This paper provides new insights on the convergence of a locally connected network of pulse coupled oscillator (PCOs) (i.e., a bioinspired model for communication networks) to synchronous and desynchronous states, and their implication in terms of the decentralized synchronization and scheduling in communication networks. Bioinspired 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 paper 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 three 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.