Calculus for Cognitive Scientists

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.

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

Auteur
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.

Contenu
Introduction.- Graham - Schmidt Orthogonalization.- Numerical Dierential Equations.- Biological Molecules.- Ion Movement.- Lumped and Distributed Cell Models.- Time Independent Solutions to Innite Cables.- Time Independent Solutions to Finite and Half-Innite Space Cables.- A Primer On Series Solutions.- Linear Partial Dierential Equations.- Simplied Dendrite - Soma Axon Information Processing.- The Basic Hodgkin - Huxley Model.- Final Thoughts.- Background Reading.