Reconciling emergences: An information-theoretic approach to identify causal emergence in multivariate data

Is artificial consciousness achievable? Lessons from the human brain

We here analyse the question of developing artificial consciousness from an evolutionary perspective, taking the evolution of the human brain and its relation with consciousness as a reference model or as a benchmark. This kind of analysis reveals several structural and functional features of the human brain that appear to be key for reaching human-like complex conscious experience and that current research on Artificial Intelligence (AI) should take into account in its attempt to develop systems capable of human-like conscious processing. We argue that, even if AI is limited in its ability to emulate human consciousness for both intrinsic (i.e., structural and architectural) and extrinsic (i.e., related to the current stage of scientific and technological knowledge) reasons, taking inspiration from those characteristics of the brain that make human-like conscious processing possible and/or modulate it, is a potentially promising strategy towards developing conscious AI. Also, it cannot be theoretically excluded that AI research can develop partial or potentially alternative forms of consciousness that are qualitatively different from the human form, and that may be either more or less sophisticated depending on the perspectives. Therefore, we recommend neuroscience-inspired caution in talking about artificial consciousness: since the use of the same word "consciousness" for humans and AI becomes ambiguous and potentially misleading, we propose to clearly specify which level and/or type of consciousness AI research aims to develop, as well as what would be common versus differ in AI conscious processing compared to human conscious experience.

Synergy as the Failure of Distributivity

The concept of emergence, or synergy in its simplest form, is widely used but lacks a rigorous definition. Our work connects information and set theory to uncover the mathematical nature of synergy as the failure of distributivity. For the trivial case of discrete random variables, we explore whether and how it is possible to get more information out of lesser parts. The approach is inspired by the role of set theory as the fundamental description of part-whole relations. If taken unaltered, synergistic behavior is forbidden by the set-theoretic axioms. However, random variables are not a perfect analogy of sets: we formalize the distinction, highlighting a single broken axiom-union/intersection distributivity. Nevertheless, it remains possible to describe information using Venn-type diagrams. The proposed multivariate theory resolves the persistent self-contradiction of partial information decomposition and reinstates it as a primary route toward a rigorous definition of emergence. Our results suggest that non-distributive variants of set theory may be used to describe emergent physical systems.

An information-theoretic approach to build hypergraphs in psychometrics

Psychological network approaches propose to see symptoms or questionnaire items as interconnected nodes, with links between them reflecting pairwise statistical dependencies evaluated on cross-sectional, time-series, or panel data. These networks constitute an established methodology to visualise and conceptualise the interactions and relative importance of nodes/indicators, providing an important complement to other approaches such as factor analysis. However, limiting the representation to pairwise relationships can neglect potentially critical information shared by groups of three or more variables (higher-order statistical interdependencies). To overcome this important limitation, here we propose an information-theoretic framework to assess these interdependencies and consequently to use hypergraphs as representations in psychometrics. As edges in hypergraphs are capable of encompassing several nodes together, this extension can thus provide a richer account on the interactions that may exist among sets of psychological variables. Our results show how psychometric hypergraphs can highlight meaningful redundant and synergistic interactions on either simulated or state-of-the-art, re-analysed psychometric datasets. Overall, our framework extends current network approaches while leading to new ways of assessing the data that differ at their core from other methods, enriching the psychometrics toolbox, and opening promising avenues for future investigation.

A deep learning classification framework for research methods of marine protected area management

The latest emerging transdisciplinary marine protected area (MPA) research scheme requires efficient approaches for theoretically based and data-driven method integration. However, due to the rapid development and diversification of research methods, it is growingly difficult to locate new methods in methodological dimensions and integrate them to the utmost utility. This study proposes a deep learning-based classification framework for MPA management methods focused particularly on data and theory capabilities using natural language processing (NLP). It extracted keywords from academic sources and performed clustering based on semantic similarity, generating benchmark texts for abstract labeling. By training the deep learning NLP model and analyzing the abstracts of 9049 MPA management empirical research articles from 1986 to 2024, the data and theory scores were attributed to each article, and a total of 19 major method categories and 110 segment branches were identified in qualitative, quantitative, and mixed genres. Combination types of research methods were summarized, yielding the data-theory neutralization principle where the average data and theory scores tend to approximate 0.50. Applying the principle broadens traditional boundaries for method integration and extends method synthesis to higher numbers, generating a practical research 2paradigm for future MPA research. Implications include bridging social and ecological data, theorizing emergent challenges in complex systems and integrating theory construction and data science. The framework is applicable to quantification of other environmental management disciplines and can serve as guidance for multidisciplinary method integration. © 2017 Elsevier Inc. All rights reserved.

An Exact Theory of Causal Emergence for Linear Stochastic Iteration Systems

After coarse-graining a complex system, the dynamics of its macro-state may exhibit more pronounced causal effects than those of its micro-state. This phenomenon, known as causal emergence, is quantified by the indicator of effective information. However, two challenges confront this theory: the absence of well-developed frameworks in continuous stochastic dynamical systems and the reliance on coarse-graining methodologies. In this study, we introduce an exact theoretic framework for causal emergence within linear stochastic iteration systems featuring continuous state spaces and Gaussian noise. Building upon this foundation, we derive an analytical expression for effective information across general dynamics and identify optimal linear coarse-graining strategies that maximize the degree of causal emergence when the dimension averaged uncertainty eliminated by coarse-graining has an upper bound. Our investigation reveals that the maximal causal emergence and the optimal coarse-graining methods are primarily determined by the principal eigenvalues and eigenvectors of the dynamic system's parameter matrix, with the latter not being unique. To validate our propositions, we apply our analytical models to three simplified physical systems, comparing the outcomes with numerical simulations, and consistently achieve congruent results.

