A generalized scaling law to characterize coordinated brain activity

Abstract

Complex ecological systems obey universal scaling laws such as Taylors law, which links population variance to the sample mean. A natural question is whether other complex systems with integrated and segregated functional networks, such as the human brain, follow analogous scaling laws, and how these laws behave under synchronized activity. Addressing this is challenging because neural signals at all measurement scales contain both positive and negative values, whereas Taylors law was developed for non-negative ecological data. To overcome this limitation, we replace the sample mean with the root-mean-square (RMS) of detrended neural activity and formulate a generalized spatial scaling law applicable to signed data. This framework extends Taylor-like scaling to neural systems and other complex systems across disciplines and allows examination of how coordinated, synchronized activity shapes these laws. Synchrony is quantified using a dedicated metric of functional coordination. Analytical derivations and numerical simulations using multivariate Poisson, binomial, uniform, and gamma distributions demonstrate that the scaling exponent is inversely related to synchrony. Applying this framework to human fMRI data from three large lifespan cohorts (N =840, ages 18-88) reveals distinct age-related trajectories of the scaling exponent. Healthy aging is characterized by a substantial synchrony-induced reduction in the exponent. The synchrony-scaling relationship is stable during resting-state activity but progressively shifts during naturalistic tasks across the lifespan. Limbic, subcortical, and cerebellar regions show particularly strong synchrony-scaling coupling, indicating preservation at subnetwork levels. Finally, individuals with ADHD exhibit disrupted synchrony-scaling relationships, underscoring the potential clinical utility of this novel metric.

Significance statementWe introduce a generalized scaling law that extends Taylors law from strictly non-negative ecological counts to signals with positive and negative fluctuations, such as detrended neural activity and other real-time measurements. This revision enables systematic characterization of scale-invariant variability in a broad range of complex systems. By applying this framework to large fMRI datasets spanning the adult lifespan, during both rest and naturalistic tasks, we show that synchrony-driven changes in the scaling exponent capture robust, age-dependent reconfigurations of brain-wide coordination. The synchrony-scaling relationship thus offers a compact, interpretable, and modality-agnostic marker of healthy brain aging and establishes a principled link between coordination dynamics and variability scaling in neural systems.