Proceedings of the National Academy of Sciences

Defining and measuring proximity to criticality based on univariate time series

J. Samuel Sooter, Antonio J. Fontenele, Andrea K. Barreiro, Cheng Ly, Keith B. Hengen, Woodrow L. Shew

Published Oct 28, 2026• 11 citations

Abstract

Diverse social, engineered, and biological systems are made up of multiple interacting dynamical components; to work properly, certain components must be carefully regulated so that their dynamics avoid instability. However, many desirable properties, like optimized computation and control, emerge close to the tipping point between stability and instability, i.e., near criticality. Claims that such dynamical systems operate near criticality have been controversial because it is difficult to assess proximity to criticality when control parameters are unknown, hard to manipulate experimentally, and fluctuating in response to changing environmental or internal conditions. Moreover, observations of system dynamics are often limited to time series measurements of one or a few components. Here we develop a temporal renormalization group theory for defining criticality for univariate time series models and an information theoretic definition of proximity to criticality that is independent of model parameterization. We apply this theory to order-n autoregressive models resulting in a data-analytic strategy for time-resolved estimates of proximity to criticality based on measured univariate time series. After benchmarking our approach on diverse ground-truth cases, we apply it to recordings of spiking activity in the mammalian brain. Our estimate of proximity to criticality predicts behavior and suggests that brain dynamics shift closer to criticality during wakefulness and shift away during deep sleep.

DOI: 10.1073/pnas.2526984123

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