Time Stepping for the cable equation, Part 1: Serial performance

Electrical propagation (action potentials) in excitable tissue, such as nerve cells and cardiac myocytes, is described by a parabolic diffusion-reaction equation for the transmembrane potential (voltage) V (x, t), known as the cable equation. It is driven by a rapidly varying, highly nonlinear (and expensive to evaluate) ionic source term Iion(V, t) representing the total ionic current across the cell membrane, plus a stimulus current. The ionic term is governed by the Hodgkin-Huxley or some other more complicated ionic model, appropriate to the tissue; its evaluation requires solving a system of ODEs for the “gates”. We compare the performance of eleven time-stepping numerical schemes on the 1D cable equation with Luo-Rudy I (1991) ionic source (the evaluation of which involves seven ODEs). The time-steppers include Euler, Super-Time-Stepping, DuFort-Frankel, as well as low and high order, explicit and implicit, non-adaptive and adaptive Runge-Kutta integrators.