Tracking the distance to criticality in systems with unknown noise

Many real-world systems undergo abrupt changes in dynamics as they move across critical points, often with dramatic consequences. Much existing theory on identifying the time-series signatures of nearby critical points -- such as increased variance and slower timescales -- is derived for the case of fixed, low-amplitude noise. However, real-world systems are often corrupted by unknown levels of noise that can distort these temporal signatures. Here we aimed to develop noise-robust indicators of the distance to criticality (DTC) for systems affected by dynamical noise in two cases: when the noise amplitude is fixed, or is unknown and variable across recordings. To approach this problem, we compare the ability of over 7000 candidate time-series features to track the DTC in the vicinity of a supercritical Hopf bifurcation. We recover existing theory in the fixed-noise case, highlighting conventional time-series features that accurately track the DTC. But in the variable-noise setting, where these conventional indicators perform poorly, we highlight new types of high-performing time-series features and show that their success is accomplished by capturing the shape of the invariant density (which depends on both the DTC and the noise amplitude) relative to the spread of fast fluctuations (which depends on the noise amplitude). We introduce a new high-performing time-series statistic, the Rescaled Auto-Density (RAD), that combines these two algorithmic components. We then use RAD to provide new evidence that brain regions higher in the visual hierarchy are positioned closer to criticality, supporting existing hypotheses about patterns of brain organization that are not detected using conventional metrics of the DTC. Our results demonstrate how large-scale algorithmic comparison can yield theoretical insights that can motivate new theory and interpretable algorithms for real-world problems.