Cognitive function requires the coordination of neural activity across many scales, from neurons and circuits to large-scale networks. As such, it is unlikely that an explanatory framework focused upon any single scale will yield a comprehensive theory of brain activity and cognitive function. Modelling and analysis methods for neuroscience should aim to accommodate multiscale phenomena. Emerging research now suggests that multi-scale processes in the brain arise from so-called critical phenomena that occur very broadly in the natural world. Criticality arises in complex systems perched between order and disorder, and is marked by fluctuations that do not have any privileged spatial or temporal scale. We review the core nature of criticality, the evidence supporting its role in neural systems and its explanatory potential in brain health and disease.Keywords: Bifurcations; metastability; multistability; dynamics; power-law; cognition. Highlights Criticality is a wide-spread phenomenon in natural systems  Criticality provides a unifying framework to model and understand brain activity and cognitive function  Substantial evidence now supports the hypothesis that the brain operates near criticality  We review the role of criticality in healthy and pathological brain dynamics  Caveats and pitfalls regarding the assessment of criticality in the brain are discussed *

Brain structure reflects the influence of evolutionary processes that pit the costs of its anatomical wiring against the computational advantages conferred by its complexity. We show that cost-neutral 'mutations' of the human connectome almost inevitably degrade its complexity and disconnect high-strength connections to prefrontal network hubs. Conversely, restoring the peripheral location and strong connectivity of empirically observed hubs confers a wiring cost that the brain appears to minimize. Progressive cost-neutral randomization yields daughter networks that differ substantially from one another and results in a topologically unstable phenomenon consistent with a phase transition in complex systems. The fragility of hubs to disconnection shows a significant association with the acceleration of gray matter loss in schizophrenia. Together with effects on wiring cost, we suggest that fragile prefrontal hub connections and topological volatility act as evolutionary influences on brain networks whose optimal set point may be perturbed in neuropsychiatric disorders.

Traveling patterns of neuronal activity -brain waves -have been observed across a breadth of neuronal recordings, states of awareness, and species, but their emergence in the human brain lacks a firm understanding. Here, we analyze the complex nonlinear dynamics that emerge from modeling large-scale spontaneous neural activity on a whole-brain network derived from human tractography. We find a rich array of three-dimensional wave patterns, including traveling waves, spiral waves, sources, and sinks. These patterns are metastable, such that multiple spatiotemporal wave patterns are visited in sequence. Transitions between states correspond to reconfigurations of underlying phase flows, characterized by nonlinear instabilities. These metastable dynamics accord with empirical data from multiple imaging modalities, including electrical waves in cortical tissue, sequential spatiotemporal patterns in resting-state MEG data, and large-scale waves in human electrocorticography. By moving the study of functional networks from a spatially static to an inherently dynamic (wave-like) frame, our work unifies apparently diverse phenomena across functional neuroimaging modalities and makes specific predictions for further experimentation.

Rapid reconfigurations of brain activity support efficient neuronal communication and flexible behaviour. Suboptimal brain dynamics is associated to impaired adaptability, possibly leading to functional deficiencies. We hypothesize that impaired flexibility in brain activity can lead to motor and cognitive symptoms of Parkinson’s disease (PD). To test this hypothesis, we studied the ‘functional repertoire’—the number of distinct configurations of neural activity—using source-reconstructed magnetoencephalography in PD patients and controls. We found stereotyped brain dynamics and reduced flexibility in PD. The intensity of this reduction was proportional to symptoms severity, which can be explained by beta-band hyper-synchronization. Moreover, the basal ganglia were prominently involved in the abnormal patterns of brain activity. Our findings support the hypotheses that: symptoms in PD relate to impaired brain flexibility, this impairment preferentially involves the basal ganglia, and beta-band hypersynchronization is associated with reduced brain flexibility. These findings highlight the importance of extensive functional repertoires for correct behaviour.