Spatially structured waves and oscillations in neuronal networks with synaptic depression and adaptation

We analyze the spatiotemporal dynamics of systems of nonlocal integro-differential equations, which all represent neuronal networks with synaptic depression and spike frequency adaptation. These networks support a wide range of spatially structured waves, pulses, and oscillations, which are suggestive of phenomena seen in cortical slice experiments and in vivo. In a one-dimensional network with synaptic depression and adaptation, we study traveling waves, standing bumps, and synchronous oscillations. We find that adaptation plays a minor role in determining the speed of waves; the dynamics are dominated by depression. Spatially structured oscillations arise in parameter regimes when the space-clamped version of the network supports limit cycles. Analyzing standing bumps in the network with only depression, we find the stability of bumps is determined by the spectrum of a piecewise smooth operator. We extend these results to a two-dimensional network with only depression.