  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. 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. 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. 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.
