Analysis of the quasi-static approximation for calculating potentials generated by neural stimulation

In models of electrical stimulation of the nervous system, the electric potential is typically calculated using the quasi-static approximation. The quasi-static approximation allows Maxwell's equations to be simplified by ignoring capacitive, inductive and wave propagation contributions to the potential. While this simplification has been validated for bioelectric sources, its application to rapid stimulation pulses, which contain more high-frequency power, may not be appropriate. We compared the potentials calculated using the quasi-static approximation with those calculated from the exact solution to the inhomogeneous Helmholtz equation. The mean absolute errors between the two potential calculations were limited to 5-13% for pulse widths commonly used for neural stimulation (25 micros-1 ms). We also quantified the excitation properties of extracellular point source stimulation of a myelinated nerve fiber model using potentials calculated from each method. Deviations between the strength-duration curves for potentials calculated using the quasi-static (sigma = 0.105 S m(-1)) and Helmholtz approaches ranged from 3 to 16%, with the minimal error occurring for 100 micros pulses. Differences in the threshold-distance curves for the two calculations ranged from 0 to 9%, for the same value of quasi-static conductivity. A sensitivity analysis of the material parameters revealed that the potential was much more strongly dependent on the conductivity than on the permittivity. These results indicate that for commonly used stimulus pulse parameters, the exact solution for the potential can be approximated by quasi-static simplifications only for appropriate values of conductivity.