Mixed-Mode Oscillations in a Stochastic, Piecewise-Linear System

TL;DR

This paper investigates how small noise induces mixed-mode oscillations in a stochastic piecewise-linear FitzHugh-Nagumo model, revealing robustness of MMOs and their easier occurrence compared to the classical model.

Contribution

It introduces a piecewise-linear FitzHugh-Nagumo model to analytically study noise-induced MMOs and compares its behavior with the classical cubic model.

Findings

MMOs are robust to parameter changes that preserve noise-to-time-scale ratio.

Piecewise-linear model exhibits MMOs more readily than the classical cubic model.

Analytical expressions effectively explain the occurrence of MMOs.

Abstract

We analyze a piecewise-linear FitzHugh-Nagumo model. The system exhibits a canard near which both small amplitude and large amplitude periodic orbits exist. The addition of small noise induces mixed-mode oscillations (MMOs) in the vicinity of the canard point. We determine the effect of each model parameter on the stochastically driven MMOs. In particular we show that any parameter variation (such as a modification of the piecewise-linear function in the model) that leaves the ratio of noise amplitude to time-scale separation unchanged typically has little effect on the width of the interval of the primary bifurcation parameter over which MMOs occur. In that sense, the MMOs are robust. Furthermore we show that the piecewise-linear model exhibits MMOs more readily than the classical FitzHugh-Nagumo model for which a cubic polynomial is the only nonlinearity. By studying a piecewise-linear model we are able to explain results using analytical expressions and compare these with numerical investigations.