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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.
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.
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...
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, ...
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.
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.
Understanding Vision
Vision science has grown hugely in the past decades, but there have been few books showing readers how to adopt a computional approach to understanding visual perception, along with the underlying mechanisms in the brain. This book explains the computational principles and models of biological visual processing, and in particular, primate vision.
The NEURON Book
The authoritative reference on NEURON, the simulation environment for modeling biological neurons and neural networks that enjoys wide use in the experimental and computational neuroscience communities. This book shows how to use NEURON to construct and apply empirically based models. Written primarily for neuroscience investigators, teachers, and students, it assumes no previous knowledge of computer programming or numerical methods. Readers with a background in the physical sciences or mathematics, who have some knowledge about brain cells and circuits and are interested in computational modeling, will also find it helpful. The NEURON Book covers material that ranges from the inner workings of this program, to practical considerations involved in specifying the anatomical and biophysical properties that are to be represented in models. It uses a problem-solving approach, with many working examples that readers can try for themselves.
Tutorials in Mathematical Biosciences I
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.
