Speaker: Theodore Vo
Abstract. Canards are special solutions of slow/fast systems of differential equations that alternately spend long times following attracting and repelling equilibria of the fast subsystem. Canards ]are ubiquitous in applications and have been used to explain the firing patterns of electrically excitable cells in neuro-science, the sudden change in amplitude and period of oscillatory behaviour in chemical reactions, and the anomalous delays in response to exogenous pulses of inositol triphosphate in calcium signalling. In the first part of this colloquium, we survey some of the exciting new developments in canard theory and its applications to bursting in nerve and endocrine cells.In the second part of the colloquium, we focus on the torus canard problem. Torus canards are solutions of slow/fast systems that alternate between attracting and repelling manifolds of limit cycles of the fast subsystem. First discovered in 2008 in a cerebellar Purkinje cell model, torus canards have been found in several paradigm computational neural models to mediate the transition from rapid spiking to bursting. So far, torus canards have only been studied numerically, and their behaviour inferred based on averaging and geometric singular perturbation methods. This approach, however, is not rigorously justified since the averaging method breaks down near a fold of periodics - exactly where the torus canards originate. We develop a novel averaging method for folded manifolds of limit cycles, and devise an analytic scheme for the detection and classification of torus canards based on a new class of singularities for differential equations. We demonstrate our torus canard theory in a model for intracellular calcium dynamics, where we discover a new class of bursting rhythms and explain the mechanisms that underlie them.