Nygren A, Halter J A
Department of Electrical and Computer Engineering, Division of Neuroscience, Rice University, Houston, TX 77030, USA.
J Theor Biol. 1999 Aug 7;199(3):329-58. doi: 10.1006/jtbi.1999.0962.
This paper discusses mathematical approaches for modeling the propagation of the action potential and ion concentration dynamics in a general class of excitable cells and cell assemblies of concentric cylindrical geometry. Examples include myelinated and unmyelinated axons, single strands of interconnected cardiac cells and outer hair cells. A key feature in some of the cells is the presence of a small working volume such as the periaxonal space between the myelin sheath and the axon in the myelinated axon and the extracisternal space between the plasma membrane and the subsurface cisterna of the outer hair cell. Proper treatment of these cell types requires a modeling approach which can readily address these anatomical properties and the non-uniform biophysical properties of the concentric membranes and the ionic composition of the volumes between the membranes. An electrodiffusion approach is first developed in which the Nernst-Planck equation is used to characterize axial ion fluxes. It is then demonstrated that this "full" model can be stepwise reduced, eventually becoming equivalent to the standard cable equation formulation. This is done in a manner that permits direct comparisons between the full and simplified models by running simulations using a single parameter set. An intermediate approach where the contributions of the axial currents to ion concentration changes and the effect of varying ion concentrations on solution conductivities are ignored is derived and is found adequate in many cases. Two application examples are given: a "cardiac strand" model, for which the intermediate formulation is shown sufficient and a model of the myelinated axon, for which the full electrodiffusion formulation is clearly necessary. The latter finding is due to spatial inhomogeneities in the anatomy and distribution of ion channels and transporters in the myelinated axon and the restricted periaxonal space between the myelin sheath and the axon.
本文讨论了用于模拟一般类型的可兴奋细胞以及具有同心圆柱几何形状的细胞集合体中动作电位传播和离子浓度动态变化的数学方法。示例包括有髓鞘和无髓鞘轴突、相互连接的单链心肌细胞以及外毛细胞。某些细胞的一个关键特征是存在小的工作体积，例如有髓鞘轴突中髓鞘与轴突之间的轴周间隙以及外毛细胞的质膜与表面下池之间的池外间隙。对这些细胞类型进行恰当处理需要一种建模方法，该方法能够轻松应对这些解剖学特性以及同心膜的非均匀生物物理特性和膜间体积的离子组成。首先开发了一种电扩散方法，其中使用能斯特 - 普朗克方程来表征轴向离子通量。然后证明这个“完整”模型可以逐步简化，最终等同于标准电缆方程公式。通过使用单个参数集进行模拟，以一种允许对完整模型和简化模型进行直接比较的方式完成这一过程。推导了一种中间方法，其中忽略了轴向电流对离子浓度变化的贡献以及离子浓度变化对溶液电导率的影响，并且发现在许多情况下这种方法是足够的。给出了两个应用示例：一个“心肌链”模型，对于该模型中间公式已证明是足够的；以及一个有髓鞘轴突模型，对于该模型完整的电扩散公式显然是必要的。后一个发现是由于有髓鞘轴突中离子通道和转运体的解剖结构和分布存在空间不均匀性以及髓鞘与轴突之间受限的轴周间隙。
