The heart functions as a mechanical pump that propels blood through the circulatory system. Blood flows from the body into the right atrium, and then into the right ventricle where it gets pumped into the lungs through the pulmonary artery. Blood gets oxygenated in the lungs, flows into the left atrium, and then into the left ventricle where it gets pumped through the aorta to different organs of the body. Ventricular fibrillation (VF), identified as disorganized and ineffective quiver of the ventricles, is the most serious ventricular arrhythmias. When VF occurs, the heart is unable to pump blood. Sudden cardiac arrest follows. About 220,000 deaths from heart attacks in the United States each year are thought to be caused by VF. Alternans, a periodic beat-to-beat short-long alternation in action potential duration (APD), is considered to be a precursor of VF. In extended cardiac tissue, electrical alternans can be either spatially concordant (SCA, all cells oscillate in phase) or spatially discordant (SDA, cells in different regions oscillate out of phase). SDA gives rise to an increase in the spatial dispersion of repolarization, which is thought to be proarrhythmic. In this dissertation, I use a mapping model with two beats of memory, a simple but powerful tool to study cardiac dynamics and their relation to VF at the level of single cell and a spatially coupled cell strand. Ventricular myocytes are connected electrically by gap junction channels formed mainly by connexin 43 (Cx43) and, to a smaller extent, Cx45. Decreased and heterogeneous expression of Cx43 is a common feature in animal heart failure models. Ephaptic coupling is an electric field mechanism of propagation. It relies on the presence of narrow cleft space between neighboring cells, which is resistively connected to the extracellular space. Emerging evidence suggests a more active role for ephaptic coupling in mediating intercellular electrical communication when gap junctions are reduced in a homogeneous manner. Here, I explore the effect of ephaptic coupling when Cx43 expression is reduced in a heterogeneous fashion. For this purpose, I use a physiologically detailed ionic channel model with their gating kinetics applied in a two dimensional tissue model of electrically coupled cardiac cells.