Systems of individual oscillators that are coupled via regularly emitted pulses are ubiquitous in biology. Examples include fireflies that synchronize their blinking light pattern and neurons that communicate via action potentials. We aim to understand the emergence of synchronous behavior in these systems mathematically using cellular automata. We generalize the fireflies cellular automaton developed by Lyu to study the impact of coupling range on inhibitory pulse-coupled oscillators in one-dimension. From numerical simulation, we discover four ordered phase-regimes: phase-fixating, clustering, over-coupled, and chaotic. We adapt mathematical techniques developed by Fisch for the cyclic cellular automaton to prove that synchronization does not occur when the coupling range is small. This complements a recent result by Lyu and Sivakoff, who show that local synchronization (clustering) occurs at intermediate coupling range.