Towards a further understanding of the dynamics in NNLIF models: blow-up, global existence and coupled networks.

Pierre Roux (1) , Delphine Salort (2)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 LCQB - Biologie Computationnelle et Quantitative = Laboratory of Computational and Quantitative Biology

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 study the impact of excitatory-inhibitory coupling on finite time blow-up. Finally, we show that a relatively weak control on the firing rate suffices to obtain global-in-time existence of classical solutions.

Mots clés: global existence; delay; neural networks; blow-up; noise; Leaky integrate and fire models

Domaines: Mathématiques [math]

Pierre Roux, Delphine Salort. Towards a further understanding of the dynamics in NNLIF models: blow-up, global existence and coupled networks.. 2021. ⟨hal-02508412v2⟩