  Hallmarks of criticality, such as power-laws and scale invariance, have been
empirically found in cortical networks and it has been conjectured that
operating at criticality entails functional advantages, such as optimal
computational capabilities, memory, and large dynamical ranges. As critical
behavior requires a high degree of fine tuning to emerge, some type of
self-tuning mechanism needs to be invoked. Here we show that, taking into
account the complex hierarchical-modular architecture of cortical networks, the
singular critical point is replaced by an extended critical-like region which
corresponds --in the jargon of statistical mechanics-- to a Griffiths phase.
Using computational and analytical approaches, we find Griffiths phases in
synthetic hierarchical networks and also in empirical brain networks such as
the human connectome and the caenorhabditis elegans one. Stretched critical
regions, stemming from structural disorder, yield enhanced functionality in a
generic way, facilitating the task of self-organizing, adaptive, and
evolutionary mechanisms selecting for criticality.
