Open AccessTowards a further understanding of the dynamics in the excitatory NNLIF neuron model: Blow-up and global existence
Author(s) -
Pierre Le Roux,
Delphine Salort
Publication year - 2021
Publication title -
kinetic and related models
Language(s) - English
Resource type - Journals
eISSN - 1937-5093
pISSN - 1937-5077
DOI - 10.3934/krm.2021025
Subject(s) - limit (mathematics) , nonlinear system , excitatory postsynaptic potential , convergence (economics) , stochastic differential equation , dynamics (music) , computer science , order (exchange) , statistical physics , mathematics , artificial neural network , physics , mathematical analysis , inhibitory postsynaptic potential , artificial intelligence , finance , quantum mechanics , neuroscience , economics , acoustics , biology , economic growth
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.