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

## TL;DR

This paper demonstrates that power-law scaling observed in neural activity data can arise from stochastic processes and does not necessarily indicate self-organized criticality, emphasizing the need for rigorous statistical testing.

## Contribution

It provides analytical and empirical evidence that stochastic processes can produce spurious power-law scaling, challenging the interpretation of such patterns as evidence of criticality.

## Key findings

- Power-law scaling can be observed in surrogate stochastic signals.
- Rigorous statistical tests can distinguish true criticality from spurious scaling.
- Artificial critical systems show clear power-law distributions confirmed by statistical analysis.

## Abstract

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.