# Hierarchical Wilson-Cowan Models and Connection Matrices

arXiv: https://arxiv.org/abs/2305.00120 · HTML: https://mlsanity.com/paper/2305.00120/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Apr 2023.

## Built on by (most cited first, 2 of 2)

- [Hierarchical Neural Networks, p-Adic PDEs, and Applications to Image Processing](https://mlsanity.com/paper/2406.07790/) (2024)
- [Deep Neural Networks: A Formulation Via Non-Archimedean Analysis](https://mlsanity.com/paper/2402.00094/) (2024)

## Abstract (arXiv)

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.