We propose a novel generalized cellular automaton (GCA) model for discrete-time pulse-coupled oscillators and study the emergence of synchrony. Our discrete model brings the study of coupled oscillators into the realm of combinatorics and probability. Given a finite simple graph and an integer $n\ge 3$, we place identical $n$-state oscillators on vertices with the following weak coupling along the edges: each oscillator inhibits its phase update if it has at least one neighboring oscillator at a particular "blinking" state and if its state is ahead of this blinking state. We obtain conditions on initial configurations and on network topologies for which states of all vertices eventually synchronize. For finite graphs, we show that our GCA model synchronizes arbitrary initial configurations on paths, trees, and, with random perturbation, any connected graph. In particular, we obtain the following local-global principle for tree networks: for $n\in \{3,4,5,6\}$, any $n$-periodic network on a tree synchronizes arbitrary initial configuration if and only if the maximum degree of the tree is less than $n$.

For infinite graphs, we study our discrete dynamics on the integer lattice $\mathbb{Z}^{d}$ using probabilistic methods. For $d=1$, our model is related to annihilating particle systems and random walks, and by this connection, we show that starting with uniform product measure, the probability of not having synchrony on a finite fixed interval at time $t$ decays to zero in the order of $O(t^{-1/2+o(1)})$. For $d\geq 2$ and for $n\neq 4$, our model shows spiraling autowave behavior, and hence non-synchrony, in resemblance of the famous Belousov–Zhabotinsky reaction. While this behavior is common to similar models for excitable media, the four-color model is critical and characteristic behavior: we seem to have clustering on $\mathbb{Z}^{d}$ through a complex wave process, regardless of dimension. This may be thought of as a four-state Ising spin system with deterministic dynamics, with convergence toward global ferromagnetism.

If time allows, we also discuss an asynchronous and continuous-state generalization of our GCA model. Despite being continuous model, we can prove similar synchronization theorems for trees. As an application, we combine this model with a distributed spanning tree algorithm and obtain a distributed self-stabilizing algorithm for clock synchronization in a system of processors with identical local clocks on any connected graph with $v$ nodes and diameter $d$, which runs in $O(d+\sqrt{v}\log^{*}v)$ time.