Complex harmonics reveal low-dimensional manifolds of critical brain dynamics

The brain needs to perform time-critical computations to ensure survival. A potential solution lies in the nonlocal, distributed computation at the whole-brain level made possible by criticality and amplified by the rare long-range connections found in the brain's unique anatomical structure. This nonlocality can be captured by the mathematical structure of Schrödinger's wave equation, which is at the heart of the complex harmonics decomposition (CHARM) framework that performs the necessary dimensional manifold reduction able to extract nonlocality in critical spacetime brain dynamics. Using a large neuroimaging dataset of over 1000 people, CHARM captured the critical, nonlocal and long-range nature of brain dynamics and the underlying mechanisms were established using a precise whole-brain model. Equally, CHARM revealed the significantly different critical dynamics of wakefulness and sleep. Overall, CHARM is a promising theoretical framework for capturing the low-dimensionality of the complex network dynamics observed in neuroscience and provides evidence that networks of brain regions rather than individual brain regions are the key computational engines of critical brain dynamics.