Nonparametric estimation of the jump rate in mean field interacting systems of neurons

Abstract

We consider finite systems of $N$ interacting neurons described by non-linear Hawkes processes in a mean field frame. Neurons are described by their membrane potential. They spike randomly, at a rate depending on their potential. In between successive spikes, their membrane potential follows a deterministic flow. We estimate the spiking rate function based on the observation of the system of $N$ neurons over a fixed time interval $[0,t]$. Asymptotic are taken as $N,$ the number of neurons, tends to infinity. We introduce a kernel estimator of Nadaraya-Watson type and discuss its asymptotic properties with help of the deterministic dynamical system describing the mean field limit. We compute the minimax rate of convergence in an $L^2-$error loss over a range of Hölder classes and obtain the classical rate of convergence $N^{-2\beta /(2\beta +1)},$ where $\beta$ is the regularity of the unknown spiking rate function.

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