Mixed-mode dynamics and the canard phenomenon: Towards a classification

Abstract

Mixed-mode dynamics is a complex type of oscillatory behavior that is characterized by an alternation of small-amplitude oscillations and large-amplitude excursions. In this overview article, we focus on one particular mechanism that has been shown to generate mixed-mode oscillations (MMOs) in multiple-scale systems: the generalized canard mechanism. After a brief review of the classical canard phenomenon, we present a model problem that was proposed in [23] as a canonical form for a family of three-dimensional three time-scale systems, and we reiterate some of the results obtained there. In particular, we discuss how that canonical form can be placed in the context of the well-developed geometric theory of canards in three dimensions. Finally, we introduce two examples of problems from mathematical neuroscience that fit into the framework of our model problem, and we discuss the implications of our results for the mixed-mode dynamics observed in these two examples. Our results are intended as a first step towards a more general classification of the mixed-mode dynamics that can arise via the generalized canard mechanism, with the long-term goal of constructing a 'toolbox' of prototypical minimal models.