  Many of the amazing functional capabilities of the brain are collective
properties stemming from the interactions of large sets of individual neurons.
In particular, the most salient collective phenomena in brain activity are
oscillations, which require the synchronous activation of many neurons. Here,
we analyse parsimonious dynamical models of neural synchronisation running on
top of synthetic networks that capture essential aspects of the actual brain
anatomical connectivity such as a hierarchical-modular and core-periphery
structure. These models reveal the emergence of complex collective states with
intermediate and flexible levels of synchronisation, halfway in the
synchronous-asynchronous spectrum. These states are best described as broad
Griffiths-like phases, i.e. an extension of standard critical points that
emerge in structurally heterogeneous systems. We analyse different routes
(bifurcations) to synchronisation and stress the relevance of 'hybrid-type
transitions' to generate rich dynamical patterns. Overall, our results
illustrate the complex interplay between structure and dynamics, underlining
key aspects leading to rich collective states needed to sustain brain
functionality.
