Mixed-mode oscillations in a stochastic, piecewise-linear system

Abstract

We analyse 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.