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.
Self-organized bistability is the counterpart of "self-organized criticality" (SOC), for systems tuning themselves to the edge of bistability of a discontinuous/first-order phase transition, rather than to the critical point of a continuous/second order one. The equations defining the theory of SOB turn out to be very similar to a mesoscopic (Landau-Ginzburg) theory recently proposed to describe the dynamics in the cerebral cortex. This theory describes the bistable/oscillating neuronal activity of coupled mesoscopic patches of cortex, homeostatically regulated by short-term synaptic plasticity. However, the theory for cortex dynamics entails significant differences with respect to SOB, including the lack of a (bulk) conservation law, the absence of a perfect separation of timescales between driving and dissipation and, the fact that in the former there is a parameter that controls the overall system state (in blatant contrast with the very idea of self-organization). Here, we scrutinize -by employing a combination of analytical and computational tools-the analogies and differences between both theories and explore whether in some limit SOB could play an important role to explain the emergence of scale-invariant neuronal avalanches observed empirically in the cortex. We conclude that actually, in the limit of infinitely slow synaptic-dynamics, the two theories behave identically, but the separation of timescales required for the self-organization mechanism to be effective does not seem to be biologically plausible. We discuss the key differences between selforganization mechanisms with or without conservation and separated timescales, and in particular, we scrutinize the implications of our findings in neuroscience, hopefully shedding new light into the problem of scale invariance in cortical dynamics. arXiv:1911.05382v1 [cond-mat.stat-mech]
Many of the amazing functional capabilities of the brain are collective properties stemming from the interactions of large sets of individual neurons. In particular, the most salient collective phenomena in brain activity are oscillations, which require the synchronous activation of many neurons. Here, we analyse parsimonious dynamical models of neural synchronization running on top of synthetic networks that capture essential aspects of the actual brain anatomical connectivity such as a hierarchical-modular and core-periphery structure. These models reveal the emergence of complex collective states with intermediate and flexible levels of synchronization, halfway in the synchronous–asynchronous spectrum. These states are best described as broad Griffiths-like phases, i.e. an extension of standard critical points that emerge in structurally heterogeneous systems. We analyse different routes (bifurcations) to synchronization and stress the relevance of ‘hybrid-type transitions’ to generate rich dynamical patterns. Overall, our results illustrate the complex interplay between structure and dynamics, underlining key aspects leading to rich collective states needed to sustain brain functionality. This article is part of the theme issue ‘Emergent phenomena in complex physical and socio-technical systems: from cells to societies’.
Cortical networks are shaped by the combined action of excitatory and inhibitory interactions. Among other important functions, inhibition solves the problem of the all-or-none type of response that comes about in purely excitatory networks, allowing the network to operate in regimes of moderate or low activity, between quiescent and saturated regimes. Here, we elucidate a noise-induced effect that we call “Jensen’s force” –stemming from the combined effect of excitation/inhibition balance and network sparsity– which is responsible for generating a phase of self-sustained low activity in excitation-inhibition networks. The uncovered phase reproduces the main empirically-observed features of cortical networks in the so-called asynchronous state, characterized by low, un-correlated and highly-irregular activity. The parsimonious model analyzed here allows us to resolve a number of long-standing issues, such as proving that activity can be self-sustained even in the complete absence of external stimuli or driving. The simplicity of our approach allows for a deep understanding of asynchronous states and of the phase transitions to other standard phases it exhibits, opening the door to reconcile, asynchronous-state and critical-state hypotheses, putting them within a unified framework. We argue that Jensen’s forces are measurable experimentally and might be relevant in contexts beyond neuroscience.
Pathogen transmission and virulence are main evolutionary variables broadly assumed to be linked through trade-offs. In well-mixed populations, these trade-offs are often ascribed to physiological restrictions, while populations with spatial self-structuring might evolve emergent trade-offs. Here, we reexamine a spatially-explicit, SIR model of the latter kind proposed by Ballegooijen and Boerlijst with the aim of characterising the mechanisms causing the emergence of the trade-off and its structural robustness. Using invadability criteria, we establish the conditions under which an evolutionary feedback between transmission and virulence mediated by pattern formation can poise the system to a critical boundary separating a disordered state (without emergent trade-off) from a self-structured phase (where the trade-off emerges), and analytically calculate the functional shape of the boundary in a certain approximation. Beyond evolutionary parameters, the success of an invasion depends on the size and spatial structure of the invading and invaded populations. Spatial self-structuring is often destroyed when hosts are mobile, changing the evolutionary dynamics to those of a well-mixed population. In a metapopulation scenario, the systematic extinction of the pathogen in the disordered phase may counteract the disruptive effect of host mobility, favour pattern formation and therefore recover the emergent trade-off.
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