Statistical estimation of a mean-field FitzHugh-Nagumo model

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.