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2023 | Buch

Overview Kapitel lesen Erstes Kapitel lesen

Neurodynamics

An Applied Mathematics Perspective

verfasst von: Stephen Coombes, Kyle C. A. Wedgwood

Verlag: Springer International Publishing

Buchreihe : Texts in Applied Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book is about the dynamics of neural systems and should be suitable for those with a background in mathematics, physics, or engineering who want to see how their knowledge and skill sets can be applied in a neurobiological context. No prior knowledge of neuroscience is assumed, nor is advanced understanding of all aspects of applied mathematics! Rather, models and methods are introduced in the context of a typical neural phenomenon and a narrative developed that will allow the reader to test their understanding by tackling a set of mathematical problems at the end of each chapter. The emphasis is on mathematical- as opposed to computational-neuroscience, though stresses calculation above theorem and proof. The book presents necessary mathematical material in a digestible and compact form when required for specific topics. The book has nine chapters, progressing from the cell to the tissue, and an extensive set of references. It includes Markov chain models for ions, differential equations for single neuron models, idealised phenomenological models, phase oscillator networks, spiking networks, and integro-differential equations for large scale brain activity, with delays and stochasticity thrown in for good measure. One common methodological element that arises throughout the book is the use of techniques from nonsmooth dynamical systems to form tractable models and make explicit progress in calculating solutions for rhythmic neural behaviour, synchrony, waves, patterns, and their stability. This book was written for those with an interest in applied mathematics seeking to expand their horizons to cover the dynamics of neural systems. It is suitable for a Masters level course or for postgraduate researchers starting in the field of mathematical neuroscience.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Overview
Abstract
The book is organised in chapters that progress from the scale of a single cell up to an entire brain. The use of mathematical methodologies does not follow a natural hierarchy, and instead we introduce these as and when needed. On this note, we rely on some familiarity of the reader with techniques from applied mathematics. However, for pragmatic purposes, we have also included a number of Boxes to act as self-contained expositions or reminders of useful pieces of mathematical theory. Nonetheless, we do not expect a typical reader to come armed with the perfect mathematical repertoire to absorb the story in this book. Indeed, we hope that some of the topics covered, such as those associated with non-smooth dynamical systems, will be of interest in their own right. In a similar vein, the reader need not have any prior background in neuroscience to understand this book, and at the risk of reducing the whole of this field to a few glib paragraphs, we first mention some of the more common facts and terminology that you will come across in the pages ahead.
Stephen Coombes, Kyle C. A. Wedgwood
Chapter 2. Single neuron models
Abstract
Few phenomena in neuroscience are as evocative as the pulses of electrical activity produced by individual cells. Indeed, the discovery by Galvani of electrical activity in nervous tissue began a centuries long fascination with the origin and purpose of this behaviour, and motivated the initial design of the battery by Volta. Half a century later, du Bois-Reymond demonstrated that these pulses follow a prototypical temporal pattern, now referred to as an action potential. The pioneering work of Ramón y Cajal highlighted that, rather than being a syncytium, the brain is a large network of individual cells acting in concert to produce the diverse range of behaviours seen at the organism level. This shift in perspective gave new meaning to action potentials as units of information that could be exchanged between cells.
Stephen Coombes, Kyle C. A. Wedgwood
Chapter 3. Phenomenological models and their analysis
Abstract
The role of mathematical neuroscience, of which neurodynamics is a subset, is to explore the mechanisms underlying the behaviour observed in real neural tissues using mathematics as the primary tool.
Stephen Coombes, Kyle C. A. Wedgwood
Chapter 4. Axons, dendrites, and synapses
Abstract
