Efficient Numerical Solution of the EMI Model Representing the Extracellular Space (E), Cell Membrane (M) and Intracellular Space (I) of a Collection of Cardiac Cells

The EMI model represents excitable cells in a more accurate manner than traditional homogenized models at the price of increased computational complexity. The increased complexity of solving the EMI model stems from a significant increase in the number of computational nodes and from the form of the linear systems that need to be solved. Here, we will show that the latter problem can be solved by careful use of operator splitting of the spatially coupled equations. By using this method, the linear systems can be broken into sub-problems that are of the classical type of linear, elliptic boundary value problems. Therefore, the vast collection of methods for solving linear, elliptic partial differential equations can be used. We demonstrate that this enables us to solve the systems using shared-memory parallel computers. The computing time scales perfectly with the number of physical cells. For a collection of 512×256 cells, we manage to solve linear systems with about 2.5×10^8 unknows. Since the computational effort scales linearly with the number of physical cells, we believe that larger computers can be used to simulate millions of excitable cells and thus allow careful analysis of physiological systems of great importance.