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.

翻译：我们提出一种边界元方法，用于精确求解电生理学中的逐细胞双域模型。逐细胞模型（也称为胞外-膜-胞内模型）是一个反应-扩散方程组，描述每个域（细胞内空间、细胞外空间及细胞膜）中电势的演化。该方程组属于抛物型但具有退化性，因为时间导数仅存在于膜域中。在本工作中，我们采用边界积分公式来消除系统的退化性，并将其重新表述为膜上的抛物型方程。该公式在数值计算上也具有优势，因为与原模型相比，自由度数量显著减少。具体而言，我们证明EMI模型的边界元离散等价于一个常微分方程组，并采用基于多率显式稳定龙格-库塔方法的时间离散方案。数值实验表明，我们的方法在单细胞情形下具有空间指数收敛性。最后，我们提供若干具有生物学意义的数值实验结果。