Modeling the effects of electrical conductivity of the head on the induced electric field in the brain during magnetic simulation

Objective: The objective of this document is to quantify the effect of changing conductivity within the brain in transcranial magnetic stimulation. Methods: Extreme examples of white and grey matter distributions as well as cerebral spinal fluid are analyzed with numerical boundary element methods to show that the induced E fields for these various distributions vary little from the homogeneous case. Results: Models representative of the brain that demarcate regions of white matter and grey matter add an unnecessary level of complexity to the design and analysis of magnetic stimulators. The induced E field varies little between a precise model with exact placement of white and grey matter from that of its homogeneous counterpart. The E field will increase in white matter, and decrease in grey, but the variation is small. The contour integral of the E field around a closed path is dictated by the flux change through that contour. Discussion: The maximum value of the variation of the electric field between a fully homogeneous medium, and one filled with different conductivity media is 1/2 the conductivity ratio of the media involved. Neuronal stimulation is more likely at the interface between dissimilar mediums, the greatest being between white matter and cerebral spinal fluid. The interface location where no normal electric field exists will witness a localized electric field 51% greater than the homogeneous E field on the white matter side of that interface. White–grey matter interfaces will have a maximum localized increase in the E field 22.9% greater than the homogeneous case. Conclusions: Variations in neural intracellular potential during a magnetic stimulation pulse will be small among patients. The most efficient modeling will follow by assuming the medium homogeneous, and noting that perturbations from this result will exist.