Brain stimulation using electromagnetic sources: theoretical aspects

Abstract

We prove that, at the frequencies generally proposed for extracranial stimulation of the brain, it is not possible, using any superposition of external current sources, to produce a three-dimensional local maximum of the electric field strength inside the brain. The maximum always occurs on a boundary where the conductivity jumps in value. Nevertheless, it may be possible to achieve greater two-dimensional focusing and shaping of the electric field than is currently available. Towards this goal we have used the reciprocity theorem to present a uniform treatment of the electric field inside a conducting medium produced by a variety of sources: an external magnetic dipole (current loop), an external electric dipole (linear antenna), and surface and depth electrodes. This formulation makes use of the lead fields from magneto- and electroencephalography. For the special case of a system with spherically symmetric conductivity, we derive a simple analytic formula for the electric field due to an external magnetic dipole. This formula is independent of the conductivity profile and therefore embraces spherical models with any number of shells. This explains the "insensitivity" to the skull's conductivity that has been described in numerical studies. We also present analytic formulas for the electric field due to an electric dipole, and also surface and depth electrodes, for the case of a sphere of constant conductivity.

Recurrent Processing in V1/V2 Contributes to Categorization of Natural Scenes

“…To obtain a comfortable level of stimulation for all of the participants, the intensity was decreased to 65% for one participant in the V1/V2 condition, and to 60 -67% for four participants in the LO condition. The TMS-induced electric field distribution in the V1/V2 and LO target areas were modeled with the eXimia NBS system that estimates the E-field strength in the brain by using spherical conductor model (Sarvas, 1987;Heller and van Hulsteyn, 1992). The estimated E-field strength did not differ between V1/V2 (128 V/m, SD ϭ 25) and LO (122 V/m; SD ϭ 24) target areas (t (12) ϭ 0.83; not significant).…”

EEG and MEG: forward solutions for inverse methods

“…In the forward problem, the source and hence the function is known, and the task is to determine , such that the residual is as small as possible. The standard method of weighted residuals solves this problem using a weighting function , i.e., we solve the related problem (28) or equivalently (29) where denotes the inner product of the two functions. The integration is over the domain of the unknown potential function , which in (5) is restricted to the two-dimensional surfaces.…”

“…For the biological signals of interest in E/MEG, the time-derivatives of the associated electric and magnetic fields are sufficiently small that they can be ignored in Maxwell's equations. Recent discussions and details of this quasi-static approximation can be found in [27], [28], and [49]. The static magnetic field equations are and i.e., the curl of the magnetic field at location is proportional to the current density, and the divergence of the magnetic field is zero.…”

Multiple Signal Classification Algorithm Based Electric Dipole Source Localization Method in an Underwater Environment

“…The electric dipole source in the underwater localization problem is equivalent to a quasi-static electric field in a closed body, which is shown in Figure 1, and the discussions and details of the quasi-static electric field approximation can be found in [18]. The potential of each point in region Ω is ϕ, and the boundary is defined as ∂Ω.…”