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

P. Hodara (), N. Krell () and E. Löcherbach ()
Additional contact information
P. Hodara: UMR 8088, CNRS, Université de Cergy-Pontoise
N. Krell: CNRS-UMR 6625, Université de Rennes 1
E. Löcherbach: UMR 8088, CNRS, Université de Cergy-Pontoise

Statistical Inference for Stochastic Processes, 2018, vol. 21, issue 1, No 4, 111 pages

Abstract: 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 $${\mathbb {R}}^N, $$ 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 $$\frac{1}{N}.$$ 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 $$L^2 .$$ L 2 . This rate of convergence is shown to be optimal for a given Hölder 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.

Keywords: Piecewise deterministic Markov processes; Kernel estimation; Nonparametric estimation; Biological neural nets; 62G05; 60J75; 62M05 (search for similar items in EconPapers)
Date: 2018
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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DOI: 10.1007/s11203-016-9150-4

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