Closed-form expressions for the directions of maximum modulation depth in temporal interference electrical brain stimulation

Abstract

Objective: In temporal interference (TI) transcranial electrical stimulation (tES), an emergingbrain stimulation technique, the interference of two high-frequency currents with a smallfrequency difference is used to target specific brain regions with better focality than instandard tES. While the magnitude of the modulation depth has been previously investigated,an explicit formula for the direction in which this modulation is maximized has been lacking.This work provides a novel closed-form analytical expression for the orientation of maximummodulation depth in TI tES. We also found a secondary orientation where the modulationdepth has a local maximum. Moreover, we provide closed-form analytical formulas for thisorientation as well as for the modulation depth along this orientation. To our knowledge,these closed-form expressions and the presence of the secondary maximum have not beenpreviously reported. Approach: We derive compact analytical expressions and validate themthrough comprehensive computational simulations using a realistic human head model. Wealso provide a complete analytical derivation of the widely used formula for the maximummodulation depth magnitude stated in Grossman et al, 2017. Main results: Our simulationsdemonstrate that the modulation depth predicted with our new analytical direction formula isindeed the maximum compared to other directions. The derived closed-form expressionprovides a faster and more accurate alternative to iterative numerical optimization methodsused in previous studies to estimate this direction. Furthermore, we found that due tointerference in 3D, the modulation depth along the secondary maximum orientation can be ofsimilar strength to the maximum modulation depth intensity when interfering electric fieldvectors are significantly misaligned. Finally, we show that by modifying the ratio of theinjected current strengths, it is possible to steer these directions and fine-tune the stimulationalong a desired direction of interest. Significance: Overall, this work provides a detailedtreatment of TI electric fields in 3D. The presented closed-form expressions for the directionsof maximum and secondary maximum modulation depths are relevant for the betterinterpretation of both simulated and experimental results in TI studies by allowingcomparison with neuronal orientations in the brain.