Rounding of abrupt phase transitions in brain networks

The observation of critical-like behavior in cortical networks represents a major step forward in elucidating how the brain manages information. Understanding the origin and functionality of critical-like dynamics, as well as their robustness, is a major challenge in contemporary neuroscience. Here, we present an extensive numerical study of a family of simple dynamic models, which describe activity propagation in brain networks through the integration of different neighboring spiking potentials, mimicking basic neural interactions. The requirement of signal integration may lead to discontinuous phase transitions in networks that are well described by the mean-field approximation, thus preventing the emergence of critical points in such systems. Here we show that criticality in the brain is instead robust, as a consequence of the hierarchical organization of the higher layers of cortical networks, which signals a departure from the mean-field paradigm. We show that, in finitedimensional hierarchical networks, discontinuous phase transitions exhibit a rounding phenomenon and turn continuous for values of the topological dimension D ≤ 2, due to the presence of structural or topological disorder. Our results may prove significant in explaining the observation of traits of critical behavior in large-scale measurements of brain activity. Rounding of abrupt phase transitions in brain networks 2