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

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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 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.

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Notes

  1. A short remark concerning the continuous time observation scheme : Presumably, if we deal with discrete time samples, observed at sufficiently high frequency such that with huge probability at most one jump can take place during one sampling step, it would be possible to reconstruct the continuous trajectory of the process with hight probability and to perform our estimation procedure also in this frame.

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Acknowledgements

We thank an anonymous referee for helpful comments and suggestions. This research has been conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01), as part of the Agence Nationale de la Recherche PIECE 12-JS01-0006-01 and as part of the activities of FAPESP Research, Dissemination and Innovation Center for Neuromathematics (Grant 2013/07699-0, S. Paulo Research Foundation).

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Authors and Affiliations

  1. Département de Mathématiques, UMR 8088, CNRS, Université de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex, France

    P. Hodara & E. Löcherbach

  2. Institut de Recherche mathématique de Rennes, CNRS-UMR 6625, Université de Rennes 1, Campus de Beaulieu, Bâtiment 22, 35042, Rennes Cedex, France

    N. Krell

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  1. P. Hodara
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  2. N. Krell
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  3. E. Löcherbach
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Correspondence to P. Hodara.

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Hodara, P., Krell, N. & Löcherbach, E. Non-parametric estimation of the spiking rate in systems of interacting neurons. Stat Inference Stoch Process 21, 81–111 (2018). https://doi.org/10.1007/s11203-016-9150-4

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  • DOI: https://doi.org/10.1007/s11203-016-9150-4

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