Traveling Wave Solutions to a Neural Field Model With Oscillatory Synaptic Coupling Types

Abstract

In this paper, we investigate the existence, uniqueness, and spectral stability of traveling waves arising from a single threshold neural field model with one spatial dimension, a Heaviside firing rate function, axonal propagation delay, and biologically motivated oscillatory coupling types. Neuronal tracing studies show that long-ranged excitatory connections form stripe-like patterns throughout the mammalian cortex; thus, we aim to generalize the notions of pure excitation, lateral inhibition, and lateral excitation by allowing coupling types to spatially oscillate between excitation and inhibition. In turn, we hope to analyze traveling fronts and pulses with novel shapes. With fronts as our main focus, we exploit Heaviside firing rate functions in order to establish existence and utilize speed index functions with at most one critical point as a tool for showing uniqueness of wave speed. We are able to construct Evans functions, the so-called stability index functions, in order to provide positive spectral stability results. Finally, we show that by incorporating slow linear feedback, we can compute fast pulses numerically with phase space dynamics that are similar to their corresponding singular homoclinical orbits; hence, our work answers open problems and provides insight into new ones.