James K. Peterson

Calculus for Cognitive Scientists

Partial Differential Equation Models

• Gebundenes Buch

Produktbeschreibung

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.

Produktdetails

• Cognitive Science and Technology 0
• Verlag: Springer / Springer Nature Singapore / Springer, Berlin
• Artikelnr. des Verlages: 978-981-287-878-6
• 1st ed. 2016
• Seitenzahl: 568
• Erscheinungstermin: 19. Februar 2016
• Englisch
• Abmessung: 241mm x 160mm x 36mm
• Gewicht: 953g
• ISBN-13: 9789812878786
• ISBN-10: 9812878785
• Artikelnr.: 43445933

Autorenporträt

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.

Inhaltsangabe

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

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Introductory Remarks.- Linear Algebra.- Numerical Methods Order One ODEs.- Multivariable Calculus.- Integration.- Complex Numbers.- Linear Second Order ODEs.- Systems.- Numerical Methods Systems of ODEs.- Predator Prey Models.- Predator Prey Models With Self Interaction.- Disease Models.- A Cancer Model.- Nonlinear Differential Equations.- An Insulin Model.- Series Solutions.- Final Thoughts.- Background Reading.

Rezensionen

"The introductory remarks in this book address the concern that many presentations of mathematics focus so heavily on traditional content that the usefulness of mathematics in other fields can be overlooked. ... the approach is both demanding and thorough, but the brief exercise sets remain focused on providing opportunities for straightforward practice of the principles at hand. ... Summing Up: Recommended. Lower- and upper-division undergraduates." (N. W. Schillow, Choice, Vol. 54 (4), December, 2016)