Abstract
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.
Highlights
The FitzHugh-Nagumo (FHN) model [21, 30] focuses on the evolution of the membrane electrical potential of a nerve cell depending on the input it receives
The main contribution of this article is the rigorous justification of the kinetic model (1.5) we considered in [16] as the mean-field limit of the FHN system (1.3)
In this paper, we have proved the mean-field limit of the deterministic spatially-extended FHN model for neural networks towards a nonlocal kinetic equation as the number of neurons goes to infinity
Summary
The FitzHugh-Nagumo (FHN) model [21, 30] focuses on the evolution of the membrane electrical potential of a nerve cell depending on the input it receives. We mention the work of Dobrushin [17], who introduced some classical methods to prove the mean-field limit of an individual-based model of interacting particles with a bounded globally Lipschitz continuous interaction kernel towards a Vlasov equation, using a stability result of the solutions to the kinetic model with respect to their initial data, in a suitable topology of probability measures. We prove the following lemma, which yields the existence and uniqueness of the solution of a system of equations approximating (2.2), in which we consider the contribution of the interactions as a source term. One of the motivations to study the mean-field model is the analysis of the macroscopic quantities computed from a measure solution to (1.5), though the equation formally satisfied by the average membrane potential in the network is not closed. We mention that in the presence of noise as in [36], the authors showed some numerical simulations of the FHN system (1.3) with homogeneous interactions, where the finite number of neurons can cause the emergence of relaxation cycles near the transition from the excitable regime to the oscillatory regime, whereas the deterministic model still presents a unique stable fixed point
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