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Calculus for Cognitive Scientists: Partial Differential Equation Models

✍ Scribed by James K. Peterson (auth.)

Publisher
Springer Singapore
Year
2016
Tongue
English
Leaves
546
Series
Cognitive Science and Technology
Edition
1
Category
Library
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✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-xxxi
Front Matter....Pages 1-1
Introduction....Pages 3-9
Front Matter....Pages 11-11
Graham–Schmidt Orthogonalization....Pages 13-46
Numerical Differential Equations....Pages 47-61
Front Matter....Pages 63-63
Biological Molecules....Pages 65-96
Ion Movement....Pages 97-147
Lumped and Distributed Cell Models....Pages 149-169
Time Independent Solutions to Infinite Cables....Pages 171-197
Time Independent Solutions to Finite and Half-Infinite Space Cables....Pages 199-225
Front Matter....Pages 227-227
A Primer on Series Solutions....Pages 229-308
Linear Partial Differential Equations....Pages 309-359
Simplified Dendrite—Soma—Axon Information Processing....Pages 361-399
The Basic Hodgkin–Huxley Model....Pages 401-484
Front Matter....Pages 485-485
Final Thoughts....Pages 487-488
Front Matter....Pages 489-489
Background Reading....Pages 491-495
Back Matter....Pages 497-534

✦ Subjects

Computational Intelligence; Theoretical, Mathematical and Computational Physics; Mathematical Models of Cognitive Processes and Neural Networks; Artificial Intelligence (incl. Robotics); Computer Imaging, Vision, Pattern Recognition and G

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