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.
From: Eva Löcherbach [view email]
[v1]
Mon, 30 Jun 2025 17:15:14 UTC (148 KB)
[v2]
Mon, 8 Jun 2026 11:44:33 UTC (608 KB)
[v3]
Mon, 17 Aug 2026 08:12:27 UTC (604 KB)
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