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

TL;DR

This paper develops a nonparametric kernel estimator for the spiking rate in large systems of interacting neurons modeled by mean field Hawkes processes, analyzing its asymptotic properties and convergence rates.

Contribution

It introduces a Nadaraya-Watson type estimator for the spiking rate in mean field neuron models and derives its asymptotic convergence rates.

Findings

Estimator achieves classical minimax convergence rate of N^{-2β/(2β+1)}.

Asymptotic properties are linked to the deterministic mean field limit.

The method applies to systems with regularity characterized by Hölder classes.

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 H\"older 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.