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An Introduction To The Mathematics Of Neurons
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Book Synopsis An Introduction to the Mathematics of Neurons by : Hoppensteadt
Download or read book An Introduction to the Mathematics of Neurons written by Hoppensteadt and published by Cambridge University Press. This book was released on 1986-05-31 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: Neurons, or nerve cells, are basic timers in our bodies; they also play a central role in storing and processing information in our brains. This book introduces neuron physiology and some mathematical methods that can help us to understand how neurons work. The author's aim is to uncover frequency-response properties of neurons and to show that neural networks can support stable patterns of synchronized firing. He does this using a novel electrical circuit model of a neuron, called VCON, which shares many features with the Hodgkin-Huxley model, though it is much simpler to study. This makes the book suitable for advanced undergraduate or new graduate students studying mathematical biology. Indeed the book grew from such a course taught at the University of Utah. The only prerequisites are basic calculus, differential equations and matrix algebra. Problems (some with solutions) are provided at the end of each chapter; they range from simple illustrative exercises to more challenging extensions of the text. Some projects, often involving microcomputers, are also suggested.
Book Synopsis An Introduction to the Mathematics of Neurons by : F. C. Hoppensteadt
Download or read book An Introduction to the Mathematics of Neurons written by F. C. Hoppensteadt and published by Cambridge University Press. This book was released on 1997-06-28 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book describes the signal processing aspects of neural networks. It begins with a presentation of the necessary background material in electronic circuits, mathematical modeling and analysis, signal processing, and neurosciences, and then proceeds to applications. These applications include small networks of neurons, such as those used in control of warm-up and flight in moths and control of respiration during exercise in humans. Next, a theory of mnemonic surfaces is developed and studied and material on pattern formation and cellular automata is presented. Finally, large networks are studied, such as the thalamus-reticular complex circuit, believed to be involved in focusing attention, and the development of connections in the visual cortex. Additional material is also provided about nonlinear wave propagation in networks. This book will serve as an excellent text for advanced undergraduates and graduates in the physical sciences, mathematics, engineering, medicine and life sciences.
Book Synopsis An Introduction to Modeling Neuronal Dynamics by : Christoph Börgers
Download or read book An Introduction to Modeling Neuronal Dynamics written by Christoph Börgers and published by Springer. This book was released on 2017-04-17 with total page 457 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Introduction to Neural Dynamics and Signal Transmission Delay by : Jianhong Wu
Download or read book Introduction to Neural Dynamics and Signal Transmission Delay written by Jianhong Wu and published by Walter de Gruyter. This book was released on 2011-07-20 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the design of a neural network, either for biological modeling, cognitive simulation, numerical computation or engineering applications, it is important to investigate the network's computational performance which is usually described by the long-term behaviors, called dynamics, of the model equations. The purpose of this book is to give an introduction to the mathematical modeling and analysis of networks of neurons from the viewpoint of dynamical systems.
Book Synopsis Mathematics for Neuroscientists by : Fabrizio Gabbiani
Download or read book Mathematics for Neuroscientists written by Fabrizio Gabbiani and published by Academic Press. This book was released on 2017-03-21 with total page 628 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Book Synopsis Dynamical Systems in Neuroscience by : Eugene M. Izhikevich
Download or read book Dynamical Systems in Neuroscience written by Eugene M. Izhikevich and published by MIT Press. This book was released on 2010-01-22 with total page 459 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis An Introduction to Neural Networks by : Kevin Gurney
Download or read book An Introduction to Neural Networks written by Kevin Gurney and published by CRC Press. This book was released on 2018-10-08 with total page 234 pages. Available in PDF, EPUB and Kindle. Book excerpt: Though mathematical ideas underpin the study of neural networks, the author presents the fundamentals without the full mathematical apparatus. All aspects of the field are tackled, including artificial neurons as models of their real counterparts; the geometry of network action in pattern space; gradient descent methods, including back-propagation; associative memory and Hopfield nets; and self-organization and feature maps. The traditionally difficult topic of adaptive resonance theory is clarified within a hierarchical description of its operation. The book also includes several real-world examples to provide a concrete focus. This should enhance its appeal to those involved in the design, construction and management of networks in commercial environments and who wish to improve their understanding of network simulator packages. As a comprehensive and highly accessible introduction to one of the most important topics in cognitive and computer science, this volume should interest a wide range of readers, both students and professionals, in cognitive science, psychology, computer science and electrical engineering.
Book Synopsis Mathematical Foundations of Neuroscience by : G. Bard Ermentrout
Download or read book Mathematical Foundations of Neuroscience written by G. Bard Ermentrout and published by Springer Science & Business Media. This book was released on 2010-07-01 with total page 434 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis An Introduction to Neural Network Methods for Differential Equations by : Neha Yadav
Download or read book An Introduction to Neural Network Methods for Differential Equations written by Neha Yadav and published by Springer. This book was released on 2015-02-26 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces a variety of neural network methods for solving differential equations arising in science and engineering. The emphasis is placed on a deep understanding of the neural network techniques, which has been presented in a mostly heuristic and intuitive manner. This approach will enable the reader to understand the working, efficiency and shortcomings of each neural network technique for solving differential equations. The objective of this book is to provide the reader with a sound understanding of the foundations of neural networks and a comprehensive introduction to neural network methods for solving differential equations together with recent developments in the techniques and their applications. The book comprises four major sections. Section I consists of a brief overview of differential equations and the relevant physical problems arising in science and engineering. Section II illustrates the history of neural networks starting from their beginnings in the 1940s through to the renewed interest of the 1980s. A general introduction to neural networks and learning technologies is presented in Section III. This section also includes the description of the multilayer perceptron and its learning methods. In Section IV, the different neural network methods for solving differential equations are introduced, including discussion of the most recent developments in the field. Advanced students and researchers in mathematics, computer science and various disciplines in science and engineering will find this book a valuable reference source.
