Waves in Neural Media: From Single Neurons to Neural Fields

ISBN-13: 9781461488651 / Angielski / Miękka / 2013 / 436 str.

Paul C. Bressloff

Opis

​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.

Recenzje

"This is a summary of the mathematical modeling of the dendrites, single and multi- neuronal units. ... I think this book will appeal to neuromathematicians, researchers, graduate students, fellows, and theoretical neuroscientists." (Joseph J. Grenier, Amazon.com, November, 2015)

Spis treści

Preface.- Part I Neurons.- Single Neuron Modeling.- Traveling Waves in One-Dimensional Excitable Media.- Wave Propagation Along Spiny Dendrites.- Calcium Waves and Sparks.- Part II Networks.- Waves in Synaptically-Coupled Spiking Networks.- Population Models and Neural Fields.- Waves in Excitable Neural Fields.- Neural Field Model of Binocular Rivalry Waves.- Part III Development and Disease.- Waves in the Developing and the Diseased Brain.- Index.

Blurb

Waves in Neural Media: From Single Cells 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 and partial differential equations, making this an accessible and unique contribution to the field of mathematical biology.