Geometric singular perturbation analysis of mixed-mode dynamics in pituitary cells

Abstract

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. In a recent model for the electrical activity in a pituitary lactotroph, two types of pseudo-plateau bursts were discovered: one in which the calcium drives the bursts and another in which the calcium simply follows them. Multiple methods from dynamical systems theory have been used to understand the bursting. The classic 2-timescale approach treats the calcium concentration as a slowly varying parameter and considers a parametrized family of fast subsystems. A more novel and successful 2-timescale approach divides the system so that there is only one fast variable and shows that the bursting arises from canard dynamics. Both methods can be effective analytic tools but there has been little justification for one approach over the other. In the first part of this thesis, we demonstrate that the two analysis techniques are different unfoldings of a 3-timescale system. We show that elementary applications of geometric singular perturbation theory and bifurcation theory in the 2-timescale and 3- timescale methods provides us with substantial predictive power. We use that predictive power to explain the transient and long-term dynamics of the pituitary lactotroph model. The canard phenomenon occurs generically in singular perturbation problems with at least two slow variables. 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 in 3 dimensions. The folded saddle-node (FSN) is the codimension-1 bifurcation that gives rise to folded nodes and folded saddles. There are two types of FSN. In the FSN type I, the center manifold of the FSN is tangent to the curve of fold bifurcations of the fast subsystem. In the FSN II, the center manifold of the FSN is transverse to the curve of fold bifurcations of the fast subsystem. Both types of FSN bifurcation are ubiquitous in applications and are typically the organizing centers for delay phenomena. In particular, the FSN I and FSN II demarcate the bursting regions in parameter space. Their dynamics however, are not completely understood. Recent studies have unravelled the local dynamics of the FSN II. In the second part of this thesis, we extend canard theory into the FSN I 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). We prove the existence of canards and faux canards near the FSN I, and study the associated delayed loss of stability.