A generalized scaling law to characterize coordinated brain activity

Complex ecological systems obey universal scaling laws such as Taylors law, which links population variance to the sample mean. 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, remains unknown. Progress has been hindered because neural signals at all measurement scales contain both positive and negative values, whereas Taylors law was developed for non-negative ecological data. Here we introduce a generalized spatial scaling law that replaces the sample mean with the root-mean-square (RMS) of detrended activity, extending Taylor-like scaling to signed multivariate time series in neural and other complex systems. Within this framework, we quantify synchrony using a dedicated metric of functional coordination and show analytically, and via numerical simulations using multivariate Poisson, binomial, uniform and gamma distributions, that the scaling exponent is inversely related to synchrony. Applying this approach to human fMRI data from three large lifespan cohorts (N = 840, ages 18-88) reveals distinct age-related trajectories of the scaling exponent, with healthy aging characterized by a substantial synchrony-induced reduction in the exponent. The synchrony-scaling relationship remains stable during resting-state activity but progressively shifts during naturalistic tasks across the lifespan, and is particularly strong in limbic, subcortical, and cerebellar subnetworks, indicating preservation at subnetwork levels. Finally, individuals with ADHD exhibit disrupted synchrony-scaling coupling, highlighting the potential clinical and translational utility of this generalized scaling metric.