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

by Eva Löcherbach

Monday Jul 7, 2025, 9:30 AM → 10:00 AM Europe/Paris

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 L2 -error loss over a range of Hölder classes and obtain the same rate of convergence as for density estimation in the classical iid setting. This is a joint work with Aline Duarte (USP), Kadmo Laxa (USP) and Dasha Loukianova (Evry).