Synchronization of stochastic mean field networks of Hodgkin-Huxley neurons with noisy channels

TL;DR

This paper studies the synchronization behavior of large stochastic Hodgkin-Huxley neuron networks with noisy channels, proving exponential synchronization under strong electrical coupling and analyzing the propagation of chaos.

Contribution

It provides rigorous proofs of exponential synchronization and propagation of chaos for mean field Hodgkin-Huxley networks with noise, extending understanding of collective neural dynamics.

Findings

Neurons synchronize exponentially fast under strong electrical coupling.

Propagation of chaos holds regardless of interaction strength.

Infinite network behavior concentrates around a single neuron dynamics.

Abstract

In this work we are interested in a mathematical model of the collective behavior of a fully connected network of finitely many neurons, when their number and when time go to infinity. We assume that every neuron follows a stochastic version of the Hodgkin-Huxley model, and that pairs of neurons interact through both electrical and chemical synapses, the global connectivity being of mean field type. When the leak conductance is strictly positive, we prove that if the initial voltages are uniformly bounded and the electrical interaction between neurons is strong enough, then, uniformly in the number of neurons, the whole system synchronizes exponentially fast as time goes to infinity, up to some error controlled by (and vanishing with) the channels noise level. Moreover, we prove that if the random initial condition is exchangeable, on every bounded time interval the propagation of chaos property for this system holds (regardless of the interaction intensities). Combining these results, we deduce that the nonlinear McKean-Vlasov equation describing an infinite network of such neurons concentrates, as time goes to infinity, around the dynamics of a single Hodgkin-Huxley neuron with chemical neurotransmitter channels. Our results are illustrated and complemented with numerical simulations.