Desynchronization of pulse-coupled oscillators with delayed excitatory coupling

Collective behavior of pulse-coupled oscillators has been investigated widely. As an example of pulse-coupled networks, fireflies display many kinds of flashing patterns. Mirollo and Strogatz (1990) proposed a pulse-coupled oscillator model to explain the synchronization of South East Asian fireflies ({\itshape Pteroptyx malaccae}). However, transmission delays were not considered in their model. In fact, the presence of transmission delays can lead to desychronization. In this paper, pulse-coupled oscillator networks with delayed excitatory coupling are studied. Our main result is that under reasonable assumptions, pulse-coupled oscillator networks with delayed excitatory coupling can not achieve complete synchronization, which can explain why another species of fireflies ({\itshape Photinus pyralis}) rarely synchronizes flashing. Finally, two numerical simulations are given. In the first simulation, we illustrate that even if all the initial phases are very close to each other, there could still be big variations in the times to process the pulses in the pipeline. It implies that asymptotical synchronization typically also cannot be achieved. In the second simulation, we exhibit a phenomenon of clustering synchronization

A firefly-inspired scheme for energy-efficient transmission scheduling using a self-organizing method in a wireless sensor network

Various types of natural phenomena are regarded as primary sources of information for artificial occurrences that involve spontaneous synchronization. Among the artificial occurrences that mimic natural phenomena are Wireless Sensor Networks (WSNs) and the Pulse Coupled Oscillator (PCO), which utilizes firefly synchronization for attracting mating partners. However, the PCO model was not appropriate for wireless sensor networks because sensor nodes are typically not capable to collect sensor data packets during transmission (because of packet collision and deafness). To avert these limitations, this study proposed a self-organizing time synchronization algorithm that was adapted from the traditional PCO model of fireflies flashing synchronization. Energy consumption and transmission delay will be reduced by using this method. Using the proposed model, a simulation exercise was performed and a significant improvement in energy efficiency was observed, as reflected by an improved transmission scheduling and a coordinated duty cycling and data gathering ratio. Therefore, the energy-efficient data gathering is enhanced in the proposed model than in the original PCO-based wave-traveling model. The battery lifetime of the Sensor Nodes (SNs) was also extended by using the proposed model

Phase locking in networks of synaptically coupled McKean relaxation oscillators

We use geometric dynamical systems methods to derive phase equations for networks of weakly connected McKean relaxation oscillators. We derive an explicit formula for the connection function when the oscillators are coupled with chemical synapses modeled as the convolution of some input spike train with an appropriate synaptic kernel. The theory allows the systematic investigation of the way in which a slow recovery variable can interact with synaptic time scales to produce phase-locked solutions in networks of pulse coupled neural relaxation oscillators. The theory is exact in the singular limit that the fast and slow time scales of the neural oscillator become effectively independent. By focusing on a pair of mutually coupled McKean oscillators with alpha function synaptic kernels, we clarify the role that fast and slow synapses of excitatory and inhibitory type can play in producing stable phase-locked rhythms. In particular we show that for fast excitatory synapses there is coexistence of a stable synchronous, a stable anti-synchronous, and one stable asynchronous solution. For slower synapses the anti-synchronous solution can lose stability, whilst for even slower synapses it can regain stability. The case of inhibitory synapses is similar up to a reversal of the stability of solution branches. Using a return-map analysis the case of strong pulsatile coupling is also considered. In this case it is shown that the synchronous solution can co-exist with a continuum of asynchronous states