  We propose a boundary element method for the accurate solution of the
cell-by-cell bidomain model of electrophysiology. The cell-by-cell model, also
called Extracellular-Membrane-Intracellular (EMI) model, is a system of
reaction-diffusion equations describing the evolution of the electric potential
within each domain: intra- and extra-cellular space and the cellular membrane.
The system is parabolic but degenerate because the time derivative is only in
the membrane domain. In this work, we adopt a boundary-integral formulation for
removing the degeneracy in the system and recast it to a parabolic equation on
the membrane. The formulation is also numerically advantageous since the number
of degrees of freedom is sensibly reduced compared to the original model.
Specifically, we prove that the boundary-element discretization of the EMI
model is equivalent to a system of ordinary differential equations, and we
consider a time discretization based on the multirate explicit stabilized
Runge-Kutta method. We numerically show that our scheme convergences
exponentially in space for the single-cell case. We finally provide several
numerical experiments of biological interest.
