Mathematicians explain how some fireflies flash in sync

Settle into Pennsylvania’s Cook State Forest at the right time of year and you can see one of nature’s great light shows: swarms of fireflies that synchronize their flashes like garlands of lights from Christmas in the dark.

A new study by Pitt’s mathematicians shows that math borrowed from neuroscience can describe how swarms of these unique insects coordinate their light show, capturing key details about their behavior in the wild.

“This firefly has a quick sequence of flashes, then a big pause before the next burst,” said Jonathan Rubin, professor and chair of the mathematics department at the Kenneth P. Dietrich School of Arts and Sciences. “We knew of a good framework to model this that could capture a lot of functionality, and we were curious how far we could push it.”

Male fireflies produce a glow from their abdomens to call out potential mates, sending out flashing patterns in the dark to woo females of their own species. Synchronous fireflies of the species Photinus carolinus go one step further by coordinating their flashing through entire swarms. It’s a rare trait – there are only a handful of such species in North America – and the striking lights they produce draw crowds to places where insects are known to congregate.

They have also aroused the interest of mathematicians seeking to understand how they synchronize their blinks. This is just one example of how timing can evolve from chance, a process that has puzzled mathematicians for centuries. A famous example from the 1600s showed that pendulum clocks hung next to each other synchronize themselves through vibrations passing through the wall, and the same branch of mathematics can be used to describe everything from the action of the intestines to applause from members of the public.

“Synchrony is important for many things, good and bad,” said co-author Bard Ermentrout, a distinguished professor of mathematics at the Dietrich School. “Physicists, mathematicians, we are all interested in synchronization.”

To crack the fireflies’ light show, the Pitt team used a more complex model called an “elliptical spark gap” that is used to describe the behavior of brain cells. The duo, along with then-undergraduate student Madeline McCrea (A&S ’22), released details of their model Oct. 26 in the Royal Society Interface Journal.

The first step was to simulate the blinks of a single firefly, then grow a pair to see how they matched their blink rates. Next, the team moved to a larger swarm of simulated insects to see how number, distance and flight speed affect the resulting blinks.

Varying the distances that each firefly could “see” each other and respond to each other changed the insects’ light show, they found: By adjusting the settings, they could produce blinking patterns that resembled waves or spirals.

The results match several recently published observations of real-life synchronous fireflies, for example, that individual fireflies are inconsistent while groups blink more regularly, and that when new fireflies join the swarm they are already perfectly in time.

“It captured a lot of the finer detail that they saw in biology, which was cool,” Ermentrout said. “We didn’t expect that.”

Mathematics also makes predictions that could inform firefly research – for example, light pollution and the time of day can alter the patterns produced by fireflies by changing how they can see each other’s blinks .

McCrea worked on the research as an undergraduate student supported by the department’s Painter Fellowship, which provided her with funding to work on the project throughout the summer. “She was awesome working on this project and really persistent,” Rubin said.

The team is the first to use this particular framework of brain cells to model fireflies, which several research teams are trying to understand using different types of math. “It’s more of a Wild West research topic,” Ermentrout said. “This is the beginning, and who knows where things will go from here?”

Ermentrout and Rubin also hope the math will capture the imagination of those inspired by the glow of fireflies. In the midst of this project, Rubin himself decided to go to Cook State Forest to see if he could spot his research subjects first hand.

“I convinced my wife to go on a trip for a few days in high season,” he said. “It’s not clear that we ever saw any synchronized activity, but there were all kinds of fireflies around us. It was amazing.”