An understanding of the physical solutions and the blow-up phenomenon for Nonlinear Noisy Leaky Integrate and Fire neuronal models

TL;DR

This paper investigates the behavior and blow-up phenomena in nonlinear noisy leaky integrate-and-fire neuron models through numerical analysis of particle systems, comparing stochastic and mean-field approaches.

Contribution

It provides a detailed numerical study of classical and physical solutions, enhancing understanding of blow-up phenomena and solution behaviors in neural network models.

Findings

Weakly connected systems converge to a steady state.

Strongly connected systems may tend to a plateau distribution.

Blow-up occurs in finite time under certain conditions.

Abstract

The Nonlinear Noisy Leaky Integrate and Fire neuronal models are mathematical models that describe the activity of neural networks. These models have been studied at a microscopic level, using Stochastic Differential Equations, and at a mesoscopic/macroscopic level, through the mean field limits using Fokker-Planck type equations. The aim of this paper is to improve their understanding, using a numerical study of their particle systems. We analyse in depth the behaviour of the classical and physical solutions of the Stochastic Differential Equations and, we compare it with what is already known about the Fokker-Planck equation. This allows us to better understand what happens in the neural network when an explosion occurs in finite time. After firing all neurons at the same time, if the system is weakly connected, the neural network converges towards its unique steady state. Otherwise, its behaviour is more complex, because it can tend towards a stationary state or a "plateau" distribution.