Electrical Restitution, Critical Mass, and the Riddle of Fibrillation

The elusive riddle of ventricular tibrillation <VF) has been approached from many creative angles. The latest, presented by Wu et al.' in this issue of JCE, examines the effects of restitution properties on the "critical mass," i.e., the minimum size of cardiac tissue capable of sustaining tibrillation. 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, because 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 dynamic consequence of the slope of the restitution relation for action potential duration (APD). If the slojx: 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 < I. 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, 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 by Glass and Mac key. •* The difference equation method subsequently was adapted further to explain periodic and cha-