Duality between predictability and reconstructability in complex systems

Predicting the evolution of a large system of units using its structure of interaction is a fundamental problem in complex system theory. And so is the problem of reconstructing the structure of interaction from temporal observations. Here, we find an intricate relationship between predictability and reconstructability using an information-theoretical point of view. We use the mutual information between a random graph and a stochastic process evolving on this random graph to quantify their codependence. Then, we show how the uncertainty coefficients, which are intimately related to that mutual information, quantify our ability to reconstruct a graph from an observed time series, and our ability to predict the evolution of a process from the structure of its interactions. We provide analytical calculations of the uncertainty coefficients for many different systems, including continuous deterministic systems, and describe a numerical procedure when exact calculations are intractable. Interestingly, we find that predictability and reconstructability, even though closely connected by the mutual information, can behave differently, even in a dual manner. We prove how such duality universally emerges when changing the number of steps in the process. Finally, we provide evidence that predictability-reconstruction dualities may exist in dynamical processes on real networks close to criticality.

Integrated information decomposition unveils major structural traits of in silico and in vitro neuronal networks

The properties of complex networked systems arise from the interplay between the dynamics of their elements and the underlying topology. Thus, to understand their behavior, it is crucial to convene as much information as possible about their topological organization. However, in large systems, such as neuronal networks, the reconstruction of such topology is usually carried out from the information encoded in the dynamics on the network, such as spike train time series, and by measuring the transfer entropy between system elements. The topological information recovered by these methods does not necessarily capture the connectivity layout, but rather the causal flow of information between elements. New theoretical frameworks, such as Integrated Information Decomposition (Φ-ID), allow one to explore the modes in which information can flow between parts of a system, opening a rich landscape of interactions between network topology, dynamics, and information. Here, we apply Φ-ID on in silico and in vitro data to decompose the usual transfer entropy measure into different modes of information transfer, namely, synergistic, redundant, or unique. We demonstrate that the unique information transfer is the most relevant measure to uncover structural topological details from network activity data, while redundant information only introduces residual information for this application. Although the retrieved network connectivity is still functional, it captures more details of the underlying structural topology by avoiding to take into account emergent high-order interactions and information redundancy between elements, which are important for the functional behavior, but mask the detection of direct simple interactions between elements constituted by the structural network topology.

A synergetic turn in cognitive neuroscience of brain diseases

Despite significant improvements in our understanding of brain diseases, many barriers remain. Cognitive neuroscience faces four major challenges: complex structure-function associations; disease phenotype heterogeneity; the lack of transdiagnostic models; and oversimplified cognitive approaches restricted to the laboratory. Here, we propose a synergetics framework that can help to perform the necessary dimensionality reduction of complex interactions between the brain, body, and environment. The key solutions include low-dimensional spatiotemporal hierarchies for brain-structure associations, whole-brain modeling to handle phenotype diversity, model integration of shared transdiagnostic pathophysiological pathways, and naturalistic frameworks balancing experimental control and ecological validity. Creating whole-brain models with reduced manifolds combined with ecological measures can improve our understanding of brain disease and help identify novel interventions. Synergetics provides an integrated framework for future progress in clinical and cognitive neuroscience, pushing the boundaries of brain health and disease toward more mature, naturalistic approaches.

Emergence and Causality in Complex Systems: A Survey of Causal Emergence and Related Quantitative Studies

Emergence and causality are two fundamental concepts for understanding complex systems. They are interconnected. On one hand, emergence refers to the phenomenon where macroscopic properties cannot be solely attributed to the cause of individual properties. On the other hand, causality can exhibit emergence, meaning that new causal laws may arise as we increase the level of abstraction. Causal emergence (CE) theory aims to bridge these two concepts and even employs measures of causality to quantify emergence. This paper provides a comprehensive review of recent advancements in quantitative theories and applications of CE. It focuses on two primary challenges: quantifying CE and identifying it from data. The latter task requires the integration of machine learning and neural network techniques, establishing a significant link between causal emergence and machine learning. We highlight two problem categories: CE with machine learning and CE for machine learning, both of which emphasize the crucial role of effective information (EI) as a measure of causal emergence. The final section of this review explores potential applications and provides insights into future perspectives.

Decomposing Neural Circuit Function into Information Processing Primitives

It is challenging to measure how specific aspects of coordinated neural dynamics translate into operations of information processing and, ultimately, cognitive functions. An obstacle is that simple circuit mechanisms-such as self-sustained or propagating activity and nonlinear summation of inputs-do not directly give rise to high-level functions. Nevertheless, they already implement simple the information carried by neural activity. Here, we propose that distinct functions, such as stimulus representation, working memory, or selective attention, stem from different combinations and types of low-level manipulations of information or information processing primitives. To test this hypothesis, we combine approaches from information theory with simulations of multi-scale neural circuits involving interacting brain regions that emulate well-defined cognitive functions. Specifically, we track the information dynamics emergent from patterns of neural dynamics, using quantitative metrics to detect where and when information is actively buffered, transferred or nonlinearly merged, as possible modes of low-level processing (storage, transfer and modification). We find that neuronal subsets maintaining representations in working memory or performing attentional gain modulation are signaled by their boosted involvement in operations of information storage or modification, respectively. Thus, information dynamic metrics, beyond detecting which network units participate in cognitive processing, also promise to specify how and when they do it, that is, through which type of primitive computation, a capability that may be exploited for the analysis of experimental recordings.

