  This work provides new insights on the convergence of a locally connected
network of pulse coupled oscillator (PCOs) (i.e., a bio-inspired model for
communication networks) to synchronous and desynchronous states, and their
implication in terms of the decentralized synchronization and scheduling in
communication networks. Bio-inspired techniques have been advocated by many as
fault-tolerant and scalable alternatives to produce self-organization in
communication networks. The PCO dynamics in particular have been the source of
inspiration for many network synchronization and scheduling protocols. However,
their convergence properties, especially in locally connected networks, have
not been fully understood, prohibiting the migration into mainstream standards.
This work provides further results on the convergence of PCOs in locally
connected networks and the achievable convergence accuracy under propagation
delays. For synchronization, almost sure convergence is proved for $3$ nodes
and accuracy results are obtained for general locally connected networks
whereas, for scheduling (or desynchronization), results are derived for locally
connected networks with mild conditions on the overlapping set of maximal
cliques. These issues have not been fully addressed before in the literature.
