Observing scale-invariance in non-critical dynamical systems

TL;DR

This paper clarifies that observed scale invariance in neural systems may not indicate true criticality, emphasizing the importance of sampling biases and distinguishing between critical states and critical systems.

Contribution

It distinguishes between critical states and critical systems, introduces the concept of observational criticality, and analyzes models showing scale invariance without being critical.

Findings

Scale invariance can arise from sampling biases in non-critical systems.

Critical dynamical systems can appear scale-invariant under certain observational conditions.

The paper clarifies the difference between criticality and critical states in neural dynamics.

Abstract

Recent observation for scale invariant neural avalanches in the brain have been discussed in details in the scientific literature. We point out, that these results do not necessarily imply that the properties of the underlying neural dynamics are also scale invariant. The reason for this discrepancy lies in the fact that the sampling statistics of observations and experiments is generically biased by the size of the basins of attraction of the processes to be studied. One has hence to precisely define what one means with statements like `the brain is critical'. We recapitulate the notion of criticality, as originally introduced in statistical physics for second order phase transitions, turning then to the discussion of critical dynamical systems. We elucidate in detail the difference between a 'critical system', viz a system on the verge of a phase transition, and a 'critical state', viz state with scale-invariant correlations, stressing the fact that the notion of universality is linked to critical states. We then discuss rigorous results for two classes of critical dynamical systems, the Kauffman net and a vertex routing model, which both have non-critical states. However, an external observer that samples randomly the phase space of these two critical models, would find scale invariance. We denote this phenomenon as 'observational criticality' and discuss its relevance for the response properties of critical dynamical systems.