Limits and dynamics of stochastic neuronal networks with random heterogeneous delays

TL;DR

This paper investigates the behavior of large stochastic neuronal networks with heterogeneous random delays, revealing how delay distributions influence network dynamics and synchronization through convergence to a delayed McKean-Vlasov equation.

Contribution

It introduces a framework for analyzing large stochastic multi-population networks with random delays, deriving limit equations that incorporate delay distributions and noise effects.

Findings

Delay distribution affects propagation of chaos in networks

Limit equations are Gaussian with parameters satisfying delay differential equations

Noise and delay dispersion significantly influence network synchronization

Abstract

Realistic networks display heterogeneous transmission delays. We analyze here the limits of large stochastic multi-populations networks with stochastic coupling and random interconnection delays. We show that depending on the nature of the delays distributions, a quenched or averaged propagation of chaos takes place in these networks, and that the network equations converge towards a delayed McKean-Vlasov equation with distributed delays. Our approach is mostly fitted to neuroscience applications. We instantiate in particular a classical neuronal model, the Wilson and Cowan system, and show that the obtained limit equations have Gaussian solutions whose mean and standard deviation satisfy a closed set of coupled delay differential equations in which the distribution of delays and the noise levels appear as parameters. This allows to uncover precisely the effects of noise, delays and coupling on the dynamics of such heterogeneous networks, in particular their role in the emergence of synchronized oscillations. We show in several examples that not only the averaged delay, but also the dispersion, govern the dynamics of such networks.