  We consider a network of identical pulse-coupled oscillators with delay and
all-to-all coupling. We demonstrate that the discontinuous nature of the
dynamics induces the appearance of isochronous regions---subsets of the phase
space filled with periodic orbits having the same period. For fixed values of
the network parameters each such isochronous region corresponds to a subset of
initial states on an appropriate surface of section with non-zero dimension
such that all periodic orbits in this set have qualitatively similar dynamical
behaviour. We analytically and numerically study in detail such an isochronous
region, give a proof of its existence, and describe its properties. We further
describe other isochronous regions that appear in the system.
