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Computational Modeling Methods for Neuroscientists$
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Erik De Schutter

Print publication date: 2009

Print ISBN-13: 9780262013277

Published to MIT Press Scholarship Online: August 2013

DOI: 10.7551/mitpress/9780262013277.001.0001

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Differential Equations

Differential Equations

Chapter:
(p.1) 1 Differential Equations
Source:
Computational Modeling Methods for Neuroscientists
Author(s):

Bard Ermentrout

John Rinzel

Publisher:
The MIT Press
DOI:10.7551/mitpress/9780262013277.003.0002

This chapter, which addresses the partial differential equations (PDEs) with the example of finding the speed and profile of a propagating impulse for a Hodgkin-Huxley-like cable equation, highlights a few properties of differential equations and concepts for understanding them. It mentions that the notions of stability are a crucial aspect of linear autonomous differential equations, and shows that linear autonomous systems have solutions which are sums of exponentials. The chapter suggests that PDEs are important when spatial differences matter—they require both initial and boundary conditions; and certain forms of solutions to PDEs can be reduced to ordinary differential equations (ODEs)—and reviews the methods for solving ODEs using a one-dimensional model.

Keywords:   partial differential equations, cable equation, notions of stability, autonomous differential equations, boundary conditions, ordinary differential equations, exponentials

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