​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.

excitable media
mathematical neuroscience
neural fields
reaction-diffusion equations
traveling waves
ordinary differential equations

• Front Matter
• Neurons
  Front Matter
• Single Neuron Modeling
• Traveling Waves in One-Dimensional Excitable Media
• Wave Propagation Along Spiny Dendrites
• Calcium Waves and Sparks
• Networks
  Front Matter
• Waves in Synaptically Coupled Spiking Networks
• Population Models and Neural Fields
• Waves in Excitable Neural Fields
• Neural Field Model of Binocular Rivalry Waves
• Development and Disease
  Front Matter
• Waves in the Developing and the Diseased Brain
• Back Matter

Department of Mathematics, University of Utah, Salt Lake City, USA
Paul C. Bressloff

Book Title
Waves in Neural Media

Authors
Paul C. Bressloff

Series Title
Lecture Notes on Mathematical Modelling in the Life Sciences

DOI
https://doi.org/10.1007/978-1-4614-8866-8

Softcover ISBN
978-1-4614-8865-1
Published: 17 October 2013

eBook ISBN
978-1-4614-8866-8
Published: 17 October 2013

Series ISSN
2193-4789

Series E-ISSN
2193-4797

Edition Number
1

Number of Pages
XIX, 436

Number of Illustrations
133 b/w illustrations, 18 illustrations in colour

Topics
Mathematical Models of Cognitive Processes and Neural Networks, Physiological, Cellular and Medical Topics, Neurosciences, Mathematical and Computational Biology, Probability Theory and Stochastic Processes, Ordinary Differential Equations