Noise-induced Canard and Mixed-Mode Oscillations in Large-Scale Stochastic Networks

Abstract

Slow-fast dynamical systems, such as those describing the excitable dynamics of nerve cells, can present a variety of very delicate phenomena, such as canard explosions or mixed-mode oscillations (MMOs), in which dramatic transitions occur upon very small variation of the parameters. In biological situations, these excitable systems are coupled and subject to intense noise, which may have the effect of destroying these phenomena. While this is true for single cells or small networks, we show that large-scale networks do display reliable canard explosions and MMOs in the presence of noise, and these transitions can even occur upon variation of the noise standard deviation. We show that this phenomenon can be related to averaging effects occurring in large populations by investigating a network of interconnected Wilson--Cowan rate neurons which converges to a stochastic process whose mean satisfies a slow-fast dynamical system in which noise is a parameter. In this system, we show using a combination of analytic and numerical arguments that generic canard explosions and MMOs occur as noise is varied. This result sheds new light on the qualitative effects of noise and sensitivity to precise noise values in large stochastic networks. We further investigate finite-sized networks and show that systematic differences with the mean-field limits arise in bistable regimes (where random switches between different attractors occur) or in MMOs, where the finite-sized effects induce early jumps due to the sensitivity of the attractor.