The effect of noise on mixed-mode oscillations

Many neuronal systems and models display so-called mixed-mode oscillations (MMOs) consisting of small-amplitude oscillations alternating with large-amplitude oscillations. Different mechanisms have been identified which may cause this type of behaviour. In this talk, we will focus on MMOs in a slow-fast dynamical system with one fast and two slow variables, containing a folded-node singularity. The main question we will address is whether and how noise may change the dynamics.

We will first outline a general approach to stochastic slow-fast systems which allows

(1) to construct small sets in which the sample paths are typically concentrated, and
(2) to give precise bounds on the exponentially small probability to observe atypical behaviour.

Applying this method to our model system shows the existence of a critical noise intensity beyond which the small-amplitude oscillations become hidden by noise. Furthermore, we will show that in the presence of noise sample paths are likely to jump away from so-called canard solutions earlier than the corresponding deterministic orbits. This early-jump mechanism can drastically change the mixed-mode patterns, even for rather small noise intensities. The methods used to derive the results range from deterministic bifurcation theory and averaging to martingale techniques and estimates on Markov transition kernels.

Joint work with Nils Berglund (Université d’Orléans) and Christian Kuehn (TU Wien).