Towards a further understanding of the dynamics in the excitatory NNLIF neuron model: Blow-up and global existence

Pierre Roux; Delphine Salort

doi:10.3934/krm.2021025

Article Contents

2021, Volume 14, Issue 5: 819-846. Doi: 10.3934/krm.2021025

This issue Previous Article Pencil-beam approximation of fractional Fokker-Planck Next Article A criterion for asymptotic preserving schemes of kinetic equations to be uniformly stationary preserving

Towards a further understanding of the dynamics in the excitatory NNLIF neuron model: Blow-up and global existence

Pierre Roux^1, , and
Delphine Salort^2,

1.
Laboratory Jacques-Louis Lions (LJLL), UMR 7238 CNRS, Sorbonne Université, 75205 Paris Cedex 06, France
2.
Laboratory of Computational and Quantitative Biology (LCQB), UMR 7238 CNRS, Sorbonne Université, 75205 Paris Cedex 06, France

^* Corresponding author: Pierre Roux
^* Corresponding author: Pierre Roux

Received: February 29, 2020

Revised: May 31, 2021

Early access: August 2021

Published: October 2021

Delphine Salort was supported by the grant ANR ChaMaNe, ANR-19-CE40-0024

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Abstract

The Nonlinear Noisy Leaky Integrate and Fire (NNLIF) model is widely used to describe the dynamics of neural networks after a diffusive approximation of the mean-field limit of a stochastic differential equation. In previous works, many qualitative results were obtained: global existence in the inhibitory case, finite-time blow-up in the excitatory case, convergence towards stationary states in the weak connectivity regime. In this article, we refine some of these results in order to foster the understanding of the model. We prove with deterministic tools that blow-up is systematic in highly connected excitatory networks. Then, we show that a relatively weak control on the firing rate suffices to obtain global-in-time existence of classical solutions.

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Mathematics Subject Classification: Primary: 35K60, 82C31, 92B20; Secondary: 35Q84.

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Figure 1. Comparison between the lower bound (10) in [4] and the lower bound in Theorem 3.1 for non-existence of stationary states, for different values of a.The new bound is always better when $ V_F-V_R $ is large enough.We set $ V_R =-1 $ for convenience but it does not impact the results

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