The EMI (Extracellular-Membrane-Intracellular) model describes electrical activity in excitable tissue, where the extracellular and intracellular spaces and cellular membrane are explicitly represented. The model couples a system of partial differential equations in the intracellular and extracellular spaces with a system of ordinary differential equations on the membrane. A key challenge for the EMI model is the generation of high-quality meshes conforming to the complex geometries of brain cells. To overcome this challenge we propose a novel cut finite element method (CutFEM) where the membrane geometry can be represented independently of a structured and easy-to-generated background mesh for the remaining computational domain. Starting from a Godunov splitting scheme, the EMI model is split into separate PDE and ODE parts. The resulting PDE part is a non-standard elliptic interface problem, for which we devise two different CutFEM formulations: one single-dimensional formulation with the intra/extracellular electrical potentials as unknowns, and a multi-dimensional formulation which also introduces the electrical current over the membrane as an additional unknown leading to a generalized saddle point problem with a penalty-like term. Both formulations are augmentied by suitably designed ghost penalties to ensure that the stability and convergence properties of the resulting discretizations are insensitive to how the membrane surface mesh cuts the background mesh.For the ODE part, we introduce a new unfitted discretization which is based on a stabilized mass matrix approach and allows us to solve the membrane bound ODEs even if the membrane interface is not aligned with the background mesh. Finally, we perform extensive numerical to demonstrate that CutFEM is a promising approach to efficiently simulate electrical activity in geometrically resolved brain cells.

翻译：EMI（细胞外-膜-细胞内）模型描述了可兴奋组织中的电活动，其中细胞外空间、细胞内空间和细胞膜被明确表示。该模型将细胞内和细胞外空间中的偏微分方程组与膜上的常微分方程组耦合在一起。EMI模型的一个关键挑战是生成符合脑细胞复杂几何结构的高质量网格。为克服这一挑战，我们提出了一种新型切割有限元方法（CutFEM），其中膜几何结构可以独立于剩余计算域的结构化且易于生成的背景网格进行表示。基于Godunov分裂格式，我们将EMI模型分解为独立的偏微分方程（PDE）和常微分方程（ODE）部分。得到的PDE部分是一个非标准椭圆界面问题，我们为其设计了两种不同的CutFEM公式：一种是以细胞内/外电势为未知量的单维度公式，另一种是引入膜上电流作为额外未知量的多维度公式，后者导致了一个带有惩罚项类型的广义鞍点问题。两种公式均通过适当设计的鬼影惩罚项进行增广，以确保所得离散化的稳定性和收敛性对膜表面网格如何切割背景网格不敏感。对于ODE部分，我们引入了一种基于稳定质量矩阵方法的新型非拟合离散化，即使膜界面与背景网格不对齐，也能求解膜上的常微分方程。最后，我们进行了广泛的数值实验，证明CutFEM是一种在几何解析的脑细胞中高效模拟电活动的有前景方法。