Concentration profiles in FitzHugh-Nagumo neural networks: A Hopf-Cole approach

TL;DR

This paper analyzes how neuron density concentrates in a spatially extended FitzHugh-Nagumo model with strong local interactions, using a Hopf-Cole approach to derive precise estimates and describe the limiting Gaussian profile.

Contribution

It introduces a Hopf-Cole framework to obtain explicit $L^ Infty$ estimates and convergence rates for the concentration profile in the FitzHugh-Nagumo model.

Findings

Neuronal density concentrates into a Dirac distribution under strong local interactions.

The blow-up profile is shown to be Gaussian.

Explicit convergence rates are established.

Abstract

In this paper we focus on a spatially extended FitzHugh-Nagumo model with interactions. In the regime where strong and local interactions dominate, we quantify how the probability density of neurons concentrates into a Dirac distribution. Previous work investigating this question have provided relative bounds in integrability spaces. Using a Hopf-Cole framework, we derive precise $L^\infty$ estimates using subtle explicit sub- and super- solutions which prove, with rates of convergence, that the blow up profile is Gaussian.