Synchronization of finite-state pulse-coupled oscillators

We propose a simple cellular automata model for discrete-time pulse-coupled oscillators and study its network behavior. Given a finite simple graph and an integer n ≥ 3, each vertex will be considered as an identical oscillator of period n with the following weak coupling along the edges: any oscillator with a particular "blinking" state will postpone the update of its neighbors whose states are "counterclockwise" to the blinking state. We obtain various conditions on initial configurations or on network topologies for which the states of all vertices eventually synchronize. In particular, we show that our system exhibits properties of self-stabilization, which is a central concept in distributed control. We first show that the network on any finite path synchronizes regardless of the period and initial configuration, with at most linear time in the size of the path and quadratic time in the period n. This leads to a stronger synchronization property of a randomized version of our model, which ensures synchrony with high probability regardless of the period, initial configuration, and network structure, only given the connectedness of the network. The main result is the following local-global principle for tree networks: for any n ∈ {3,4,5,6}, any n-periodic network on a tree synchronizes arbitrary initial configuration if and only if the maximum degree of the tree is less than the period n. We also present a simple neural network construction for each individual oscillator whose network behavior obeys the 6-periodic network on graphs. KeywordsSynchronization, self-stabilization, distributed control, pulse-coupled oscillators, cellular automata, trees, and neural networks