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Introduced by the late Per Bak and his colleagues, self-organized criticality (SOC) has been one of the most stimulating concepts to come out of statistical mechanics and condensed matter theory in the last few decades, and has played a significant role in the development of complexity science. SOC, and more generally fractals and power laws, have attracted much comment, ranging from the very positive to the polemical. The other papers (Aschwanden et al. in Space Sci. Rev., 2014 , this issue; McAteer et al. in Space Sci. Rev., 2015 , this issue; Sharma et al. in Space Sci. Rev. 2015 , in preparation) in this special issue showcase the considerable body of observations in solar, magnetospheric and fusion plasma inspired by the SOC idea, and expose the fertile role the new paradigm has played in approaches to modeling and understanding multiscale plasma instabilities. This very broad impact, and the necessary process of adapting a scientific hypothesis to the conditions of a given physical system, has meant that SOC as studied in these fields has sometimes differed significantly from the definition originally given by its creators. In Bak’s own field of theoretical physics there are significant observational and theoretical open questions, even 25 years on (Pruessner 2012 ). One aim of the present review is to address the dichotomy between the great reception SOC has received in some areas, and its shortcomings, as they became manifest in the controversies it triggered. Our article tries to clear up what we think are misunderstandings of SOC in fields more remote from its origins in statistical mechanics, condensed matter and dynamical systems by revisiting Bak, Tang and Wiesenfeld’s original papers.

The broad concept of emergence is instrumental in various of the most challenging open scientific questions—yet, few quantitative theories of what constitutes emergent phenomena have been proposed. This article introduces a formal theory of causal emergence in multivariate systems, which studies the relationship between the dynamics of parts of a system and macroscopic features of interest. Our theory provides a quantitative definition of downward causation , and introduces a complementary modality of emergent behaviour—which we refer to as causal decoupling . Moreover, the theory allows practical criteria that can be efficiently calculated in large systems, making our framework applicable in a range of scenarios of practical interest. We illustrate our findings in a number of case studies, including Conway’s Game of Life, Reynolds’ flocking model, and neural activity as measured by electrocorticography.

•Limits of drug models of psychosis investigated with MEG of subjects under LSD and ketamine.•Changes elicited by drugs and schizophrenia are similar for complexity, opposite for information transfer.•Experimental findings are reproduced in a predictive processing model.•Both drugs and schizophrenia are characterised by stronger “bottom-up” signalling of qualitatively different kinds. Schizophrenia and states induced by certain psychotomimetic drugs may share some physiological and phenomenological properties, but they differ in fundamental ways: one is a crippling chronic mental disease, while the others are temporary, pharmacologically-induced states presently being explored as treatments for mental illnesses. Building towards a deeper understanding of these different alterations of normal consciousness, here we compare the changes in neural dynamics induced by LSD and ketamine (in healthy volunteers) against those associated with schizophrenia, as observed in resting-state M/EEG recordings. While both conditions exhibit increased neural signal diversity, our findings reveal that this is accompanied by an increased transfer entropy from the front to the back of the brain in schizophrenia, versus an overall reduction under the two drugs. Furthermore, we show that these effects can be reproduced via different alterations of standard Bayesian inference applied on a computational model based on the predictive processing framework. In particular, the effects observed under the drugs are modelled as a reduction of the precision of the priors, while the effects of schizophrenia correspond to an increased precision of sensory information. These findings shed new light on the similarities and differences between schizophrenia and two psychotomimetic drug states, and have potential implications for the study of consciousness and future mental health treatments.

Experiences of collective creative activities play an essential role in human societies, yet these experiences are particularly hard to capture, making their scientific study challenging. In a classical music concert-experiment performed by a string quartet, we contrast a Let-go performance mode, characterised by a more creative and improvisatory approach that encourages risk-taking and spontaneous expression, with a more Strict mode which requires adhering closely to the score, common in many Western classical music performance environments. We investigate the experience of audience members by analysing their subjective reports and movement patterns. Our results show that during performances in Let-go mode, movement synchronization was reduced between performers and audience members in shorter timescales, while the synchronization and its temporal variability were enhanced in longer timescales. Furthermore, these differences in the synchronization dynamics are predictive of changes in the audience’s perception of music. These results provide a first step towards the quantification of some of the fundamental aspects of collective music experiences. Specifically, the reported findings demonstrate the relevance of the often-neglected multiscale coordination between audiences and performers, and explain how this rich tapestry of physical behaviour is connected with the quality of the collective music experience.

