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Author: Richard A. Clement
Publisher: Springer Nature
ISBN: 3030984958
Category : Mathematics
Languages : en
Pages : 168
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Book Description
This book provides a brief but accessible introduction to a set of related, mathematical ideas that have proved useful in understanding the brain and behaviour. If you record the eye movements of a group of people watching a riverside scene then some will look at the river, some will look at the barge by the side of the river, some will look at the people on the bridge, and so on, but if a duck takes off then everybody will look at it. How come the brain is so adept at processing such biological objects? In this book it is shown that brains are especially suited to exploiting the geometric properties of such objects. Central to the geometric approach is the concept of a manifold, which extends the idea of a surface to many dimensions. The manifold can be specified by collections of n-dimensional data points or by the paths of a system through state space. Just as tangent planes can be used to analyse the local linear behaviour of points on a surface, so the extension to tangent spaces can be used to investigate the local linear behaviour of manifolds. The majority of the geometric techniques introduced are all about how to do things with tangent spaces. Examples of the geometric approach to neuroscience include the analysis of colour and spatial vision measurements and the control of eye and arm movements. Additional examples are used to extend the applications of the approach and to show that it leads to new techniques for investigating neural systems. An advantage of following a geometric approach is that it is often possible to illustrate the concepts visually and all the descriptions of the examples are complemented by comprehensively captioned diagrams. The book is intended for a reader with an interest in neuroscience who may have been introduced to calculus in the past but is not aware of the many insights obtained by a geometric approach to the brain. Appendices contain brief reviews of the required background knowledge in neuroscience and calculus.
Author: Fabrizio Gabbiani
Publisher: Academic Press
ISBN: 0128019069
Category : Mathematics
Languages : en
Pages : 630
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Book Description
Mathematics for Neuroscientists, Second Edition, presents a comprehensive introduction to mathematical and computational methods used in neuroscience to describe and model neural components of the brain from ion channels to single neurons, neural networks and their relation to behavior. The book contains more than 200 figures generated using Matlab code available to the student and scholar. Mathematical concepts are introduced hand in hand with neuroscience, emphasizing the connection between experimental results and theory. - Fully revised material and corrected text - Additional chapters on extracellular potentials, motion detection and neurovascular coupling - Revised selection of exercises with solutions - More than 200 Matlab scripts reproducing the figures as well as a selection of equivalent Python scripts
Author: Richard A. Clement
Publisher: Springer Nature
ISBN: 3030984958
Category : Mathematics
Languages : en
Pages : 168
Get Book Here
Book Description
This book provides a brief but accessible introduction to a set of related, mathematical ideas that have proved useful in understanding the brain and behaviour. If you record the eye movements of a group of people watching a riverside scene then some will look at the river, some will look at the barge by the side of the river, some will look at the people on the bridge, and so on, but if a duck takes off then everybody will look at it. How come the brain is so adept at processing such biological objects? In this book it is shown that brains are especially suited to exploiting the geometric properties of such objects. Central to the geometric approach is the concept of a manifold, which extends the idea of a surface to many dimensions. The manifold can be specified by collections of n-dimensional data points or by the paths of a system through state space. Just as tangent planes can be used to analyse the local linear behaviour of points on a surface, so the extension to tangent spaces can be used to investigate the local linear behaviour of manifolds. The majority of the geometric techniques introduced are all about how to do things with tangent spaces. Examples of the geometric approach to neuroscience include the analysis of colour and spatial vision measurements and the control of eye and arm movements. Additional examples are used to extend the applications of the approach and to show that it leads to new techniques for investigating neural systems. An advantage of following a geometric approach is that it is often possible to illustrate the concepts visually and all the descriptions of the examples are complemented by comprehensively captioned diagrams. The book is intended for a reader with an interest in neuroscience who may have been introduced to calculus in the past but is not aware of the many insights obtained by a geometric approach to the brain. Appendices contain brief reviews of the required background knowledge in neuroscience and calculus.
