Here is the introductory section of the manuscript:
Ordinary first-order differential equations come up repeatedly in neuroscience. They are used to model many fundamental processes such as passive membrane dynamics and gating kinetics in individual ion channels. When the equations come up, most electrophysiology texts provide the solution, but do not provide any explanation. This manuscript tries to fill the gap, providing an introduction to some of the mathematical facets of first-order differential equations. While the main goal of this manuscript is to examine the equations neurophysiology texts present without justification, I pushed the proofs to the end of the manuscript so that those who wish to skip them don't have to waste time working out which pages to skip. The manuscript, including the proofs, presupposes a little knowledge of first-year calculus, much of which is reviewed when needed.
The manuscript has three sections. Section One provides a brief statement of the general problem and its solution. Section Two examines the solution for a special case that comes up quite frequently in practice, and also examines a concrete example, the equivalent circuit model of a patch of neuronal membrane. Section Three contains a simple derivation of the general solution given in Section One.
There is a Matlab function that accompanies it which you can use to simulate the equivalent circuit model I go over in the document. You can get the code here: equiv_circuit_plot.m. Feel free to suggest improvements to the code.
I welcome any criticisms of content, clarity, or formatting. I hope it can serve as a useful reference to explain all those taus and infinities in Hille's treatment of the Hodgkin-Huxley equations in Chapter 2, which I'll post about next week.
Update 8/27/05: Version 0.2 of the document is up. I have retooled the document so that the connection with Hodgkin-Huxley is more transparent, and so those who want to avoid proofs can do so while still learning some differential equations (NB: skipping proofs causes warts).