Spatiotemporal patterns in the wake of traveling wave solutions to the Morris-Lecar model of neural tissue
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In this dissertation, we discuss spatiotemporal patterns in the wake of traveling waves in a microelectrode array (MEA) recording of a human epileptic seizure. In chapter two, we describe a method for estimating the direction of planar waves found in the last third of the seizure. We categorize the different phenomena that occur during those waves when projected along a one-dimensional slice in the domain. In chapter three, we summarize known examples of patterns in the wake of traveling wave solutions to reaction-diffusion systems. A brief review of results regarding the spectral stability of traveling wave solutions to reaction-diffusion equations is provided in chapter four. We review the essential spectrum and absolute spectrum, and summarize results about glued front-and-back pulse solutions. Using a reaction-diffusion model with Morris-Lecar dynamics, we present numerical experiments on a one-dimensional domain that exhibit spatiotemporal patterns in the wake of traveling waves. These patterns are precipitated by “backfiring” waves emitted from the primary wave in the opposite direction of initial travel, and qualitatively reproduce many of the features found in the last third of the seizure. A review of the model is given in chapter five. and a description of the phenomena found over an exhaustive set in a relevant parameter space of the model is given in chapter six. We compute branches of solutions in the parameter plane using numerical continuation in chapter seven. We describe the different types of solutions found along these branches. We present results on a curve of solutions where two branches of homoclinic orbits to equilibria in the moving coordinate frame meet at a heteroclinic loop, or T-point. We analyze the linear stability of solutions along this branch and draw comparisons to a known model that exhibits backfiring behavior. In chapter eight, we discuss seizure behavior in two spatial dimensions and present numerical experiments of the Morris-Lecar model in two dimensions. We describe results from backfiring waves initiated by a single point source and by two point sources in a two-dimensional domain. We show examples of simulations generated by two point sources that mimic the patterns in the empirical data.