  We explore systems of pulse-coupled oscillators beyond the mean-field limit
[R.E. Mirollo and S.H. Strogatz, {SIAM J. Appl. Math.} {\bf 50}, 1645 (1990)]
by means of a manageable description which leads to a great simplification of
the dynamics. Sufficient conditions for synchronization are exactly obtained
for a ring with nearest-neighbor directed interactions, turning out to be the
same as in the all-to-all case. Our analysis shows new synchronization
mechanisms occurring when the interactions are local.
