http://www.maths.qmul.ac.uk/%7Elatora/report_06.pdf

Complex networks: Structure and dynamics

I have developed "tennis elbow" from lugging this book around the past four weeks, but it is worth the pain, the effort, and the aspirin. It is also worth the (relatively speaking) bargain price. Including appendixes, this book contains 894 pages of text. The entire panorama of the neural sciences is surveyed and examined, and it is comprehensive in its scope, from genomes to social behaviors. The editors explicitly state that the book is designed as "an introductory text for students of biology, behavior, and medicine," but it is hard to imagine any audience, interested in any fragment of neuroscience at any level of sophistication, that would not enjoy this book. The editors have done a masterful job of weaving together the biologic, the behavioral, and the clinical sciences into a single tapestry in which everyone from the molecular biologist to the practicing psychiatrist can find and appreciate his or

Principles of Neural Science

This book explains the relationship of electrophysiology, nonlinear dynamics, and the computational properties of neurons, with each concept presented in terms of both neuroscience and mathematics and illustrated using geometrical intuition In order to model neuronal behavior or to interpret the results of modeling studies, neuroscientists must call upon methods of nonlinear dynamics This book offers an introduction to nonlinear dynamical systems theory for researchers and graduate students in neuroscience It also provides an overview of neuroscience for mathematicians who want to learn the basic facts of electrophysiology "Dynamical Systems in Neuroscience" presents a systematic study of the relationship of electrophysiology, nonlinear dynamics, and computational properties of neurons It emphasizes that information processing in the brain depends not only on the electrophysiological properties of neurons but also on their dynamical properties The book introduces dynamical systems starting with one- and two-dimensional Hodgkin-Huxley-type models and continuing to a description of bursting systems Each chapter proceeds from the simple to the complex, and provides sample problems at the end The book explains all necessary mathematical concepts using geometrical intuition; it includes many figures and few equations, making it especially suitable for non-mathematicians Each concept is presented in terms of both neuroscience and mathematics, providing a link between the two disciplines Nonlinear dynamical systems theory is at the core of computational neuroscience research, but it is not a standard part of the graduate neuroscience curriculum - or taught by math or physics department in a way that is suitable for students of biology This book offers neuroscience students and researchers a comprehensive account of concepts and methods increasingly used in computational neuroscience

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Dynamical Systems in Neuroscience

Neurons in the brain communicate by short electrical pulses, the so-called action potentials or spikes. How can we understand the process of spike generation? How can we understand information transmission by neurons? What happens if thousands of neurons are coupled together in a seemingly random network? How does the network connectivity determine the activity patterns? And, vice versa, how does the spike activity influence the connectivity pattern? These questions are addressed in this 2002 introduction to spiking neurons aimed at those taking courses in computational neuroscience, theoretical biology, biophysics, or neural networks. The approach will suit students of physics, mathematics, or computer science; it will also be useful for biologists who are interested in mathematical modelling. The text is enhanced by many worked examples and illustrations. There are no mathematical prerequisites beyond what the audience would meet as undergraduates: more advanced techniques are introduced in an elementary, concrete fashion when needed.

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Spiking Neuron Models: Single Neurons, Populations, Plasticity

Bifurcation mechanisms involved in the generation of action potentials (spikes) by neurons are reviewed here. We show how the type of bifurcation determines the neuro-computational properties of the cells. For example, when the rest state is near a saddle-node bifurcation, the cell can fire all-or-none spikes with an arbitrary low frequency, it has a well-defined threshold manifold, and it acts as an integrator; i.e. the higher the frequency of incoming pulses, the sooner it fires. In contrast, when the rest state is near an Andronov–Hopf bifurcation, the cell fires in a certain frequency range, its spikes are not all-or-none, it does not have a well-defined threshold manifold, it can fire in response to an inhibitory pulse, and it acts as a resonator; i.e. it responds preferentially to a certain (resonant) frequency of the input. Increasing the input frequency may actually delay or terminate its firing. We also describe the phenomenon of neural bursting, and we use geometric bifurcation theory to extend the existing classification of bursters, including many new types. We discuss how the type of burster defines its neuro-computational properties, and we show that different bursters can interact, synchronize and process information differently.

