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

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 charge-balanced pulses, and this is disregarded by the quasi-static assumption. Secondly, relative errors for current-distance curves were below 3%, while for strength-duration curves these 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, retaining its simplicity, which can be a valuable resource in computational neuroengineering.

A partially averaged system to model neuron responses to interferential current stimulation.

The interferential current (IFC) therapy is a noninvasive electrical neurostimulation technique intended to activate deep neurons using surface electrodes. In IFC, two independent kilohertz-frequency currents purportedly intersect where an interference field is generated. However, the effects of IFC on neurons within and outside the interference field are not completely understood, and it is unclear whether this technique can reliable activate deep target neurons without side effects. In recent years, realistic computational models of IFC have been introduced to quantify the effects of IFC on brain cells, but they are often complex and computationally costly. Here, we introduce a simplified model of IFC based on the FitzHugh-Nagumo (FHN) model of a neuron. By considering a modified averaging method, we obtain a non-autonomous approximated system, with explicit representation of relevant IFC parameters. For this approximated system we determine conditions under which it reliably approximates the complete FHN system under IFC stimulation, and we mathematically prove its ability to predict nonspiking states. In addition, we perform numerical simulations that show that the interference effect is observed only for a narrow set of IFC parameters and, in particular, for a beat frequency no higher than about 100 [Hz]. Our novel model tailored to the IFC technique contributes to the understanding of neurostimulation modalities using this type of signals, and can have implications in the design of noninvasive electrical stimulation therapies.