  This work aims to study the interplay between the Wilson-Cowan model and the
connection matrices. These matrices describe the cortical neural wiring, while
the Wilson-Cowan equations provide a dynamical description of neural
interaction. We formulate the 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.
