Large coupling in a FitzHugh-Nagumo neural network: quantitative and strong convergence results

TL;DR

This paper analyzes a neural network model based on FitzHugh-Nagumo equations, showing that under certain conditions, the potential distribution concentrates into a Gaussian profile, with rigorous quantitative convergence results.

Contribution

It provides the first rigorous proof of Gaussian concentration profiles in a neural network model using novel entropy and regularity techniques.

Findings

Potential density concentrates into a Dirac distribution

The concentration profile is proven to be Gaussian

Two strong convergence estimates are established

Abstract

We consider a mesoscopic model for a spatially extended FitzHugh-Nagumo neural network and prove that in the regime where short-range interactions dominate, the probability density of the potential throughout the network concentrates into a Dirac distribution whose center of mass solves the classical non-local reaction-diffusion FitzHugh-Nagumo system. In order to refine our comprehension of this regime, we focus on the blow-up profile of this concentration phenomenon. Our main purpose here consists in deriving two quantitative and strong convergence estimates proving that the profile is Gaussian: the first one in a L1 functional framework and the second in a weighted L2 functional setting. We develop original relative entropy techniques to prove the first result whereas our second result relies on propagation of regularity.