Abstract or Additional Information
Pseudo-plateau bursting is a type of oscillatory waveform associated with mixed mode dynamics in slow/fast systems and commonly found in neural bursting models. Multiple methods from dynamical systems theory have been used to understand these bursting rhythms, which are typically treated as 2-timescale problems. In the first part of this talk, we demonstrate that the two most common analysis techniques are different unfoldings of a 3-timescale system. Our analysis shows that canards are a key feature of these systems that locally organise the dynamics in phase space.
Canards are closely associated with folded singularities and in the case of folded nodes, lead to a local twisting of invariant manifolds. Folded node canards and folded saddle canards (and their bifurcations) have been studied extensively. The folded saddle-node (FSN) is the codimension-1 bifurcation that gives rise to folded nodes and folded saddles. Their dynamics however, are not well-understood. In the second part of this talk, we extend canard theory into the FSN regime by combining methods from geometric singular perturbation theory (blow-up), and the theory of dynamic bifurcations (analytic continuation into the plane of complex time).