Chapter 2 and chapter 3 considered single neuron models that idealised the nerve cell as a point or patch of cell membrane in which voltage is the same everywhere. However, the classical notion of a neuron is of a specialised cell with a body, an axon, and dendrites [802]. The cell body or soma, contains the nucleus and cytoplasm, the axon extends from the soma and ultimately branches before ending at nerve terminals, and dendrites are branched structures connected to the soma that receive signals from other neurons. Axons and dendrites make contact with each other at axo-dendritic synapses, and dendro-dendritic synapses are also possible. These structures allow neurons to communicate in different ways, and to transmit information to other nerve cells, muscle, or gland cells. In this chapter, we consider idealised mathematical models of these processes and methods for their analysis, which now include a spatial aspect. In later chapters, we make use of these structures and their dynamics to construct model neuronal networks.
Stephen Coombes, Kyle C. A. Wedgwood
Chapter 5. Response properties of single neurons
Abstract
Neurons are often viewed as biological filters that convert input signals to output signals. This perspective allows specific neural systems to be studied in isolation. For example, one may study sensory areas by assuming that they process and relay information from the sensory periphery to higher brain areas.
Stephen Coombes, Kyle C. A. Wedgwood
Chapter 6. Weakly coupled oscillator networks
Abstract
The complexity and functionality of the human brain comes from the joint action of populations of cells acting in concert. The preceding chapters have presented the key building blocks required to construct networks composed of either single neurons, or of spatially localised neural populations. The remaining chapters focus on modelling and analysing such neural networks, taking into consideration how neurons communicate with one another.
Stephen Coombes, Kyle C. A. Wedgwood
Chapter 7. Strongly coupled spiking networks
Abstract
In Chap. 6, we explored how collective behaviour, such as synchronous network oscillations and phase waves, are generated in weakly coupled networks. For networks of spiking networks, the assumption of weak interactions allows us to reduce the full network model down to one that describes each neuron by its phase along an underlying spiking orbit. In this way, the dynamics for the system are considered only on a reduced phase space given by the Cartesian product of each neuron’s periodic orbit. Whilst the individual postsynaptic currents induced by spiking events may be small, their sum over time (since these currents have a characteristic decay timescale) and over the set of pre-synaptic neurons to which a given cell is coupled, may not be. In addition, the phase reduction approach can only be used when the uncoupled dynamics of each neuron is oscillatory. Hence, the weakly coupled framework is unable to describe networks of excitable cells. In this chapter, we relax the assumption of weak coupling and present results for collective dynamics in networks with arbitrary coupling strength. It is important to stress that we currently lack a general theory for strongly coupled networks and that results presented here are mostly for specific model choices.
Stephen Coombes, Kyle C. A. Wedgwood
Chapter 8. Population models
Abstract
Ever since the first recordings of the human electroencephalogram (EEG) in 1924 by Hans Berger [72], electrophysiological brain recordings have been shown to be dominated by oscillations (rhythmic activity in cell assemblies) across a wide range of temporal scales. From a behavioural perspective, these oscillations can be grouped into five main frequency bands: delta (\(1-4\) Hz), theta (\(4-8\) Hz), alpha (\(8-13\) Hz), beta (\(13-30\) Hz), and  gamma (\(30-200\) Hz). Alpha  rhythms are associated with awake resting states and REM sleep, delta with deep sleep, theta with drowsiness, whilst beta and gamma rhythms are associated with task-specific responses [32, 59, 133, 796].
Stephen Coombes, Kyle C. A. Wedgwood
Chapter 9. Firing rate tissue models
Abstract
Despite the success of population models, such as the neural mass models described in Chap. 8, in describing brain rhythms that can be measured, e.g., with localised electroencephalogram (EEG) scalp electrodes, they do not provide large-scale models of brain activity on their own. Rather, they can be seen as building blocks for this larger endeavour. Through their axons and dendrites, which may span many hundreds of microns, neural populations are able to extend their influence over many times the scale of the cell soma.
Stephen Coombes, Kyle C. A. Wedgwood
Backmatter
Metadaten
Titel
Neurodynamics
verfasst von
Stephen Coombes
Kyle C. A. Wedgwood
Copyright-Jahr
2023
Electronic ISBN
978-3-031-21916-0
Print ISBN
978-3-031-21915-3
DOI
https://doi.org/10.1007/978-3-031-21916-0