From random Poincar\'e maps to stochastic mixed-mode-oscillation patterns

We quantify the effect of Gaussian white noise on fast--slow dynamical systems with one fast and two slow variables, which display mixed-mode oscillations owing to the presence of a folded-node singularity. The stochastic system can be described by a continuous-space, discrete-time Markov chain, recording the returns of sample paths to a Poincar\'e section. We provide estimates on the kernel of this Markov chain, depending on the system parameters and the noise intensity. These results yield predictions on the observed random mixed-mode oscillation patterns. Our analysis shows that there is an intricate interplay between the number of small-amplitude oscillations and the global return mechanism. In combination with a local saturation phenomenon near the folded node, this interplay can modify the number of small-amplitude oscillations after a large-amplitude oscillation. Finally, sufficient conditions are derived which determine when the noise increases the number of small-amplitude oscillations and when it decreases this number.Comment: 56 pages, 14 figures; revised versio

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

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

Relaxation oscillations in slow-fast systems beyond the standard form

Relaxation oscillations are highly non-linear oscillations, which appear to feature many important biological phenomena such as heartbeat, neuronal activity, and population cycles of predator-prey type. They are characterized by repeated switching of slow and fast motions and occur naturally in singularly perturbed ordinary differential equations, which exhibit dynamics on different time scales. Traditionally, slow-fast systems and the related oscillatory phenomena -- such as relaxation oscillations -- have been studied by the method of the matched asymptotic expansions, techniques from non-standard analysis, and recently a more qualitative approach known as geometric singular perturbation theory. It turns out that relaxation oscillations can be found in a more general setting; in particular, in slow-fast systems, which are not written in the standard form. Systems in which separation into slow and fast variables is not given a priori, arise frequently in applications. Many of these systems include additionally various parameters of different orders of magnitude and complicated (non-polynomial) non-linearities. This poses several mathematical challenges, since the application of singular perturbation arguments is not at all straightforward. For that reason most of such systems have been studied only numerically guided by phase-space analysis arguments or analyzed in a rather non-rigorous way. It turns out that the main idea of singular perturbation approach can also be applied in such non-standard cases. This thesis is concerned with the application of concepts from geometric singular perturbation theory and geometric desingularization based on the blow-up method to the study of relaxation oscillations in slow-fast systems beyond the standard form. A detailed geometric analysis of oscillatory mechanisms in three mathematical models describing biochemical processes is presented. In all the three cases the aim is to detect the presence of an isolated periodic movement represented by a limit cycle. By using geometric arguments from the perspective of dynamical systems theory and geometric desingularization based on the blow-up method analytic proofs of the existence of limit cycles in the models are provided. This work shows -- in the context of non-trivial applications -- that the geometric approach, in particular the blow-up method, is valuable for the understanding of the dynamics of systems with no explicit splitting into slow and fast variables, and for systems depending singularly on several parameters