  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.
