Frequency adjustment and synchrony in networks of delayed pulse-coupled oscillators

We introduce a system of pulse-coupled oscillators that can change both their phases and frequencies and prove that when there is a separation of time scales between phase and frequency adjustment the system converges to exact synchrony on strongly connected graphs with time delays. The analysis involves decomposing the network into a forest of tree-like structures that capture causality. These results provide a robust method of sensor net synchronization as well as demonstrate a new avenue of possible pulse-coupled oscillator research.

Figure 1

Numerical simulations demonstrate robustness (bottom left) to large frequency response over 100 trials on a random geometric graph (top left) with phase resetting curves $\left(B=0.5\right)$ and frequency response curves (top right), $\tau =0.05$, initial frequency range $[1,1.2]$, and $q=0.5$. The system remains robust (bottom right) to the addition of distance based delays ${\tau }_{ij}\in [0.5,0.1]$.

Figure 2

The proof of lemma t3 involves an induction on nodes in an order consistent with the signal reception (black arrows). The result of the induction establishes that all nodes fit into two different classes, represented here as black and gray nodes.

Figure 3

The relationship between oscillator firing times and network topology reveals key structure. Since an oscillator processes only a single signal each round, a round of firing induces a tree-like set of dependencies, in this case with $l{\to }^{n}k{\to }^{n}j{\to }^{n}i$ and $k{\to }^{n}l$.

Figure 4

From round to round firing trees may change. However, lemmas t8 and t9 establish bounds for pair differences across any of the three types of edges. For instance, in round $n$, lemma t9 can apply to edge $(8,10)$ with dependencies on oscillators 2, 7, and 11, as well as edge (9,13) with dependencies on oscillators 7, 8, 11, and 14.

Figure 5

Under the same parameters as described in the legend of Fig. 1, but when oscillators are additionally subjected to random errors in frequencies, the system is still able to approximate synchrony (results from 100 simulations shown). For small error rates, the system can display a troubling continual decrease in frequency (top left), whereas for larger error rates (bottom) and for errors that are mean-reverting with $\eta =0.1\%$ (right), the oscillator frequency range stabilizes.