  For networks of pulse-coupled oscillators with complex connectivity, we
demonstrate that in the presence of coupling heterogeneity precisely timed
periodic firing patterns replace the state of global synchrony that exists in
homogenous networks only. With increasing disorder, these patterns persist
until they reach a critical temporal extent that is of the order of the
interaction delay. For stronger disorder these patterns cease to exist and only
asynchronous, aperiodic states are observed. We derive self-consistency
equations to predict the precise temporal structure of a pattern from the
network heterogeneity. Moreover, we show how to design heterogenous coupling
architectures to create an arbitrary prescribed pattern.
