Arthur T. Winfree

Spiral waves of chemical activity.

The Zhabotinsky-Zaikin reagent propagates waves of chemical activity. Reaction kinetics remain to be fully resolved, but certain features of wave behavior are determined by purely geometrical considerations. If a wave is broken, then spiral waves, resembling involutes of the circle, appear, persist, and eventually exclude all concentric ring waves.

Spatiotemporal evolution of ventricular fibrillation.

Sudden cardiac death is the leading cause of death in the industrialized world, with the majority of such tragedies being due to ventricular fibrillation. Ventricular fibrillation is a frenzied and irregular disturbance of the heart rhythm that quickly renders the heart incapable of sustaining life. Rotors, electrophysiological structures that emit rotating spiral waves, occur in several systems that all share with the heart the functional properties of excitability and refractoriness. These re-entrant waves, seen in numerical solutions of simplified models of cardiac tissue, may occur during ventricular tachycardias,. It has been difficult to detect such forms of re-entry in fibrillating mammalian ventricles. Here we show that, in isolated perfused dog hearts, high spatial and temporal resolution mapping of optical transmembrane potentials can easily detect transiently erupting rotors during the early phase of ventricular fibrillation. This activity is characterized by a relatively high spatiotemporal cross-correlation. During this early fibrillatory interval, frequent wavefront collisions and wavebreak generation are also dominant features. Interestingly, this spatiotemporal pattern undergoes an evolution to a less highly spatially correlated mechanism that lacks the epicardial manifestations of rotors despite continued myocardial perfusion.

Integrated view of resetting a circadian clock.

Abstract This report extends and systematizes the observations of Pittendrigh and his coworkers on the phase resetting of the circadian rhythm of pupal eclosion in populations of the fruitfly, Drosophila pseudoobscura, in response to a single discrete perturbation. Using a pulse of dim blue light as the resetting stimulus, the daily eclosion time is plotted as a function of the duration of the stimulus and the time at which it is given. The resulting three-dimensional graph, covering four days of eclosion, resembles four turns of a corkscrew linking tilted planes. The corkscrew axis points to a unique stimulus time and duration, small variations of which result in radically altered phase resetting. Approximations to this stimulus result in unusual broadening of the daily eclosion peaks, even to the extreme of obscuring the circadian rhythm. The resetting data published by investigators of other circadian systems suggest that this behavior in fruitflies is typical of circadian clocks in many phyla. Abolition of the steady-state rhythm following exposure to the unique stimulus might be attributed to diversification of phases within the pupal population. Further experiments are reported which seem to exclude this interpretation. The circadian clock may have been “turned off” in each individual pupa, thus returning them to the state characteristic of darkreared populations.

RE-ENTRANT ROTATING WAVES IN A BEELER–REUTER BASED MODEL OF TWO-DIMENSIONAL CARDIAC ELECTRICAL ACTIVITY

Propagation of cardiac electrical activity is simulated in a two-dimensional sheet of cells using the cable equation and the Beeler–Reuter membrane model for ventricular muscle cells. Re-entrant patterns are produced using the original Beeler–Reuter equations and two modified versions involving changes in the calcium and sodium channel dynamics. Both stable rotating waves and irregular activity are observed. The model is shown to exhibit conduction block due to stationary repolarization fronts. The origin and properties of these fronts are described.

A SURVEY OF SPIRAL-WAVE BEHAVIORS IN THE OREGONATOR MODEL

We carried out a numerical exploration of spiral waves in a typical excitable medium, emphasizing the variety of behaviors encountered while changing two parameters of local excitability: the threshold for starting an excitation and the excitation rate at the wavefront. Within this parameter plane we found domains in which: 1) propagation is impossible, 2) propagation succeeds but there are no spiral waves, 3) spiral waves are stable and strictly periodic, 4) spiral waves exhibit two-period quasiperiodicity, and 5) spiral waves exhibit complex behavior that might be associated with the well-known instability of three-period flows on the three-torus. The boundary curves (bifurcation loci) separating these domains run parallel to the propagation boundary over much of their extent.