A role for the serotonin 2A receptor in the expansion and functioning of human transmodal cortex

Integrating independent but converging lines of research on brain function and neurodevelopment across scales, this article proposes that serotonin 2A receptor (5-HT2AR) signalling is an evolutionary and developmental driver and potent modulator of the macroscale functional organization of the human cerebral cortex. A wealth of evidence indicates that the anatomical and functional organization of the cortex follows a unimodal-to-transmodal gradient. Situated at the apex of this processing hierarchy-where it plays a central role in the integrative processes underpinning complex, human-defining cognition-the transmodal cortex has disproportionately expanded across human development and evolution. Notably, the adult human transmodal cortex is especially rich in 5-HT2AR expression and recent evidence suggests that, during early brain development, 5-HT2AR signalling on neural progenitor cells stimulates their proliferation-a critical process for evolutionarily-relevant cortical expansion. Drawing on multimodal neuroimaging and cross-species investigations, we argue that, by contributing to the expansion of the human cortex and being prevalent at the apex of its hierarchy in the adult brain, 5-HT2AR signalling plays a major role in both human cortical expansion and functioning. Owing to its unique excitatory and downstream cellular effects, neuronal 5-HT2AR agonism promotes neuroplasticity, learning and cognitive and psychological flexibility in a context-(hyper)sensitive manner with therapeutic potential. Overall, we delineate a dual role of 5-HT2ARs in enabling both the expansion and modulation of the human transmodal cortex.

Neural Causal Information Extractor for Unobserved Causes

Causal inference aims to faithfully depict the causal relationships between given variables. However, in many practical systems, variables are often partially observed, and some unobserved variables could carry significant information and induce causal effects on a target. Identifying these unobserved causes remains a challenge, and existing works have not considered extracting the unobserved causes while retaining the causes that have already been observed and included. In this work, we aim to construct the implicit variables with a generator-discriminator framework named the Neural Causal Information Extractor (NCIE), which can complement the information of unobserved causes and thus provide a complete set of causes with both observed causes and the representations of unobserved causes. By maximizing the mutual information between the targets and the union of observed causes and implicit variables, the implicit variables we generate could complement the information that the unobserved causes should have provided. The synthetic experiments show that the implicit variables preserve the information and dynamics of the unobserved causes. In addition, extensive real-world time series prediction tasks show improved precision after introducing implicit variables, thus indicating their causality to the targets.

Simultaneity of consciousness with physical reality: the key that unlocks the mind-matter problem

The problem of explaining the relationship between subjective experience and physical reality remains difficult and unresolved. In most explanations, consciousness is epiphenomenal, without causal power. The most notable exception is Integrated Information Theory (IIT), which provides a causal explanation for consciousness. However, IIT relies on an identity between subjectivity and a particular type of physical structure, namely with an information structure that has intrinsic causal power greater than the sum of its parts. Any theory that relies on a psycho-phyiscal identity must eventually appeal to panpsychism, which undermines that theory's claim to be fundamental. IIT has recently pivoted towards a strong version of causal emergence, but macroscopic structures cannot be stronger causally than their microphysical parts without some new physical law or governing principle. The approach taken here is designed to uncover such a principle. The decisive argument is entirely deductive from initial premises that are phenomenologically certain. If correct, the arguments prove that conscious experience is sufficient to create additional degrees of causal freedom independently of the content of experience, and in a manner that is unpredictable and unobservable by any temporally sequential means. This provides a fundamental principle about consciousness, and a conceptual bridge between it and the physics describing what is experienced. The principle makes testable predictions about brain function, with notable differences from IIT, some of which are also empirically testable.

Inference of time-ordered multibody interactions

We introduce time-ordered multibody interactions to describe complex systems manifesting temporal as well as multibody dependencies. First, we show how the dynamics of multivariate Markov chains can be decomposed in ensembles of time-ordered multibody interactions. Then, we present an algorithm to extract those interactions from data capturing the system-level dynamics of node states and a measure to characterize the complexity of interaction ensembles. Finally, we experimentally validate the robustness of our algorithm against statistical errors and its efficiency at inferring parsimonious interaction ensembles.

Dynamical independence: Discovering emergent macroscopic processes in complex dynamical systems

We introduce a notion of emergence for macroscopic variables associated with highly multivariate microscopic dynamical processes. Dynamical independence instantiates the intuition of an emergent macroscopic process as one possessing the characteristics of a dynamical system "in its own right," with its own dynamical laws distinct from those of the underlying microscopic dynamics. We quantify (departure from) dynamical independence by a transformation-invariant Shannon information-based measure of dynamical dependence. We emphasize the data-driven discovery of dynamically independent macroscopic variables, and introduce the idea of a multiscale "emergence portrait" for complex systems. We show how dynamical dependence may be computed explicitly for linear systems in both time and frequency domains, facilitating discovery of emergent phenomena across spatiotemporal scales, and outline application of the linear operationalization to inference of emergence portraits for neural systems from neurophysiological time-series data. We discuss dynamical independence for discrete- and continuous-time deterministic dynamics, with potential application to Hamiltonian mechanics and classical complex systems such as flocking and cellular automata.

Is information the other face of causation in biological systems?

Is information the other face of causation? This issue cannot be clarified without discussing how these both are related to physical laws, logic, computation, networks, bio-signaling, and the mind-body problem. The relation between information and causation is also intrinsically linked to many other concepts in complex systems theory such as emergence, self-organization, synergy, criticality, and hierarchy, which in turn involve various notions such as observer-dependence, dimensionality reduction, and especially downward causation. A canonical example proposed for downward causation is the collective behavior of the whole system at a macroscale that may affect the behavior of each its member at a microscale. In neuroscience, downward causation is suggested as a strong candidate to account for mental causation (free will). However, this would be possible only on the condition that information might have causal power. After introducing the Causal Equivalence Principle expanding the relativity principle for coarse-grained and fine-grained linear causal chains, and a set-theoretical definition of multiscale nested hierarchy composed of modular ⊂-chains, it is shown that downward causation can be spurious. It emerges only in the eyes of an observer, though, due to information that could not be obtained by "looking" exclusively at the behavior of a system at a microscale. On the other hand, since biological systems are hierarchically organized, this information gain is indicative of how information can be a function of scale in these systems and a prerequisite for scale-dependent emergence of cognition and consciousness in neural networks.

