Feedback Mechanisms for Self-Organization to the Edge of a Phase Transition

Victor Buendía,Serena Di Santo,Juan A Bonachela,Miguel A Muñoz

doi:10.3389/fphy.2020.00333

Abstract

Scale-free outbursts of activity are commonly observed in physical, geological, and biological systems. The idea of self-organized criticality (SOC), introduced back in 1987 by Bak, Tang and Wiesenfeld suggests that, under certain circumstances, natural systems can seemingly self-tune to a critical state with its concomitant power-laws and scaling. Theoretical progress allowed for a rationalization of how SOC works by relating its critical properties to those of a standard non-equilibrium second-order phase transition that separates an active state in which dynamical activity reverberates indefinitely, from an absorbing or quiescent state where activity eventually ceases. Here, we briefly review these ideas as well as a recent closely-related concept: self-organized bistability (SOB). In SOB, the very same type of feedback operates in a system characterized by a discontinuos phase transition, which has no critical point but instead presents bistability between active and quiescent states. SOB also leads to scale-invariant avalanches of activity but, in this case, with a different type of scaling and coexisting with anomalously large outbursts. Moreover, SOB explains experiments with real sandpiles more closely than SOC. We review similarities and differences between SOC and SOB by presenting and analyzing them under a common theoretical framework, covering recent results as well as possible future developments. We also discuss other related concepts for "imperfect" self-organization such as "self-organized quasi-criticality" and "self-organized collective oscillations", of relevance in e.g. neuroscience, with the aim of providing an overview of feedback mechanisms for self-organization to the edge of a phase transition.

Highlights

The seminal work of Bak, Tang, and Wiesenfeld in which the idea of “self-organized criticality” was first introduced [1], which has been cited thousands of times in the scientific literature and beyond, opened a whole research field and triggered a huge avalanche of scientific excitement in Statistical physics
In the context of neurodynamics, models of neuronal activity regulated by the level of synaptic resources—very similar in essence to self-organized bistability (SOB)—can reproduce scale-free avalanches coexisting with anomalous large waves of activity in agreement with empirical observations [116]
Note that the feedback mechanism is “just” a way to reach the neighborhood of a phase transition, but it is the intrinsic dynamics that determines the universality class that the system belongs to

Summary

INTRODUCTION

The seminal work of Bak, Tang, and Wiesenfeld in which the idea of “self-organized criticality” was first introduced [1], which has been cited thousands of times in the scientific literature and beyond, opened a whole research field and triggered a huge avalanche of scientific excitement in Statistical physics. These values coincide with those of the mean-field branching process, which is equivalent to compact directed percolation and the voter model [64,65,66, 72, 110] This type of scaling emerges because the system becomes self-organized to the Maxwell point EM (see Figure 3, upper-right panel), where the two phases are stable (or “neutral” [111, 112]). In this way, clusters of active sites in a non-active environment are likely to expand or shrink through fluctuations; this marginality is tantamount to criticality and generates scale invariance. In the context of neurodynamics, models of neuronal activity regulated by the level of synaptic resources—very similar in essence to SOB—can reproduce scale-free avalanches coexisting with anomalous large waves of activity in agreement with empirical observations [116] (see sections for more details on neural dynamics)

THEORIES FOR IMPERFECT

Theory of Self-Organized Quasi

Theory of Self-Organized Collective

SUMMARY AND DISCUSSION