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Mathematical Foundations of Neuroscience

Mathematical Foundations of Neuroscience Book
Author : G. Bard Ermentrout,David H. Terman
Publisher : Springer Science & Business Media
Release : 2010-07-01
ISBN : 0387877088
Language : En, Es, Fr & De

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Book Description :

This book applies methods from nonlinear dynamics to problems in neuroscience. It uses modern mathematical approaches to understand patterns of neuronal activity seen in experiments and models of neuronal behavior. The intended audience is researchers interested in applying mathematics to important problems in neuroscience, and neuroscientists who would like to understand how to create models, as well as the mathematical and computational methods for analyzing them. The authors take a very broad approach and use many different methods to solve and understand complex models of neurons and circuits. They explain and combine numerical, analytical, dynamical systems and perturbation methods to produce a modern approach to the types of model equations that arise in neuroscience. There are extensive chapters on the role of noise, multiple time scales and spatial interactions in generating complex activity patterns found in experiments. The early chapters require little more than basic calculus and some elementary differential equations and can form the core of a computational neuroscience course. Later chapters can be used as a basis for a graduate class and as a source for current research in mathematical neuroscience. The book contains a large number of illustrations, chapter summaries and hundreds of exercises which are motivated by issues that arise in biology, and involve both computation and analysis. Bard Ermentrout is Professor of Computational Biology and Professor of Mathematics at the University of Pittsburgh. David Terman is Professor of Mathematics at the Ohio State University.

Mathematical Neuroscience

Mathematical Neuroscience Book
Author : Stanislaw Brzychczy,Roman R. Poznanski
Publisher : Academic Press
Release : 2013-08-16
ISBN : 0124104827
Language : En, Es, Fr & De

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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

Tutorials in Mathematical Biosciences I

Tutorials in Mathematical Biosciences I Book
Author : Alla Borisyuk,G. Bard Ermentrout,Avner Friedman,David H. Terman
Publisher : Springer Science & Business Media
Release : 2005-02-18
ISBN : 9783540238584
Language : En, Es, Fr & De

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Book Description :

This volume introduces some basic theories on computational neuroscience. Chapter 1 is a brief introduction to neurons, tailored to the subsequent chapters. Chapter 2 is a self-contained introduction to dynamical systems and bifurcation theory, oriented towards neuronal dynamics. The theory is illustrated with a model of Parkinson's disease. Chapter 3 reviews the theory of coupled neural oscillators observed throughout the nervous systems at all levels; it describes how oscillations arise, what pattern they take, and how they depend on excitory or inhibitory synaptic connections. Chapter 4 specializes to one particular neuronal system, namely, the auditory system. It includes a self-contained introduction, from the anatomy and physiology of the inner ear to the neuronal network that connects the hair cells to the cortex, and describes various models of subsystems.

Mathematical Neuroscience

Mathematical Neuroscience Book
Author : Anonim
Publisher : Unknown
Release : 2005
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Download Mathematical Neuroscience book written by , available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Neuroscience of Mathematical Cognitive Development

Neuroscience of Mathematical Cognitive Development Book
Author : Rhonda Douglas Brown
Publisher : Springer
Release : 2018-04-13
ISBN : 3319764098
Language : En, Es, Fr & De

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Book Description :

​This book examines the neuroscience of mathematical cognitive development from infancy into emerging adulthood, addressing both biological and environmental influences on brain development and plasticity. It begins by presenting major theoretical frameworks for designing and interpreting neuroscience studies of mathematical cognitive development, including developmental evolutionary theory, developmental systems approaches, and the triple-code model of numerical processing. The book includes chapters that discuss findings from studies using neuroscience research methods to examine numerical and visuospatial cognition, calculation, and mathematical difficulties and exceptionalities. It concludes with a review of mathematical intervention programs and recommendations for future neuroscience research on mathematical cognitive development. Featured neuroscience research methods include: Functional Magnetic Resonance Imaging (fMRI). Diffusion Tensor Imaging (DTI). Event Related Potentials (ERP). Transcranial Magnetic Stimulation (TMS). Neuroscience of Mathematical Cognitive Development is an essential resource for researchers, clinicians and related professionals, and graduate students in child and school psychology, neuroscience, educational psychology, neuropsychology, and mathematics education.

Mathematical and Theoretical Neuroscience

Mathematical and Theoretical Neuroscience Book
Author : Giovanni Naldi,Thierry Nieus
Publisher : Springer
Release : 2018-03-20
ISBN : 3319682970
Language : En, Es, Fr & De

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Book Description :

This volume gathers contributions from theoretical, experimental and computational researchers who are working on various topics in theoretical/computational/mathematical neuroscience. The focus is on mathematical modeling, analytical and numerical topics, and statistical analysis in neuroscience with applications. The following subjects are considered: mathematical modelling in Neuroscience, analytical and numerical topics; statistical analysis in Neuroscience; Neural Networks; Theoretical Neuroscience. The book is addressed to researchers involved in mathematical models applied to neuroscience.

