# Asymptotic limit of a spatially-extended mean-field FitzHugh-Nagumo model

## TL;DR

This paper proves that a spatially extended mean-field FitzHugh-Nagumo neural network model converges to a reaction-diffusion system in the strong local interaction regime, using a modulated energy approach.

## Contribution

It establishes the asymptotic limit of a spatially extended mean-field FitzHugh-Nagumo model to a reaction-diffusion system, addressing regularity and nonlocal dissipation challenges.

## Key findings

- Convergence of the mean-field model to reaction-diffusion equations.
- Development of a modulated energy method for this context.
- Handling of nonlocal dissipation in the analysis.

## Abstract

We consider a spatially extended mean-field model of a FitzHugh-Nagumo neural network, with a rescaled interaction kernel. Our main purpose is to prove that its asymptotic limit in the regime of strong local interactions converges toward a system of reaction-diffusion equations taking account for theaverage quantities of the network. Our approach is based on a modulated energy argument, to compare the macroscopic quantities computed from the solution of the transport equation, and the solution of the limit system. The main difficulty, compared to the literature, lies in the need of regularity in space of the solutions of the limit system and a careful control of an internal nonlocal dissipation.