NeuroEngineering Laboratory, Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, VIC 3010, Australia.
J Neural Eng. 2012 Dec;9(6):065005. doi: 10.1088/1741-2560/9/6/065005. Epub 2012 Nov 27.
Neuroprosthetic devices, such as cochlear and retinal implants, work by directly stimulating neurons with extracellular electrodes. This is commonly modeled using the cable equation with an applied extracellular voltage. In this paper a framework for modeling extracellular electrical stimulation is presented. To this end, a cylindrical neurite with confined extracellular space in the subthreshold regime is modeled in three-dimensional space. Through cylindrical harmonic expansion of Laplace's equation, we derive the spatio-temporal equations governing different modes of stimulation, referred to as longitudinal and transverse modes, under types of boundary conditions. The longitudinal mode is described by the well-known cable equation, however, the transverse modes are described by a novel ordinary differential equation. For the longitudinal mode, we find that different electrotonic length constants apply under the two different boundary conditions. Equations connecting current density to voltage boundary conditions are derived that are used to calculate the trans-impedance of the neurite-plus-thin-extracellular-sheath. A detailed explanation on depolarization mechanisms and the dominant current pathway under different modes of stimulation is provided. The analytic results derived here enable the estimation of a neurite's membrane potential under extracellular stimulation, hence bypassing the heavy computational cost of using numerical methods.
神经假体设备，如耳蜗和视网膜植入物，通过使用细胞外电极直接刺激神经元来工作。这通常使用带有外加细胞外电压的电缆方程进行建模。本文提出了一种用于建模细胞外电刺激的框架。为此，在亚阈值状态下，在三维空间中对具有受限细胞外空间的圆柱形神经突进行建模。通过拉普拉斯方程的圆柱谐展开，我们推导出在不同类型的边界条件下，控制刺激不同模式的时空方程，这些模式称为纵向和横向模式。纵向模式由著名的电缆方程描述，然而，横向模式由一个新的常微分方程描述。对于纵向模式，我们发现，在两种不同的边界条件下，应用了不同的电紧张长度常数。推导出了将电流密度与电压边界条件联系起来的方程，用于计算神经突加薄细胞外鞘的跨阻抗。提供了对不同刺激模式下去极化机制和主导电流途径的详细解释。这里推导的解析结果可以估计细胞外刺激下神经突的膜电位，从而避免使用数值方法的巨大计算成本。
