Analytical solutions for the Extracellular-Membrane-Intracellular model

The Extracellular-Membrane-Intracellular (EMI) model is a novel mathematical framework for cardiac electrophysiology simulations. The EMI model provides a more detailed description of the heart's electrical activity compared to traditional monodomain and bidomain models, potentially making it better-suited for understanding the electrical dynamics of the heart under pathological conditions. In this paper, we derive and verify several analytical solutions for the EMI model. Specifically, we obtain a family of solutions for a single two-dimensional cell in polar coordinates and for a pair of coupled three-dimensional cells in spherical coordinates. We also introduce a manufactured solution for N three-dimensional cells in Cartesian coordinates. To verify the analytical solutions, we conduct numerical experiments using the mortar finite element method combined with operator splitting. The results demonstrate that the analytical solutions are effective for verifying the accuracy of numerical simulations of the EMI model.