Non-parametric estimation of the spiking rate in systems of interacting neurons

TL;DR

This paper develops a non-parametric kernel estimator for the spiking rate of neurons in a network modeled by a multidimensional PDMP, proving its optimal convergence rate and establishing a central limit theorem.

Contribution

It introduces a new kernel estimator for neuron spiking rates in interacting neuron models and proves its optimal convergence and asymptotic normality.

Findings

The estimator achieves the optimal convergence rate in L^2.

A central limit theorem for the estimator's error is established.

The method is based on the uniform ergodicity and invariant measure analysis.

Abstract

We consider a model of interacting neurons where the membrane potentials of the neurons are described by a multidimensional piecewise deterministic Markov process (PDMP) with values in ${\mathbb R}^N, $ where $ N$ is the number of neurons in the network. A deterministic drift attracts each neuron's membrane potential to an equilibrium potential $m.$ When a neuron jumps, its membrane potential is reset to $0,$ while the other neurons receive an additional amount of potential $\frac{1}{N}.$ We are interested in the estimation of the jump (or spiking) rate of a single neuron based on an observation of the membrane potentials of the $N$ neurons up to time $t.$ We study a Nadaraya-Watson type kernel estimator for the jump rate and establish its rate of convergence in $L^2 .$ This rate of convergence is shown to be optimal for a given H\"older class of jump rate functions. We also obtain a central limit theorem for the error of estimation. The main probabilistic tools are the uniform ergodicity of the process and a fine study of the invariant measure of a single neuron.