  Physiological networks are usually made of a large number of biological
oscillators evolving on a multitude of different timescales. Phase oscillators
are particularly useful in the modelling of the synchronization dynamics of
such systems. If the coupling is strong enough compared to the heterogeneity of
the internal parameters, synchronized states might emerge where phase
oscillators start to behave coherently. Here, we focus on the case where
synchronized oscillators are divided into a fast and a slow component so that
the two subsets evolve on separated timescales. We assess the resilience of the
slow component by reducing the dynamics of the fast one and evaluating the
variance of the phase deviations when the oscillators in the two components are
subject to noise with possibly distinct correlation times. From the general
expression for the variance, we consider specific network structures and show
how the noise transmission between the fast and slow components is affected.
Interestingly, we find that oscillators that are among the most robust when
there is only a single timescale, might become the most vulnerable when the
system separates into two timescales. We also find that layered networks seem
to be insensitive to timescale separations when the noise has homogeneous
correlation time.
