Predicting neural responses to intra- and extracranial electric brain stimulation by means of the reciprocity theorem
Torbjørn V. Ness*¹, Christof Koch²,Gaute T. Einevoll¹,³
¹ Department of Physics, Norwegian University of Life Sciences, Ås, Norway
² Allen Institute, Seattle, WA, USA
³ Department of Physics, University of Oslo, Oslo, Norway
*Email: gaute.einevoll@nmbu.no
Introduction
Neural activity can be modulated through electric stimulation (ES), which is extensively used in both science and the clinic, including deep brain stimulation and temporal interference stimulation. While ES is grounded in well-established biophysics, it has proven difficult to gain a solid understanding of ES and its sensitivity to features like location, orientation, different cell types, and the ES frequency-content. This represents a major obstacle to the applications of ES.
Here, we show that the reciprocity theorem (RT) can be applied more broadly than previously recognized [1], offering a whole new perspective on ES which reproduces known features, explains surprising observations, and makes new predictions.
Methods
The effect of ES on different biophysically detailed cell models is simulated with NEURON [2] and LFPy [3]. The ES is treated as a current point source which sets up an extracellular potential that is used as a boundary condition at each cellular compartment. The somatic membrane potential-responseVmis calculated. In the RT-based approach the current is inserted intracellularly in the soma, and the resulting extracellular potentialVecalculated. According to the RT the two approaches should give identical results for passive cell models (Vm=Ve, Fig. 1). For transcranial electric stimulation (tES), we used a detailed head model to estimate membrane potential responses to tES deep in the brain.
Results
In all tested cases the RT-based approach to simulating ES introduces zero error for passive cell models, and below a few percent error for subthreshold active cell models [1].
By leveraging the RT, we show that the effect of ES has a 1/rdecay for nearby neurons and 1/r² for distant neurons. Furthermore, for nearby neurons the ES response is approximately cell-type and frequency-independent, while for distant neurons (e.g., tES), pyramidal neurons are most strongly targeted at low frequencies, and interneurons at high frequencies, but with a less synchronous effect [1]. Finally, tES at conventional safety limits (<4 mA) induces subthreshold potential changes of ~40-200 µV, far below the threshold for direct neuronal firing [1].
Discussion
By applying RT, we provide a framework for understanding neural responses to ES, by leveraging our good understanding of extracellular potentials [4]. Our results indicate that conventional tES primarily affects neural activity via subtle subthreshold effects, suggesting indirect network-level mechanisms such as synchronization or stochastic resonance. The weak frequency dependence of subthreshold responses explains recent experimental findings [5], reinforcing RT as a powerful tool for modeling ES. Future work should incorporate network-level dynamics to assess the broader implications of these findings for neuromodulation and brain stimulation therapies.
Figure 1. Reciprocity theorem in the context of electrical brain stimulation: The somatic membrane potential response to an extracellular current injection at position r (panel A) corresponds to the extracellular potential at location r from the same current source injected into the soma .
Acknowledgements
T.V.N. and G.T.E. received funding from the European Union Horizon 2020 Research and Innovation Programme under Grant Agreement No. 101147319 [EBRAINS 2.0]. C.K. thanks the Allen Institute founder, Paul G. Allen, for his vision, encouragement, and support.
References
[1] Ness et al. (2025) bioRxivhttps://doi.org/10.1101/2024.08.04.603691
[2] The NEURON bookhttps://doi.org/10.1017/CBO9780511541612
[3] Hagen et al. (2018)https://doi.org/10.3389/fninf.2018.00092
[4] Halnes et al. (2024)https://doi.org/10.1017/9781009039826
[5] Lee et al. (2024) Neuronhttps://doi.org/10.1016/j.neuron.2024.05.009
