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

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• 作者：Javier Baladron (3)
  Diego Fasoli (3)
  Olivier Faugeras (7) (8)
  Jonathan Touboul (7) (7) (8) (8)
  
• 关键词：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
• 刊名：The Journal of Mathematical Neuroscience (JMN)
• 出版年：2012
• 出版时间：December 2012
• 年：2012
• 卷：2
• 期：1
• 全文大小：2085KB
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• 作者单位：Javier Baladron (3)
  Diego Fasoli (3)
  Olivier Faugeras (7) (8)
  Jonathan Touboul (7) (7) (8) (8)
  
  3. NeuroMathComp Laboratory, INRIA, Sophia-Antipolis Méditerranée, 06902, Kragujevac, France
  7. CNRS/UMR 7241-INSERM U1050, Université Pierre et Marie Curie, ED 158, Paris, 75005, France
  8. MEMOLIFE Laboratory of Excellence and Paris Science Lettre, 11, Place Marcelin Berthelot, Paris, 75005, France
  
• ISSN：2190-8567

文摘

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.