Journal of Statistical Physics

, Volume 166, Issue 5, pp 1310–1333 | Cite as

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

  • Daniel Gandolfo
  • Roger Rodriguez
  • Henry C. Tuckwell


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.


Computational neuroscience Conductance-based neural models Fitzhugh–Nagumo model Stochastic differential equations Klimontovich method Mean field limits Neural networks 

Mathematics Subject Classification

82C32 35Q83 35Q84 45K05 65C30 65C05 



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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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Daniel Gandolfo
    • 1
  • Roger Rodriguez
    • 1
  • Henry C. Tuckwell
    • 2
    • 3
  1. 1.Aix Marseille Univ, Université de Toulon, CNRS, CPTMarseilleFrance
  2. 2.School of Electrical and Electronic EngineeringUniversity of AdelaideAdelaideAustralia
  3. 3.School of Mathematical SciencesMonash UniversityClaytonAustralia

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