Cut finite element discretizations of cell-by-cell EMI electrophysiology models

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.