# American Institute of Mathematical Sciences

October  2021, 14(5): 819-846. doi: 10.3934/krm.2021025

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

 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

Received  March 2020 Revised  June 2021 Published  October 2021 Early access  August 2021

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

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.

Citation: Pierre Roux, Delphine Salort. Towards a further understanding of the dynamics in the excitatory NNLIF neuron model: Blow-up and global existence. Kinetic & Related Models, 2021, 14 (5) : 819-846. doi: 10.3934/krm.2021025
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##### References:
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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