Emergent periodicity in the collective synchronous flashing of fireflies
- BioFrontiers Institute, University of Colorado Boulder, United States
- Department of Physics and Astronomy, Purdue University, United States
- Department of Computer Science, University of New Mexico, United States
- Santa Fe Institute, United States
Figures
Figure 1
Schematic representation of the proposed principle and its theoretical implication: the emergence of periodicity in stochastically flashing P. carolinus fireflies.
(A) Long exposure photograph illustrating flashes in a P. carolinus natural swarm. (B) Overlaid time series of three isolated individual fireflies emitting flash bursts which appear random. The inset (C) shows the burst-like nature of P. carolinus flash events. (D) Interburst distributions for one firefly (purple) and 20 fireflies (blue) insulated from the rest of the swarm. (E) Twenty P. carolinus fireflies flashing in a tent exhibiting the periodic nature of their collective flashing.
Figure 2
Schematic representation of the proposed principle and its theoretical implication: the emergence of periodicity in stochastically flashing P. carolinus fireflies.
(A) A schematic of the flashing pattern of a single isolated firefly. State 0 corresponds to no flashes, and state 1 corresponds to a burst of consecutive flashes. The durations between bursts of single isolated fireflies are highly irregular. (B) In a system with more than one firefly, if a non-flashing firefly sees another one flash, it too starts flashing. Thus, for a system with two fireflies, their bursts are synchronized. After each burst, the time to next burst is determined by which firefly flashes first. Thus, on average, the interburst interval is lower, and hence slightly more regular, than that for a single isolated firefly. (C) As the number of fireflies increases, the probability increases that at least one of them will flash with an interburst interval near the minimum of the distribution for isolated fireflies. This minimum value is expected to be set by the refractory period of the fireflies, which is expected to be similar for all fireflies. Thus, the overall behavior becomes highly periodic with a period approaching this minimum value.
Figure 3
Experimental data vis-à-vis results from analytic theory (no fitting parameters) and computational approach (wherein is a fitting parameter as explained in accompanying text).
Experimental data for each value of N come from three repetitions of experiments at that density. (A) The experimentally measured single firefly interburst distribution (Figure 1D, purple, represented here also in purple). The smoothed version of this distribution (blue curve, detailed methods outlined in the Methods Section) is used as an input in analytical theory and, in conjunction with values, in the computational approach. The inset shows the region between 0–160 s within which most firefly values lie. (B–E) show the interburst distributions for different numbers of fireflies. Our theoretical framework accurately predicts the sharpening of the interburst distribution as increases, without the need of fitting parameters. The value atop each figure is fit by minimizing the two-sided Kolmogorov-Smirnov test between the simulation and experimental distributions (see Figure 7 for a full sensitivity analysis). (F) demonstrates that the standard deviation of the interburst interval distribution decreases with N as predicted by analytic theory (no fitting parameter; see theory section) and the computational approach (using the respective value of best-fit shown with the corresponding distribution in B–E).
Figure 4
A schematic illustrating the computational approach.
The dynamics proceed as follows: for each flashing firefly , follow three simple steps at each timestep. (1) Update according to voltage value. If == 1, update = 0; if is 0, update is 1. (2) If = 0, flash. (3) Update their own voltage based on Equation 6. (A) A single firefly ’s dynamics. Dark bars indicate voltage values from 0 to 1. The start-to-start interflash interval , end-to-start interflash interval , and quiet period , each of which is a random variable for each individual and subject to resampling after each flash event, are indicated below the trace. Flashing state is indicated above, along with the times at which a flash is being actively emitted by the firefly. (B) Schematic of a second firefly , with different parameters, interacting with firefly via integrate-and-fire donation. For simplicity, we only show a one-way interaction here, where donations occur from firefly to firefly and not the reverse. Note the non-linearity in the voltage trace as a flash by firefly triggers a larger gain in voltage between t=4 and t=5 and t=5 and t=6, indicated by the green bars. Firefly ’s second flash is ignored by firefly since it is already flashing (t=11, t=12).
Figure 5
Emergence of collective periodicity in large swarms.
(A–E) Visual demonstration of the emergence of a collective periodicity above as ranges between 0–1 for each value of N, including (E) , a value outside the scope of our experimental observations but that is relevant for the theoretical analysis.
The lack of coupling in the first few rows produces noisy and cluttered collective interburst intervals as flashes from any individual are uncorrelated with those from its neighbors. As the coupling constant increases, a consistent interburst interval emerges at the peak of each distribution. (F) The relationship between the most probable interburst interval (the distribution peak) as and vary. The shaded regions represent the standard error of the distributions for each density. For small values of beta, the collective produces noisy distributions where the pulsatile coupling of flashes is not quite enough to pull the starts of bursts into alignment. However, as the coupling constant increases, individual flashes begin to trigger subsequent flashes in neighboring fireflies, causing the quiet periods of the individuals to line up and the emergence of a collective frequency at the fastest interval in each burst cycle. Each higher density simulated causes the peak of the distribution to both shift slightly downwards and become less variant, as it is progressively more likely for one individual in the swarm to drive the collective frequency towards intervals on the short end of the input distribution. A cartoon of this effect is shown in .
Figure 6
Schematic illustration demonstrating the evolution of the collective burst distribution, i.e., the distribution of time intervals between collective bursts, , with increasing number of fireflies, .
corresponds to the intrinsic burst distribution of a single firefly, . Evidently, the distribution of time intervals between collective bursts becomes a sharply peaked distribution with maximum probability peaked at a value larger than .
Figure 7
Two-sided Kolmogorov–Smirnov test results between the simulation results and experimental results at each and .
For the two-sided Kolmogorov–Smirnov test, the null hypothesis states that the two compared distributions can be drawn from the same underlying distribution: effectively, accepting the null hypothesis accepts the statistical probability that the distributions do not differ. All distributions were generated from ten simulations, each of 200000 simulation timesteps/30 min of real time. The best values for each are .
Additional files
Download links
A two-part list of links to download the article, or parts of the article, in various formats.
Downloads (link to download the article as PDF)
Open citations (links to open the citations from this article in various online reference manager services)
Cite this article (links to download the citations from this article in formats compatible with various reference manager tools)
Emergent periodicity in the collective synchronous flashing of fireflies
eLife 12:e78908.
https://doi.org/10.7554/eLife.78908
