A review about Nonlinear Noisy Leaky Integrate and Fire models for neural networks

Abstract

In these notes we analyse a family of self-contained mean field systems, called nonlinear noisy leaky integrate and fire (NNLIF) models, which describe the activity of neural networks by means of the membrane potential. These models are based on nonlinear systems of PDEs of Fokker-Planck type. We study the wide range of phenomena that appear in this kind of models: blow-up/global existence, asynchronous/synchronous solutions, instability/stability of the steady states... This review is intended to collect the main results of papers in collaboration with José A. Carrillo, Benoît Perthame, Pierre Roux, Ricarda Schneider and Delphine Salort.