Griffiths phases and the stretching of criticality in brain networks.

Hallmarks of criticality, such as power-laws and scale invariance, have been empirically found in cortical-network dynamics and it has been conjectured that operating at criticality entails functional advantages, such as optimal computational capabilities, memory and large dynamical ranges. As critical behaviour 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 that 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 that of Caenorhabditis elegans. 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.