Bursting

The following sections are included:

• PREFACE

• CONTENTS

We review the development of the Hindmarsh-Rose equations of 1982 and 1984. After a brief description of bifurcations in a model for subthreshold oscillations, that can generate bursting, we show that similar bifurcations will not be found in the Hindmarsh-Rose nor in other similar equations.

We trace the development of negative feedback by Ca²⁺ as a driving mechanism for bursting in the Chay-Keizer model for insulin-secreting pancreatic β-cells and successor models. We focus on the role of the endoplasmic reticulum in providing the slow kinetic components needed to obtain bursting that lasts more than a few seconds. The fast/slow analysis applied by Rinzel to Chay-Keizer is generalized to elucidate the properties of bursting when there are two slow, negative feedback variables. Citations are provided to the current literature on further ramifications to Chay-Keizer that can be dissected by similar techniques.

The following sections are included:

• Introduction

• Electrical Properties of AVP Cells

• Mathematical Model

• Firing Patterns

• Burst Structure

• The Role of Calcium

• The Action of Dynorphin

• The Bursting Mechanism

• The Dynamics of Dynorphin

• Analysis of Bursting
  • FAST
  • SLOW

• Discussion
  • A dual role for calcium
  • Alternative mechanisms for the plateau potential
  • Excitable bursting

• Acknowledgments

• References

Singular perturbation analysis of models for bursting neural rhythms relies upon decomposition of models into slow and fast time scales. Transitions between spiking and quiescent dynamics correspond to bifurcations of the fast subsystems. We observe that the fast subsystems of bursting models also have two time scales. Consequently, the bifurcation analysis of the fast subsystems is surprisingly complex. This is illustrated by a complete two parameter bifurcation analysis of a model that combines just sodium and leak currents. The very fast time scale in bursting models is frequently dominated by a sodium current, suggesting that the bifurcation diagram presented here is representative of that found for large classes of systems. The second part of this chapter explores further the relationship between transitions in several bursting niodels with bifurcations of reduced subsystems. Slow currents are combined into a pair of parameters for an “effective” leak conductance, and the bifurcations of these reduced models are compared with the transitions in the bursting rhythms.

The two-compartment Pinsky-Rinzel model of a hippocampal CA3 pyramidal neuron consists of electrically coupled soma and dendritic compartments, each with active ionic conductances. This model has been widely used in a variety of contexts, but little analysis has been performed on its bursting solutions. In this chapter, we provide a geometric framework to study the Pinsky-Rinzel model. Using a combination of analysis and simulations, we identify neuronal mechanisms responsible for certain salient features of the model such as burst initiation and somatic-dendritic ping-pong. Some of these mechanisms are then demonstrated in a conceptually simpler, two-compartment Morris-Lecar model.

We give an overview of “ghostbursting”, a novel type of bursting observed in sensory processing neurons of weakly electric fish. We discuss the steps taken to understand the bursting mechanism, including experimental manipulations and the development of multi-compartmental and minimal neuron models. Using dynamical systems theory, we emphasise the main differences between this type of bursting and other previous models. We finally review results showing how electrosensory neurons can process different components of dynamic stimuli in parallel with both bursts and isolated spikes.

Closed circuits composed of bursting neurons were analysed using the phase resetting observed in a postsynaptic neuron in response to a burst in the presynaptic neuron in the open loop condition without feedback. Phase resetting before (first order) and after (second order) burst initiation was considered separately. Existence and stability criteria were derived for small networks composed of endogenously bursting neurons assuming that the effect of a perturbation is the same under open or closed loop conditions, postulating only that the resetting is complete by the time a subsequent input is received. Synaptic delays could be accomodated as long as the above postulate was satisfied. The method had excellent success in small networks of model neurons, and good success in hybrid circuits composed of one biological and one model neuron. The latter success was limited somewhat by the inherent variability in biological neurons and by changes in burst durations resulting from feedback in the closed circuit. Stability criteria were also derived for variable burst duration, and the dependence of first order resetting in a biological bursting neuron on duration was elucidated as the first step into accommodating variability in burst duration into the theoretical framework.

Periodic bursting in fast-slow systems can be viewed as closed paths through the unfolding parameters of degenerate singularities. Using this approach we show that bursting in coupled systems can have interesting behavior. We focus on two identical cell systems and use the Z₂ symmetry present in such systems to illustrate interesting bursting phenomena. In particular, we show that Hopf/Hopf mode interactions can lead to bursting between in phase and out of phase periodic solutions and symmetry-breaking Takens-Bogdanov singularities can lead to bursting that randomly chooses between two (symmetrically related) limit cycles.

