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The EMI (Extracellular-Membrane-Intracellular) model describes electricalactivity in excitable tissue, where the extracellular and intracellular spacesand cellular membrane are explicitly represented. The model couples a system ofpartial differential equations in the intracellular and extracellular spaceswith a system of ordinary differential equations on the membrane. A keychallenge for the EMI model is the generation of high-quality meshes conformingto the complex geometries of brain cells. To overcome this challenge, wepropose a novel cut finite element method (CutFEM) where the membrane geometrycan be represented independently of a structured and easy-to-generatedbackground mesh for the remaining computational domain.  Starting from a Godunov splitting scheme, the EMI model is split intoseparate PDE and ODE parts. The resulting PDE part is a non-standard ellipticinterface problem, for which we devise two different CutFEM formulations: onesingle-dimensional formulation with the intra/extracellular electricalpotentials as unknowns, and a multi-dimensional formulation that alsointroduces the electrical current over the membrane as an additional unknownleading to a penalized saddle point problem. Both formulations are augmented bysuitably designed ghost penalties to ensure stability and convergenceproperties that are insensitive to how the membrane surface mesh cuts thebackground mesh. For the ODE part, we introduce a new unfitted discretizationto solve the membrane bound ODEs on a membrane interface that is not alignedwith the background mesh. Finally, we perform extensive numerical experimentsto demonstrate that CutFEM is a promising approach to efficiently simulateelectrical activity in geometrically resolved brain cells.

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