Multivariate information theory uncovers synergistic subsystems of the human cerebral cortex

One of the most well-established tools for modeling the brain is the functional connectivity network, which is constructed from pairs of interacting brain regions. While powerful, the network model is limited by the restriction that only pairwise dependencies are considered and potentially higher-order structures are missed. Here, we explore how multivariate information theory reveals higher-order dependencies in the human brain. We begin with a mathematical analysis of the O-information, showing analytically and numerically how it is related to previously established information theoretic measures of complexity. We then apply the O-information to brain data, showing that synergistic subsystems are widespread in the human brain. Highly synergistic subsystems typically sit between canonical functional networks, and may serve an integrative role. We then use simulated annealing to find maximally synergistic subsystems, finding that such systems typically comprise ≈10 brain regions, recruited from multiple canonical brain systems. Though ubiquitous, highly synergistic subsystems are invisible when considering pairwise functional connectivity, suggesting that higher-order dependencies form a kind of shadow structure that has been unrecognized by established network-based analyses. We assert that higher-order interactions in the brain represent an under-explored space that, accessible with tools of multivariate information theory, may offer novel scientific insights.

Reduced emergent character of neural dynamics in patients with a disrupted connectome

High-level brain functions are widely believed to emerge from the orchestrated activity of multiple neural systems. However, lacking a formal definition and practical quantification of emergence for experimental data, neuroscientists have been unable to empirically test this long-standing conjecture. Here we investigate this fundamental question by leveraging a recently proposed framework known as "Integrated Information Decomposition," which establishes a principled information-theoretic approach to operationalise and quantify emergence in dynamical systems - including the human brain. By analysing functional MRI data, our results show that the emergent and hierarchical character of neural dynamics is significantly diminished in chronically unresponsive patients suffering from severe brain injury. At a functional level, we demonstrate that emergence capacity is positively correlated with the extent of hierarchical organisation in brain activity. Furthermore, by combining computational approaches from network control theory and whole-brain biophysical modelling, we show that the reduced capacity for emergent and hierarchical dynamics in severely brain-injured patients can be mechanistically explained by disruptions in the patients' structural connectome. Overall, our results suggest that chronic unresponsiveness resulting from severe brain injury may be related to structural impairment of the fundamental neural infrastructures required for brain dynamics to support emergence.

Decomposing past and future: Integrated information decomposition based on shared probability mass exclusions

A core feature of complex systems is that the interactions between elements in the present causally constrain their own futures, and the futures of other elements as the system evolves through time. To fully model all of these interactions (between elements, as well as ensembles of elements), it is possible to decompose the total information flowing from past to future into a set of non-overlapping temporal interactions that describe all the different modes by which information can be stored, transferred, or modified. To achieve this, I propose a novel information-theoretic measure of temporal dependency (Iτsx) based on the logic of local probability mass exclusions. This integrated information decomposition can reveal emergent and higher-order interactions within the dynamics of a system, as well as refining existing measures. To demonstrate the utility of this framework, I apply the decomposition to spontaneous spiking activity recorded from dissociated neural cultures of rat cerebral cortex to show how different modes of information processing are distributed over the system. Furthermore, being a localizable analysis, Iτsx can provide insight into the computational structure of single moments. I explore the time-resolved computational structure of neuronal avalanches and find that different types of information atoms have distinct profiles over the course of an avalanche, with the majority of non-trivial information dynamics happening before the first half of the cascade is completed. These analyses allow us to move beyond the historical focus on single measures of dependency such as information transfer or information integration, and explore a panoply of different relationships between elements (and groups of elements) in complex systems.

Flickering Emergences: The Question of Locality in Information-Theoretic Approaches to Emergence

"Emergence", the phenomenon where a complex system displays properties, behaviours, or dynamics not trivially reducible to its constituent elements, is one of the defining properties of complex systems. Recently, there has been a concerted effort to formally define emergence using the mathematical framework of information theory, which proposes that emergence can be understood in terms of how the states of wholes and parts collectively disclose information about the system's collective future. In this paper, we show how a common, foundational component of information-theoretic approaches to emergence implies an inherent instability to emergent properties, which we call flickering emergence. A system may, on average, display a meaningful emergent property (be it an informative coarse-graining, or higher-order synergy), but for particular configurations, that emergent property falls apart and becomes misinformative. We show existence proofs that flickering emergence occurs in two different frameworks (one based on coarse-graining and another based on multivariate information decomposition) and argue that any approach based on temporal mutual information will display it. Finally, we argue that flickering emergence should not be a disqualifying property of any model of emergence, but that it should be accounted for when attempting to theorize about how emergence relates to practical models of the natural world.

Neural Information Squeezer for Causal Emergence

Conventional studies of causal emergence have revealed that stronger causality can be obtained on the macro-level than the micro-level of the same Markovian dynamical systems if an appropriate coarse-graining strategy has been conducted on the micro-states. However, identifying this emergent causality from data is still a difficult problem that has not been solved because the appropriate coarse-graining strategy can not be found easily. This paper proposes a general machine learning framework called Neural Information Squeezer to automatically extract the effective coarse-graining strategy and the macro-level dynamics, as well as identify causal emergence directly from time series data. By using invertible neural network, we can decompose any coarse-graining strategy into two separate procedures: information conversion and information discarding. In this way, we can not only exactly control the width of the information channel, but also can derive some important properties analytically. We also show how our framework can extract the coarse-graining functions and the dynamics on different levels, as well as identify causal emergence from the data on several exampled systems.

Virtual cells in a virtual microenvironment recapitulate early development-like patterns in human pluripotent stem cell colonies

The mechanism by which morphogenetic signals engage the regulatory networks responsible for early embryonic tissue patterning is incompletely understood. Here, we developed a minimal gene regulatory network (GRN) model of human pluripotent stem cell (hPSC) lineage commitment and embedded it into "cellular" agents that respond to a dynamic morphogenetic signaling microenvironment. Simulations demonstrated that GRN wiring had significant non-intuitive effects on tissue pattern order, composition, and dynamics. Experimental perturbation of GRN connectivities supported model predictions and demonstrated the role of OCT4 as a master regulator of peri-gastrulation fates. Our so-called GARMEN strategy provides a multiscale computational platform to understand how single-cell-based regulatory interactions scale to tissue domains. This foundation provides new opportunities to simulate the impact of network motifs on normal and aberrant tissue development.

