Firefly Synchronization on Cayley Trees

Please note: The following text and some of the images is not my own. It is taken directly from different sources (references provided at the end). However, the project, its implementation, source code and output is my own.

Introduction
Fireflies provide on of the most spectacular examples of synchronization in nature. At night in certain parts of southeast asia (e.g. Thailand) thousands of male fireflies of some species (e.g. Pteroptyx cribellata, Luciola papilla, etc.) congregate in trees and flash in synchrony. Renato E. Mirrolo and Steven H. Strogatz were the first researchers to model the synchronization of fireflies. I have implemented this model only in my project.

Fireflies can simply be abstracted as oscillators that emit a pulse of light periodically. This type of oscillators is referred to as “pulse coupled oscillators”, and are also used to study biological systems such as neurons and earthquakes. This section describes how time synchronization is achieved in a decentralized fashion between these oscillators.

Mathematical Model
Pulse-coupled oscillators refer to systems that oscillate periodically in time and interact each time they complete an oscillation. This interaction takes the form of a pulse that is perceived by neighboring oscillators.

As a simple mathematical representation, a pulse-coupled oscillator is completely described by its phase function phi(t). This function evolves linearly over time until it reaches the threshold value phi_th. When this happens, the oscillator is said to fire, meaning that it will transmit a pulse and reset its phase. If not coupled to any other oscillator, it will naturally oscillate and fire with a period equal to T. Fig. 1(a) plots the evolution of the phase function during one period when the oscillator is isolated.

The phase function encodes the remaining time until the next firing, which corresponds to an emission of light for a firefly.

Synchronization of Pulse-Coupled Oscillators
Mirollo and Strogatz analyzed spontaneous synchronization phenomena and also derived a theoretical framework based on pulse-coupled oscillators for the convergence of synchrony. When coupled to others, an oscillator is receptive to the pulses of its neighbors. When receiving such a pulse, it will instantly increment its phase by an amount that depends on the current value:

when receiving a pulse

Fig. 1(b) plots the time evolution of the phase when receiving a pulse. The received pulse causes the oscillator to fire early.
The phase increment, delta(phi), depends on the current phase, and it is determined by the Phase Response Curve (PRC), which was chosen to be linear:

Where b is the dissipation factor and epsilon is the amplitude increment. Both factors determine the coupling between oscillators.

The threshold is normalized to one.

It was shown by Mirrolo and Strogatz that if the network is fully meshed and b > 0 and epsilon > 0, the system always converges, i.e. all oscillators will fire as one independently of initial conditions. The time to synchrony is inversely proportional to the product b.epsilon

Cayley Trees
It is a tree in which non-leaf node has a constant number of branches = 3. I modeled the fireflies sitting on each of the node of the cayley tree and saw if their flashes synchronize. The answer was yes and no. Overall, the fireflies did not synchronize but local groups of synchornized fireflies did emerge.

Output

Cayley tree of degree 2

Cayley tree of degree 3

References