Hierarchical Wilson–Cowan Models and Connection Matrices

Abstract

This work aims to study the interplay between the Wilson–Cowan model and connection matrices. These matrices describe cortical neural wiring, while Wilson–Cowan equations provide a dynamical description of neural interaction. We formulate Wilson–Cowan equations on locally compact Abelian groups. We show that the Cauchy problem is well posed. We then select a type of group that allows us to incorporate the experimental information provided by the connection matrices. We argue that the classical Wilson–Cowan model is incompatible with the small-world property. A necessary condition to have this property is that the Wilson–Cowan equations be formulated on a compact group. We propose a p-adic version of the Wilson–Cowan model, a hierarchical version in which the neurons are organized into an infinite rooted tree. We present several numerical simulations showing that the p-adic version matches the predictions of the classical version in relevant experiments. The p-adic version allows the incorporation of the connection matrices into the Wilson–Cowan model. We present several numerical simulations using a neural network model that incorporates a p-adic approximation of the connection matrix of the cat cortex.