Cut Finite Element Discretizations of Cell-by-Cell EMI Electrophysiology Models

SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page B527-B553, August 2024.
Abstract. 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 (PDEs) in the intracellular and extracellular spaces with a system of ordinary differential equations (ODEs) 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-generate 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 nonstandard 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 that also introduces the electrical current over the membrane as an additional unknown leading to a penalized saddle point problem. Both formulations are augmented by suitably designed ghost penalties to ensure stability and convergence properties that are insensitive to how the membrane surface mesh cuts the background mesh. For the ODE part, we introduce a new unfitted discretization to solve the membrane bound ODEs on a membrane interface that is not aligned with the background mesh. Finally, we perform extensive numerical experiments to demonstrate that CutFEM is a promising approach to efficiently simulate electrical activity in geometrically resolved brain cells. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://zenodo.org/record/8068506.