LAN property for the parameter of the jump rate in mean field interacting systems of neurons

TL;DR

This paper investigates the statistical properties of parameter estimation in large-scale neuron models with spike interactions, establishing LAN and asymptotic optimality of the MLE even with neuron resets.

Contribution

It extends existing theory to include neuron reset dynamics, proving the LAN property and optimality of the MLE in more realistic neural models.

Findings

Proved LAN property for the jump rate parameter estimation.

Established the asymptotic efficiency and minimax optimality of the MLE.

Extended theoretical results to models with neuron resets.

Abstract

In the context of a large system of $N$ neurons interacting through spike events in a mean-field regime as $N\rightarrow \infty$, we characterize the estimation of a multidimensional parameter in the spiking rate, when the neural states are observed over a fixed time horizon. We first prove the local asymptotic normality (LAN) property and leverage classical theory to establish the asymptotic efficiency of the maximum likelihood estimator. While the theory of Ibragimov and Hasminski yields strong results, up to global asymptotic minimax bound, its applicability appears currently limited to models without state resets at spike times. Following then H\"{o}pfner's classical approach, we nevertheless derive, in a general setting including neuron reset, the consistency, asymptotic normality and local asymptotic minimax optimality of the estimator. Keywords: Local Asymptotic Normality (LAN); Mean-field regime; Interacting particle system; Multidimensional parameter estimation; Jump rate estimation; Maximum likelihood estimator (MLE); Asymptotic minimax optimality