On the Nature of Information: How FAIR Digital Objects are Building-up Semantic Space

In this paper, we are concerned about the nature of information and how to gather and compose data with the help of so called FAIR digital objects (FDOs) in order to transform them to knowledge. FDOs are digital surrogates of real objects. The nature of information is intrinsically linked to the kind of questions one is asking. One might not ask a question or get philosophical about it. Answers depend on the data different disciplines gather about their objects of study. In Statistical Physics, classical Shannon entropy measures system order which in equilibrium just equals the heat exchanged with the environment. In cell biology, each protein carries certain functions which create specific information. Cognitive science describes how organisms perceive their environment via functional sensors and control behavior accordingly. Note that one can have function and control without meaning. In contrast, psychology is concerned with the assessment of our perceptions by assigning meaning and ensuing actions. Finally, philosophy builds logical constructs and formulates principles, in effect transforming facts into complex knowledge. All these statements make sense, but there is an even more concise way. Indeed, Luciano Floridi provides a precise and thorough classification of information in his central oeuvre On the Philosophy of Information (Floridi 2013). Especially, he performs a sequential construction to develop the attributes which data need to have in order to count as knowledge. Semantic information is necessarily well-formed, meaningful and truthful. Well-formed data becomes meaningful by action based-semantics of an autonomous-agent solving the symbol grounding problem (Taddeo and Floridi 2005) interacting with the environment. Knowledge is created then by being informed through relevant data accounted for. We notice that the notion of agency is crucial for defining meaning. The apparent gap between Sciences and Humanities (Bawden and Robinson 2020) is created by the very existence of meaning. Further, meaning depends on interactions & connotations which are commensurate with the effective complexity of the environment of a particular agent resulting in an array of possible definitions.

In his classical paper More is different (Anderson 1972) discussed verbatim the hierarchical nature of science. Each level is made of and obeys the laws of its constituents from one level below with the higher-level exhibiting emergent properties like wetness of water assignable only to the whole system. As we rise through the hierarchies, there is a branch of science for each level of complexity; on each complexity level there are objects for which it is appropriate and fitting to build up vocabulary for the respective levels of description leading to formation of disciplinary languages. It is the central idea of causal emergence that on each level there is an optimal degree of coarse graining to define those objects in such a way that causality becomes maximal between them. This means there is emergence of informative higher scales in complex materials extending to biological systems and into the brain with its neural networks representing our thoughts in a hierarchy of neural correlates. A computational toolkit for optimal level prediction and control has been developed (Hoel and Levin 2020) which was conceptually extended to integrated information theory of consciousness (Albantakis et al. 2019). The large gap between sciences and humanities discussed above exhibits itself in a series of small gaps connected to the emergence of informative higher scales. It has been suggested that the origin of life may be identified as a transition in causal structure and information flow (Walker 2014). Integrated information measures globally how much the causal mechanisms of a system reduce the uncertainty about the possible causes for a given state. A measure of “information flow” that accurately captures causal effects has been proposed (Ay and Polani 2008). The state of the art is presented in (Ay et al. 2022) where the link between information and complexity is discussed. Ay et al single out hierarchical systems and interlevel causation. Even further, (Rosas et al. 2020) reconcile conflicting views of emergence via an exact information-theoretic approach to identify causal emergence in multivariate data. As information becomes differentially richer one eventually needs complexity measures beyond {Rn}. One may define generalized metrices on these spaces (Pirr&oacute 2009) measuring information complexity on ever higher hierarchical levels of information. As one rises through hierarchies, information on higher scale is usually gained by coarse graining to arrive at an effective, nevertheless exact description, on the higher scale. It is repeated coarse graining of syntactically well-ordered information layers which eventually leads to semantic information in a process which I conjecture to be reminiscent of renormalization group flow leading to a universal classification scheme. Thus, we identify scientific disciplines and their corresponding data sets as dual universality classes of physical and epistemic structure formation, respectively. Above the semantic gap, we may call this process quantification of the qualitative by semantic metrics. Indeed, (Kolchinsky and Wolpert 2018) explored for the first time quantitative semantic concepts in Physics in their 2018 seminal paper entitled Semantic information, autonomous agency and non-equilibrium statistical physics. Their measures are numeric variants of entropy. Semantic information is identified with ‘the information that a physical system has about its environment that is causally necessary for the system to maintain its own existence over time’.

FDOs are employed in these processes in two fundamental ways. For practical implementations of FDO technology, see accompanying abstract (Wittenburg et al. 2022). First, the FAIR principles (Wilkinson et al. 2016) ensure that unconnected pieces of data may be percolated into an integrated data space. Percolation creates the information density needed to feed AI-driven built up of semantic space. Without FDOs we wouldn't have the gravity for this to occur. Second, the very structure of FDOs, capable of symmetry preserving or breaking fusion events into composed entities, makes them homologous to mathematical categories. This will proof to be a powerful tool to unravel the nature of information via analyzing its topological structure algebraically, especially when considering our conjecture concerning universality, classes of information and their possible instantiations on vastly different length and time scales, in effect explaining analogous structure formation.

