BIOLOGICAL OSCILLATORS GENERATE patterns that range from the waves of peristalsis in the intestines to the walk, trot, canter and gallop of a horse. One of the most spectacular effects of coupled oscillation is the mass synchronization of thousands of fireflies of certain species. Steven H. Strogatz and Renato Mirollo proved how such synchrony arises by postulating a mathematical system based on an electrical circuit known as a relaxation oscillator [see "Coupled Oscillators and Biological Synchronization," by Steven H. Strogatz and Ian Stewart, page 98]. It is a fairly simple matter to build such an oscillator and watch the same phenomenon on a tabletop in a darkened room.

We wanted our oscillators to be as faithful as possible to our conception of how real fireflies synchronize as dusk deepens. Consequently, they are coupled by flashes of light. In daylight, their photodetectors are swamped by ambient light, and so each blinks at its own rate, but in a dark room they respond to each other's flashes and eventually flash in unison.

Our firefly works by pumping charge into a capacitor until the voltage across it reaches a threshold. The capacitor then discharges through a switch, the firefly flashes and the cycle repeats. If the firefly receives a flash from a neighbor, the amount of charge flowing into the capacitor briefly increases by an amount proportional to the strength of the flash. This increase makes the firefly complete its own cycle more quickly and thus brings the time of its firing closer to that of the one whose flash it received [see Figure 2]. After some number of cycles, the two will flash in synchrony. (This simple analysis ignores the firefly's effect on its neighbor's cycle, but as long as the charging curve slows as it nears the firing threshold, the two will in fact synchronize.)

What goes for two likely applies for a larger number, and so it would be natural to expect a large collection of firefly oscillators to synchronize as well. You will need to build at least two fireflies to observe synchronization, but three, four or even nine are better.

Probably the easiest experiment to conduct with the artificial fireflies is determining the time they take to synchronize as a function of the coupling between them. The stronger the signal that the phototransistors receive, the faster the devices reach lockstep. When you turn on the fireflies, they will be flashing at random. Place two fireflies a few centimeters apart with detectors and light-emitting diodes (LEDs) facing each other and turn off the light [see a in illustration below]; in a few seconds they will synchronize. The precise time required for synchrony depends on how far out of phase the fireflies are when the lights go off; for the most accurate results, you should make several trials.

To change the strength of the coupling between the fireflies, adjust the distance between them. The amount of light from an LED that reaches its neighbor's detector falls off as the square of the distance between them. Does the average time for synchronization follow a similar curve?

With additional units, you can conduct more complex experiments. If you arrange nine fireflies in a grid, for example, the center one is coupled to four neighbors, the four edge ones are coupled to three neighbors each and the corner ones are coupled to two each [see b in Figure 3]. This difference can affect the rate at which they become synchronized. You can also change the grid spacing or interpose opaque barriers to change the number of units each firefly is coupled to.

Because of the way signals pass between our artificial fireflies, each one is effectively coupled only to its neighbors. As a result, a group of fireflies can oscillate in a rhythmic pattern other than simple synchrony. Arrange the units in a straight line about four centimeters apart, and turn out the lights to allow them to synchronize [see c in Figure 3]. Then use a piece of cardboard to break the coupling between the firefly at one end of the line and its neighbor; it will quickly go out of sync with the rest. When you remove the card, the resulting disturbance will propagate rapidly down the line. Changing the distance between fireflies reveals that this propagation speed depends very strongly on the strength of the coupling. (You can do the same experiment with eight oscillators arranged in a ring, in which case the disturbance propagates either clockwise or counterclockwise.)

All the experiments described thus far are based on the assumption that the oscillators' natural frequencies are similar enough that differences between them can be ignored-as in the synchronization proof of Strogatz and Mirollo If you deliberately alter the frequency of one of the fireflies so that it differs significantly from those of the others, however, you can investigate yet another class of phenomena. One "oddball" firefly in the corner of an array of nine, for example, will delay the onset of synchronization for the rest of the array. Moreover, after a while the oddball will pull first a subgroup of its neighbors and then the entire array out of synchrony. In this case, the coupling between oscillators ultimately works to destroy order rather than to create it.

Building Electronic Fireflies

The heart of our device is an LM555 timer [see schematic at bottom]. The 555 is a workhorse for projects requiring periodic behavior. It has an internal switch that closes when the voltage across its control pins exceeds two thirds of its power-supply voltage and opens when the voltage falls below one third of the power-supply voltage. This dependence on voltage rations, rather than on absolute voltages, renders the device insensitive to minor variations, a crucial characteristic for a battery-operated circuit. The capacitor charges through resistor R1 and discharges through both R1 and R2. Four infrared phototransistors (one for each direction) are connected in parallel with R1. When they "see" a flash of light, they conduct charge to the capacitor, quickly increasing the voltage across it and shortening the charging cycle. We have included a 50-kilohm variable resistor in the design so that the blinking of each firefly can be adjusted to roughly the same frequency. A rate of about one flash per second in the dark is best for most experiments.

When the capacitor discharges, the 555's digital output drives four infrared light-emitting diodes (LEDs), which send light to other electronic fireflies to bring them into synchronization. A green LED mimics the color of a natural firefly and tells the human experimenter that the firefly is flashing.

Although it is possible to wire a firefly together on a breadboard, concerns for reproducibility suggest a printed circuit. We used a kit available from Newark Electronics (4801 North Ravenswood, Chicago, IL 60640, (312) 784-5100) to transfer the circuit pattern (reproduced at right) to a copper surface for etching. One kit is sufficient for five fireflies. The cost of the parts for the nine we built, including the kits, was about $180.

Bibliography

SYNCHRONOUS FIREFLIES. J. Buck and E. Buck in Scientific American, Vol. 234, No. 5, pages 74-85; May 1976.

TALKNG TO STRANGERS. David Attenborough's Trials of Life. BBC Television, 1991. Distributed by Time-Life Videos.

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