Resilience of the slow component in timescale separated synchronized oscillators Provisionally Accepted

Melvyn Tyloo^1*

• ¹Los Alamos National Laboratory (DOE), United States

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, first, reducing the dynamics of the fast one using Mori-
Zwanzig formalism. Second, we evaluate 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 undergoes a timescale separation. We also find that layered networks
seem to be insensitive to such timescale separations.