Rigorous derivation of the nonlocal reaction-diffusion FitzHugh-Nagumo system

TL;DR

This paper rigorously derives the nonlocal reaction-diffusion FitzHugh-Nagumo system from a spatially extended kinetic model using a relative entropy method, establishing a rigorous link between kinetic and reaction-diffusion models.

Contribution

It introduces a kinetic FitzHugh-Nagumo model with local interactions and proves its hydrodynamic limit converges to the classical nonlocal reaction-diffusion system.

Findings

Hydrodynamic limit converges to the reaction-diffusion system

Relative entropy method effectively links kinetic and macroscopic models

Provides a rigorous mathematical foundation for the derivation

Abstract

We introduce a spatially extended transport kinetic FitzHugh-Nagumo model with forced local interactions and prove that its hydrodynamic limit converges towards the classical nonlocal reaction-diffusion FitzHugh-Nagumo system. Our approach is based on a relative entropy method, where the macroscopic quantities of the kinetic model are compared with the solution to the nonlocal reaction-diffusion system. This approach allows to make the rigorous link between kinetic and reaction-diffusion models.