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

Joachim Crevat (1)

1 UT3 - Université Toulouse III - Paul Sabatier

We introduce a spatially-extended model of a neural network of interacting neurons of FitzHugh-Nagumo type without noise, and we establish the mean-field limit of this system towards a nonlocal kinetic equation as the number of neurons in the network goes to infinity. We also state a well-posedness result of this kinetic equation. We work in a space of measures equiped with the Wasserstein distance of order $2$ in order to use optimal transport theory. Our approach is based on deterministic methods, and on an argument of stability of solutions of the kinetic equation in their initial data. The difficulty, as compared to other mean-field models, lies in the spatially-extended aspect of this model, and in the fact that the interaction kernel is not globally Lipschitz continuous. This result rigorously provides a link between the microscopic and mesoscopic scales of observation of the network, which can be used as an intermediary step to derive macroscopic models.

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

Domaines: Equations aux dérivées partielles [math.AP]

Joachim Crevat. Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network. 2018. ⟨hal-01897746v1⟩