2011/10/19 by Javier Baladron, Baladron, Javier, Diego Fasoli +5 · 1 citation
Computer Science · Neuroscience · Physics and Astronomy · #35Q80 #60B10 #60F99 #82C32 #82C80 #92B20 #Advanced Thermodynamics and Statistical Mechanics #FOS: Biological sciences #FOS: Mathematics #Neural dynamics and brain function #Neurons and Cognition (q-bio.NC) #Nonlinear Dynamics and Pattern Formation #Probability (math.PR) #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.1110.4294
openalex publication_date 2011/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 places, 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 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 these equations, i.e. the existence and uniqueness of a solution. We also show the results of some preliminary 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 experiment 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.