 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.
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Figure 1: ARTIFICIAL
FIREFLIES synchronize their flashes as each is affected by the other's
blinking. They are controlled by a simple oscillator consisting of a
timing circuit, a light-emitting diode and a photoresistor to sense
incoming light. Multiple fireflies can be used for experiments that
exploit rhythms that go beyond simple synchrony. |
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?
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Figure 2: Advancing
a flash |
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.
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Figure 3: Some
arrangements of artificial fireflies |
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
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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.
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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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