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.

翻译：许多现实系统在跨越临界点时动力学行为会发生突变，常伴随严重后果。现有关于识别邻近临界点时间序列特征（如方差增大和时间尺度变慢）的理论大多建立在固定低振幅噪声的假设基础上。然而实际系统常受未知水平的噪声干扰，这些噪声会扭曲时间序列特征。本研究旨在开发对噪声鲁棒的"到临界态距离"（DTC）指标，适用于两种动态噪声场景：噪声振幅固定或未知且随记录变化。为此，我们比较了7000余种候选时间序列特征在超临界霍普夫分岔邻域内追踪DTC的能力。在固定噪声条件下，我们复现了现有理论，揭示了精确追踪DTC的经典时间序列特征。但在噪声振幅变化场景中，当这些经典指标失效时，我们发现了新型高性能时间序列特征，并证明其成功源于捕捉了不变密度（由DTC和噪声振幅共同决定）相对于快涨落扩散范围（仅由噪声振幅决定）的形状。我们提出了新型高性能时间序列统计量——重标自密度（RAD），该指标整合了这两类算法组件。利用RAD，我们发现视觉层级较高脑区的定位更接近临界态，这一发现支持了关于脑组织模式的新假说，而传统DTC指标无法检测到该模式。本研究证明，大规模算法比较可产生理论洞见，为真实世界问题催生新理论与可解释算法。