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

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

Pseudo-plateau bursting and mixed-mode oscillations in a model of developing inner hair cells

This is the final version. Available on open access from Elsevier via the DOI in this recordInner hair cells (IHCs) are excitable sensory cells in the inner ear that encode acoustic information. Before the onset of hearing IHCs fire calcium-based action potentials that trigger transmitter release onto developing spiral ganglion neurones. There is accumulating experimental evidence that these spontaneous firing patterns are associated with maturation of the IHC synapses and hence involved in the development of hearing. The dynamics organising the IHCs’ electrical activity are therefore of interest. Building on our previous modelling work we propose a three-dimensional, reduced IHC model and carry out non-dimensionalisation. We show that there is a significant range of parameter values for which the dynamics of the reduced (three-dimensional) model map well onto the dynamics observed in the original biophysical (four-dimensional) IHC model. By estimating the typical time scales of the variables in the reduced IHC model we demonstrate that this model could be characterised by two fast and one slow or one fast and two slow variables depending on biophysically relevant parameters that control the dynamics. Specifically, we investigate how changes in the conductance of the voltage-gated calcium channels as well as the parameter corresponding to the fraction of free cytosolic calcium concentration in the model affect the oscillatory model bahaviour leading to transition from pseudo-plateau bursting to mixed-mode oscillations. Hence, using fast-slow analysis we are able to further our understanding of this model and reveal a path in the parameter space connecting pseudo-plateau bursting and mixed-mode oscillations by varying a single parameter in the model.Engineering and Physical Sciences Research Council (EPSRC

Classification of bursting patterns: A tale of two ducks

Bursting is one of the fundamental rhythms that excitable cells can generate either in response to incoming stimuli or intrinsically. It has been a topic of intense research in computational biology for several decades. The classification of bursting oscillations in excitable systems has been the subject of active research since the early 1980s and is still ongoing. As a by-product it establishes analytical and numerical foundations for studying complex temporal behaviors in multiple-timescale models of cellular activity. In this review, we first present the seminal works of Rinzel and Izhikevich in classifying bursting patterns of excitable systems. We recall a complementary mathematical classification approach by Bertram et al., and then by Golubitsky et al., which together with the Rinzel-Izhikevich proposals provide the state-of-the-art foundations to these classifications. Beyond classical approaches, we review a recent bursting example that falls outside the previous classification systems. Generalizing this example leads us to propose an extended classification, which requires the analysis of both fast and slow subsystems of an underlying slow-fast model and allows the dissection of a larger class of bursters. Namely, we provide a general framework for bursting systems with both subthreshold and superthreshold oscillations. A new class of bursters with at least two slow variables is then added, which we denote folded-node bursters, to convey the idea that the bursts are initiated or annihilated via a folded-node singularity. Key to this mechanism are so-called canard or duck orbits, organizing the underpinning excitability structure. We describe the two main families of folded-node bursters, depending upon the phase (active/spiking or silent/non-spiking) of the bursting cycle during which folded-node dynamics occurs. We classify both families and give examples of minimal systems displaying these novel bursting patterns. Finally, we provide a biophysical example by reinterpreting a generic conductance-based episodic burster as a folded-node burster, showing that the associated framework can explain its subthreshold oscillations over a larger parameter region than the fast-subsystem approach

Bifurcations and Slow-Fast Analysis in a Cardiac Cell Model for Investigation of Early Afterdepolarizations

In this study, we teased out the dynamical mechanisms underlying the generation of arrhythmogenic early afterdepolarizations (EADs) in a three-variable model of a mammalian ventricular cell. Based on recently published studies, we consider a 1-fast, 2-slow variable decomposition of the system describing the cellular action potential. We use sweeping techniques, such as the spike-counting method, and bifurcation and continuation methods to identify parametric regions with EADs. We show the existence of isolas of periodic orbits organizing the different EAD patterns and we provide a preliminary classification of our fast-slow decomposition according to the involved dynamical phenomena. This investigation represents a basis for further studies into the organization of EAD patterns in the parameter space and the involved bifurcations