Some theoretical results for a class of neural mass equations

TL;DR

This paper provides new theoretical insights into neural mass equations modeling visual cortex processes, demonstrating existence, uniqueness, and stability of solutions within a hyperbolic geometric framework, supported by numerical simulations.

Contribution

It introduces a rigorous mathematical analysis of nonlinear integro-differential equations on hyperbolic space, including existence, uniqueness, and stability results, and studies localized solutions using hyperbolic analysis.

Findings

Existence and uniqueness of solutions established

Stability analysis of stationary solutions performed

Numerical simulations support theoretical results

Abstract

We study the neural field equations introduced by Chossat and Faugeras in their article to model the representation and the processing of image edges and textures in the hypercolumns of the cortical area V1. The key entity, the structure tensor, intrinsically lives in a non-Euclidean, in effect hyperbolic, space. Its spatio-temporal behaviour is governed by nonlinear integro-differential equations defined on the Poincar\'e disc model of the two-dimensional hyperbolic space. Using methods from the theory of functional analysis we show the existence and uniqueness of a solution of these equations. In the case of stationary, i.e. time independent, solutions we perform a stability analysis which yields important results on their behavior. We also present an original study, based on non-Euclidean, hyperbolic, analysis, of a spatially localised bump solution in a limiting case. We illustrate our theoretical results with numerical simulations.