Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network

Joachim Crevat (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

We consider a spatially-extended model for a network of interacting FitzHugh-Nagumo neurons without noise, and rigorously establish its mean-field limit towards a nonlocal kinetic equation as the number of neurons goes to infinity. Our approach is based on deterministic methods, and namely on the stability of the solutions of the kinetic equation with respect to their initial data. The main difficulty lies in the adaptation in a deterministic framework of arguments previously introduced for the mean-field limit of stochastic systems of interacting particles with a certain class of locally Lipschitz continuous interaction kernels. This result establishes a rigorous link between the microscopic and mesoscopic scales of observation of the network, which can be further used as an intermediary step to derive macroscopic models. We also propose a numerical scheme for the discretization of the solutions of the kinetic model, based on a particle method, in order to study the dynamics of its solutions, and to compare it with the microscopic model.

Mots clés: FitzHugh-Nagumo; Wasserstein distance; Mean-field limit; neural network

Domaines: Equations aux dérivées partielles [math.AP]

Joachim Crevat. Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network. Kinetic and Related Models , 2019, 12 (6), pp.1329-1358. ⟨hal-01897746v2⟩