Untangling Synergistic Effects of Intersecting Social Identities with Partial Information Decomposition

The theory of intersectionality proposes that an individual's experience of society has aspects that are irreducible to the sum of one's various identities considered individually, but are "greater than the sum of their parts". In recent years, this framework has become a frequent topic of discussion both in social sciences and among popular movements for social justice. In this work, we show that the effects of intersectional identities can be statistically observed in empirical data using information theory, particularly the partial information decomposition framework. We show that, when considering the predictive relationship between various identity categories such as race and sex, on outcomes such as income, health and wellness, robust statistical synergies appear. These synergies show that there are joint-effects of identities on outcomes that are irreducible to any identity considered individually and only appear when specific categories are considered together (for example, there is a large, synergistic effect of race and sex considered jointly on income irreducible to either race or sex). Furthermore, these synergies are robust over time, remaining largely constant year-to-year. We then show using synthetic data that the most widely used method of assessing intersectionalities in data (linear regression with multiplicative interaction coefficients) fails to disambiguate between truly synergistic, greater-than-the-sum-of-their-parts interactions, and redundant interactions. We explore the significance of these two distinct types of interactions in the context of making inferences about intersectional relationships in data and the importance of being able to reliably differentiate the two. Finally, we conclude that information theory, as a model-free framework sensitive to nonlinearities and synergies in data, is a natural method by which to explore the space of higher-order social dynamics.

Emergence and algorithmic information dynamics of systems and observers

One of the challenges of defining emergence is that one observer's prior knowledge may cause a phenomenon to present itself as emergent that to another observer appears reducible. By formalizing the act of observing as mutual perturbations between dynamical systems, we demonstrate that the emergence of algorithmic information does depend on the observer's formal knowledge, while being robust vis-a-vis other subjective factors, particularly: the choice of programming language and method of measurement; errors or distortions during the observation; and the informational cost of processing. This is called observer-dependent emergence (ODE). In addition, we demonstrate that the unbounded and rapid increase of emergent algorithmic information implies asymptotically observer-independent emergence (AOIE). Unlike ODE, AOIE is a type of emergence for which emergent phenomena will be considered emergent no matter what formal theory an observer might bring to bear. We demonstrate the existence of an evolutionary model that displays the diachronic variant of AOIE and a network model that displays the holistic variant of AOIE. Our results show that, restricted to the context of finite discrete deterministic dynamical systems, computable systems and irreducible information content measures, AOIE is the strongest form of emergence that formal theories can attain. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.

Emergence as the conversion of information: a unifying theory

Is reduction always a good scientific strategy? The existence of the special sciences above physics suggests not. Previous research has shown that dimensionality reduction (macroscales) can increase the dependency between elements of a system (a phenomenon called 'causal emergence'). Here, we provide an umbrella mathematical framework for emergence based on information conversion. We show evidence that coarse-graining can convert information from one 'type' to another. We demonstrate this using the well-understood mutual information measure applied to Boolean networks. Using partial information decomposition, the mutual information can be decomposed into redundant, unique and synergistic information atoms. Then by introducing a novel measure of the synergy bias of a given decomposition, we are able to show that the synergy component of a Boolean network's mutual information can increase at macroscales. This can occur even when there is no difference in the total mutual information between a macroscale and its underlying microscale, proving information conversion. We relate this broad framework to previous work, compare it to other theories, and argue it complexifies any notion of universal reduction in the sciences, since such reduction would likely lead to a loss of synergistic information in scientific models. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.

Understanding brain states across spacetime informed by whole-brain modelling

In order to survive in a complex environment, the human brain relies on the ability to flexibly adapt ongoing behaviour according to intrinsic and extrinsic signals. This capability has been linked to specific whole-brain activity patterns whose relative stability (order) allows for consistent functioning, supported by sufficient intrinsic instability needed for optimal adaptability. The emergent, spontaneous balance between order and disorder in brain activity over spacetime underpins distinct brain states. For example, depression is characterized by excessively rigid, highly ordered states, while psychedelics can bring about more disordered, sometimes overly flexible states. Recent developments in systems, computational and theoretical neuroscience have started to make inroads into the characterization of such complex dynamics over space and time. Here, we review recent insights drawn from neuroimaging and whole-brain modelling motivating using mechanistic principles from dynamical system theory to study and characterize brain states. We show how different healthy and altered brain states are associated to characteristic spacetime dynamics which in turn may offer insights that in time can inspire new treatments for rebalancing brain states in disease. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.

Greater than the parts: a review of the information decomposition approach to causal emergence

Emergence is a profound subject that straddles many scientific disciplines, including the formation of galaxies and how consciousness arises from the collective activity of neurons. Despite the broad interest that exists on this concept, the study of emergence has suffered from a lack of formalisms that could be used to guide discussions and advance theories. Here, we summarize, elaborate on, and extend a recent formal theory of causal emergence based on information decomposition, which is quantifiable and amenable to empirical testing. This theory relates emergence with information about a system's temporal evolution that cannot be obtained from the parts of the system separately. This article provides an accessible but rigorous introduction to the framework, discussing the merits of the approach in various scenarios of interest. We also discuss several interpretation issues and potential misunderstandings, while highlighting the distinctive benefits of this formalism. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.

Emergence and algorithmic information dynamics of systems and observers

One of the challenges of defining emergence is that one observer's prior knowledge may cause a phenomenon to present itself as emergent that to another observer appears reducible. By formalizing the act of observing as mutual perturbations between dynamical systems, we demonstrate that the emergence of algorithmic information does depend on the observer's formal knowledge, while being robust vis-a-vis other subjective factors, particularly: the choice of programming language and method of measurement; errors or distortions during the observation; and the informational cost of processing. This is called observer-dependent emergence (ODE). In addition, we demonstrate that the unbounded and rapid increase of emergent algorithmic information implies asymptotically observer-independent emergence (AOIE). Unlike ODE, AOIE is a type of emergence for which emergent phenomena will be considered emergent no matter what formal theory an observer might bring to bear. We demonstrate the existence of an evolutionary model that displays the diachronic variant of AOIE and a network model that displays the holistic variant of AOIE. Our results show that, restricted to the context of finite discrete deterministic dynamical systems, computable systems and irreducible information content measures, AOIE is the strongest form of emergence that formal theories can attain. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.

