  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 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.
