Mathematical models for excitable cells are commonly based on cable theory, which considers a homogenized domain and spatially constant ionic concentrations. Although such models provide valuable insight, the effect of altered ion concentrations or detailed cell morphology on the electrical potentials cannot be captured. In this paper, we discuss an alternative approach to detailed modeling of electrodiffusion in neural tissue. The mathematical model describes the distribution and evolution of ion concentrations in a geometrically-explicit representation of the intra- and extracellular domains. As a combination of the electroneutral Kirchhoff-Nernst-Planck (KNP) model and the Extracellular-Membrane-Intracellular (EMI) framework, we refer to this model as the KNP-EMI model. Here, we introduce and numerically evaluate a new, finite element-based numerical scheme for the KNP-EMI model, capable of efficiently and flexibly handling geometries of arbitrary dimension and arbitrary polynomial degree. Moreover, we compare the electrical potentials predicted by the KNP-EMI and EMI models. Finally, we study ephaptic coupling induced in an unmyelinated axon bundle and demonstrate how the KNP-EMI framework can give new insights in this setting.

中文翻译：

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细胞几何中离子电扩散的有限元模拟


可兴奋细胞的数学模型通常基于电缆理论，该理论考虑均匀化域和空间恒定的离子浓度。尽管此类模型提供了有价值的见解，但无法捕获改变的离子浓度或详细的细胞形态对电势的影响。在本文中，我们讨论了神经组织中电扩散详细建模的另一种方法。该数学模型以细胞内和细胞外域的几何显式表示形式描述了离子浓度的分布和演变。作为电中性基尔霍夫-能斯特-普朗克 (KNP) 模型和细胞外-膜-细胞内 (EMI) 框架的组合，我们将该模型称为 KNP-EMI 模型。在这里，我们引入并数值评估了一种新的、基于有限元的 KNP-EMI 模型数值方案，该方案能够高效、灵活地处理任意维度和任意多项式次数的几何形状。此外，我们还比较了 KNP-EMI 和 EMI 模型预测的电势。最后，我们研究了无髓鞘轴突束中诱导的触觉耦合，并演示了 KNP-EMI 框架如何在这种情况下提供新的见解。