  Network couplings of oscillatory large-scale systems, such as the brain, have
a space-time structure composed of connection strengths and signal transmission
delays. We provide a theoretical framework, which allows treating the spatial
distribution of time delays with regard to synchronization, by decomposing it
into patterns and therefore reducing the stability analysis into the tractable
problem of a finite set of delay-coupled differential equations. We analyse
delay-structured networks of phase oscillators and we find that, depending on
the heterogeneity of the delays, the oscillators group in phase-shifted,
anti-phase, steady, and non-stationary clusters, and analytically compute their
stability boundaries. These results find direct application in the study of
brain oscillations.
