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Waves in Neural Media

From Single Neurons to Neural Fields

  • Paul C.¬†Bressloff

Table of contents

  1. Front Matter
    Pages i-xix
  2. Neurons

    1. Front Matter
      Pages 1-1
    2. Paul C. Bressloff
      Pages 3-62
    3. Paul C. Bressloff
      Pages 101-136
    4. Paul C. Bressloff
      Pages 137-181
  3. Networks

    1. Front Matter
      Pages 183-183
    2. Paul C. Bressloff
      Pages 185-231
    3. Paul C. Bressloff
      Pages 233-269
    4. Paul C. Bressloff
      Pages 271-318
    5. Paul C. Bressloff
      Pages 319-345
  4. Development and Disease

    1. Front Matter
      Pages 347-347
    2. Paul C. Bressloff
      Pages 349-404
  5. Back Matter
    Pages 405-436

About this book

Introduction

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. 

Keywords

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

Authors and affiliations

  • Paul C.¬†Bressloff
    • 1
  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4614-8866-8
  • Copyright Information Springer Science+Business Media New York 2014
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4614-8865-1
  • Online ISBN 978-1-4614-8866-8
  • Series Print ISSN 2193-4789
  • Series Online ISSN 2193-4797
  • Buy this book on publisher's site