Author: Stanislaw Brzychczy
Publisher: Academic Press
ISBN: 0124104827
Category : Mathematics
Languages : en
Pages : 201
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Book Description
Mathematical Neuroscience is a book for mathematical biologists seeking to discover the complexities of brain dynamics in an integrative way. It is the first research monograph devoted exclusively to the theory and methods of nonlinear analysis of infinite systems based on functional analysis techniques arising in modern mathematics. Neural models that describe the spatio-temporal evolution of coarse-grained variables—such as synaptic or firing rate activity in populations of neurons —and often take the form of integro-differential equations would not normally reflect an integrative approach. This book examines the solvability of infinite systems of reaction diffusion type equations in partially ordered abstract spaces. It considers various methods and techniques of nonlinear analysis, including comparison theorems, monotone iterative techniques, a truncation method, and topological fixed point methods. Infinite systems of such equations play a crucial role in the integrative aspects of neuroscience modeling. - The first focused introduction to the use of nonlinear analysis with an infinite dimensional approach to theoretical neuroscience - Combines functional analysis techniques with nonlinear dynamical systems applied to the study of the brain - Introduces powerful mathematical techniques to manage the dynamics and challenges of infinite systems of equations applied to neuroscience modeling
Author: G. Bard Ermentrout
Publisher: Springer Science & Business Media
ISBN: 038787707X
Category : Mathematics
Languages : en
Pages : 434
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Book Description
Arising from several courses taught by the authors, this book provides a needed overview illustrating how dynamical systems and computational analysis have been used in understanding the types of models that come out of neuroscience.
Author: Grace Lindsay
Publisher: Bloomsbury Publishing
ISBN: 1472966457
Category : Science
Languages : en
Pages : 401
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Book Description
The human brain is made up of 85 billion neurons, which are connected by over 100 trillion synapses. For more than a century, a diverse array of researchers searched for a language that could be used to capture the essence of what these neurons do and how they communicate – and how those communications create thoughts, perceptions and actions. The language they were looking for was mathematics, and we would not be able to understand the brain as we do today without it. In Models of the Mind, author and computational neuroscientist Grace Lindsay explains how mathematical models have allowed scientists to understand and describe many of the brain's processes, including decision-making, sensory processing, quantifying memory, and more. She introduces readers to the most important concepts in modern neuroscience, and highlights the tensions that arise when the abstract world of mathematical modelling collides with the messy details of biology. Each chapter of Models of the Mind focuses on mathematical tools that have been applied in a particular area of neuroscience, progressing from the simplest building block of the brain – the individual neuron – through to circuits of interacting neurons, whole brain areas and even the behaviours that brains command. In addition, Grace examines the history of the field, starting with experiments done on frog legs in the late eighteenth century and building to the large models of artificial neural networks that form the basis of modern artificial intelligence. Throughout, she reveals the value of using the elegant language of mathematics to describe the machinery of neuroscience.
Author: Jürgen Jost
Publisher: Springer Science & Business Media
ISBN: 1447163532
Category : Mathematics
Languages : en
Pages : 233
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Book Description
Mathematical models can be used to meet many of the challenges and opportunities offered by modern biology. The description of biological phenomena requires a range of mathematical theories. This is the case particularly for the emerging field of systems biology. Mathematical Methods in Biology and Neurobiology introduces and develops these mathematical structures and methods in a systematic manner. It studies: • discrete structures and graph theory • stochastic processes • dynamical systems and partial differential equations • optimization and the calculus of variations. The biological applications range from molecular to evolutionary and ecological levels, for example: • cellular reaction kinetics and gene regulation • biological pattern formation and chemotaxis • the biophysics and dynamics of neurons • the coding of information in neuronal systems • phylogenetic tree reconstruction • branching processes and population genetics • optimal resource allocation • sexual recombination • the interaction of species. Written by one of the most experienced and successful authors of advanced mathematical textbooks, this book stands apart for the wide range of mathematical tools that are featured. It will be useful for graduate students and researchers in mathematics and physics that want a comprehensive overview and a working knowledge of the mathematical tools that can be applied in biology. It will also be useful for biologists with some mathematical background that want to learn more about the mathematical methods available to deal with biological structures and data.
