Article Contents

2024, Volume 29, Issue 4: 2018-2042. Doi: 10.3934/dcdsb.2023165

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

Alain Blaustein^1, , and
Emeric Bouin^{1, 2,}

1.
Pennsylvania State University, Department of Mathematics, State College, PA 16802, USA
2.
CEREMADE - Université Paris-Dauphine, UMR CNRS 7534, 75775 Paris Cedex 16, France

^*Corresponding author: Emeric Bouin
^*Corresponding author: Emeric Bouin

Received: July 2022

Revised: March 2023

Early access: October 2023

Published: April 2024

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Abstract

In this paper we focus on a mean-field model for a spatially extended FitzHugh-Nagumo neural network. In the regime where strong and local interactions dominate, we quantify how the probability density of voltage concentrates into a Dirac distribution. Previous work investigating this question has 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.

Keywords:

Mathematics Subject Classification: Primary: 35Q92; Secondary: 35B40, 35C20, 35K57, 35G20, 53C35.

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