Heller L, van Hulsteyn D B
Theoretical Division, Los Alamos National Laboratory, New Mexico 87545.
Biophys J. 1992 Jul;63(1):129-38. doi: 10.1016/S0006-3495(92)81587-4.
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.
我们证明，在通常提议用于颅外大脑刺激的频率下，使用外部电流源的任何叠加都不可能在大脑内部产生电场强度的三维局部最大值。最大值总是出现在电导率值发生跃变的边界上。然而，有可能实现比目前更好的二维电场聚焦和成形。为了实现这一目标，我们利用互易定理对由各种源产生的导电介质内部的电场进行了统一处理：外部磁偶极子（电流环）、外部电偶极子（线性天线）以及表面和深度电极。这种表述利用了来自脑磁图和脑电图的导联场。对于具有球对称电导率的系统的特殊情况，我们推导出了外部磁偶极子产生的电场的简单解析公式。该公式与电导率分布无关，因此涵盖了具有任意数量壳层的球形模型。这解释了数值研究中所描述的对颅骨电导率的“不敏感性”。我们还给出了恒定电导率球体情况下电偶极子以及表面和深度电极产生的电场的解析公式。
