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

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. The basic mechanism underlying SOC is the alternation of a slow driving process and fast dynamics with dissipation, which generates a feedback loop that tunes the system to the critical point of an absorbing-active continuous phase transition. 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 discontinuous 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.

Mechanisms of Self-Organized Quasicriticality in Neuronal Network Models

“…Such models have distinct features from conservative systems. A large fraction of them, in particular neuronal networks, have been described as displaying self-organized quasi-criticality (SOqC) [5][6][7] or weak criticality [8,9], which is the subject of this review.…”

“…Since 2003, however, the study of criticality in neuronal networks developed itself as a research paradigm, with a large literature, diverse experimental approaches, and several problems addressed theoretically and computationally (some reviews include Refs. [7,[21][22][23][24][25][26][27]). One of the main results is that information processing seems to be optimized at a secondorder absorbing phase transition [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42].…”

“…This transition occurs between no activity (the absorbing phase) and nonzero steadystate activity (the active phase). Such transition is familiar from the SOC literature and pertains to the directed percolation (DP) or the conservative-DP (C-DP or Manna) universality classes [7,[42][43][44][45].…”

“…The literature about these homeostatic mechanisms is vast, and we do not intend to present an exhaustive review. However, we discuss here some prototypical mechanisms and try to connect them to self-organized quasicriticality (SOqC), a concept developed to account for nonconservative systems that hover around but do not exactly sit on the critical point [5][6][7].…”

“…Last, before we begin, a few words about the fine-tuning problem. Even perfect SOC systems are in a sense fine-tuned: they must be conservative and require infinite separation of time scales with driving rate 1/τ → 0 + and dissipation rate u → 0 + with 1/(τu) → 0 [3,4,7,43,45]. For homeostatic systems, we turn a control parameter like the coupling W into a timedependent slow variable W[t] 〈W ij [t]〉 by imposing a local dynamics in the individual W ij .…”