May the 4C's be with you: an overview of complexity-inspired frameworks for analysing resting-state neuroimaging data

Competing and complementary models of resting-state brain dynamics contribute to our phenomenological and mechanistic understanding of whole-brain coordination and communication, and provide potential evidence for differential brain functioning associated with normal and pathological behaviour. These neuroscientific theories stem from the perspectives of physics, engineering, mathematics and psychology and create a complicated landscape of domain-specific terminology and meaning, which, when used outside of that domain, may lead to incorrect assumptions and conclusions within the neuroscience community. Here, we review and clarify the key concepts of connectivity, computation, criticality and coherence-the 4C's-and outline a potential role for metastability as a common denominator across these propositions. We analyse and synthesize whole-brain neuroimaging research, examined through functional magnetic imaging, to demonstrate that complexity science offers a principled and integrated approach to describe, and potentially understand, macroscale spontaneous brain functioning.

The strength of weak integrated information theory

The integrated information theory of consciousness (IIT) is divisive: while some believe it provides an unprecedentedly powerful approach to address the 'hard problem', others dismiss it on grounds that it is untestable. We argue that the appeal and applicability of IIT can be greatly widened if we distinguish two flavours of the theory: strong IIT, which identifies consciousness with specific properties associated with maxima of integrated information; and weak IIT, which tests pragmatic hypotheses relating aspects of consciousness to broader measures of information dynamics. We review challenges for strong IIT, explain how existing empirical findings are well explained by weak IIT without needing to commit to the entirety of strong IIT, and discuss the outlook for both flavours of IIT.

Naturalising Agent Causation

The idea of agent causation—that a system such as a living organism can be a cause of things in the world—is often seen as mysterious and deemed to be at odds with the physicalist thesis that is now commonly embraced in science and philosophy. Instead, the causal power of organisms is attributed to mechanistic components within the system or derived from the causal activity at the lowest level of physical description. In either case, the ‘agent’ itself (i.e., the system as a whole) is left out of the picture entirely, and agent causation is explained away. We argue that this is not the right way to think about causation in biology or in systems more generally. We present a framework of eight criteria that we argue, collectively, describe a system that overcomes the challenges concerning agent causality in an entirely naturalistic and non-mysterious way. They are: (1) thermodynamic autonomy, (2) persistence, (3) endogenous activity, (4) holistic integration, (5) low-level indeterminacy, (6) multiple realisability, (7) historicity, (8) agent-level normativity. Each criterion is taken to be dimensional rather than categorical, and thus we conclude with a short discussion on how researchers working on quantifying agency may use this multidimensional framework to situate and guide their research.

A Novel Approach to the Partial Information Decomposition

We consider the “partial information decomposition” (PID) problem, which aims to decompose the information that a set of source random variables provide about a target random variable into separate redundant, synergistic, union, and unique components. In the first part of this paper, we propose a general framework for constructing a multivariate PID. Our framework is defined in terms of a formal analogy with intersection and union from set theory, along with an ordering relation which specifies when one information source is more informative than another. Our definitions are algebraically and axiomatically motivated, and can be generalized to domains beyond Shannon information theory (such as algorithmic information theory and quantum information theory). In the second part of this paper, we use our general framework to define a PID in terms of the well-known Blackwell order, which has a fundamental operational interpretation. We demonstrate our approach on numerous examples and show that it overcomes many drawbacks associated with previous proposals.

Causal Emergence: When Distortions in a Map Obscure the Territory

We provide a critical assessment of the account of causal emergence presented in Erik Hoel’s 2017 article “When the map is better than the territory”. The account integrates causal and information theoretic concepts to explain under what circumstances there can be causal descriptions of a system at multiple scales of analysis. We show that the causal macro variables implied by this account result in interventions with significant ambiguity, and that the operations of marginalization and abstraction do not commute. Both of these are desiderata that, we argue, any account of multi-scale causal analysis should be sensitive to. The problems we highlight in Hoel’s definition of causal emergence derive from the use of various averaging steps and the introduction of a maximum entropy distribution that is extraneous to the system under investigation.

Network Analysis of Time Series: Novel Approaches to Network Neuroscience

In the last two decades, there has been an explosion of interest in modeling the brain as a network, where nodes correspond variously to brain regions or neurons, and edges correspond to structural or statistical dependencies between them. This kind of network construction, which preserves spatial, or structural, information while collapsing across time, has become broadly known as "network neuroscience." In this work, we provide an alternative application of network science to neural data: network-based analysis of non-linear time series and review applications of these methods to neural data. Instead of preserving spatial information and collapsing across time, network analysis of time series does the reverse: it collapses spatial information, instead preserving temporally extended dynamics, typically corresponding to evolution through some kind of phase/state-space. This allows researchers to infer a, possibly low-dimensional, "intrinsic manifold" from empirical brain data. We will discuss three methods of constructing networks from nonlinear time series, and how to interpret them in the context of neural data: recurrence networks, visibility networks, and ordinal partition networks. By capturing typically continuous, non-linear dynamics in the form of discrete networks, we show how techniques from network science, non-linear dynamics, and information theory can extract meaningful information distinct from what is normally accessible in standard network neuroscience approaches.

A network information theoretic framework to characterise muscle synergies in space and time

Objective Current approaches to muscle synergy extraction rely on linear dimensionality reduction algorithms that make specific assumptions on the underlying signals. However, to capture nonlinear time varying, large-scale but also muscle-specific interactions, a more generalised approach is required. Approach Here we developed a novel framework for muscle synergy extraction that relaxes model assumptions by using a combination of information- and network theory and dimensionality reduction. We first quantify informational dynamics between muscles, time-samples or muscle-time pairings using a novel mutual information formulation. We then model these pairwise interactions as multiplex networks and identify modules representing the network architecture. We employ this modularity criterion as the input parameter for dimensionality reduction, which verifiably extracts the identified modules, and also to characterise salient structures within each module. Main results This novel framework captures spatial, temporal and spatiotemporal interactions across two benchmark datasets of reaching movements, producing distinct spatial groupings and both tonic and phasic temporal patterns. Readily interpretable muscle synergies spanning multiple spatial and temporal scales were identified, demonstrating significant task dependence, ability to capture trial-to-trial fluctuations and concordance across participants. Furthermore, our framework identifies submodular structures that represent the distributed networks of co-occurring signal interactions across scales. Significance The capabilities of this framework are illustrated through the concomitant continuity with previous research and novelty of the insights gained. Several previous limitations are circumvented including the extraction of functionally meaningful and multiplexed pairwise muscle couplings under relaxed model assumptions. The extracted synergies provide a holistic view of the movement while important details of task performance are readily interpretable. The identified muscle groupings transcend biomechanical constraints and the temporal patterns reveal characteristics of fundamental motor control mechanisms. We conclude that this framework opens new opportunities for muscle synergy research and can constitute a bridge between existing models and recent network-theoretic endeavours.

