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Mean Field Analysis of Large-Scale Interacting Populations of Stochastic Conductance-Based Spiking Neurons Using the Klimontovich Method

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Abstract

We investigate the dynamics of large-scale interacting neural populations, composed of conductance based, spiking model neurons with modifiable synaptic connection strengths, which are possibly also subjected to external noisy currents. The network dynamics is controlled by a set of neural population probability distributions (PPD) which are constructed along the same lines as in the Klimontovich approach to the kinetic theory of plasmas. An exact non-closed, nonlinear, system of integro-partial differential equations is derived for the PPDs. As is customary, a closing procedure leads to a mean field limit. The equations we have obtained are of the same type as those which have been recently derived using rigorous techniques of probability theory. The numerical solutions of these so called McKean–Vlasov–Fokker–Planck equations, which are only valid in the limit of infinite size networks, actually shows that the statistical measures as obtained from PPDs are in good agreement with those obtained through direct integration of the stochastic dynamical system for large but finite size networks. Although numerical solutions have been obtained for networks of Fitzhugh–Nagumo model neurons, which are often used to approximate Hodgkin–Huxley model neurons, the theory can be readily applied to networks of general conductance-based model neurons of arbitrary dimension.

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Acknowledgements

The authors would like to thank the referees for their helpful criticisms and suggestions which led to a much improved manuscript.

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Authors and Affiliations

  1. Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

    Daniel Gandolfo & Roger Rodriguez

  2. School of Electrical and Electronic Engineering, University of Adelaide, Adelaide, SA, 5005, Australia

    Henry C. Tuckwell

  3. School of Mathematical Sciences, Monash University, Clayton, VIC, 3800, Australia

    Henry C. Tuckwell

Authors
  1. Daniel Gandolfo
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  2. Roger Rodriguez
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  3. Henry C. Tuckwell
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Correspondence to Henry C. Tuckwell.

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Gandolfo, D., Rodriguez, R. & Tuckwell, H.C. Mean Field Analysis of Large-Scale Interacting Populations of Stochastic Conductance-Based Spiking Neurons Using the Klimontovich Method. J Stat Phys 166, 1310–1333 (2017). https://doi.org/10.1007/s10955-016-1702-x

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  • DOI: https://doi.org/10.1007/s10955-016-1702-x

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