  We study the synchronization behavior of a noisy network in which each system
is driven by two sources of state-dependent noise: (1) an intrinsic noise which
is common among all systems and can be generated by the environment or any
internal fluctuations, and (2) a coupling noise which is generated by
interactions with other systems. After providing sufficient conditions that
foster synchronization in networks of general noisy systems, we focus on weakly
coupled networks of noisy oscillators and, using the first- and second-order
phase response curves (PRCs), we derive a reduced order stochastic differential
equation to describe the corresponding phase evolutions. Finally, we derive
synchronization conditions based on the PRCs and illustrate the theoretical
results on a couple of models.
