By Christopher P. Fall, Eric S. Marland, John M. Wagner, John J. Tyson
This textbook presents an advent to dynamic modeling in molecular telephone biology, taking a computational and intuitive technique. designated illustrations, examples, and routines are incorporated in the course of the textual content. Appendices containing mathematical and computational strategies are supplied as a reference tool.
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A complete user’s manual for the public domain ODE package XPP [Ermentrout, 2002]. • Mathematical Biology, James Murray. While not especially didactic, this volume is recognized as an essential handbook describing models throughout mathematical biology [Murray, 1989]. 20 1: Dynamic Phenomena in Cells Exercises 1. Find the XPPAUT site on the World Wide Web (see Appendix B). Install XPPAUT on your computer and follow the web-based tutorial. 2. 14). 3. 19) and explore what happens to the solution when the intial condition and characteristic time τ are changed.
5 Type I and Type II Spiking Many investigators are now interested in applying biophysical models of neurons and other spike—generating mechanisms to the study of information transfer using information—theoretic measures. It is particularly important, therefore, that the dynamical behavior of a model be characterized over the space of parameters and that the characteristics of the spike generating model match the intended use. Both the Hodgkin—Huxley model, described below, and the Morris—Lecar model produce trains of action potentials with commonly used parameter sets and when suﬃciently depolarized.
This type of plot is called a phase portrait, in one dimension. Phase portraits are particularly useful for analyzing ODEs with two variables, where they are typically called phase plane diagrams. Phase plane diagrams are discussed in Appendix A. Because phase portrait diagrams will be used extensively in the remainder of the text, it would be a useful digression to review that material now. 2 Introduction to Numerical Packages While it is important to understand the limitations of whichever numerical algorithm is used for the solution of a problem, fortunately it is not necessary to face the task of implementing these algorithms on a computer from scratch.