Spherical harmonics representation of the steady-state membrane potential shift induced by tDCS in realistic neuron models

Objective We provide a systematic framework for quantifying the effect of externally applied weak electric fields on realistic neuron compartment models as captured by physiologically relevant quantities such as the membrane potential or transmembrane current as a function of the orientation of the field. Approach We define a response function as the steady-state change of the membrane potential induced by a canonical external field of 1 V/m as a function of its orientation. We estimate the function values through simulations employing reconstructions of the rat somatosensory cortex from the Blue Brain Project. The response of different cell types is simulated using the NEURON simulation environment. We represent and analyze the angular response as an expansion in spherical harmonics. Main results We report membrane perturbation values comparable to those in the literature, extend them to different cell types, and provide their profiles as spherical harmonic coefficients. We show that at rest, responses are dominated by their dipole terms (ℓ = 1), in agreement with experimental findings and compartment theory. Indeed, we show analytically that for a passive cell, only the dipole term is nonzero. However, while minor, other terms are relevant for states different from resting. In particular, we show how ℓ = 0 and ℓ = 2 terms can modify the function to induce asymmetries in the response. Significance This work provides a practical framework for the representation of the effects of weak electric fields on different neuron types and their main regions—an important milestone for developing micro- and mesoscale models and optimizing brain stimulation solutions.