The Weakly Nonlocal Limit Of A One - Population Wilson - Cowan Model

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. Based on the shape of pseudopotential we also conjecture that the stationary periodic solutions are unstable. We develop a numerical method for the weakly nonlocal limit of the Wilson - Cowan type model based on the wavelets - Galerkin approach. The method is illustrated by a testing example.