Analysis of Complex Bursting Patterns in Multiple Timescale Respiratory Neuron Models

Many physical systems feature interacting components that evolve on disparate timescales. Significant insights about the dynamics of such systems have resulted from grouping timescales into two classes and exploiting the timescale separation between classes through the use of geometric singular perturbation theory. It is natural to expect, however, that some dynamic phenomena cannot be captured by a two timescale decomposition. One example is the mixed burst firing mode, observed in both recordings and model pre-B\"{o}tzinger neurons, which appears to involve at least three timescales based on its time course. With this motivation, we construct a model system consisting of a pair of Morris-Lecar systems coupled so that there are three timescales in the full system. We demonstrate that the approach previously developed in the context of geometric singular perturbation theory for the analysis of two timescale systems extends naturally to the three timescale setting. To elucidate which characteristics truly represent three timescale features, we investigate certain reductions to two timescales and the parameter dependence of solution features in the three timescale framework. Furthermore, these analyses and methods are extended and applied to understand multiple timescale bursting dynamics in a realistic single pre-B\"{o}tzinger complex neuron and a heterogeneous population of these neurons, both of which can generate a novel mixed bursting (MB) solution, also observed in pre-B\"{o}tC neuron recordings. Rather surprisingly, we discover that a third timescale is not actually required to generate mixed bursting solution in the single neuron model, whereas at least three timescales should be involved in the latter model to yield a similar mixed bursting pattern. Through our analysis of timescales, we also elucidate how the single pre-B\"{o}tC neuron model can be tuned to improve the robustness of the MB solution

Mixed-mode Oscillations in Pyramidal Neurons Under Antiepileptic Drug Conditions

Subthreshold oscillations in combination with large-amplitude oscillations generate mixedmode oscillations (MMOs), which mediate various spatial and temporal cognition and memory processes and behavioral motor tasks. Although many studies have shown that canard theory is a reliable method to investigate the properties underlying the MMOs phenomena, the relationship between the results obtained by applying canard theory and conductancebased models of neurons and their electrophysiological mechanisms are still not well understood. The goal of this study was to apply canard theory to the conductance-based model of pyramidal neurons in layer V of the Entorhinal Cortex to investigate the properties of MMOs under antiepileptic drug conditions (i.e., when persistent sodium current is inhibited). We investigated not only the mathematical properties of MMOs in these neurons, but also the electrophysiological mechanisms that shape spike clustering. Our results show that pyramidal neurons can display two types of MMOs and the magnitude of the slow potassium current determines whether MMOs of type I or type II would emerge. Our results also indicate that slow potassium currents with large time constant have significant impact on generating the MMOs, as opposed to fast inward currents. Our results provide complete characterization of the subthreshold activities in MMOs in pyramidal neurons and provide explanation to experimental studies that showed MMOs of type I or type II in pyramidal neurons under antiepileptic drug conditions

Geometric Singular Perturbation Theory and Averaging: Analysing Torus Canards in Neural Models

Neuronal bursting, an oscillatory pattern of repeated spikes interspersed with periods of rest, is a pervasive phenomenon in brain function which is used to relay information in the body. Mathematical models of bursting typically consist of singularly perturbed systems of ordinary differential equations, which are well suited to analysis by geometric singular perturbation theory (GSPT). There are numerous types of bursting models, which are classified by a slow/fast decomposition and identification of fast subsystem bifurcation structures. Of interest are so-called fold/fold-cycle bursters, where burst initiation (termination) occurs at a fold of equilibria (periodic orbits), respectively. Such bursting models permit torus canards, special solutions which track a repelling fast subsystem manifold of periodic orbits. In this thesis we analyse the Wilson-Cowan-Izhikevich (WCI) and Butera models, two fold/fold-cycle bursters. Using numerical averaging and GSPT, we construct an averaged slow subsystem and identify the bifurcations corresponding to the transitions between bursting and spiking activity patterns. In both models we find that the transition involves toral folded singularities (TFS), averaged counterparts of folded singularities. In the WCI model, we show that the transition occurs at a degenerate TFS, resulting in a torus canard explosion, reminiscent of a classic canard explosion in the van der Pol oscillator. The TFS identified in the Butera model are generic, and using numerical continuation methods, we continue them and construct averaged bifurcation diagrams. We find three types of folded-saddle node (FSN) bifurcations which mediate transitions between activity patterns: FSN type I, II, and III. Type III is novel and studied here for the first time. We utilise the blow-up technique and dynamic bifurcation theory to extend current canard theory to the FSN III

Activity-dependent compensation of cell size is vulnerable to targeted deletion of ion channels.

