On numerical integration of the Hodgkin and Huxley equations for a membrane action potential

Several methods of numerical integration of the Hodgkin and Huxley equations (6·3°C) for a membrane action potential have been compared for speed and accuracy of computations. Conversational languages make it possible to write compact programs for these equations which will run on minicomputers with only 4000 words of core memory.The shape of the computed action potential was found to be rather method-invariant but the latency (for a given stimulus) was found to depend on the method of integration and the size of the time increment. For stimuli near threshold, the computed response was found to be especially sensitive to these two variables. A new threshold value independent of integration method, has been determined to be ? 6·50790 mV by linear extrapolations of threshold-time step relations to a zero step size.The simple Euler method was found to be fast but gave large latency errors compared to the true solution. The Runge-Kutta (RK) method (in the sequential mode, defined in the section on numerical integration methods) was slower but gave smaller latency errors. The Adams corrector-predictor method was still slower, gave less stable solutions, and had large latency errors. A hybrid program which combined a RK numerical integration for the membrane potential with analytic expressions for the variables m, n, and h gave errors indistinguishable from the RK method, but ran much faster. Modified Euler methods run in the parallel mode gave very accurate solutions with moderate computation times. Solutions run with the RK method in the parallel mode took about twice as long but gave no appreciable improvement in accuracy.We were able to develop a simple method for correcting the Euler integration, resulting in quite accurate solutions which were essentially independent of step size and stimuli. Because this method is not only accurate but also runs about as fast as the simple Euler, it appears to be the method of choice for most purposes.