Mathematical and Theoretical Neuroscience

Mathematical and Theoretical Neuroscience Book
Author : Giovanni Naldi,Thierry Nieus
Publisher : Springer
Release : 2018-03-21
ISBN : 9783319682969
Language : En, Es, Fr & De

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Book Description :

This volume gathers contributions from theoretical, experimental and computational researchers who are working on various topics in theoretical/computational/mathematical neuroscience. The focus is on mathematical modeling, analytical and numerical topics, and statistical analysis in neuroscience with applications. The following subjects are considered: mathematical modelling in Neuroscience, analytical and numerical topics; statistical analysis in Neuroscience; Neural Networks; Theoretical Neuroscience. The book is addressed to researchers involved in mathematical models applied to neuroscience.

Neuroscience

Neuroscience Book
Author : Alwyn Scott
Publisher : Springer Science & Business Media
Release : 2007-12-14
ISBN : 0387224637
Language : En, Es, Fr & De

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Book Description :

This book will be of interest to anyone who wishes to know what role mathematics can play in attempting to comprehend the dynamics of the human brain. It also aims to serve as a general introduction to neuromathematics. The book gives the reader a qualitative understanding and working knowledge of useful mathematical applications to the field of neuroscience. The book is readable by those who have little knowledge of mathematics for neuroscience but are committed to begin acquiring such knowledge.

Mathematics for Neuroscientists

Mathematics for Neuroscientists Book
Author : Fabrizio Gabbiani,Steven James Cox
Publisher : Academic Press
Release : 2010-09-16
ISBN : 9780080890494
Language : En, Es, Fr & De

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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

Mathematical Foundations of Neuroscience

Mathematical Foundations of Neuroscience Book
Author : G. Bard Ermentrout,David H. Terman
Publisher : Unknown
Release : 2010
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

This book applies methods from nonlinear dynamics to problems in neuroscience. It uses modern mathematical approaches to understand patterns of neuronal activity seen in experiments and models of neuronal behavior. The intended audience is researchers interested in applying mathematics to important problems in neuroscience, and neuroscientists who would like to understand how to create models, as well as the mathematical and computational methods for analyzing them. The authors take a very broad approach and use many different methods to solve and understand complex models of neurons and circuits. They explain and combine numerical, analytical, dynamical systems and perturbation methods to produce a modern approach to the types of model equations that arise in neuroscience. There are extensive chapters on the role of noise, multiple time scales and spatial interactions in generating complex activity patterns found in experiments. The early chapters require little more than basic calculus and some elementary differential equations and can form the core of a computational neuroscience course. Later chapters can be used as a basis for a graduate class and as a source for current research in mathematical neuroscience. The book contains a large number of illustrations, chapter summaries and hundreds of exercises which are motivated by issues that arise in biology, and involve both computation and analysis. Bard Ermentrout is Professor of Computational Biology and Professor of Mathematics at the University of Pittsburgh. David Terman is Professor of Mathematics at the Ohio State University.

An Introduction to Modeling Neuronal Dynamics

An Introduction to Modeling Neuronal Dynamics Book
Author : Christoph Börgers
Publisher : Springer
Release : 2017-04-17
ISBN : 3319511718
Language : En, Es, Fr & De

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Book Description :

This book is intended as a text for a one-semester course on Mathematical and Computational Neuroscience for upper-level undergraduate and beginning graduate students of mathematics, the natural sciences, engineering, or computer science. An undergraduate introduction to differential equations is more than enough mathematical background. Only a slim, high school-level background in physics is assumed, and none in biology. Topics include models of individual nerve cells and their dynamics, models of networks of neurons coupled by synapses and gap junctions, origins and functions of population rhythms in neuronal networks, and models of synaptic plasticity. An extensive online collection of Matlab programs generating the figures accompanies the book.

Stochastic Methods in Neuroscience

Stochastic Methods in Neuroscience Book
Author : Carlo Laing,Gabriel J Lord
Publisher : OUP Oxford
Release : 2009-09-24
ISBN : 0191607983
Language : En, Es, Fr & De

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Book Description :

Great interest is now being shown in computational and mathematical neuroscience, fuelled in part by the rise in computing power, the ability to record large amounts of neurophysiological data, and advances in stochastic analysis. These techniques are leading to biophysically more realistic models. It has also become clear that both neuroscientists and mathematicians profit from collaborations in this exciting research area. Graduates and researchers in computational neuroscience and stochastic systems, and neuroscientists seeking to learn more about recent advances in the modelling and analysis of noisy neural systems, will benefit from this comprehensive overview. The series of self-contained chapters, each written by experts in their field, covers key topics such as: Markov chain models for ion channel release; stochastically forced single neurons and populations of neurons; statistical methods for parameter estimation; and the numerical approximation of these stochastic models. Each chapter gives an overview of a particular topic, including its history, important results in the area, and future challenges, and the text comes complete with a jargon-busting index of acronyms to allow readers to familiarize themselves with the language used.

