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

Joachim Crevat

doi:10.3934/krm.2019052

Article Contents

2019, Volume 12, Issue 6: 1329-1358. Doi: 10.3934/krm.2019052

This issue Previous Article Focusing solutions of the Vlasov-Poisson system Next Article A note on two species collisional plasma in bounded domains

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

Joachim Crevat^,

Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, UPS IMT, F-31062 Toulouse Cedex 9 France

^* Corresponding author: joachim.crevat@math.univ-toulouse.fr
^* Corresponding author: joachim.crevat@math.univ-toulouse.fr

Received: March 2019

Revised: June 2019

Published: September 2019

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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.

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Mathematics Subject Classification: Primary: 92B20, 35Q92, 35B40, 82C22; Secondary: 35A01, 35A02.

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Figure 1. Bistable regime. (A)-(B) Spatio-temporal evolution of the macroscopic function $ V_f $ computed from the solution $ f $ of the kinetic equation (5) with $ \tau = 0 $, and different values of the parameter $ \varepsilon $, fixed at (A) $ 10^{-1} $, (B) $ 10^{-3} $. (C) Profile of the macroscopic function $ V_f(t, \cdot) $ at different fixed times, computed with $ \varepsilon = 10^{-3} $ and with $ \tau = 0 $

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Figure 2. Bistable regime. Numerical approximation of the density function $ f $ solution of the kinetic equation (5) at fixed time (A) $ t = 0 $, (B) $ t = 75 $ and (C) $ t = 150 $, computed with the parameters $ \varepsilon = 10^{-3} $ and $ \tau = 0 $

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Figure 3. Oscillatory regime. (A)-(B)-(C) Spatio-temporal evolution of the macroscopic function $ V_f $ computed from the solution $ f $ of the kinetic equation (5) with three different values of the parameter $ \varepsilon $, fixed at (A) $ \varepsilon = 10^{-1} $, (B) $ \varepsilon = 10^{-3} $ and (C) $ \varepsilon = 10^{-5} $. (D)-(E) Profile of the macroscopic function $ V_f(t, \cdot) $ computed with $ \varepsilon = 10^{-5} $ at time $ t = 60 $ and $ t = 400 $ respectively. (F) Trajectory in the phase space $ (v, w) $ of the couple $ (V_f, W_f) $ at fixed position $ \mathbf{x} = 0.2 $ between times $ 0 $ and $ t = 400 $ computed with $ \varepsilon = 10^{-5} $. The other parameters are fixed at $ a = -0.25 $, $ b = 3 $, and $ \tau = 0.02 $

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Figure 4. Excitable regime. (A) Spatio-temporal evolution of the macroscopic function $ V_f $ computed from the solution $ f $ of the kinetic equation (5) with $ \varepsilon = 10^{-5} $. (B) Corresponding profile of the macroscopic function $ V_f(t, \cdot) $ computed at different times. (C) Trajectory in the phase space $ (v, w) $ of the couple $ (V_f, W_f) $ at fixed position $ \mathbf{x} = 0.2 $ between times $ 0 $ and $ t = 1000 $ computed with $ \varepsilon = 10^{-5} $. The other parameters are fixed at $ a = 0 $, $ b = 7 $, and $ \tau = 0.002 $

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Figure 5. Profile of the macroscopic function $ V_f $ computed from the solution $ f $ of the kinetic equation (5) with $ \varepsilon = 10^{-4} $, and with the points $ ( \mathbf{x}_i, v_i)_{1\leq i \leq n} $ from the solution of the FHN system (3), at fixed time $ t = 225 $. The other parameters are the same as in Figure 3

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