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.

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

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.

The second part of this two-volume set contains advanced aspects of the quantitative theory of the dynamics of neurons. It begins with an introduction to the effects of reversal potentials on response to synaptic input. It then develops the theory of action potential generation based on the seminal Hodgkin-Huxley equations and gives methods for their solution in the space-clamped and nonspaceclamped cases. The remainder of the book discusses stochastic models of neural activity and ends with a statistical analysis of neuronal data with emphasis on spike trains. The mathematics is more complex in this volume than in the first volume and involves numerical methods of solution of partial differential equations and the statistical analysis of point processes.

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.

Learn to use computational modelling techniques to understand the nervous system at all levels, from ion channels to networks.

Mathematical Biology is a richly illustrated textbook in an exciting and fast growing field. Providing an in-depth look at the practical use of math modeling, it features exercises throughout that are drawn from a variety of bioscientific disciplines - population biology, developmental biology, physiology, epidemiology, and evolution, among others. It maintains a consistent level throughout so that graduate students can use it to gain a foothold into this dynamic research area.

A guide to computational modeling methods in neuroscience, covering a range of modeling scales from molecular reactions to large neural networks. This book offers an introduction to current methods in computational modeling in neuroscience. The book describes realistic modeling methods at levels of complexity ranging from molecular interactions to large neural networks. A “how to” book rather than an analytical account, it focuses on the presentation of methodological approaches, including the selection of the appropriate method and its potential pitfalls. It is intended for experimental neuroscientists and graduate students who have little formal training in mathematical methods, but it will also be useful for scientists with theoretical backgrounds who want to start using data-driven modeling methods. The mathematics needed are kept to an introductory level; the first chapter explains the mathematical methods the reader needs to master to understand the rest of the book. The chapters are written by scientists who have successfully integrated data-driven modeling with experimental work, so all of the material is accessible to experimentalists. The chapters offer comprehensive coverage with little overlap and extensive cross-references, moving from basic building blocks to more complex applications. Contributors Pablo Achard, Haroon Anwar, Upinder S. Bhalla, Michiel Berends, Nicolas Brunel, Ronald L. Calabrese, Brenda Claiborne, Hugo Cornelis, Erik De Schutter, Alain Destexhe, Bard Ermentrout, Kristen Harris, Sean Hill, John R. Huguenard, William R. Holmes, Gwen Jacobs, Gwendal LeMasson, Henry Markram, Reinoud Maex, Astrid A. Prinz, Imad Riachi, John Rinzel, Arnd Roth, Felix Schürmann, Werner Van Geit, Mark C. W. van Rossum, Stefan Wils

Computational Neuroscience - A First Course provides an essential introduction to computational neuroscience and equips readers with a fundamental understanding of modeling the nervous system at the membrane, cellular, and network level. The book, which grew out of a lecture series held regularly for more than ten years to graduate students in neuroscience with backgrounds in biology, psychology and medicine, takes its readers on a journey through three fundamental domains of computational neuroscience: membrane biophysics, systems theory and artificial neural networks. The required mathematical concepts are kept as intuitive and simple as possible throughout the book, making it fully accessible to readers who are less familiar with mathematics. Overall, Computational Neuroscience - A First Course represents an essential reference guide for all neuroscientists who use computational methods in their daily work, as well as for any theoretical scientist approaching the field of computational neuroscience.