Spike Adding and the Role of Saddle Slow Manifolds

In this thesis, we study spike adding in slow-fast systems, i.e., systems in which some variables evolve significantly faster than others. In slow-fast systems, there exist so-called slow manifolds, along which the flow is much slower than the dynamics far from them. Slow manifolds, together with the invariant manifolds of saddle equilibria and periodic orbits organise the dynamics of slow-fast systems. The slow manifolds can be attracting, repelling or of saddle type, depending on the stability of their associated set of equilibria in the so-called fast subsystem when the slow variables are treated as parameters. In this thesis, we focus on slow manifolds of saddle type and their associated (un)stable manifolds in three-dimensional systems with one slow and two fast variables. In this case, the saddle slow manifold (SSM) exists as a (nonunique) one-dimensional curve and the associated stable and unstable manifolds are (nonunique) two-dimensional surfaces. Numerical methods for computing these manifolds are scarce, so that the precise nature of their role is an open question. We give a precise definition for an SSM and its (un)stable manifold, and then design an algorithm for computing the stable manifold of an SSM. Our computational method is formulated as a two-point boundary value problem and uses pseudo-arclength continuation in Auto. Our set-up is complementary to existing approaches for computing the stable manifold of an SSM. It is very e cient and we test the accuracy of the method using several heuristic methods. Recent studies have unveiled that the SSM and, more specifically, its stable manifold play an important role in the organisation of the number of spikes that arise in periodic (bursting) oscillations and are key to the underlying and spike-adding mechanism. We explain how the stable manifold of an SSM acts as a separatrix and organises the number of spikes in a response to a piecewise-constant parameter perturbation. The number of spikes is determined by the location of the response with respect to the manifold at the moment when the perturbation is removed. The onset of a new spike occurs when the response lies on the manifold at the moment when the perturbation is removed. We use our algorithm to investigate spike adding in four di↵erent models of slow-fast systems, each of which mimics dynamics of a neuron. Three of these models are devoted to the study of spike adding in a transient response to a brief perturbation when each system iii iv Abstract has a unique attracting equilibrium. For one of the systems, we also study the spike-adding mechanism of the transient response when the system has two additional equilibria of saddle type. In contrast to the case that the system has a unique stable equilibrium, at least one of the saddle equilibria lies on the SSM. This implies that the SSM is part of the (invariant) stable manifold of the equilibrium which determines the number of spikes in the transient response. The onset of a new spike is identified by a connection between the only attracting stable equilibrium and the saddle equilibrium on the SSM. We also discuss how the curves of spike onset in the parameter plane are organised by bifurcations of the fast subsystem and the full system. In fact, the curves of spike onset accumulate on a curve of Hopf bifurcations of the full system when a parameter of the system decreases. Also, at one end, the curves of spike onset terminate when a saddle-node on invariant cycle (SNIC) bifurcation occurs in the fast subsystem. Finally, we investigate spike adding for bursting periodic orbits. In this case, the equilibrium of the system is of saddle type and has stable and unstable invariant manifolds. We confirm that the stable manifold of an SSM also determines the number of spikes for the bursting periodic orbit with the largest period. Moreover, the periodicity of a solution is organised by a return mechanism that arises from the interactions between the stable manifold of the SSM and the (invariant) manifolds of the saddle equilibrium of the full system.