Mathematics for neuroscientists / Fabrizio Gabbiani, Steven J. Cox.

This book provides 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. The book alternates between mathematical chapters, introducing important concepts and numerical methods, and neurobiological chapters, applying these concepts and methods to specific topics. It covers topics ranging from classical cellular biophysics and proceeding up to systems level neuroscience. Starting at an introductory mathematical level, presuming no more than calculus through elementary differential equations, the level will build up as increasingly complex techniques are introduced and combined with earlier ones. Each chapter includes a comprehensive series of exercises with solutions, taken from the set developed by the authors in their course lectures. MATLAB code is included for each computational figure, to allow the reader to reproduce them. Biographical notes referring the reader to more specialized literature and additional mathematical material that may be needed either to deepen the reader's understanding or to introduce basic concepts for less mathematically inclined readers completes each chapter. 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 Provides implementation examples in MATLAB code, also included for download on the accompanying support website (which will be updated with additional code and in line with major MATLAB releases) Allows the mathematical novice to analyze their results in more sophisticated ways, and consider them in a broader theoretical framework.

Includes bibliographical references (pages 473-482) and index.

Passive isopotential cell -- Differential equations -- Active isopotential cell -- Quasi-active isopotential cell -- Passive cable -- Fourier series and transforms -- Passive dendritic tree -- Active dendritic tree -- Reduced single neuron models -- Probability and random variables -- Synaptic transmission and quantal release -- Neuronal calcium signaling -- Singular value decomposition and applications -- Quantification of spike train variability -- Stochastic processes -- Membrane noise -- Power and cross spectra -- Natural light signals and phototransduction -- Firing rate codes and early vision -- Models of simple and complex cells -- Stochastic estimation theory -- Reverse-correlation and spike train decoding -- Signal detection theory -- Relating neuronal responses and psychophysics -- Population codes -- Neuronal networks -- Solutions to selected exercises.

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