation. As a first step, the statistics of spontaneous and evoked Ca2+ durations (On dwell times) and inter-event intervals (Off dwell times) can be used to include noise in the deterministic biophysical model of Chapters 2 and 3. Using this probabilistic model, the type and level of noise can be incorporated into appropriate equations based on experimental evidence (for instance, added directly to the differential equation that describes cytosolic Ca2+ levels, to account for spontaneous extracellular Ca2+ influx (Rungta et al., 2016; Srinivasan et al., 2015)). This more complete biophysical model of Ca2+ dynamics, which would include spontaneous activity and noise in its evoked responses (e.g. in latency, reliability, etc.), can then be used to examine astrocyte responses to multiple stimulations with varying frequencies, while accounting for the biological mechanisms proposed by our probabilistic model (e.g. refractory mechanisms). This model would have the potential to truly move the astrocyte modeling field forward as it would be the first biophysical model to incorporate the full extent of astrocyte Ca2+ heterogeneity and be based on in situ experimental data. To study astrocyte-neuron interactions, this astrocyte model can be further extended to include communication with neurons. This would require adding several components: glutamate and GABA transporters, Kir4.1 channels, NXC and NKA pump mechanisms, neurons (e.g. either one neuron or both presynaptic and postsynaptic 151 neurons), and an extracellular space. While the parameters for these various mechanisms can be obtained from previous modeling studies (of astrocytes, e.g. (Sibille et al., 2015b), and other cells), many can also be obtained from or restricted based on results from experiments, such as the theta rhythm experiments proposed above. Using such a model, researchers would be able to test several hypotheses in the astrocyte-neuron field and generate new testable hypotheses that can guide future experiments on astrocyte-neuron interactions. 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