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

Author

Listed:
  • 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)

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.

Suggested Citation

  • P. Hodara & N. Krell & E. Löcherbach, 2018. "Non-parametric estimation of the spiking rate in systems of interacting neurons," Statistical Inference for Stochastic Processes, Springer, vol. 21(1), pages 81-111, April.
    • Handle: RePEc:spr:sistpr:v:21:y:2018:i:1:d:10.1007_s11203-016-9150-4
    DOI: 10.1007/s11203-016-9150-4
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    Cited by:

    1. Cormier, Quentin & Tanré, Etienne & Veltz, Romain, 2020. "Long time behavior of a mean-field model of interacting neurons," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 2553-2595.
    2. Evgeny Pchelintsev & Serguei Pergamenshchikov & Maria Povzun, 2022. "Efficient estimation methods for non-Gaussian regression models in continuous time," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(1), pages 113-142, February.

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