Department of Psychiatry and Behavioral Sciences, Duke University, Durham, North Carolina.
Department of Biomedical Engineering, Duke University, Durham, North Carolina; Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina; Department of Neurobiology, Duke University, Durham, North Carolina; Department of Neurosurgery, Duke University, Durham, North Carolina.
Biophys J. 2018 Jul 3;115(1):95-107. doi: 10.1016/j.bpj.2018.06.004.
We present a theory and computational models to couple the electric field induced by magnetic stimulation to neuronal membranes. Based on the characteristics of magnetically induced electric fields and the modified cable equation that we developed previously, quasipotentials are derived as a simple and accurate approximation for coupling of the electric fields to neurons. The conventional and modified cable equations are used to simulate magnetic stimulation of long peripheral nerves by circular and figure-8 coils. Activation thresholds are obtained over a range of lateral and vertical coil positions for two nonlinear membrane models representing unmyelinated and myelinated straight axons and also for undulating myelinated axons. For unmyelinated straight axons, the thresholds obtained with the modified cable equation are significantly lower due to transverse polarization, and the spatial distributions of thresholds as a function of coil position differ significantly from predictions by the activating function. However, the activation thresholds of unmyelinated axons obtained with either cable equation are very high and beyond the output capabilities of conventional magnetic stimulators. For myelinated axons, threshold values are similar for both cable equations and within the range of magnetic stimulators. Whereas the transverse field contributes negligibly to the activation thresholds of myelinated fibers, axonal undulation can significantly increase or decrease thresholds depending on coil position. The analysis provides a rigorous theoretical foundation and implementation methods for the use of the cable equation to model neuronal response to magnetically induced electric fields. Experimentally observed stimulation with the electric fields perpendicular to the nerve trunk cannot be explained by transverse polarization and is likely due to nerve fiber undulation and other geometrical inhomogeneities.
我们提出了一种理论和计算模型，将磁场刺激产生的电场与神经元膜耦合。基于磁场诱导电场的特性和我们之前开发的修正电缆方程，我们推导出了准电势，作为将电场与神经元耦合的简单而准确的近似。我们使用传统和修正的电缆方程来模拟圆形和 8 字形线圈对长外周神经的磁刺激。对于代表无髓和有髓直轴突的两种非线性膜模型，以及对于波动的有髓轴突，我们获得了在横向和垂直线圈位置的一系列激活阈值。对于无髓直轴突，由于横向极化，修正电缆方程得到的阈值显著降低，阈值的空间分布与激活函数的预测有很大不同。然而，修正电缆方程得到的无髓轴突的激活阈值非常高，超出了传统磁刺激器的输出能力。对于有髓轴突，两种电缆方程得到的阈值值相似，并且在磁刺激器的范围内。虽然横向场对有髓纤维的激活阈值贡献可以忽略不计，但轴突波动可以根据线圈位置显著增加或降低阈值。该分析为使用电缆方程来模拟磁场诱导电场对神经元的响应提供了严格的理论基础和实现方法。与神经干垂直的电场的实验观察到的刺激不能用横向极化来解释，可能是由于神经纤维波动和其他几何不均匀性所致。