Self-organisation lies at the core of fundamental but still unresolved scientific questions, and holds the promise of de-centralised paradigms crucial for future technological developments. While self-organising processes have been traditionally explained by the tendency of dynamical systems to evolve towards specific configurations, or attractors, we see self-organisation as a consequence of the interdependencies that those attractors induce. Building on this intuition, in this work we develop a theoretical framework for understanding and quantifying self-organisation based on coupled dynamical systems and multivariate information theory. We propose a metric of global structural strength that identifies when self-organisation appears, and a multi-layered decomposition that explains the emergent structure in terms of redundant and synergistic interdependencies. We illustrate our framework on elementary cellular automata, showing how it can detect and characterise the emergence of complex structures.

Understanding the relation between patterns of human mobility and the scaling of dynamical features of urban environments is a great importance for today's society. Although recent advancements have shed light on the characteristics of individual mobility, the role and importance of emerging human collective phenomena across time and space are still unclear. In this Article, we show by using two independent data-analysis techniques that the traffic in London is a combination of intertwined clusters, spanning the whole city and effectively behaving as a single correlated unit. This is due to algebraically decaying spatio-temporal correlations, that are akin to those shown by systems near a critical point. We describe these correlations in terms of Taylor's law for fluctuations and interpret them as the emerging result of an underlying spatial synchronisation. Finally, our results provide the first evidence for a large-scale spatial human system reaching a self-organized critical state.

At what level does natural selection occur? When considering the reproductive dynamics of interacting and mutating agents, it has long been debated whether selection is better understood by focusing on the individual or if hierarchical selection emerges as a consequence of joint adaptation. Despite longstanding efforts in theoretical ecology, there is still no consensus on this fundamental issue, most likely due to the difficulty in obtaining adequate data spanning a sufficient number of generations and the lack of adequate tools to quantify the effect of hierarchical selection. Here, we capitalise on recent advances in information-theoretic data analysis to advance this state of affairs by investigating the emergence of high-order structures- such as groups of species- in the collective dynamics of the Tangled Nature model of evolutionary ecology. Our results show that evolutionary dynamics can lead to clusters of species that act as a self-perpetuating group that exhibits greater information-theoretic agency than a single species for a broad range of stable mutation rates. However, this higher-order organization breaks down for mutation rates close to the error threshold, where increased information processing is observed at the level of a single species. For mutation rates higher than the error threshold, no stable population of species are observed in time, and all individuality is lost in the ecosystem. Overall, our findings provide quantitative evidence supporting the emergence of higher-order structures in evolutionary ecology from relatively simple processes of adaptation and reproduction. Quantifying information-theoretic individuality as persistence of information in time reveals the emergence of higher-order organization in evolutionary systems, where groups of species exhibit enhanced collective persistence.

Nowadays the challenge for humanity is to find pathways towards sustainable development. Decision makers require a set of sustainability indicators to know if the sustainability strategies are following those pathways. There are more than one hundred sustainability indicators but they differ on their relative importance according to the size of the locality and change on time. The resources needed to follow these sustainability indicators are scarce and in some instances finite, especially in smaller regions. Therefore strategies to select set of these indicators are useful for decision makers responsible for monitoring sustainability. In this paper we propose a model for the identification and selection of a set of sustainability indicators that adequately represents human systems. In developing this model, we applied evolutionary dynamics in a space where sustainability indicators are fundamental entities interconnected by an interaction matrix. we used a fixed interaction that simulates the current context for the city of Cuernavaca, México as an example. We were able to identify and define relevant sets indicators for the system by using the Pareto principle. In this case we identified a set of sixteen sustainability indicators with more than 80% of the total strength. This set presents resilience to perturbations. For the Tangled Nature framework we provided a manner of treating different contexts (i.e., cities, counties, states, regions, countries, continents or the whole planet), dealing with small dimensions. This model provides decision makers with a valuable tool to select sustainability indicators set for towns, cities, regions, countries, continents or the entire planet according to a coevolutionary framework. The social legitimacy can arise from the fact that each individual indicator must be selected from those that are most important for the subject community.

Understanding the fundamental mechanisms behind the complex landscape of corporate mergers and acquisitions is of crucial importance to economies across the world. Adapting ideas from the fields of complexity and evolutionary dynamics to analyse business ecosystems, we show here that ancestry, i.e. the cumulative sum of historical mergers across all ancestors, is the key characteristic to company mergers and acquisitions. We verify this by comparing an agent-based model to an extensive range of business data, covering the period from the 1830s to the present day and a range of industries and geographies. This seemingly universal mechanism leads to imbalanced business ecosystems, with the emergence of a few very large, but sluggish ‘too big to fail’ entities, and very small, niche entities, thereby creating a paradigm where a configuration akin to effective oligopoly or monopoly is a likely outcome for free market systems.