Fireflies, synchronization

Fireflies provide one of the most spectacular examples of synchronization in nature. In some parts of southeast Asia, thousands of male fireflies gather in trees at night and flash on and off in unison. Meanwhile the female fireflies cruise overhead, looking formales with a handsome light. 

To really appreciate this amazing display, you have to see a movie or videotape of it. A good example is shown in David Attenborough s (1992) television series The Trials of Life, in the episode called Talking to Strangers. See Buck and Buck (1976) for a beautifully written introduction to synchronous fireflies, and Buck (1988) for a more recent review. For mathematical models of synchronous fireflies, see Mirollo and Strogatz (1990) and Ermentrout (1991). 

How does the synchrony occur Certainly the fireflies don t start out synchronized they arrive in the trees at dusk, and the synchrony builds up gradually as the night goes on. The key is that the fireflies influence each other. When one firefly sees the flash of another, it slows down or speeds up so as to flash more nearly in phase on the next cycle. 

Hanson (1978) studied this effect experimentally, by periodically flashing a light at a firefly and watching it try to synchronize. For a range of periods close to the firefly s natural period (about 0.9 sec), the firefly was able to match its frequency to the periodic stimulus. In this case, one says that the firefly had been entrained by the stimulus. However, if the stimulus was too fast or too slow, the firefly could not keep up and entrainment was lost—then a kind of beat phenomenon occurred. But in contrast to the simple beat phenomenon of Section 4.2, the phase difference between stimulus and firefly did not increase uniformly. The phase difference increased slowly during part of the beat cycle, as the firefly struggled in vain to synchronize, and then it increased rapidly through 2 r, after which... 

Actually, the biological reality about synchronous fireflies is more complicated. The model presented here is reasonable for certain species, such as Pteroptyx cribellata, which behave as if A and co were fixed. However, the species that is best at synchronizing, Pteroptyx malaccae, is actually able to shift its frequency co toward the drive frequency Q (Hanson 1978). In this way it is able to achieve nearly zero phase difference, even when driven at periods that differ from its natural period by 15 percent A model of this remarkable effect has been presented by Ermen trout (1991). 

Attenborough, D. (1992) The Trials of Life. For synchronous fireflies, see the episode entitled Talking to Strangers, available on videotape from Ambrose Video Publishing, 1290 Avenue of the Americas, Suite 2245, New York, NY 10104. 

Buck, J. (1988) Synchronous rhythmic flashing of fireflies. II. Quart. Rev. Biol. 63, 265. 

Ohba N. Synchronous flashing in the Japanese firefly, Luciola cruciata (Coleoptera Lampyridae). Sci Rept Yokosuka City Mus 1984 32 23-32. 

We discuss now how the synchronization transition occurs, taking the applause in an audience as an example (experimental study of synchronous clapping is reported in [35]). Initially, each person claps with an individual frequency, and the sound they all produce is noisy.As long as this sound is weak, and contains no characteristic frequency, it does not essentially affect the ensemble. Each oscillator has its own frequency oJk, each person applauds and each firefly flashes with its individual rate, but there always exists some value of it that is preferred by the majority. Definitely, some elements behave in a very individualistic manner, but the main part of the population tends to be like the neighbor . So, the frequencies u>k are distributed over some range, and this distribution has a maximum around the most probable frequency. Therefore, there are always at least two oscillators that have very close frequencies and, hence, easily synchronize. As a result, the contribution to the mean field at the frequency of these synchronous oscillations increases. This increased component of the driving force naturally entrains other elements that have close frequencies, this leads to the growth of the synchronized cluster and to a further increase of the component of the mean field at a certain frequency. This process develops (quickly for relaxation oscillators, relatively slow for quasilinear ones), and eventually almost all elements join the majority and oscillate in synchrony, and their common output - the mean field - is not noisy any more, but rhythmic. 

J. Buck and E. Buck. Mechanism of rhythmic synchronous flashing of fireflies. Science, 159 1319-1327, 1968. 

Synchronization or concurrence between two or more oscillators is a fascinating phenomenon that is common in systems of coupled nonlinear oscillators. The phenomenon of synchronization is very common in nature and occurs in a wide variety of real world systems [1-4]. Some examples include the emulsion of light pulses by a population of fireflies in a synchronous manner, synchronization in populations of electrochemical oscillators, synchronous chirping by populations of crickets, synchronous clapping of audiences in auditoria, synchronous firing in populations of neurons, synchronized pacemaker oscillations in cardiac cells, etc. The aforanen-tioned instances of synchronization can be explained by systems of coupled oscillators, which are represented typically as... 

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