We consider a spatially extended mean-field model of a FitzHugh–Nagumo neural network, with a rescaled interaction kernel. Our main purpose is to prove that its asymptotic limit in the regime of strong local interactions converges toward a system of reaction–diffusion equations taking account for the average quantities of the network. Our approach is based on a modulated energy argument, to compare the macroscopic quantities computed from the solution of the transport equation, and the solution of the limit system. The main difficulty, compared to the literature, lies in the need of regularity in space of the solutions of the limit system and a careful control of an internal nonlocal dissipation.

中文翻译：

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空间扩展平均场 FitzHugh-Nagumo 模型的渐近极限

我们考虑了 FitzHugh-Nagumo 神经网络的空间扩展平均场模型，具有重新调整的交互内核。我们的主要目的是证明其在强局部相互作用范围内的渐近极限收敛于考虑网络平均数量的反应扩散方程系统。我们的方法基于调制的能量参数，比较从输运方程的解计算的宏观量和极限系统的解。与文献相比，主要困难在于极限系统解的空间规律性和对内部非局部耗散的仔细控制。