Can power-law scaling and neuronal avalanches arise from stochastic dynamics?

The presence of self-organized criticality in biology is often evidenced by a power-law scaling of event size distributions, which can be measured by linear regression on logarithmic axes. We show here that such a procedure does not necessarily mean that the system exhibits self-organized criticality. We first provide an analysis of multisite local field potential (LFP) recordings of brain activity and show that event size distributions defined as negative LFP peaks can be close to power-law distributions. However, this result is not robust to change in detection threshold, or when tested using more rigorous statistical analyses such as the Kolmogorov-Smirnov test. Similar power-law scaling is observed for surrogate signals, suggesting that power-law scaling may be a generic property of thresholded stochastic processes. We next investigate this problem analytically, and show that, indeed, stochastic processes can produce spurious power-law scaling without the presence of underlying self-organized criticality. However, this power-law is only apparent in logarithmic representations, and does not survive more rigorous analysis such as the Kolmogorov-Smirnov test. The same analysis was also performed on an artificial network known to display self-organized criticality. In this case, both the graphical representations and the rigorous statistical analysis reveal with no ambiguity that the avalanche size is distributed as a power-law. We conclude that logarithmic representations can lead to spurious power-law scaling induced by the stochastic nature of the phenomenon. This apparent power-law scaling does not constitute a proof of self-organized criticality, which should be demonstrated by more stringent statistical tests.

💡 Research Summary

The paper critically examines the widespread practice of inferring self‑organized criticality (SOC) in neural systems from apparent power‑law scaling of event‑size distributions, typically visualized as straight lines on log‑log plots. Using multi‑site local field potential (LFP) recordings, the authors first define “events” as negative LFP peaks that exceed a chosen threshold and compute event sizes as the total duration (or summed amplitude) of consecutive suprathreshold excursions. When these sizes are plotted on logarithmic axes, the distribution often appears linear over a limited range, yielding high R² values and exponents reminiscent of neuronal avalanches reported in the literature. However, the authors demonstrate that this apparent scaling is highly sensitive to the detection threshold: modest changes (e.g., from 2 σ to 3 σ) dramatically alter the fitted exponent and the extent of the linear region, suggesting that the visual fit is not robust.

To move beyond visual inspection, the authors apply rigorous statistical tools. They fit several candidate distributions—pure power‑law, exponential, and log‑normal—using maximum‑likelihood estimation (MLE) and compare them with the Kolmogorov‑Smirnov (KS) goodness‑of‑fit test. For the real LFP data, the KS test frequently rejects the power‑law hypothesis, while exponential or log‑normal models often provide a superior description according to Akaike and Bayesian information criteria (AIC/BIC). Thus, a straight line on a log‑log plot does not constitute statistical evidence for a true power‑law.

The authors then construct surrogate “thresholded stochastic” signals by applying the same detection rule to pure Gaussian white noise. Remarkably, these surrogate data also produce apparent straight‑line segments on log‑log plots, yet the KS test again rejects the power‑law model. This demonstrates that thresholding a stochastic process can generate spurious scaling that is an artifact of the logarithmic representation rather than a property of the underlying dynamics.

A theoretical analysis underpins these observations. The authors model threshold crossings as a Poisson process with exponentially distributed inter‑arrival times. The lengths of consecutive suprathreshold intervals (the event sizes) follow a geometric‑like decay, which, when plotted on log‑log axes, mimics a power‑law over a limited range. The apparent exponent depends on the threshold level and the sampling resolution, confirming that the phenomenon is purely statistical and not indicative of critical dynamics.

Finally, the authors test an artificial neural network known to exhibit genuine SOC (e.g., a branching‑process model tuned to the critical point). In this case, both the graphical log‑log representation and the rigorous statistical tests (KS, MLE, AIC/BIC) consistently support a power‑law distribution of avalanche sizes, with no ambiguity. This contrast underscores that true critical systems survive both visual and quantitative scrutiny, whereas stochastic, thresholded processes do not.

The paper concludes with methodological recommendations: (1) visual inspection of log‑log plots should be complemented by formal statistical testing; (2) threshold selection must be justified and its influence on scaling assessed; (3) model comparison using likelihood‑based criteria is essential; and (4) additional signatures of criticality—such as scale‑invariant temporal correlations, diverging correlation lengths, and finite‑size scaling—should be sought before claiming SOC in neural data. By highlighting how easily spurious power‑law scaling can arise from stochastic dynamics, the study urges the neuroscience community to adopt more stringent analytical standards when investigating critical phenomena.