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Neural excitability, spiking and bursting

Global competition and local cooperation in a network of neural oscillators

A class of differential equations that model electrical activity in pancreatic beta cells is considered. It is demonstrated that these equations must give rise to both bursting solutions and, for different values of the parameters, continuous spiking. We also consider how the number of spikes per burst changes as parameters in the equations are varied. This transition may be continuous, in which case the period of the bursting solution increases significantly and then decreases. Hence, small perturbations may cause macroscopic changes in the bursting solution. This transition may also give rise to chaotic dynamics due to the existence of a Smale horseshoe.

Chaotic spikes arising from a model of bursting in excitable membranes

In this chapter, we consider recent models for neuronal activity. We review the sorts of oscillatory behavior which may arise from the models and then discuss how geometric singular perturbation methods have been used to analyze these rhythms. We begin by discussing models for single cells which display bursting oscillations. There are, in fact, several different classes of bursting solutions; these have been classified by the geometric properties of how solutions evolve in phase space. We describe several of the bursting classes and then review related rigorous mathematical analysis. We then discuss the dynamics of small networks of neurons. We are primarily interested in whether excitatory or inhibitory synaptic coupling leads to either synchronous or desynchronous rhythms. We demonstrate that all four combinations are possible, depending on the details of the intrinsic and synaptic properties of the cells. Finally, we discuss larger networks of neuronal oscillators involving two distinct cell populations. In particular, we demonstrate how dynamical systems methods can be used to analyze recent models for sleep rhythms and other oscillations generated in the thalamus. The analysis helps to explain the generation of the different thalamic rhythms and the transitions between them, in both of which inhibition plays a crucial role.

Geometric Singular Perturbation Analysis of Neuronal Dynamics

A network of excitatory neurons within the pre-Botzinger complex (pre-BotC) of the mammalian brain stem has been found experimentally to generate robust, synchronized population bursts of activity. An experimentally calibrated model for pre-BotC cells yields typical square-wave bursting behavior in the absence of coupling, over a certain parameter range, with quiescence or tonic spiking outside of this range. Previous simulations of this model showed that the introduction of synaptic coupling extends the bursting parameter range significantly and induces complex effects on burst characteristics. In this paper, we use geometric dynamical systems techniques, predominantly a fast/slow decomposition and bifurcation analysis approach, to explain these effects in a two-cell model network. Our analysis yields the novel finding that, over a broad range of synaptic coupling strengths, the network can support two qualitatively distinct forms of synchronized bursting, which we call symmetric and asymmetric bursting, as well as both symmetric and asymmetric tonic spiking. By elucidating the dynamical mechanisms underlying the transitions between these states, we also gain insight into how relevant parameters influence burst duration and interburst intervals. We find that, in the two-cell network with synaptic coupling, the stable family of periodic orbits for the fast subsystem features spike asynchrony within otherwise synchronized bursts and terminates in a saddle-node bifurcation, rather than in a homoclinic bifurcation, over a wide parameter range. As a result, square-wave bursting is replaced by what we call top hat bursting (also known as fold/fold cycle bursting), at least for a broad range of parameter values. Further, spike asynchrony is a key ingredient in shaping the dynamic range of bursting, leading to a significant enhancement in the parameter range over which bursting occurs and an abrupt increase in burst duration as an appropriate parameter is varied.

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David Terman

Papers

Global competition and local cooperation in a network of neural oscillators

Chaotic spikes arising from a model of bursting in excitable membranes

Geometric Singular Perturbation Analysis of Neuronal Dynamics

The Dynamic Range of Bursting in a Model Respiratory Pacemaker Network

A very singular solution of the porous media equation with absorption