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Significance. Noninvasive brain stimulation (NIBS) by quasistatic electromagnetic means is presently comprised of two methods: magnetic induction methods (transcranial magnetic perturbation or TMP) and electrical contact methods (transcranial electric perturbation or TEP). Both methods couple to neuronal systems by means of the electric fields they produce. Both methods are necessarily accompanied by a scalp electric field which is of greater magnitude than anywhere within the brain. A scalp electric field of sufficient magnitude may produce deleterious effects including peripheral nerve stimulation and heating which consequently limit the spatial and temporal characteristics of the brain electric field. Presently the electromagnetic NIBS literature has produced an accurate but non-generalized understanding of the differences between the TEP and TMP methods. Objective. The aim of this work is to contribute a generalized understanding of the differences between the two methods which may open doors to novel TEP or TMP methods and translating advances, when possible, between the two methods. Approach. This article employs a three shell spherical conductor head model to calculate general analytical results showing the relationship between the spatial scale of the brain electric fields and: (1) the scalp-to-brain mean-squared electric field ratio for the two methods and (2) TEP-to-TMP scalp mean-squared electric field ratio for similar electric fields at depth. Main results. The most general result given is an asymptotic limit to the TEP-to-TMP ratio of scalp mean-squared electric fields for similar electric fields at depth. Specific example calculations for these ratios are also given for typical TEP electrode and TMP coil configurations. While TMP has favorable mean-squared electric field ratios compared to TEP this advantage comes at an energetic cost which is briefly elucidated in this work.

Comparing the electric fields of transcranial electric and magnetic perturbation