In many species, excitable cells preserve their physiological properties despite significant variation in physical size across time and in a population. For example, neurons in crustacean central pattern generators generate similar firing patterns despite several-fold increases in size between juveniles and adults. This presents a biophysical problem because the electrical properties of cells are highly sensitive to membrane area and channel density. It is not known whether specific mechanisms exist to sense membrane area and adjust channel expression to keep a consistent channel density, or whether regulation mechanisms that sense activity alone are capable of compensating cell size. We show that destabilising effects of growth can be specifically compensated by feedback mechanism that senses average calcium influx and jointly regulate multiple conductances. However, we further show that this class of growth-compensating regulation schemes is necessarily sensitive to perturbations that alter the expression of subsets of ion channel types. Targeted perturbations of specific ion channels can trigger a pathological response of the regulation mechanism and a failure of homeostasis. Our findings suggest that physiological regulation mechanisms that confer robustness to growth may be specifically vulnerable to deletions or mutations that affect subsets of ion channels

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

Rhythmogenic and Premotor Functions of Dbx1 Interneurons in the Pre-Bötzinger Complex and Reticular Formation: Modeling and Simulation Studies

Breathing in mammals depends on rhythms that originate from the preBötzinger complex (preBötC) of the ventral medulla and a network of brainstem and spinal premotor neurons. The rhythm-generating core of the preBötC, as well as some premotor circuits, consists of interneurons derived from Dbx1-expressing precursors but the structure and function of these networks remain incompletely understood. We previously developed a cell-specific detection and laser ablation system to interrogate respiratory network structure and function in a slice model of breathing that retains the preBötC, premotor circuits, and the respiratory related hypoglossal (XII) motor nucleus such that in spontaneously rhythmic slices, cumulative ablation of Dbx1 preBötC neurons decreased XII motor output by half after only a few cell deletions, and then decelerated and terminated rhythmic function altogether as the tally increased. In contrast, cumulatively deleting Dbx1 premotor neurons decreased XII motor output monotonically, but did not affect frequency nor stop functionality regardless of the ablation tally. This dissertation presents several network modeling and cellular modeling studies that would further our understanding of how respiratory rhythm is generated and transmitted to the XII motor nucleus. First, we propose that cumulative deletions of Dbx1 preBötC neurons preclude rhythm by diminishing the amount of excitatory inward current or disturbing the process of recurrent excitation rather than structurally breaking down the topological network. Second, we establish a feasible configuration for neural circuits including an Erdős-Rényi preBötC network and a small-world reticular premotor network with interconnections following an anti-preferential attachment rule, which is the only configuration that produces consistent outcomes with previous experimental benchmarks. Furthermore, since the performance of neuronal network simulations is, to some extent, affected by the nature of the cellular model, we aim to develop a more realistic cellular model based on the one we adopted in previous network studies, which would account for some recent experimental findings on rhythmogenic preBötC neurons

A study of bursting in the preBotzinger Complex

The preBotzinger complex (PBC) of the mammalian brainstem is a heterogeneous neuronal network underlying the inspiration phase of the respiratory rhythm. Through excitatory synapses and a nontrivial network architecture, a synchronous, network-wide bursting rhythm emerges. On the other hand, during synaptic isolation, PBC neurons display three types of intrinsic dynamics: quiescence, bursting, or tonic activity. This work seeks to shed light on how the network rhythm emerges from the challenging architecture and heterogeneous population. Recent debate surrounding the role of intrinsically bursting neurons in the rhythmogenesis of the PBC inspires us to evaluate its role in a three-cell network. We found no advantage for intrinsically bursting neurons in forming synchronous network bursting; instead, intrinsically quiescent neurons were identified as a key mechanism. This analysis involved only studying the persistent sodium (NaP) current. Another important current for the PBC is the calcium-activated nonspecific cationic (CAN) current, which, when combined with a Na/K pump, was previously shown to be capable of producing bursts in coupled tonically active cells. In the second part of this study, we explore the interactions of the NaP and CAN currents, both currents are ubiquitous in the PBC. Using geometric singular perturbation theory and bifurcation analysis, we established the mechanisms through which reciprocally coupled pairs of neurons can generate various activity patterns. In particular, we highlighted how the NaP current could enhance the range of the strength of the CAN current for which bursts occur. We also were able to detail a novel bursting pattern seen in data, but not seen in previous models. With a foundation of understanding heterogeneity in the NaP and CAN currents, we again turned out attention to networks. For the third portion of the dissertation, we examine the effects that heterogeneity in the neuronal dynamics and coupling architecture can impose upon synchronous bursting of the entire network. We again found no significant advantage to including intrinsically bursting neurons in the network, and the best networks were characterized by an increased presence of quiescent neurons. We also described the way the NaP and CAN currents interact on the network scale to promote synchronous bursting