The weakly nonlocal limit of a one-population Wilson–Cowan model

Abstract

We derive the weakly nonlocal limit of a one-population neuronal field model of the Wilson–Cowan type in one spatial dimension. By transforming this equation to an equation in the firing rate variable, it is shown that stationary periodic solutions exist by appealing to a pseudo-potential analysis. The solutions of the full nonlocal equation obey a uniform bound, and the stationary periodic solutions in the weakly nonlocal limit satisfying the same uniform bound are characterized by finite ranges of pseudo energy constants. The time dependent version of the model is reformulated as a Ginzburg–Landau–Khalatnikov type of equation in the firing rate variable where the maximum (minimum) points correspond stable (unstable) homogeneous solutions of the weakly nonlocal limit. Based on this formulation it is also conjectured that the stationary periodic solutions are unstable. We implement a numerical method for the weakly nonlocal limit of the Wilson–Cowan type of model based on the wavelet–Galerkin approach. We perform some numerical tests to illustrate the stability of homogeneous solutions and the evolution of the bumps.