Author: Fabrizio Gabbiani
Publisher: Academic Press
ISBN: 0080890490
Category : Medical
Languages : en
Pages : 505
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Book Description
Virtually all scientific problems in neuroscience require mathematical analysis, and all neuroscientists are increasingly required to have a significant understanding of mathematical methods. There is currently no comprehensive, integrated introductory book on the use of mathematics in neuroscience; existing books either concentrate solely on theoretical modeling or discuss mathematical concepts for the treatment of very specific problems. This book fills this need by systematically introducing mathematical and computational tools in precisely the contexts that first established their importance for neuroscience. All mathematical concepts will be introduced from the simple to complex using the most widely used computing environment, Matlab. This book will provide a grounded introduction to the fundamental concepts of mathematics, neuroscience and their combined use, thus providing the reader with a springboard to cutting-edge research topics and fostering a tighter integration of mathematics and neuroscience for future generations of students. - A very didactic and systematic introduction to mathematical concepts of importance for the analysis of data and the formulation of concepts based on experimental data in neuroscience - Provides introductions to linear algebra, ordinary and partial differential equations, Fourier transforms, probabilities and stochastic processes - Introduces numerical methods used to implement algorithms related to each mathematical concept - Illustrates numerical methods by applying them to specific topics in neuroscience, including Hodgkin-Huxley equations, probabilities to describe stochastic release, stochastic processes to describe noise in neurons, Fourier transforms to describe the receptive fields of visual neurons - Allows the mathematical novice to analyze their results in more sophisticated ways, and consider them in a broader theoretical framework
Author: Eugene M. Izhikevich
Publisher: MIT Press
ISBN: 0262514206
Category : Medical
Languages : en
Pages : 459
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Book Description
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. An additional chapter on synchronization, with more advanced material, can be found at the author's website, www.izhikevich.com.
Author: Pascal Wallisch
Publisher: Academic Press
ISBN: 0123838371
Category : Psychology
Languages : en
Pages : 571
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Book Description
MATLAB for Neuroscientists serves as the only complete study manual and teaching resource for MATLAB, the globally accepted standard for scientific computing, in the neurosciences and psychology. This unique introduction can be used to learn the entire empirical and experimental process (including stimulus generation, experimental control, data collection, data analysis, modeling, and more), and the 2nd Edition continues to ensure that a wide variety of computational problems can be addressed in a single programming environment. This updated edition features additional material on the creation of visual stimuli, advanced psychophysics, analysis of LFP data, choice probabilities, synchrony, and advanced spectral analysis. Users at a variety of levels—advanced undergraduates, beginning graduate students, and researchers looking to modernize their skills—will learn to design and implement their own analytical tools, and gain the fluency required to meet the computational needs of neuroscience practitioners. - The first complete volume on MATLAB focusing on neuroscience and psychology applications - Problem-based approach with many examples from neuroscience and cognitive psychology using real data - Illustrated in full color throughout - Careful tutorial approach, by authors who are award-winning educators with strong teaching experience
Author: Alexander E. Hramov
Publisher: Springer Nature
ISBN: 303075992X
Category : Science
Languages : en
Pages : 384
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Book Description
This book illustrates how modern mathematical wavelet transform techniques offer fresh insights into the complex behavior of neural systems at different levels: from the microscopic dynamics of individual cells to the macroscopic behavior of large neural networks. It also demonstrates how and where wavelet-based mathematical tools can provide an advantage over classical approaches used in neuroscience. The authors well describe single neuron and populational neural recordings. This 2nd edition discusses novel areas and significant advances resulting from experimental techniques and computational approaches developed since 2015, and includes three new topics: • Detection of fEPSPs in multielectrode LFPs recordings. • Analysis of Visual Sensory Processing in the Brain and BCI for Human Attention Control; • Analysis and Real-time Classification of Motor-related EEG Patterns; The book is a valuable resource for neurophysiologists and physicists familiar with nonlinear dynamical systems and data processing, as well as for graduate students specializing in these and related areas.
