Mathematicians explain how some fireflies blink synchronously

If you stake out in Pennsylvania’s Cook State Forest at the right time of year, you can see one of nature’s magnificent light shows: swarms of fireflies synchronizing their flashes like Christmas lights in the dark.

A new study by Pitt mathematicians shows that mathematics borrowed from neuroscience can describe how swarms of these unique insects coordinate their light show and capture important details about their behavior in the wild.

“This firefly has a rapid succession of flashes and 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 for modeling that could capture a lot of the functionality, and we were curious to see how far we could take it.”

Male fireflies produce a glow from their abdomens to call potential mates and emit flashing patterns in the dark to court females of their own kind. Synchronous fireflies of the species Photinus carolinus take this a step further and coordinate their blinking across entire swarms. It’s a rare trait — there’s only a handful of such species in North America — and the showy lights they produce draw crowds to places where the insects are known to congregate.

They’ve also piqued the interest of mathematicians who want to understand how they synchronize their blinks. This is just one example of how synchronization can evolve from chance, a process that has fascinated mathematicians for centuries. A famous 16th-century example showed that pendulum clocks hung side by side synchronized by vibrations traveling through the wall, and the same branch of mathematics can be used to describe everything from the movement of the intestines to the clapping of spectators.

“Synchronism is important for a lot of things, good and bad,” said co-author Bard Ermentrout, Distinguished Professor of Mathematics at The Dietrich School. “Physicists, mathematicians, we are all interested in synchronization.”

To crack the firefly’s light show, the Pitt team used a more complex model called the “elliptic burster,” which is used to describe how brain cells behave. The duo, along with then-student Madeline McCrea (A&S ’22), released details of their model on October 26 in the Journal of the Royal Society Interface.

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

They found that varying the distances at which the fireflies could “see” and respond to each other altered the insects’ light show: by adjusting the parameters, they could create blinking patterns that looked like either waves or spirals.

The results are consistent with several recently published observations of true synchronous fireflies — for example, that individual fireflies are inconsistent while groups blink more regularly, and that by the time new fireflies join the swarm, they are already perfectly in step.

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

The math also makes some predictions that could inform firefly research — for example, light pollution and the time of day can both change the patterns fireflies create by altering how well they can see each other’s blinks.

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

The team is the first to use this particular brain cell framework to model fireflies, which several different research teams are trying to understand using different types of mathematics. “It’s more of a Wild West research topic,” Ermentrout said. “It’s still early, 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 drive to the Cook State Forest to see if he could spot his research objects firsthand.

“I convinced my wife to go away for a few days in the middle of the season,” he said. “It’s not clear if we ever saw synchronized activity, but there were all sorts of fireflies all around us. It was amazing.”