Single-Pulse Solutions for Oscillatory Coupling Functions in Neural Networks

We study the existence and linear stability of stationary pulse solutions of an integro-differential equation modeling the coarse-grained averaged activity of a single layer of interconnected neurons. The neuronal connections considered feature lateral oscillations with an exponential rate of decay and variable period. We identify regions in the parameter space where solutions exhibit areas of excitation with single- and dimpled-pulses. When the gain function reduces to the Heaviside function, we establish existence of single-pulse solutions analytically. For a more general gain function, we include numerical support of the existence of pulse-like solutions. We then study the linear stability behavior of these solutions.