论文标题
Fitzhugh-Nagumo神经元平均野外网络中混乱的传播
Propagation of chaos in mean field networks of FitzHugh-Nagumo neurons
论文作者
论文摘要
在本文中，我们对$ n $神经元的完全连接网络的行为感兴趣，$ n $倾向于无限。我们假设神经元遵循随机的Fitzhugh-Nagumo模型，其特异性是具有立方术语的非线性。我们证明了该模型混乱时间在平均场框架中的混乱时间均匀的结果。我们还表现出明确的界限。我们使用A. eberle（Arxiv：1305.1233）最初建议的耦合方法，并最近在（1805.11387）中扩展，称为反射耦合。我们同时构建了$ n $粒子系统的解决方案和非线性McKean-Vlasov限制的$ n $独立副本，以至于考虑到适当的半属性，它考虑了该过程的各种可能行为，这两个解决方案往往会更加近距离，随着$ N $的增加，$ n $在时间上增加，并且在时间上均匀。反射耦合使我们能够处理定义网络数量的动力学中潜在潜力的非跨性别性，并在系统相互作用中与足够小的Lipschitz连续相互作用显示系统的独立性。
In this article, we are interested in the behavior of a fully connected network of $N$ neurons, where $N$ tends to infinity. We assume that the neurons follow the stochastic FitzHugh-Nagumo model, whose specificity is the non-linearity with a cubic term. We prove a result of uniform in time propagation of chaos of this model in a mean-field framework. We also exhibit explicit bounds. We use a coupling method initially suggested by A. Eberle (arXiv:1305.1233), and recently extended in (1805.11387), known as the reflection coupling. We simultaneously construct a solution of the $N$-particle system and $N$ independent copies of the non-linear McKean-Vlasov limit in such a way that, considering an appropriate semi-metric that takes into account the various possible behaviors of the processes, the two solutions tend to get closer together as $N$ increases, uniformly in time. The reflection coupling allows us to deal with the non-convexity of the underlying potential in the dynamics of the quantities defining our network, and show independence at the limit for the system in mean field interaction with sufficiently small Lipschitz continuous interactions.
