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Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons

Javier Baladron1, Diego Fasoli1, Olivier Faugeras12* and Jonathan Touboul3456

Author Affiliations

1 NeuroMathComp Laboratory, INRIA, Sophia-Antipolis Méditerranée, 06902, France

2 NeuroMathComp Laboratory, ENS, Paris, 75013, France

3 BANG Laboratory, INRIA, Paris, 75013, France

4 Mathematical Neuroscience Lab, Center of Interdisciplinary Research in Biology, Collège de France, Paris, 75005, France

5 CNRS/UMR 7241-INSERM U1050, Université Pierre et Marie Curie, ED 158, Paris, 75005, France

6 MEMOLIFE Laboratory of Excellence and Paris Science Lettre, 11, Place Marcelin Berthelot, Paris, 75005, France

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The Journal of Mathematical Neuroscience 2012, 2:10  doi:10.1186/2190-8567-2-10

Published: 31 May 2012

Abstract

We derive the mean-field equations arising as the limit of a network of interacting spiking neurons, as the number of neurons goes to infinity. The neurons belong to a fixed number of populations and are represented either by the Hodgkin-Huxley model or by one of its simplified version, the FitzHugh-Nagumo model. The synapses between neurons are either electrical or chemical. The network is assumed to be fully connected. The maximum conductances vary randomly. Under the condition that all neurons’ initial conditions are drawn independently from the same law that depends only on the population they belong to, we prove that a propagation of chaos phenomenon takes place, namely that in the mean-field limit, any finite number of neurons become independent and, within each population, have the same probability distribution. This probability distribution is a solution of a set of implicit equations, either nonlinear stochastic differential equations resembling the McKean-Vlasov equations or non-local partial differential equations resembling the McKean-Vlasov-Fokker-Planck equations. We prove the well-posedness of the McKean-Vlasov equations, i.e. the existence and uniqueness of a solution. We also show the results of some numerical experiments that indicate that the mean-field equations are a good representation of the mean activity of a finite size network, even for modest sizes. These experiments also indicate that the McKean-Vlasov-Fokker-Planck equations may be a good way to understand the mean-field dynamics through, e.g. a bifurcation analysis.

Mathematics Subject Classification (2000): 60F99, 60B10, 92B20, 82C32, 82C80, 35Q80.

Keywords:
mean-field limits; propagation of chaos; stochastic differential equations; McKean-Vlasov equations; Fokker-Planck equations; neural networks; neural assemblies; Hodgkin-Huxley neurons; FitzHugh-Nagumo neurons