Practical modelling of monopolar axonal stimulation

Abstract

Three models describing monopolar electrical stimulation of unmyelinated axons in large, uniformly conducting volumes are presented. The first is a representation of the axon as a continuous and straight electric cable of finite length with sealed ends. It has a power-series solution for the steady state which after truncation beyond the fifth term provides an accurate reflection of the imposed membrane potential for all parts of the axon except, under certain conditions, near terminals and directly opposite a closely apposed electrode. At these points, the inclusion of higher-order terms improves the accuracy quite slowly — II terms are sometimes still unsatisfactory. The second is a steady-state solution, in terms of higher transcendental functions, for the point opposite the stimulating electrode in a similar but infinitely long cable. This turns out to be practical for estimating the membrane potential at the site of cathodal excitation under the majority of realistic geometrical and electrical parameters, and consequently complements the first method. It may be calculated with ease from mathematical tables. The third, a simulation of the cable by means of discrete electrical components (compartments), can give the complete distribution of membrane potential as a function of time with potentially unlimited accuracy. However, it takes special computer programs to set up and solve either the steady-state determinant or, if transients are desired, the time-dependent differential equation. Calculation of the voltage distribution with one of the analytical methods will often be faster for a reasonable level of accuracy, since their algorithms are independent of the total number of points introduced.