We present a multiscale model and an adaptive numerical scheme for simulating cardiac action potential propagation along a linear strand of heart muscle cells. This model couples macroscale partial differential equations posed over the tissue to microscale equations posed over discrete cellular geometry. The microscopic equations are used only near action potential wave fronts, and the macroscopic equations are used everywhere else. We study the effects of gap-junctional and ephaptic coupling on conduction in the multiscale model and its fully macroscale and fully microscale analogues. Our simulations reveal that the adaptive multiscale model accurately reproduces the action potential wave forms and wave speeds of the fully microscale model. They also demonstrate that, at low gap-junctional conductivities, the accuracy of fully macroscale simulations is sensitive to numerical grid spacing. Moreover, adaptive multiscale simulations capture the effect of ephaptic coupling, whereas fully macroscale simulations do not. We propose two ways of generalizing our multiscale model to higher dimensions, and we argue that such generalizations may be necessary to obtain accurate three-dimensional simulations of cardiac conduction in certain pathophysiological parameter regimes.cardiac electrophysiology | gap junction | electric-field mechanism | mathematical model M athematical models and computer simulations of physiological systems promise to enhance understanding of biological processes and to aid development of medical treatments. One difficulty in making such simulations realistic is that biological systems are often multiscale. In cardiac electrophysiology, for example, ionic flow at the subcellular level ultimately causes excitation and contraction of muscle at the tissue level.

We derive the values for the intracellular and extracellular conductivities needed for bidomain simulations of cardiac electrophysiology using homogenization of partial differential equations. In our model, cardiac myocytes are rectangular prisms and gap junctions appear in a distributed manner as flux boundary conditions for Laplace's equation. Using directly measurable microproperties such as cellular dimensions and end-to-end and side-to-side gap junction coupling strengths, we inexpensively obtain effective conductivities close to those given by simulations with a detailed cyto-architecture (Stinstra et al. in Ann. Biomed. Eng. 33:1743-1751, 2005). This model provides a convenient framework for studying the effect on conductivities of aligned vs. brick-like arrangements of cells and the effect of different distributions of gap junctions along the myocyte membranes.

We derive a homogenized description of the electrical communication along a single strand of myocytes in the presence of gap-junctional and electric-field coupling. In the model, cells are electrically coupled through narrow clefts that are resistively connected to extracellular space. Cells are also coupled directly through gap junctions. We perform numerical simulations of this full model and its homogenization. We observe that the full and homogenized descriptions agree when gap-junctional coupling is at physiologically normal levels. When gap-junctional coupling is low, the two descriptions disagree. In this case, only the full model captures the behavior that the ephaptic mechanism can speed up action potential propagation. A strength of our homogenized description is that it is a macroscale model that can account for the preferential localization of Na+ channels at the ends of cells.