Book Synopsis Mathematical Neurobiology by : John Stanley Griffith
Download or read book Mathematical Neurobiology written by John Stanley Griffith and published by . This book was released on 1971 with total page 184 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis An Introductory Course in Computational Neuroscience by : Paul Miller
Download or read book An Introductory Course in Computational Neuroscience written by Paul Miller and published by MIT Press. This book was released on 2018-10-02 with total page 405 pages. Available in PDF, EPUB and Kindle. Book excerpt: A textbook for students with limited background in mathematics and computer coding, emphasizing computer tutorials that guide readers in producing models of neural behavior. This introductory text teaches students to understand, simulate, and analyze the complex behaviors of individual neurons and brain circuits. It is built around computer tutorials that guide students in producing models of neural behavior, with the associated Matlab code freely available online. From these models students learn how individual neurons function and how, when connected, neurons cooperate in a circuit. The book demonstrates through simulated models how oscillations, multistability, post-stimulus rebounds, and chaos can arise within either single neurons or circuits, and it explores their roles in the brain. The book first presents essential background in neuroscience, physics, mathematics, and Matlab, with explanations illustrated by many example problems. Subsequent chapters cover the neuron and spike production; single spike trains and the underlying cognitive processes; conductance-based models; the simulation of synaptic connections; firing-rate models of large-scale circuit operation; dynamical systems and their components; synaptic plasticity; and techniques for analysis of neuron population datasets, including principal components analysis, hidden Markov modeling, and Bayesian decoding. Accessible to undergraduates in life sciences with limited background in mathematics and computer coding, the book can be used in a “flipped” or “inverted” teaching approach, with class time devoted to hands-on work on the computer tutorials. It can also be a resource for graduate students in the life sciences who wish to gain computing skills and a deeper knowledge of neural function and neural circuits.
Download or read book Neuroscience written by Alwyn Scott and published by Springer Science & Business Media. This book was released on 2007-12-14 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Tutorials in Mathematical Biosciences I by : Alla Borisyuk
Download or read book Tutorials in Mathematical Biosciences I written by Alla Borisyuk and published by Springer. This book was released on 2005-01-28 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Mathematical Neuroscience by : Stanislaw Brzychczy
Download or read book Mathematical Neuroscience written by Stanislaw Brzychczy and published by Academic Press. This book was released on 2013-08-16 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Book Synopsis Waves in Neural Media by : Paul C. Bressloff
Download or read book Waves in Neural Media written by Paul C. Bressloff and published by Springer Science & Business Media. This book was released on 2013-10-17 with total page 436 pages. Available in PDF, EPUB and Kindle. Book excerpt: Waves in Neural Media: From Single Neurons to Neural Fields surveys mathematical models of traveling waves in the brain, ranging from intracellular waves in single neurons to waves of activity in large-scale brain networks. The work provides a pedagogical account of analytical methods for finding traveling wave solutions of the variety of nonlinear differential equations that arise in such models. These include regular and singular perturbation methods, weakly nonlinear analysis, Evans functions and wave stability, homogenization theory and averaging, and stochastic processes. Also covered in the text are exact methods of solution where applicable. Historically speaking, the propagation of action potentials has inspired new mathematics, particularly with regard to the PDE theory of waves in excitable media. More recently, continuum neural field models of large-scale brain networks have generated a new set of interesting mathematical questions with regard to the solution of nonlocal integro-differential equations. Advanced graduates, postdoctoral researchers and faculty working in mathematical biology, theoretical neuroscience, or applied nonlinear dynamics will find this book to be a valuable resource. The main prerequisites are an introductory graduate course on ordinary differential equations or partial differential equations, making this an accessible and unique contribution to the field of mathematical biology.
Book Synopsis Mathematics for Neuroscientists by : Fabrizio Gabbiani
Download or read book Mathematics for Neuroscientists written by Fabrizio Gabbiani and published by Academic Press. This book was released on 2010-09-16 with total page 505 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Book Synopsis An Introduction to Neural Information Processing by : Peiji Liang
Download or read book An Introduction to Neural Information Processing written by Peiji Liang and published by Springer. This book was released on 2015-12-22 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an overview of neural information processing research, which is one of the most important branches of neuroscience today. Neural information processing is an interdisciplinary subject, and the merging interaction between neuroscience and mathematics, physics, as well as information science plays a key role in the development of this field. This book begins with the anatomy of the central nervous system, followed by an introduction to various information processing models at different levels. The authors all have extensive experience in mathematics, physics and biomedical engineering, and have worked in this multidisciplinary area for a number of years. They present classical examples of how the pioneers in this field used theoretical analysis, mathematical modeling and computer simulation to solve neurobiological problems, and share their experiences and lessons learned. The book is intended for researchers and students with a mathematics, physics or informatics background who are interested in brain research and keen to understand the necessary neurobiology and how they can use their specialties to address neurobiological problems. It is also provides inspiration for neuroscience students who are interested in learning how to use mathematics, physics or informatics approaches to solve problems in their field.
