Statistical estimation of a mean-field FitzHugh-Nagumo model

TL;DR

This paper develops statistical tools for estimating parameters in a mean-field FitzHugh-Nagumo neural model, including concentration inequalities for empirical measures and minimax optimal estimators, advancing understanding of kinetic models with degenerate components.

Contribution

It extends concentration inequalities for empirical measures and introduces minimax optimal parameter estimators for the FitzHugh-Nagumo model.

Findings

Proved Bernstein concentration inequality for empirical measure fluctuations.

Established minimax optimal parameter estimation using moment estimators.

Applied methodology specifically to the FitzHugh-Nagumo neural model.

Abstract

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.