Huxley's process of numerical integration

Huxley is such an accomplished scientist that he has made innovations in a variety of fields. Here is his contribution of a method to produce an accurate simulation with the minimum effort.

The calculation of an action potential requires determining the value of the membrane voltage at discrete time steps. Here are the steps of the process:

First, one needs to know at a particular time the values of all of the factors involved :
- all of the state variables v, m, n, h
- each of the six rate constants that determine the kinetics of the variables m, n, and h
Then one can estimate, from these data, the values of each of the variables at a short time step ahead.
With these new values, one then repeats step 2 with each advance in time.

The accuracy of the calculation depends on advancing the integration with such small advances in time that the changes in m, n, h, and v are consistent with their derivatives. The shorter the time steps, the more accurate the simulation but the longer it takes to achieve it.

Huxley's error-correction method (page 523 of their fourth 1952 paper, reproduced here) allowed him to take longer steps and thus speed the process. In his method, instead of proceeding relentlessly from step to step, he paused at each step to estimate the computational accuracy of that step. He then adjusted the size of the next step to meet the accuracy required.

Surprisingly, these additional error calculations allow the integration to proceed much more rapidly than by simply keeping the step size constant but tiny.