Electrical restitution, critical mass, and the riddle of fibrillation.

The elusive riddle of ventricular fibrillation (VF) has been approached from many creative angles. The latest, presented by Wu et al in this issue of the journal [1], examines the effects of restitution properties on the “critical mass” i.e., the minimum size of cardiac tissue capable of sustaining fibrillation. What is noteworthy about this novel effort is that the very basis for the working hypothesis is rooted in deep mathematical concepts, a brief and highly selective chronological account of which we attempt here, since it has interesting aspects. The trail begins with Nolasco and Dahlen, who in 1968 used a straightforward graphical technique to demonstrate that under certain conditions electrical alternans was a dynamical consequence of the slope of the restitution relation for action potential duration (APD) [2]. If the slope of the APD restitution relation (the relation between APD and the preceding diastolic interval) was > 1, APD alternans was possible, whereas if the slope was < 1, it was not. More than a decade later Michael Guevara and colleagues took the important step of formalizing Nolasco and Dahlen’s graphical method as a one dimensional difference equation [3], thereby revealing the correspondence between APD alternans in the experimental system and period-doubling bifurcations in the equations. The strategy of reducing the problem to (the iteration of) a one dimensional difference equation pioneered by the McGill groups has had wide application, as detailed in Glass and Mackey [4]. The difference equation method subsequently was further adapted to explain periodic and chaotic rhythms in Purkinje fibers and ventricular muscle [5,6]. The approach was fruitful to the point that it was possible to predict any of a wide variety of rhythms exhibited by periodic stimulation of cardiac tissue [7-11], where the APD restitution was relevant to the dynamics. In addition, the analytical insight gained by using this approach prompted the contemporaneous observations that the potential anti-arrhythmic/proarrhythmic effects of some agents might be explained “on the basis of a decrease/increase in the slope of the APD restitution at very short coupling intervals” [10] and that “it might be possible in the future to reduce the incidence of arrhythmias by pharmacologically modifying the steep left-hand portion of the restitution curve" [8]. Once the dynamics under repetitive stimulation was understood, tackling the larger problem of fibrillation could be attempted. One such attempt occurred in 1990, when a simple question, posed at least once previously (on page 157 of [4]), became firmly entrenched in Alain Vinet’s mind during one of his many biweekly car trips through the snowbelt connecting Syracuse with Montreal. The following week he was asking colleagues on both sides of the St. Lawrence river: “Imagine this hypothetical experiment: A circulating wave of excitation is set up in a thin ring of tissue. As time passes you reduce very slowly the length of the ring. Which kind of dynamic will you observe: a) if the excitable tissue is nerve-like (no electrical restitution, i.e., APD is constant) ; and b) if the excitable tissue is cardiac-like (electrical restitution present)?”.