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

Abstract

ObjectiveWe provide a systematic framework for the quantification of 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.ApproachWe 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 resultsWe 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 in resting condition, 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, for states other than resting, other terms, while small, appear relevant. In particular, we show how ℓ = 0 and ℓ = 2 terms can modify the function to induce asymmetries in the response.SignificanceThis work provides a practical framework for the realistic representation of the effects of weak electric fields on different neuron types and their main regions—an important milestone for the development of micro- and mesoscale models and the optimization of brain stimulation solutions.

A personalizable autonomous neural mass model of epileptic seizures

“…The term λ n • E(t) represents the membrane perturbation induced by an external electric field. The coupling term λ n used-which represents the membrane perturbation induced by a unitary external electric field-has been set for convenience to λ n = 0.1 mm, a value found in the experimental [47] and modeling literature [48].…”

“…Future work includes the analysis of the effects of weak electric fields at the network level and a more precise derivation of the coupling term λ n , which requires a translation of the effects of uniform weak electric fields from the microscale to the mesoscale. Work along these lines has been carried out in [48,68], where the effects of electric fields are analyzed at the single neuron level. The emergence of exact mean field theories deriving neural mass models from microscale ones, the construction of detailed population models, as well as experimental work can all help shed some light on the proper rescaling of the coupling parameter in our models [69].…”