Waves in Neural Media
From Single Neurons to Neural Fields
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Beschreibung
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.
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.
Covers an extensive array of topics ranging from sub-cellular to tissue level biology Can serve as an entry point for biologists wishing to learn about mathematical modeling Provides an excellent introduction to mathematical neuroscience from the perspective of dymanical systems
Autor*in
Paul C. Bressloff
Themen in »Waves in Neural Media«
excitable media mathematical neuroscience neural fields reaction-diffusion equations traveling waves ordinary differential equations
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“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)
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