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Mathematical Foundations of Neuroscience
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 p...
Simulating, Analyzing, and Animating Dynamical Systems
Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students provides sophisticated numerical methods for the fast and accurate solution of a variety of equations, including ordinary differential equations, delay equations, integral equations, functional equations, and some partial differential equations, as well as boundary value problems. It introduces many modeling techniques and methods for analyzing the resulting equations. Instructors, students, and researchers will all benefit from this book, which demonstrates how to use software tools to simulate and study sets of equations that arise in a variety of applications. Instructors will learn how to use computer software in their differential equations and modeling classes, while students will learn how to create animations of their equations that can be displayed on the World Wide Web. Researchers will be introduced to useful tricks that will allow them to take full advantage of XPPAUT's capabilities.
Neural Fields
Neural field theory has a long-standing tradition in the mathematical and computational neurosciences. Beginning almost 50 years ago with seminal work by Griffiths and culminating in the 1970ties with the models of Wilson and Cowan, Nunez and Amari, this important research area experienced a renaissance during the 1990ties by the groups of Ermentrout, Robinson, Bressloff, Wright and Haken. Since then, much progress has been made in both, the development of mathematical and numerical techniques and in physiological refinement und understanding. In contrast to large-scale neural network models described by huge connectivity matrices that are computationally expensive in numerical simulations, ...
Dynamical Systems in Neuroscience
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 relati...
Computational Models of Brain and Behavior
A comprehensive Introduction to the world of brain and behavior computational models This book provides a broad collection of articles covering different aspects of computational modeling efforts in psychology and neuroscience. Specifically, it discusses models that span different brain regions (hippocampus, amygdala, basal ganglia, visual cortex), different species (humans, rats, fruit flies), and different modeling methods (neural network, Bayesian, reinforcement learning, data fitting, and Hodgkin-Huxley models, among others). Computational Models of Brain and Behavior is divided into four sections: (a) Models of brain disorders; (b) Neural models of behavioral processes; (c) Models of ne...
Methods in Neuronal Modeling
Kinetic Models of Synaptic Transmission / Alain Destexhe, Zachary F. Mainen, Terrence J. Sejnowski / - Cable Theory for Dendritic Neurons / Wilfrid Rall, Hagai Agmon-Snir / - Compartmental Models of Complex Neurons / Idan Segev, Robert E. Burke / - Multiple Channels and Calcium Dynamics / Walter M. Yamada, Christof Koch, Paul R. Adams / - Modeling Active Dendritic Processes in Pyramidal Neurons / Zachary F. Mainen, Terrence J. Sejnowski / - Calcium Dynamics in Large Neuronal Models / Erik De Schutter, Paul Smolen / - Analysis of Neural Excitability and Oscillations / John Rinzel, Bard Ermentrout / - Design and Fabrication of Analog VLSI Neurons / Rodney Douglas, Misha Mahowald / - Principles of Spike Train Analysis / Fabrizio Gabbiani, Christof Koch / - Modeling Small Networks / Larry Abbott, Eve Marder / - Spatial and Temporal Processing in Central Auditory Networks / Shihab Shamma / - Simulating Large Networks of Neurons / Alexander D. Protopapas, Michael Vanier, James M. Bower / ...
Phase Response Curves in Neuroscience
This book will track advances in the application of phase response (PR) analysis to the study of electrically excitable cells, focusing on applications of PR analysis in the computational neurosciences. This proposal was motivated by discussions with colleagues at the 2007 meeting of the Organization for Computational Neuroscience (OCNS) and further motivated by the success of a workshop at the 2008 OCNS meeting this past July. At that meeting the editors hosted a workshop entitled A dialogue for theoreticians and experimentalists: What is phase response analysis, and what can it tell us about neurons and networks? Invited speakers used mathematical, modeling, and experimental results to illustrate how phase response analysis has been used to reveal or describe neuronal and neuronal population dynamics. This was the most well-attended workshop of the meeting and was standing room only.
Bursting
Neurons in the brain communicate with each other by transmitting sequences of electrical spikes or action potentials. One of the major challenges in neuroscience is to understand the basic physiological mechanisms underlying the complex spatiotemporal patterns of spiking activity observed during normal brain functioning, and to determine the origins of pathological dynamical states, such as epileptic seizures and Parkinsonian tremors. A second major challenge is to understand how the patterns of spiking activity provide a substrate for the encoding and transmission of information, that is, how do neurons compute with spikes? It is likely that an important element of both the dynamical and computational properties of neurons is that they can exhibit bursting, which is a relatively slow rhythmic alternation between an active phase of rapid spiking and a quiescent phase without spiking. This book provides a detailed overview of the current state-of-the-art in the mathematical and computational modelling of bursting, with contributions from many of the leading researchers in the field.
Computational Systems Neurobiology
Computational neurosciences and systems biology are among the main domains of life science research where mathematical modeling made a difference. This book introduces the many different types of computational studies one can develop to study neuronal systems. It is aimed at undergraduate students starting their research in computational neurobiology or more senior researchers who would like, or need, to move towards computational approaches. Based on their specific project, the readers would then move to one of the more specialized excellent textbooks available in the field. The first part of the book deals with molecular systems biology. Functional genomics is introduced through examples o...
The Handbook of Mathematical Cognition
How does the brain represent number and make mathematical calculations? What underlies the development of numerical and mathematical abilities? What factors affect the learning of numerical concepts and skills? What are the biological bases of number knowledge? Do humans and other animals share similar numerical representations and processes? What underlies numerical and mathematical disabilities and disorders, and what is the prognosis for rehabilitation? These questions are the domain of mathematical cognition, the field of research concerned with the cognitive and neurological processes that underlie numerical and mathematical abilities. TheHandbook of Mathematical Cognition is a collection of 27 essays by leading researchers that provides a comprehensive review of this important research field.
