[Paper Review] Mean Field description of and propagation of chaos in recurrent multipopulation networks of Hodgkin-Huxley and Fitzhugh-Nagumo neurons

Javier Baladron, Diego Fasoli+2 more|arXiv (Cornell University)|Oct 19, 2011

Neural dynamics and brain function46 references19 citations

TL;DR

This paper rigorously derives mean-field equations for large networks of interacting Hodgkin-Huxley and Fitzhugh-Nagumo neurons across multiple populations, proving propagation of chaos—where finite subsets of neurons become independent in the infinite-size limit. The resulting nonlinear stochastic differential equations and non-local Fokker-Planck equations describe the system's macroscopic dynamics with well-posed solutions, validated numerically even for moderate network sizes.

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 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.

Motivation & Objective

• To derive the mean-field limit of recurrent multipopulation networks of spiking neurons with biophysically realistic models.
• To rigorously prove propagation of chaos in such networks, showing independence and identical distribution of neurons in the infinite-size limit.
• To establish the well-posedness of the resulting McKean-Vlasov-type equations and their Fokker-Planck counterparts.
• To validate the mean-field approximation numerically, demonstrating its accuracy even for finite, modest-sized networks.
• To provide a mathematical framework for analyzing neural population dynamics using statistical physics and stochastic processes.

Proposed method

• Derives mean-field equations via a limit process as the number of neurons tends to infinity, assuming i.i.d. initial conditions within each population.
• Uses locally Lipschitz coefficients and Lyapunov-type growth conditions to extend classical McKean-Vlasov theory to spiking neuron models.
• Applies Wasserstein distance and large deviation techniques to prove propagation of chaos in the network.
• Establishes existence and uniqueness of solutions to the resulting nonlinear SDEs and non-local Fokker-Planck equations.
• Employs Burkholder-Davis-Gundy and Fubini-type arguments to control fluctuations and show convergence at rate $1/ ext{N}_{\text{min}}$.
• Performs numerical experiments comparing finite-network activity to the mean-field solution, confirming accuracy for moderate $N$.

Experimental results

Research questions

• RQ1 Does the mean-field limit accurately describe the macroscopic behavior of large recurrent networks of Hodgkin-Huxley and Fitzhugh-Nagumo neurons?
• RQ2 Under what conditions does propagation of chaos occur in multipopulation networks with heterogeneous, interacting neurons?
• RQ3 Are the derived McKean-Vlasov-type SDEs and Fokker-Planck equations well-posed, with unique solutions?
• RQ4 Can the mean-field equations capture the statistical behavior of finite-size networks, even with modest neuron counts?
• RQ5 Is the convergence rate of the finite network to the mean-field limit quantitatively bounded, and does it scale as $1/\sqrt{N}$?

Key findings

• The mean-field limit is described by a system of nonlinear stochastic differential equations resembling McKean-Vlasov equations, which are well-posed with unique solutions.
• Propagation of chaos is rigorously proven: in the infinite-size limit, any finite number of neurons become independent and identically distributed within each population.
• The convergence rate of finite networks to the mean-field limit is bounded by $K_3 / N_{\text{min}}$, with $N_{\text{min}}$ being the smallest population size.
• Numerical experiments confirm that the mean-field equations accurately represent the mean activity of finite networks, even at $N \approx 100$ neurons.
• The non-local Fokker-Planck equations derived from the mean-field limit are suitable for bifurcation analysis and dynamical system studies.
• The framework extends classical mean-field theory to spiking neuron models with locally Lipschitz dynamics and non-linear interactions, overcoming standard global Lipschitz assumptions.