Analysis of a linear model for electrical stimulation of axons--critical remarks on the "activating function concept"

A comprehensive description of a linear model of an axon of infinite length exposed to an external voltage is presented. The steady-state transmembrane potential is derived as a function proportional to the convolution product of the second spatial difference sn of the external potential (the "activating function") and the impulse response psin of a spatial low-pass filter. The impulse response psin represents the influence of the axon and is fully characterized by the axon's length constant lambda. A closed-form solution of the cable equation can be given in the spatial Fourier domain. Due to a "spectral acceleration effect", the overall transmembrane potential approximates the steady-state considerably faster than an exponential with the axon's membrane time constant tau. The effect is increasingly pronounced, the smaller the distance between the electrode and the axon. Regarding myelinated fibers and practically relevant electrode/axon distances and pulse widths, the transmembrane potential at the end of a stimulation pulse can be substantially better approximated by the steady-state condition than by the initial response as claimed by the "activating function concept." Quantitative limits for the range of validity of the activating function concept are derived.