标题： 平均场FitzHugh-Nagumo模型的统计估计 显示英文标题

标题： Statistical estimation of a mean-field FitzHugh-Nagumo model

摘要： 我们考虑一个粒子相互作用系统，其取值于$\mathbb{R}^d \times \mathbb{R}^d$，该系统由第一分量上的输运和扩散所支配，可以作为具有退化分量的动能模型的代表性模型。 在第一部分，我们通过证明一个伯恩斯坦浓度不等式来控制系统经验测度围绕对应 Vlasov-Fokker-Planck 方程解的波动，从而在多个方向上扩展了 arXiv:2011.03762 的先前结果。 在第二部分，我们研究从经验测度观察中经典 Vlasov-Fokker-Planck 方程解的非参数统计估计，并使用 Goldenshluger-Lepski 方法证明了一个 oracle 不等式，同时获得了渐近最优性。 然后我们将注意力集中在神经元群体的 FitzHugh-Nagumo 模型上。 我们考虑 Mischler 等人在 arXiv:1503.00492 中提出的模型版本，并通过矩估计量最优地估计模型的$6$参数。 显示英文摘要

摘要： We consider an interacting system of particles with value in $\mathbb{R}^d \times \mathbb{R}^d$, governed by transport and diffusion on the first component, on that may serve as a representative model for kinetic models with a degenerate component. In a first part, we control the fluctuations of the empirical measure of the system around the solution of the corresponding Vlasov-Fokker-Planck equation by proving a Bernstein concentration inequality, extending a previous result of arXiv:2011.03762 in several directions. In a second part, we study the nonparametric statistical estimation of the classical solution of Vlasov-Fokker-Planck equation from the observation of the empirical measure and prove an oracle inequality using the Goldenshluger-Lepski methodology and we obtain minimax optimality. We then specialise on the FitzHugh-Nagumo model for populations of neurons. We consider a version of the model proposed in Mischler et al. arXiv:1503.00492 an optimally estimate the $6$ parameters of the model by moment estimators.