Abstract
In this note, we clarify the well-posedness of the limit equations to the mean-field N-neuron models proposed in (Baladron et al. in J. Math. Neurosci. 2:10, 2012) and we prove the associated propagation of chaos property. We also complete the modeling issue in (Baladron et al. in J. Math. Neurosci. 2:10, 2012) by discussing the well-posedness of the stochastic differential equations which govern the behavior of the ion channels and the amount of available neurotransmitters.
Highlights
The paper of Baladron et al [1] studies quite general networks of neurons and aims to prove that these networks propagate chaos in the sense originally developed by Sznitman [2] after the seminal work of Kac on mean-field limits and McKean’s work [3] on diffusion processes propagating chaos
The variants of the FitzHugh–Nagumo and Hodgkin–Huxley dynamics proposed in Baladron et al [1] to model neuron networks are all of the two types below; the only differences concern the algebraic expressions of the function Fα and the fact that the FitzHugh–Nagumo model does not depend on the variables but on the recovery variable only
The square root arises from the fact that this SDE is a Langevin approximation to a stochastic hybrid, or piecewise deterministic, model of the ion channels
Summary
The paper of Baladron et al [1] studies quite general networks of neurons and aims to prove that these networks propagate chaos in the sense originally developed by Sznitman [2] after the seminal work of Kac on mean-field limits and McKean’s work [3] on diffusion processes propagating chaos. As observed by the authors, the membrane potentials of the neurons in the networks they consider are described by interacting stochastic particle dynamics. The coefficients of these McKean–Vlasov systems are not globally Lipschitz. Our objective is two-fold: first, we discuss a modeling issue on the diffusion coefficients of the equations describing the proportions of open and closed channels that guarantees that these variables do not escape from the interval [0, 1] This was not completely achieved in [1] and can be seen as a complement to this paper. We give a rigorous proof of the propagation of the chaos property
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