On existence and stability of ring solutions to Amari neural field equation with periodic microstructure and Heaviside activation function

Abstract

In the present research, existence and stability of ring solutions to two-dimensional Amari neural field equation with periodic microstructure and Heaviside activation function are studied. Results on dependence of the inner and the outer radii of the ring solutions are obtained. Necessary conditions for existence and sufficient conditions for non-existence of radial travelling waves are formulated for homogeneous neural medium and neural media with mild periodic microstructure. Theoretical results obtained are illustrated with a concrete example based on a connectivity function commonly used in the neuroscience community.