Resetting the amplitude ofDrosophila's circadian chronometer

SummaryEven in constant temperature darkness, mature flies emerge from populations ofDrosophila pupae only in a 6-hour-wide pulse once every 24 hours. The “circadian clock” governing this behavior can be rephased by exposing the pupae to even a few seconds of dim light in otherwise constant darkness. But there exists a critical exposure which, if and only if administered at a critical hour,turns off the circadian clock instead of rephasing it.This paper describes measurements in which a rephasing light pulse is followed by a second pulse so that the once-reset clock can be examined for normal rhythmicity of its sensitivity to light. The latter is found to be rephased in step with the emergence rhythm, but additionally attenuated in a way which strongly suggests that not only phase, but also the amplitude, intensity, or vigor of circadian oscillation is lastingly reset by a light pulse. Amplitude is apparently reset to zero by the critical pulse, but to other subnormal values by nearby non-critical pulses, even those which leave phase unaffected. The results of two thousand such rephasings are fitted by a simple theoretical model to within the reproducibility of measurements.

Suppressing Drosophila Circadian Rhythm with Dim Light

Drosophila larvae were reared and allowed to pupate in continuous bright white light. The pupae were then transferred to a much dimmer blue light. In continuous blue light of intensity below 0.001 erg per square centimeter per second, adult flies emerged in pulses 24.7 hours apart, each pulse occupying about 6 hours. But in continuous light of intensity exceeding 0.1 erg per square centimeter per second, they emerged at a steady rate. This intensity range from effective darkness to effective light is roughly from starlight to moonlight. Inside this range, the emergence peaks broaden for about a week with little change of period.

Oscillatory glycolysis in yeast: The pattern of phase resetting by oxygen

Abstract Glycolytic oscillations are initiated in a yeast cell suspension by an aerobic to anaerobic transition, and followed by recording NADH fluorescence. Injection of dissolved oxygen, up to 70 μ m , is found to phase-reset the NADH rhythm, usually without effect on period or amplitude. The graphed dependence of phase-resetting on oxygen dosage and on time of administration resembles a helicoidal surface. There is a critical dosage and phase at which injection terminates the oscillation. This pattern is typical of the simplest kinds of continuous and homogeneous reaction schemes. These results seem to support limit-cycle models involving only two important interacting control variables.

Stable three-dimensional action potential circulation in the FitzHugh-Nagumo model

Abstract Excitable media generally support vortex rings of self-excitation. Depending on the exact nature of the medium, such a ring may expand or contract, possibly to a stable radius. We describe one such case encountered during numerical experiments on a simple model of electrophysiological excitability in nerve and cardiac muscle membrane. The rings rate of shrinkage depends parabolically on curvature and on proximity to other rings. The vortex period also depends on curvature, so rings of different sizes compete for territory. We associate the stability of the ring with repulsive forces which we show are present between two-dimensional rotors. The observed minimal distance for repulsion agrees with the stable radius of the vortex ring.

The Electrical Thresholds of Ventricular Myocardium

The Electrical Thresholds of Ventricular Myocardium. According to the basic principles of electrophysiology, an action potential cannot propagate three‐dimensionally if its front is too sharply curved. The critical radius of curvature is estimated for ventricular myocardium as 1/3 mm and checked against experimental determinations of the pacing threshold. An implication of the agreement found is that pacemaker electrodes can be improved by optimizing their tip curvatures. The same basic principles imply that there should exist a vortex‐like action potential, which has in fact heen found in both two‐ and three‐dimensional settings. It rotates in 120 msec and has a 2/3‐cm diameter. This diameter can he used to derive the electrical threshold for fibrillation in normal ventricular myocardium: ahout 16 mA, depending on electrode geometry. This compares favorably with ohservations. As theory suggests, the ratio of this threshold to the pacing threshold seems independent of pulse duration and depends on electrode geometry: the minimum ratio is about five for large electrodes. Electrical defihrillation in normal myocardium should require local potential gradients of about 6 V/cm or current densities near 20 mA/cm2, roughly as observed, but much more uniform deHhrillating fields are needed to achieve this theoretical minimum throughout the myocardium. It is suggested that in normal myocardium, the transition from monomorphic tachycardia or ventricular flutter to fibrillation in some cases may be a consequence of the three‐dimensional geometry of vortex‐like action potentials; the transition should take a long time in two‐dimensional preparations unless they are pervaded hy discontinuities or other nonuniformities. (J Cardiovasc Electrophysiol, Vol. 1, pp. 393–410, October 1990)