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

Abstract

We consider a model of interacting neurons where the membrane potentials of the neurons are described by a multidimensional piecewise deterministic Markov process 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 a resting potential, here 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 Holder 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.