The standard bidomain or monodomain equations model a cardiac tissue at the macroscopic scale. They represent an average of the electrical function of individual cells assumed to organize in to a regular, periodic network. Anyway, it is an important subject to decipher the role of dysfunction or disorganization of the tissue at the cellular level on the propagation of action potential at the tissue scale. For instance, in order to understand how and in which conditions conduction pathways at the cell scale may provide a substrate for arrhythmias at the tissue scale. In order to tackle these questions, we focus on computing a three-dimensional numerical solution of the bidomain equations at the microscopic, cellular, scale.

Therefore, we consider a few cells with a fixed intra-celluar conductivity coefficient, connected into a network, and included in an extracellular matrix, with a fixed extracellular conductivity coefficient. The intra and extracellular electrical fields solve an electrostatic balance, with a time-dependant transmission condition on the transmembrane voltage. This condition is precisely given by the chosen ionic model. The geometry is discretized into a tetrahedral mesh, and the equations are discretized with the P1-Lagrange finite element method, together with an implicit-explicit time-stepping method. Gap junctions are assumed to be small channels that connect the intracellular part of the cells end-to-end along isolating planes.

The numerical simulations show the expected propagation of the action potential along an individual cell. We will illustrate the interest of this model on network of cells with variable packing of cells, and with varying number of gap junctions. High performance parallel computating may be required if a large number of cells is considered to work on a bigger number of cells.