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Figure 10: Cartoons showing quatrefoil reentry produced by cathodal break (left column) and anodal break (right column) for tissue with a horizontal fiber direction. Top row depicts the experimental configuration. Black regions are tissue stimulated by cathodal excitation, gray border is initial wavefront at the edge of excited tissue, hatched regions are refractory tissue, white regions are unexcited/hyperpolarized tissue, and stars show location of phase singularities. Middle row illustrates initial numerical approximation of the experimental configuration. Black is refractory, the gray border is excited, and white is unexcited. Bottom row depicts the spatial distribution of the fast variable a short time later. The arrows show the direction of the motion of the wavefront as it passes through the plane of the ring that is defined by the singular filament that encircles the z axis and the black arrows (Bray and Wikswo, reprinted with permission ©2003 by the American Physical Society [http://www.vanderbilt.edu/lsp/abstracts/9906-Bray-PRL-2003.htm])88

Figure 10: Cartoons showing quatrefoil reentry produced by cathodal break (left column) and anodal break (right column) for tissue with a horizontal fiber direction. Top row depicts the experimental configuration. Black regions are tissue stimulated by cathodal excitation, gray border is initial wavefront at the edge of excited tissue, hatched regions are refractory tissue, white regions are unexcited/hyperpolarized tissue, and stars show location of phase singularities. Middle row illustrates initial numerical approximation of the experimental configuration. Black is refractory, the gray border is excited, and white is unexcited. Bottom row depicts the spatial distribution of the fast variable a short time later. The arrows show the direction of the motion of the wavefront as it passes through the plane of the ring that is defined by the singular filament that encircles the z axis and the black arrows (Bray and Wikswo, reprinted with permission ©2003 by the American Physical Society [http://www.vanderbilt.edu/lsp/abstracts/9906-Bray-PRL-2003.htm])88

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Figure 11: Simulations of quatrefoil reentry following a strong cathodal S2 stimulus. An Si stimulus at time zero triggered an outwardly propagating wavefront. By 280ms, the region around the electrode (center black rectangle) is in the refractory tail of the S1 action potential. The S2 stimulus lasts from 280 to 300 ms, followed by break excitation and quatrefoil reentry. The color scale for the transmembrane potential is the same as in Fig. 4 (Calculated according to Roth)54

as in Fig. 11, recover to become a steadily propagating wavefront. Sidorov et al.83 and others84'85 have examined the spatiotemporal dynamics of damped propagation in detail and concluded that the transition from a damped to a steadily propagating wavefront, illustrated in Fig. 13, is a key link in understanding defibrillation.

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Figure 12: Isochrones showing the position of the wavefront during quatrefoil reentry, following a (a) cathodal and (b) anodal S2 stimulus. Data were obtained from a rabbit heart (Lin et al.)82

Fiber Direction Frame Number

Figure 12: Isochrones showing the position of the wavefront during quatrefoil reentry, following a (a) cathodal and (b) anodal S2 stimulus. Data were obtained from a rabbit heart (Lin et al.)82

Tracing the trajectories of phase singularities during quatrefoil reentry (Fig. 14) offers an excellent model system for studying the way phase singularities interact.86-88 Gray et al.89 have used measurements of the transmembrane potential together with calcium imaging to monitor reentry. Figure 15 shows that measuring both these variables simultaneously provides additional information about the dynamics of phase singularities. An interesting application of simultaneous imaging includes determining whether a particular arrhythmic focus is driven by calcium or voltage, which affects locally the direction of rotation in the phase plane.90

A third S3 stimulus can terminate reentry induced by S1-S2 stimulation. The timing of S3 is crucial, with certain times resulting in termination ("protective zones") and other times not.91'92 These protective zones recur periodically.93 Simulations by Hildebrandt and Roth94 showed that quatrefoil reentry displays periodic protective zones that recur with the period of the quatrefoil reentrant circuit, and that the protective zones are wider for anodal than cathodal stimulation.

Traditionally, researchers have focused on the interaction of the S1 refractory gradient and the S2 stimulus gradient during the induction of reentry.76'77 In fact Winfree's original prediction of quatrefoil reentry took just this point of view.80'95 However, an S1 gradient of refractoriness is not essential for reentry induction by an S2 stimulus.96-98 Figure 16 shows the induction of quatrefoil reentry when the S1 action potential is uniform in space, so there is no refractory gradient. The S2 shock (80ms) has two roles: it creates a gradient of refractoriness during the shock by hyperpolarizing and deexciting the tissue at the virtual anode, and then initiates the wavefront by break excitation after the shock ends. This effect was experimentally verified by Cheng et al.99 who observed that the direction of S2 excitation and reentry did not depend on the direction of the S1 refractory gradient.

Our discussion of reentry began with critical point theory and the pinwheel experiment (Fig. 9c) and culminated in our claim that the S1 gradient of refractoriness is sometimes not even necessary because virtual electrodes alone are sufficient to trigger quatrefoil reentry (Fig. 16). Returning now to the pinwheel experiment, it is worthwhile to determine how it is influenced by the formation of virtual electrodes. Sidorov et al.100 recently used optical mapping to study the pinwheel experiment and found that immediately after a cathodal S2 shock delivered in the refractory period, virtual anodes formed along the fiber direction, as shown in Fig. 1. Depending on the timing of S2, they observed make excitation, transitional make-break, break excitation, or damped waves. The fate of these excitation fronts depended on the direction of S1 propagation relative to the fibers. Wavefronts initiated by virtual electrode mechanisms are shown in Fig. 13, but those wavefronts in more refractory tissue died, while wavefronts in more recovered tissue successfully propagated, consistent with the critical point hypothesis.

Sidorov et al.100 used relatively weak shocks that did not induce reentry. Using numerical simulations, Lindblom et al.101'102 performed a similar study with stronger S2 shocks (to see these results explained with an extremely simple cellular automata model, see http://sprojects.mmi.mcgill.ca/heart/pages/rot/rothom.html). Depending on the S2 timing and polarity and the direction of the S1 wave relative to the fibers, they found figure-8 reentry (consistent with the pinwheel experiment) or quatrefoil reentry. Their simulations are consistent with the observations of Sidorov et al., and these studies demonstrate how to use the pinwheel experiment to reconcile two competing views of reentry induction: the critical point hypothesis and virtual electrodes.101'103'104

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