Statistical estimation of a mean-field FitzHugh-Nagumo model
Claudia Fonte Sanchez, Marc Hoffmann
TL;DR
The paper develops finite-sample, nonasymptotic inference for a mean-field FitzHugh–Nagumo model embedded in a kinetic McKean–Vlasov framework. It first proves a Bernstein concentration inequality for the empirical measure around the Vlasov–Fokker–Planck limit under locally Lipschitz/diffusive settings, then constructs a data-driven nonparametric estimator for the time-marginal density using Goldenshluger–Lepski bandwidth selection and proves an oracle inequality with minimax rates. Specializing to the FitzHugh–Nagumo neuron population, the authors derive a moment-based parameter estimator with a controllable nonasymptotic error bound and a $\sqrt{N}$-type concentration, enabling reliable inference from population measurements. The results rest on well-posedness of the McKean–Vlasov equation under Lyapunov-type conditions and a contraction framework in Wasserstein space, yielding applicability to high-dimensional kinetic models and neuronal network applications.
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.