Analysis of Nonlinear Noisy Integrate\&Fire Neuron Models: blow-up and steady states

TL;DR

This paper investigates the mathematical properties of nonlinear noisy integrate-and-fire neuron models, focusing on steady states, blow-up phenomena, and convergence, highlighting the critical role of noise and connectivity balance.

Contribution

It provides a comprehensive analysis of the existence, uniqueness, and blow-up behavior of solutions in NNLIF models, emphasizing the impact of network parameters.

Findings

Blow-up occurs in excitatory networks with initial data near firing potential.

Number of steady states depends on network parameters.

Convergence to equilibrium analyzed in the linear case.

Abstract

Nonlinear Noisy Leaky Integrate and Fire (NNLIF) models for neurons networks can be written as Fokker-Planck-Kolmogorov equations on the probability density of neurons, the main parameters in the model being the connectivity of the network and the noise. We analyse several aspects of the NNLIF model: the number of steady states, a priori estimates, blow-up issues and convergence toward equilibrium in the linear case. In particular, for excitatory networks, blow-up always occurs for initial data concentrated close to the firing potential. These results show how critical is the balance between noise and excitatory/inhibitory interactions to the connectivity parameter.