Deadlocks in the synchronization of pulse-coupled oscillators on star graphs

Some convergence proofs for systems of oscillators with inhibitory pulse coupling assume that all initial phases reside in one half of their domain. A violation of this assumption can trigger deadlocks that prevent synchronization. We analyze the conditions for such deadlocks in star graphs, characterizing the domain of initial states leading to deadlocks and deriving its fraction of the state space. The results show that convergence is feasible from a wider range of initial phases. The same type of deadlock occurs in random graphs.

Figure 1

The center node fires. Phase constellation before the fire event (left) and after reception of the pulse by leaf nodes (right). The antipodal phase gap increases; all other gaps decrease.

Figure 2

Fraction of oscillator configurations leading to a deadlock in the synchronization of star graphs. Deadlocks occur only for a sufficiently large response parameter $a>2/n$ (small circles). They become more frequent with increasing $a$ and number of leaf nodes $n$. Shown are the curves for $n\in \{4,...,10\}$. For the maximum response $a=1$, the oscillators reset their phases straight to the beginning or end of their cycle. This also holds when all configurations with all ${\delta }_k\le \frac{1}{2}$ (dashed lines) lead to a deadlock (if $n>2$).

Figure 3

Ratio of initial configurations that do not lead to synchrony for 25 connected random graphs $G(n,p)$ with $n$ nodes and connection probability $p$ as a function of the response parameter $a$. There are many nonsynchronizing configurations for weak interactions (small $a$), all of which lead to periodic states. For strong interaction (large $a$), most configurations lead to synchrony, but the ones that do not synchronize have at least one deadlock node. The $(n,p)$-values of those graphs are given and their curves are marked. For each graph, the dot marks the smallest $a$ for which more than half of the nonsynchronizing configurations exhibit deadlocks.

Figure 4

Examples of nonsynchronizing configurations. Upper row: graph and two sets of initial phase configurations, indicated by the directions of the arrows. Lower row: pulses over a few periods after a sufficient convergence time. Left side: a periodic state that does not further synchronize despite repeated communication. Right side: some nodes in a deadlocked state [marked black in (b)], in which they are prevented from firing; the remaining nodes have converged to a periodic state.