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We propose a boundary element method for the accurate solution of thecell-by-cell bidomain model of electrophysiology. The cell-by-cell model, alsocalled Extracellular-Membrane-Intracellular (EMI) model, is a system ofreaction-diffusion equations describing the evolution of the electric potentialwithin each domain: intra- and extra-cellular space and the cellular membrane.The system is parabolic but degenerate because the time derivative is only inthe membrane domain. In this work, we adopt a boundary-integral formulation forremoving the degeneracy in the system and recast it to a parabolic equation onthe membrane. The formulation is also numerically advantageous since the numberof degrees of freedom is sensibly reduced compared to the original model.Specifically, we prove that the boundary-element discretization of the EMImodel is equivalent to a system of ordinary differential equations, and weconsider a time discretization based on the multirate explicit stabilizedRunge-Kutta method. We numerically show that our scheme convergencesexponentially in space for the single-cell case. We finally provide severalnumerical experiments of biological interest.

Boundary Integral Formulation of the Cell-by-Cell Model of Cardiac Electrophysiology