On Emerging Coherence

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Science  20 Dec 2002:
Vol. 298, Issue 5602, pp. 2336-2337
DOI: 10.1126/science.1072560

Many diverse processes cycle regularly. If coupled in pairs, such oscillating systems tend to “mutually entrain”—that is, get in step with each other or alternate. The effect is not limited to pairs: Large populations of similar oscillators can interact in this way. Rhythmic clapping in a large crowd is one example of emerging coherence of this kind. An inorganic example has been described by Vanag and Epstein (1), who showed that when nanodroplets of an oscillating aqueous Belousov-Zhabotinsky reaction solution are suspended in oil, they appear to mutually entrain each other. In a recent report (2), Kiss et al. have described another inorganic system with sufficient accuracy to test the prevailing theory.

The first tractable mathematical model of emerging coherence was developed in 1966 (3, 4). In this severely simplified model, the unit oscillator progresses along a fixed cycle of successive states that eventually lead back to a prior state. The oscillator is always somewhere on the unique continuous cycle, the shape and amplitude of which cannot change. A pendulum, for example, is not like that; although it swings with fixed period through a cycle of angular positions and velocities, both can have arbitrary amplitude, determined by the initial impetus that started the oscillation. Oscillators in general need not be “on the cycle”: They can be at rest or engaged in alternative dynamics, depending merely on initial conditions.

Is there anything in the world that is like this idealized oscillator? The answer is yes. Chemical clocks go through a fixed sequence of successive concentration states, leading back to a prior state after a fixed time. In Josephson junctions driven by a constant current, the phase shift of a quantum wave function changes progressively, eventually completing a cycle. Biochemical oscillations in living cells or organisms, such as the cell mitotic cycle or the menstrual cycle, may progress through a fixed cycle of states, although the rate at which this happens may vary.

Coherence of coupled oscillators.

The long-term collective amplitude for a population of 100 oscillators that are identical except for their native periods is shown as a function of their mutual coupling. The coupling is normalized to the range of native frequencies, such that the threshold for mutual synchronization is 0.5. Numerical simulation data replotted from figure 8 of (4).


In idealized models of such processes, the oscillators can be coupled together by weak interactions—for example, by diffusive exchange of chemical concentrations or of electrochemical potential. In the 1960s models, it was found that such oscillators not only can attract one another in phase but can also mutually repel, thus avoiding synchronization even when all are driven by an external rhythm. They only mutually entrain when the intensity of mutual coupling is sufficient compared with the range of their individual native frequencies. Below a threshold, anarchy prevails; above it, there is a collective rhythm as some oscillators lock together. If the coupling is intensified, more oscillators join in until the whole population marches as one (see the figure) (3, 4).

According to an analytical model of such dynamics (4), the collective amplitude in a population of 100 oscillators of unit amplitude should rise from 0 to 100 as a sigmoidal function of the ratio of coupling intensity to the dispersion of native periods. A mathematically more tractable special case emerged in 1975 (5), prompting an avalanche of theoretical literature exploring its ramifications (6). But no quantifiable illustrations emerged from natural science experiments.

Arrays of electronic oscillators bore out qualitative expectations (7, 8). It was shown that aggregates of beating heart cells couple electrically to beat as one (9) and that in multinucleate “slime molds,” all nuclei divide at once as a result of chemical diffusion among them (10). Suspensions of yeast cells synchronously metabolize glucose in half-minute cycles (6, 11), and suspensions of cellular slime mold cells synchronously emit pulses of cyclic AMP (adenosine 3_,5_-monophosphate) (12), if there are enough cells per milliliter to communicate adequately. Fireflies signal one another to flash together, and crickets and frogs chorus (2, 12). But for diverse technical reasons, none of these systems lent itself to the quantitative observations needed to test the theory.

Kiss et al. have produced such a system. It is electrochemical: a polished nickel electrode in sulfuric acid with an electrical potential that varies sinusoidally in less than 1 s. Neighboring electrodes interact electrically in a common bath, and all interact equally with all others through a single resistor. Their individual native frequencies can be tuned by individual resistors. When tuned to a sufficiently narrow distribution, the array is susceptible to mutual entrainment at or above a critical level of global coupling.

This is the first laboratory system in which collective amplitude has been reported quantitatively as a function of coupling intensity. The outcome resembles that shown in the figure: There is a sigmoidal relationship that rises sharply near the threshold for mutual entrainment. Wiesenfeld et al. (13) predicted the same sort of threshold effect and sigmoidal rise of collective amplitude for Josephson junctions, but a lab experiment has not yet been reported.

Can such experiments be done in vivo? Much is now known about the molecular mechanisms of circadian clocks that was completely hidden a decade ago. Circadian clocks in individual cyanobacteria, just like cells of the mammalian suprachiasmatic nucleus, synchronize with one another when closely enough coupled by proximity. Can a threshold level of coupling be established? And can the agent of communication be isolated?

Thirty years ago, both these questions were explored by biochemists interested in the regulation of glycolysis in yeast cells. Under anaerobic conditions, glycolysis oscillates with half-minute periodicity; cells in dense suspension oscillate synchronously but not in more dilute suspension. Ghosh et al. (11) found the expected threshold of dilution and enthusiastically imposed the foregoing diagrams (see the figure) on dubious biochemists. But it turned out recently (6) that the threshold was measured incorrectly by two orders of magnitude. Dano et al. (14) have refined the experimental methods and identified the coupling agent as acetaldehyde (which had seemed excluded by the first measurements), so that delicately quantitative experiments are now feasible.

The remarkable precision achieved by Kiss et al. (2) is thus within reach for glycolysis and may eventually come to the world of circadian clocks with the use of suspensions of cyanobacteria or suprachiasmatic nucleus neurons in tissue culture.

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