The pumping function of the heart is driven by an electrical wave
traversing the cardiac muscle in a well-organized manner. Perturbations to this
wave are referred to as arrhythmias. Such arrhythmias can, under unfortunate
circumstances, turn into fibrillation, which is often lethal. The only known
therapy for fibrillation is a strong electrical shock. This process, referred to as
defibrillation, is routinely used in clinical practice. Despite the importance of
this procedure and the fact that it is used frequently, the reasons for defibrillation's effectiveness are not fully understood. For instance, theoretical estimates
of the shock strength needed to defibrillate are much higher than what is actually used in practice. Several authors have pointed out that, in theoretical
models, the strength of the shock can be decreased if the cardiac tissue is modeled as a heterogeneous substrate. In this paper, we address this issue using
the bidomain model and the Courtemanche ionic model; we also consider a
linear approximation of the Courtemanche model here. We present analytical
considerations showing that for the linear model, the necessary shock strength
needed to achieve defibrillation (defined in terms of a sufficiently strong change
of the resting state) decreases as a function of an increasing perturbation of
the intracellular conductivities. Qualitatively, these theoretical results compare well with computations based on the Courtemanche model. The analysis
is based on an energy estimate of the difference between the linear solution of
the bidomain system and the equilibrium solution. The estimate states that the
difference between the linear solution and the equilibrium solution is bounded
in terms of the shock strength. Since defibrillation can be defined in terms of
a certain deviation from equilibrium, we use the energy estimate to derive a
necessary condition for the shock strength.