Calculus for Cognitive Scientists

Partial Differential Equation Models
1. Auflage 2016
ISBN: 978-981-287-878-6
Verlag: Springer Nature Singapore

Buch, Englisch, 534 Seiten, HC runder Rücken kaschiert, Format (B × H): 160 mm x 241 mm, Gewicht: 1010 g

Reihe: Cognitive Science and Technology

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.

Zielgruppe

Research

Autoren/Hrsg.

Peterson, James

Fachgebiete

• Interdisziplinäres Wissenschaften Wissenschaften Interdisziplinär Neurowissenschaften, Kognitionswissenschaft
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Naturwissenschaften Biowissenschaften Biowissenschaften Neurobiologie, Verhaltensbiologie

Inhaltsverzeichnis

Introduction.- Graham-Schmidt Orthogonalization (GSO).- Numerical ODEs.- Biological molecules.- Ion movement.- Lumped models.- Infinite cables no time.- Finite/Half infinite no time.- Series.- Linear PDE.- Neural processing.- Hodgkin-Huxley model.- Conclusions.

Über Autor(innen)

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.