Integrated information as a common signature of dynamical and information-processing complexity

The apparent dichotomy between information-processing and dynamical approaches to complexity science forces researchers to choose between two diverging sets of tools and explanations, creating conflict and often hindering scientific progress. Nonetheless, given the shared theoretical goals between both approaches, it is reasonable to conjecture the existence of underlying common signatures that capture interesting behavior in both dynamical and information-processing systems. Here, we argue that a pragmatic use of integrated information theory (IIT), originally conceived in theoretical neuroscience, can provide a potential unifying framework to study complexity in general multivariate systems. By leveraging metrics put forward by the integrated information decomposition framework, our results reveal that integrated information can effectively capture surprisingly heterogeneous signatures of complexity-including metastability and criticality in networks of coupled oscillators as well as distributed computation and emergent stable particles in cellular automata-without relying on idiosyncratic, ad hoc criteria. These results show how an agnostic use of IIT can provide important steps toward bridging the gap between informational and dynamical approaches to complex systems.

What it is like to be a bit: an integrated information decomposition account of emergent mental phenomena

A central question in neuroscience concerns the relationship between consciousness and its physical substrate. Here, we argue that a richer characterization of consciousness can be obtained by viewing it as constituted of distinct information-theoretic elements. In other words, we propose a shift from quantification of consciousness-viewed as integrated information-to its decomposition. Through this approach, termed Integrated Information Decomposition (ΦID), we lay out a formal argument that whether the consciousness of a given system is an emergent phenomenon depends on its information-theoretic composition-providing a principled answer to the long-standing dispute on the relationship between consciousness and emergence. Furthermore, we show that two organisms may attain the same amount of integrated information, yet differ in their information-theoretic composition. Building on ΦID's revised understanding of integrated information, termed ΦR, we also introduce the notion of ΦR-ing ratio to quantify how efficiently an entity uses information for conscious processing. A combination of ΦR and ΦR-ing ratio may provide an important way to compare the neural basis of different aspects of consciousness. Decomposition of consciousness enables us to identify qualitatively different 'modes of consciousness', establishing a common space for mapping the phenomenology of different conscious states. We outline both theoretical and empirical avenues to carry out such mapping between phenomenology and information-theoretic modes, starting from a central feature of everyday consciousness: selfhood. Overall, ΦID yields rich new ways to explore the relationship between information, consciousness, and its emergence from neural dynamics.

Consciousness and complexity: a consilience of evidence

Over the last years, a surge of empirical studies converged on complexity-related measures as reliable markers of consciousness across many different conditions, such as sleep, anesthesia, hallucinatory states, coma, and related disorders. Most of these measures were independently proposed by researchers endorsing disparate frameworks and employing different methods and techniques. Since this body of evidence has not been systematically reviewed and coherently organized so far, this positive trend has remained somewhat below the radar. The aim of this paper is to make this consilience of evidence in the science of consciousness explicit. We start with a systematic assessment of the growing literature on complexity-related measures and identify their common denominator, tracing it back to core theoretical principles and predictions put forward more than 20 years ago. In doing this, we highlight a consistent trajectory spanning two decades of consciousness research and provide a provisional taxonomy of the present literature. Finally, we consider all of the above as a positive ground to approach new questions and devise future experiments that may help consolidate and further develop a promising field where empirical research on consciousness appears to have, so far, naturally converged.

Bits and pieces: understanding information decomposition from part-whole relationships and formal logic

Partial information decomposition (PID) seeks to decompose the multivariate mutual information that a set of source variables contains about a target variable into basic pieces, the so-called ‘atoms of information’. Each atom describes a distinct way in which the sources may contain information about the target. For instance, some information may be contained uniquely in a particular source, some information may be shared by multiple sources and some information may only become accessible synergistically if multiple sources are combined. In this paper, we show that the entire theory of PID can be derived, firstly, from considerations of part-whole relationships between information atoms and mutual information terms, and secondly, based on a hierarchy of logical constraints describing how a given information atom can be accessed. In this way, the idea of a PID is developed on the basis of two of the most elementary relationships in nature: the part-whole relationship and the relation of logical implication. This unifying perspective provides insights into pressing questions in the field such as the possibility of constructing a PID based on concepts other than redundant information in the general n-sources case. Additionally, it admits of a particularly accessible exposition of PID theory.

A Complex Systems Perspective on Neuroimaging Studies of Behavior and Its Disorders

The study of complex systems deals with emergent behavior that arises as a result of nonlinear spatiotemporal interactions between a large number of components both within the system, as well as between the system and its environment. There is a strong case to be made that neural systems as well as their emergent behavior and disorders can be studied within the framework of complexity science. In particular, the field of neuroimaging has begun to apply both theoretical and experimental procedures originating in complexity science—usually in parallel with traditional methodologies. Here, we illustrate the basic properties that characterize complex systems and evaluate how they relate to what we have learned about brain structure and function from neuroimaging experiments. We then argue in favor of adopting a complex systems-based methodology in the study of neuroimaging, alongside appropriate experimental paradigms, and with minimal influences from noncomplex system approaches. Our exposition includes a review of the fundamental mathematical concepts, combined with practical examples and a compilation of results from the literature.