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.

We introduce a spatially extended transport kinetic FitzHugh-Nagumo model with forced local interactions and prove that its hydrodynamic limit converges towards the classical nonlocal reaction-diffusion FitzHugh-Nagumo system. Our approach is based on a relative entropy method, where the macroscopic quantities of the kinetic model are compared with the solution to the nonlocal reaction-diffusion system. This approach allows to make the rigorous link between kinetic and reaction-diffusion models. 1 All our results remain true for N (v) = v(α − β|v| 2 ) with any α, β > 0.

We consider a spatially extended kinetic model of a FitzHugh-Nagumo neural network, with a rescaled interaction kernel. Our main purpose is to prove that its diffusive limit in the regime of strong local interactions converges towards a FitzHugh-Nagumo reaction-diffusion system, taking account for the average quantities of the network. Our approach is based on a relative entropy argument, to compare the macroscopic quantities computed from the solution of the kinetic equation, and the solution of the limiting system. The main difficulty, compared to the literature, lies in the need of regularity in space of the solutions of the limiting system and a careful control of an internal nonlocal kinetic dissipation.As showed in [6], a mean-field description of this network when n goes to infinity is given by the following spatially-extended kinetic model for all ε ą 0, t ą 0, x P R d and u " pv, wq P R 2 , with the notation u 1 " pv 1 , w 1 q P R 2