  For a class of coupled limit cycle oscillators, we give a condition on a
linear coupling operator that is necessary and sufficient for exponential
stability of the synchronous solution. We show that with certain modifications
our method of analysis applies to networks with partial, time-dependent, and
nonlinear coupling schemes, as well as to ensembles of local systems with
nonperiodic attractors. We also study robustness of synchrony to noise. To this
end, we analytically estimate the degree of coherence of the network
oscillations in the presence of noise. Our estimate of coherence highlights the
main ingredients of stochastic stability of the synchronous regime. In
particular, it quantifies the contribution of the network topology. The
estimate of coherence for the randomly perturbed network can be used as means
for analytic inference of degree of stability of the synchronous solution of
the unperturbed deterministic network. Furthermore, we show that in large
networks, the effects of noise on the dynamics of each oscillator can be
effectively controlled by varying the strength of coupling, which provides a
powerful mechanism of denoising. This suggests that the organization of
oscillators in a coupled network may play an important role in maintaining
robust oscillations in random environment. The analysis is complemented with
the results of numerical simulations of a neuronal network.
  PACS: 05.45.Xt, 05.40.Ca
  Keywords: synchronization, coupled oscillators, denoising, robustness to
noise, compartmental model
