Date

2019

Document Type

Dissertation

Degree

Doctor of Philosophy

Department

Mathematics

First Adviser

Zhang, Linghai

Abstract

We study the existence, uniqueness, and stability of traveling waves in neural field models under various assumptions on the firing rates and kernels. In the case of Heaviside ring rates, we study unique fronts arising from oscillatory synaptic coupling types. We then use functional analysis to show that Heaviside firing rates can be continuously deformed into smooth, sigmoidal functions and the existence of fronts persists. Finally, we combine our results for the front with geometric singular perturbation theory to prove the existence of pulses when certain lateral inhibition kernels reduce the systems of integral equations into systems of PDEs.

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