Skip to main content
SpringerLink
Log in
Book cover

Calculus for Cognitive Scientists

Partial Differential Equation Models

  • Book
  • © 2016

Overview

Authors:
  1. James K. Peterson

    1. Dept of Mathematical Sciences, Clemson University, Clemson, USA

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • Derives the cable equation and the Hodgkin Huxley models for excitatory neurons from first principles of biophysics
  • Highlights theoretical and computational tools to enhance understanding of both the underlying science and the solutions
  • Provides insights into the theory and computation of the underlying cognitive and biological science models that are studied
  • Includes supplementary material: sn.pub/extras

Part of the book series: Cognitive Science and Technology (CSAT)

This is a preview of subscription content, log in via an institution to check access.

Access this book

Log in via an institution
eBook USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access

Licence this eBook for your library

Institutional subscriptions

Table of contents (14 chapters)

  1. Front Matter

    Pages i-xxxi
    Download chapter PDF

  2. Introductory Matter

    1. Front Matter

      Pages 1-1
      Download chapter PDF

    2. Introduction

      • James K. Peterson
      Pages 3-9

  3. Quantitative Tools

    1. Front Matter

      Pages 11-11
      Download chapter PDF

    2. Graham–Schmidt Orthogonalization

      • James K. Peterson
      Pages 13-46

    3. Numerical Differential Equations

      • James K. Peterson
      Pages 47-61

  4. Deriving the Cable Model

    1. Front Matter

      Pages 63-63
      Download chapter PDF

    2. Biological Molecules

      • James K. Peterson
      Pages 65-96

    3. Ion Movement

      • James K. Peterson
      Pages 97-147

    4. Lumped and Distributed Cell Models

      • James K. Peterson
      Pages 149-169

    5. Time Independent Solutions to Infinite Cables

      • James K. Peterson
      Pages 171-197

    6. Time Independent Solutions to Finite and Half-Infinite Space Cables

      • James K. Peterson
      Pages 199-225

  5. Excitable Neuron Models

    1. Front Matter

      Pages 227-227
      Download chapter PDF

    2. A Primer on Series Solutions

      • James K. Peterson
      Pages 229-308

    3. Linear Partial Differential Equations

      • James K. Peterson
      Pages 309-359

    4. Simplified Dendrite—Soma—Axon Information Processing

      • James K. Peterson
      Pages 361-399

    5. The Basic Hodgkin–Huxley Model

      • James K. Peterson
      Pages 401-484

  6. Summing It All Up

    1. Front Matter

      Pages 485-485
      Download chapter PDF

    2. Final Thoughts

      • James K. Peterson
      Pages 487-488

  7. Advise to the Beginner

    1. Front Matter

      Pages 489-489
      Download chapter PDF
Back to top

Keywords

About this book

This book shows cognitive scientists in training how mathematics, computer science and science can be usefully and seamlessly intertwined. It is a follow-up to the first two volumes on mathematics for cognitive scientists, and includes the mathematics and computational tools needed to understand how to compute the terms in the Fourier series expansions that solve the cable equation. The latter is derived from first principles by going back to cellular biology and the relevant biophysics.  A detailed discussion of ion movement through cellular membranes, and an explanation of how the equations that govern such ion movement leading to the standard transient cable equation are included. There are also solutions for the cable model using separation of variables, as well an explanation of why Fourier series converge and a description of the implementation of MatLab tools to compute the solutions. Finally, the standard Hodgkin - Huxley model is developed for an excitable neuron and is solved using MatLab.

Authors and Affiliations

  • Dept of Mathematical Sciences, Clemson University, Clemson, USA

    James K. Peterson

About the author

Dr. James Peterson is an Associate Professor in Mathematical Sciences and Biological Sciences at Clemson University, USA. His formal training is in mathematics but he has worked as an aerospace engineer and a software engineer also. He enjoys working on very hard problems that require multiple disciplines to make sense out of and he reads, studies and plays in cutting edge areas a lot as part of his interests.

Bibliographic Information

Publish with us

Policies and ethics

Back to top

Search

Navigation