Two bump solutions of a homogenized Wilson–Cowan model with periodic microstructure

Abstract

We study the existence and stability of 2-bump solutions of the one-population homogenized Wilson–Cowan model, where the heterogeneity is built in the connectivity functions by assuming periodic modulations in both the synaptic footprint and in the spatial scale. The existence analysis reveals that the generic picture consists of two bumps states for each admissible threshold value for the case when the solutions are independent of the local variable and the firing rate function is modeled as a Heaviside function. A framework for analyzing the stability of 2-bumps is formulated, based on spectral theory for Fredholm integral operators. The stability method deforms to the standard Evans function approach for the translationally invariant case in the limit of no heterogeneity, in a way analogous to the single bump case for the homogenized model. The numerical study of the stability problem reveals that both the broad and narrow bumps are unstable just as in the translationally invariant case when the connectivity function is modeled by means of a wizard hat function. For the damped oscillating connectivity kernel, we give a concrete example of a 2-bump solution which is stable for all admissible values of the heterogeneity parameter.