Weakly Connected Neural Networks

Weakly Connected Neural Networks Book
Author : Frank C. Hoppensteadt,Eugene M. Izhikevich
Publisher : Springer Science & Business Media
Release : 2012-12-06
ISBN : 1461218284
Language : En, Es, Fr & De

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Book Description :

Devoted to local and global analysis of weakly connected systems with applications to neurosciences, this book uses bifurcation theory and canonical models as the major tools of analysis. It presents a systematic and well motivated development of both weakly connected system theory and mathematical neuroscience, addressing bifurcations in neuron and brain dynamics, synaptic organisations of the brain, and the nature of neural codes. The authors present classical results together with the most recent developments in the field, making this a useful reference for researchers and graduate students in various branches of mathematical neuroscience.

Neural Networks with Discontinuous Impact Activations

Neural Networks with Discontinuous Impact Activations Book
Author : Marat Akhmet,Enes Yılmaz
Publisher : Springer Science & Business Media
Release : 2013-10-30
ISBN : 1461485665
Language : En, Es, Fr & De

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Book Description :

This book presents as its main subject new models in mathematical neuroscience. A wide range of neural networks models with discontinuities are discussed, including impulsive differential equations, differential equations with piecewise constant arguments, and models of mixed type. These models involve discontinuities, which are natural because huge velocities and short distances are usually observed in devices modeling the networks. A discussion of the models, appropriate for the proposed applications, is also provided.

Mathematical Foundations of Neuroscience

Mathematical Foundations of Neuroscience Book
Author : G. Bard Ermentrout,David H. Terman
Publisher : Springer Science & Business Media
Release : 2010-07-08
ISBN : 038787707X
Language : En, Es, Fr & De

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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.

Dynamical Systems in Neuroscience

Dynamical Systems in Neuroscience Book
Author : Eugene M. Izhikevich
Publisher : MIT Press
Release : 2007
ISBN : 0262090430
Language : En, Es, Fr & De

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Book Description :

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.

Mathematics for Neuroscientists

Mathematics for Neuroscientists Book
Author : Fabrizio Gabbiani,Steven James Cox
Publisher : Academic Press
Release : 2017-02-23
ISBN : 9780128018958
Language : En, Es, Fr & De

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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

Theoretical Neuroscience

Theoretical Neuroscience Book
Author : Laurence F. Abbott,Peter Dayan
Publisher : MIT Press
Release : 2005-08-12
ISBN : 0262311429
Language : En, Es, Fr & De

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Book Description :

Theoretical neuroscience provides a quantitative basis for describing what nervous systems do, determining how they function, and uncovering the general principles by which they operate. This text introduces the basic mathematical and computational methods of theoretical neuroscience and presents applications in a variety of areas including vision, sensory-motor integration, development, learning, and memory. The book is divided into three parts. Part I discusses the relationship between sensory stimuli and neural responses, focusing on the representation of information by the spiking activity of neurons. Part II discusses the modeling of neurons and neural circuits on the basis of cellular and synaptic biophysics. Part III analyzes the role of plasticity in development and learning. An appendix covers the mathematical methods used, and exercises are available on the book's Web site.

Advances in Cognitive Neurodynamics III

Advances in Cognitive Neurodynamics  III  Book
Author : Yoko Yamaguchi
Publisher : Springer Science & Business Media
Release : 2013-04-08
ISBN : 9400747926
Language : En, Es, Fr & De

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Book Description :

Within our knowledge, the series of the International Conference on Cognitive Neurodynamics (ICCN) is the only conference series dedicating to cognitive neurodynamis. This volume is the proceedings of the 3rd International Conference on Cognitive Neurodynamics held in 2011, which reviews the progress in this field since the 1st ICCN - 2007. The topics include: Neural coding and realistic neural network dynamics, Neural population dynamics, Firing Oscillations and Patterns in Neuronal Networks, Brain imaging, EEG, MEG, Sensory and Motor Dynamics, Global cognitive function, Multi-scalar Neurodynamics - from Physiology to Systems Theory, Neural computing, Emerging Technologies for Brain Computer Interfaces, Neural dynamics of brain disorders.

Quantum Information Theory

Quantum Information Theory Book
Author : Masahito Hayashi
Publisher : Springer
Release : 2016-11-03
ISBN : 3662497255
Language : En, Es, Fr & De

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Book Description :

This graduate textbook provides a unified view of quantum information theory. Clearly explaining the necessary mathematical basis, it merges key topics from both information-theoretic and quantum- mechanical viewpoints and provides lucid explanations of the basic results. Thanks to this unified approach, it makes accessible such advanced topics in quantum communication as quantum teleportation, superdense coding, quantum state transmission (quantum error-correction) and quantum encryption. Since the publication of the preceding book Quantum Information: An Introduction, there have been tremendous strides in the field of quantum information. In particular, the following topics – all of which are addressed here – made seen major advances: quantum state discrimination, quantum channel capacity, bipartite and multipartite entanglement, security analysis on quantum communication, reverse Shannon theorem and uncertainty relation. With regard to the analysis of quantum security, the present book employs an improved method for the evaluation of leaked information and identifies a remarkable relation between quantum security and quantum coherence. Taken together, these two improvements allow a better analysis of quantum state transmission. In addition, various types of the newly discovered uncertainty relation are explained. Presenting a wealth of new developments, the book introduces readers to the latest advances and challenges in quantum information. To aid in understanding, each chapter is accompanied by a set of exercises and solutions.