| Description |
We observe that the problem of cross field design is related to the Ginzburg-Landau problem from mathematical physics. Understanding this connection leads us to an efficient technique for computing boundary aligned cross fields with good singularity placement. Such cross fields are well suited for quad meshing and parameterization of geometries common in CAD applications where boundaries and sharp features are common. We prove that the separatrices of a cross field obtained via our method partition a domain into four-sided regions. We next investigate the practical problem of building coarse quad partitions on discrete curved surfaces. We extend the cross field design method developed in the first chapter to curved surfaces, develop a method for accurately computing the trajectory of streamlines in the neighborhood of a singularity, and introduce a simple and robust algorithm for simplifying the partitions obtained via naive separatrix tracing into coarse quad partitions. We conclude the first part of this dissertation with a discussion of future work. In the second part of this dissertation, we investigate the neurocomputational properties and dynamics of the external tufted cell of the mammalian olfactory bulb by developing a conductance-based model of the cell. We demonstrate that the model exhibits key dynamical properties that have been observed in biological cells, and identify the bifurcation structure and dynamics that explain bursting behavior in the model. We investigate parameter sensitivity of the model and identify parameters that control several bursting characteristics such as burst frequency and burst duration. Finally, we examine the response of the model cell to periodic inputs and corroborate observations in the literature that external tufted cells are better entrained by input signal frequencies that are higher rather than lower than the natural bursting frequency of the cell |
