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

Objective.In temporal interference (TI) transcranial electrical stimulation (tES), an emerging brain stimulation technique, the interference of two high-frequency currents with a small frequency difference is used to target specific brain regions with better focality than in standard 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 maximum modulation depth in TI tES. We also found a secondary orientation where the modulation depth has a local maximum. Moreover, we provide closed-form analytical formulas for this orientation 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 been previously reported.Approach.We derive compact analytical expressions and validate them through comprehensive computational simulations using a realistic human head model. We also provide a complete analytical derivation of the widely used formula for the maximum modulation depth magnitude stated in Grossman et al, 2017.Main results.Our simulations demonstrate that the modulation depth predicted with our new analytical direction formula is indeed the maximum compared to other directions. The derived closed-form expression provides a faster and more accurate alternative to iterative numerical optimization methods used in previous studies to estimate this direction. Furthermore, we found that due to interference in 3D, the modulation depth along the secondary maximum orientation can be of similar strength to the maximum modulation depth intensity when interfering electric field vectors are significantly misaligned. Finally, we show that by modifying the ratio of the injected current strengths, it is possible to steer these directions and fine-tune the stimulation along a desired direction of interest.Significance.Overall, this work provides a detailed treatment of TI electric fields in 3D. The presented closed-form expressions for the directions of maximum and secondary maximum modulation depths are relevant for the better interpretation of both simulated and experimental results in TI studies by allowing comparison with neuronal orientations in the brain.