Concentration phenomena in Fitzhugh-Nagumo's equations: A mesoscopic approach

TL;DR

This paper rigorously demonstrates how a mesoscopic FitzHugh-Nagumo model with strong local interactions converges to a classical nonlocal reaction-diffusion system, linking microscopic and macroscopic neural models.

Contribution

It provides a rigorous mathematical proof of the convergence from a mesoscopic to a macroscopic FitzHugh-Nagumo model using kinetic theory techniques.

Findings

Convergence of mesoscopic model to reaction-diffusion system

Use of Wasserstein distance and entropy methods

Establishment of a rigorous link between microscopic and macroscopic models

Abstract

We consider a spatially extended mesoscopic FitzHugh-Nagumo model with strong local interactions and prove that its asymptotic limit converges towards the classical nonlocal reaction-diffusion FitzHugh-Nagumo system. As the local interactions strongly dominate, the weak solution to the mesoscopic equation under consideration converges to the local equilibrium, which has the form of Dirac distribution concentrated to an averaged membrane potential. Our approach is based on techniques widely developed in kinetic theory (Wasserstein distance, relative entropy method), where macroscopic quantities of the mesoscopic model are compared with the solution to the nonlocal reaction-diffusion system. This approach allows to make the rigorous link between microscopic and reaction-diffusion models.