Introduction

This section is largely stolen from Artie Sherman's notes from Woods Hole.

The general initial value problem we want to solve is

$$\frac{dy}{dt} = f(y,t)$$ (1)

with initial condition y(0) = y₀. This is a first-order differential equation. y and f(y,t) can be vectors when we have a first-order system. For example, the Hodgkin-Huxley equations have y = (V,m,h,n). The t dependence may reflect experimental manipulations, such as turning an applied current on and off, or other external influences, such as an imposed synaptic conductance change from another cell. We will suppress the t dependence for simplicity in many cases below.

First order systems are natural in neurobiology. If confronted with a higher order system, convert it to first order, since most solution packages assume this form. For example, the second order equation  

$$z^{\prime\prime} + 101 z^{\prime} + 100 z = 0$$ (2)

can be converted by the transformation $x = z, y = z^{\prime}$ to

$$\frac{d}{dt} \left( \begin{array} {c} x\ y\end{array}\right... ...{array}\right) \left( \begin{array} {c} x\ y\end{array}\right)$$ (3)

We will make use of this equation in the discussion of stiffness below.