Towards a more accurate quasi-static approximation of the electric potential for neurostimulation with kilohertz-frequency sources

Abstract

Objective: Our goal was to determine the conditions for which a more precise calculation of the electric potential than the quasi-static approximation may be needed in models of electrical neurostimulation, particularly for signals with kilohertz-frequency components. Approach: We conducted a comprehensive quantitative study of the differences in nerve fiber activation and conduction block when using the quasi-static and Helmholtz approximations for the electric potential in a model of electrical neurostimulation. Main results: We first show that the potentials generated by sources of unbalanced pulses exhibit different transients as compared to those of energy-balanced pulses, and this is disregarded by the quasi-static assumption. Secondly, the relative errors for current-distance curves were below 3%, while for strength-duration curves the variations ranged between 1-17%, but could be improved to less than 3% across the range of pulse duration by providing a corrected quasi-static conductivity. Third, we extended our analysis to trains of pulses and reported a "congruence area" below 700 Hz, where the fidelity of fiber responses is maximal for supra-threshold stimulation. Further examination of waveforms and polarities revealed similar fidelities in the congruence area, but significant differences were observed beyond this area. However, the spike-train distance revealed differences in activation patterns when comparing the response generated by each model. Finally, in simulations of conduction-block, we found that block thresholds exhibited errors above 20% for repetition rates above 10 kHz. Yet, employing a corrected value of the conductivity improved the agreement between models, with errors no greater than 8%. Significance: Our results emphasize that the quasi-static approximation cannot be naively extended to electrical stimulation with high-frequency components, and notable differences can be observed in activation patterns. As well, we introduce a methodology to obtain more precise model responses using the quasi-static approach, which can be a valuable resource in computational neuroengineering.