We treat one-dimensional and two-dimensional maps exhibiting bursting oscillations. These maps have been proposed as a computationally efficient alternative to conventional models based on the Hodgkin-Huxley formalism in studying the collective behaviour of networks of neurons. We review cobwebbing and bifurcation analyses which are used to reveal the mechanisms underlying the formation of the bursting oscillations. New and existing results on the synchronizing effect of mean-field and diffusive coupling are brought together, compared, and contrasted. We demonstrate that synchronization comes in the form of in-phase and antiphase bursting oscillations, as well as in-phase and anti-phase spiking. We explain some coupling effects by extending the bifurcation analysis of a single cell to a two-cell system.

We show that diffusive coupling of square-wave bursters, beyond the expected effect of synchronization, can greatly influence the characteristics of the oscillations. For cells that are already bursters, coupling tends to increase burst period. Cells that are not bursters, but near a bursting regime, can be induced to burst by coupling; we call this emergent bursting. Both effects require that coupling be strong enough to synchronize the slow bursts but not the fast spikes. We therefore review in detail the transitions of the fast subsystem from out-of-phase spiking to in-phase spiking as coupling strength is increased and the consequences for the full fast-slow bursting system. The beginnings of an extension of the theory to synaptic coupling are indicated.

Rhythmic activities underly many crucial functions such as locomotion, respiration, secretion, etc. Models of network rhythmicity have generally included cellular oscillators or inhibitory connections between antagonistic centers. Here, we present a model of the spontaneous episodic activity in developing spinal cord networks that include neither of these two components. This model only represents the averaged activity and excitability in the network, assumed purely excitatory. In our model, positive feedback through excitatory connections generates episodes of activity, which are terminated by a slow, activity-dependent depression of network activity (slow negative feedback). This idealized model allowed a qualitative understanding of the network dynamics which lead to prediction and comprehension of experimental observations. Although the complexity of the system has been restricted to the interactions between fast positive and slow negative feedback, the emergent feature of the network rhythm was captured, and it applies to many developinglexcitatory networks. Finally, we show that this model of network rhythmicity behaves similarly to models of repetitive bursting in single cells.

An excitatory network of bursting neurons are postulated to be involved in generating the rhythm of breathing in the pre-Bötzinger complex of the mammalian brainstem. A subpopulation of these neurons exhibit complex voltage-dependent properties with multi-state behavior consisting of silence, oscillatory bursting and tonic spiking activity modes. In this chapter we review the development of single cell models, followed by simulation-based studies of population dynamics of a network of such model cells. In particular, we find that excitatory synaptic coupling increases the range of parameter space where bursting occurs, and that parameter heterogeneity of intrinsic and synaptic properties further increases this range, as well as the range of frequencies where bursting occurs at a network level. The mechanisms underlying these findings have been elucidated by recent mathematical analyses. We start with a geometric fast/slow approach to analyze bursting dynamics, including transitions to tonic spiking states in single model neurons as well as synaptically coupled pairs of model cells. Certain aspects of the increased dynamic range due to synaptic coupling can be understood from the effects of coupling strength on the bifurcation structure of a single, self-coupled cell. Additional complexity emerges when neuron pairs exhibit synchrony of spikes within bursts. Here synaptic coupling changes the form of bursting from square-wave bursting to a bursting class known as fold/fold cycle or “top hat bursting”. A two-variable slow averaged system analysis reveals the existence of, and transitions between, two subtypes of bursting. To conclude the chapter, we extend the fast/slow analysis to a larger cell population with heterogeneities, which further enhance the dynamic range of network bursting. We determine conditions for synchronized whole-population bursting, allowing for gradual jumps between the silent and active phases of each burst cycle. We conclude by highlighting unresolved problems in modeling and analyzing this system as well as considering limitations of our studies to date.

Three examples in which geometric singular perturbation methods have been used to analyze firing patterns in models of bursting oscillators are discussed. In the first example, we consider a single square-wave burster and review rigorous results concerned with the existence, uniqueness and stability of the bursting oscillation. The analysis demonstrates that the bursting solution may not be uniquely determined for arbitrarily small values of the singular perturbation parameter. In the second example, we consider wave-propagation in a one-dimensional array of neuronal oscillators arising in models of thalamic networks. This network may generate two types of waves: smoothly propagating waves and lurching waves which propagate in a saltatory fashion. Geometric singular perturbation methods are used to derive explicit formulas for when a particular wave exists and for how the velocity of the wave depends on parameters. Finally, in the third example, we consider transitions between activity patterns in an excitatory-inhibitory network; this network was first introduced as a model for neuronal activity within the basal ganglia. Geometric methods are used to uncover mechanisms both for generating irregular firing and for synchronizing subsets of cells into regularly bursting clusters.

Cortical slices bathed in low magnesium with reduced inhibition are able to sustain propagating waves of spikes. Intracellular recordings suggest that a possible mechanism for termination of waves is depolarization block of the sodium channels. In some instances, repeated waves occur with a frequency of 10-15 Hz. In other instances, the waves splinter into locally active regions. We describe a bursting model based on the interplay between slow synaptic depolarization and block. We show that waves can oscillate and break-up depending on parameters. A simplified normal form permits some insight into the underlying dynamics.

The following sections are included:

• INDEX