How human prefrontal and insular regions interact while maximizing rewards and minimizing punishments is unknown. Capitalizing on human intracranial recordings, we demonstrate that the functional specificity toward reward or punishment learning is better disentangled by interactions compared to local representations. Prefrontal and insular cortices display non-selective neural populations to rewards and punishments. Non-selective responses, however, give rise to context-specific interareal interactions. We identify a reward subsystem with redundant interactions between the orbitofrontal and ventromedial prefrontal cortices, with a driving role of the latter. In addition, we find a punishment subsystem with redundant interactions between the insular and dorsolateral cortices, with a driving role of the insula. Finally, switching between reward and punishment learning is mediated by synergistic interactions between the two subsystems. These results provide a unifying explanation of distributed cortical representations and interactions supporting reward and punishment learning.

Understanding how different networks relate to each other is key for understanding complex systems. We introduce an intuitive yet powerful framework to disentangle different ways in which networks can be (dis)similar and complementary to each other. We decompose the shortest paths between nodes as uniquely contributed by one source network, or redundantly by either, or synergistically by both together. Our approach considers the networks\' full topology, providing insights at multiple levels of resolution: from global statistics to individual paths. Our framework is widely applicable across scientific domains, from public transport to brain networks. In humans and 124 other species, we demonstrate the prevalence of unique contributions by long-range white-matter fibers in structural brain networks. Across species, efficient communication also relies on significantly greater synergy between long-range and short-range fibers than expected by chance. Our framework could find applications for designing network systems or evaluating existing ones.

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.

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\'.

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.

If regularity in data takes the form of higher-order functions among groups of variables, models which are biased towards lower-order functions may easily mistake the data for noise. To distinguish whether this is the case, one must be able to quantify the contribution of different orders of dependence to the total information. Recent work in information theory attempts to do this through measures of multivariate mutual information (MMI) and information decomposition (ID). Despite substantial theoretical progress, practical issues related to tractability and learnability of higher-order functions are still largely unaddressed. In this work, we introduce a new approach to information decomposition-termed Neural Information Decomposition (NID)-which is both theoretically grounded, and can be efficiently estimated in practice using neural networks. We show on synthetic data that NID can learn to distinguish higher-order functions from noise, while many unsupervised probability models cannot. Additionally, we demonstrate the usefulness of this framework as a tool for exploring biological and artificial neural networks.

The entropy of a pair of random variables is commonly depicted using a Venn diagram. This representation is potentially misleading, however, since the multivariate mutual information can be negative. This paper presents new measures of multivariate information content that can be accurately depicted using Venn diagrams for any number of random variables. These measures complement the existing measures of multivariate mutual information and are constructed by considering the algebraic structure of information sharing. It is shown that the distinct ways in which a set of marginal observers can share their information with a non-observing third party corresponds to the elements of a free distributive lattice. The redundancy lattice from partial information decomposition is then subsequently and independently derived by combining the algebraic structures of joint and shared information content.

Heart rate variability (HRV; variability of the RR interval of the electrocardiogram) results from the activity of several coexisting control mechanisms, which involve the influence of respiration (RESP) and systolic blood pressure (SBP) oscillations operating across multiple temporal scales and changing in different physiological states. In this study, multiscale information decomposition is used to dissect the physiological mechanisms related to the genesis of HRV in 78 young volunteers monitored at rest and during postural and mental stress evoked by head-up tilt (HUT) and mental arithmetics (MA). After representing RR, RESP and SBP at different time scales through a recently proposed method based on multivariate state space models, the joint information transfer T RESP , SBP → RR is decomposed into unique, redundant and synergistic components, describing the strength of baroreflex modulation independent of respiration ( U SBP → RR ), nonbaroreflex ( U RESP → RR ) and baroreflex-mediated ( R RESP , SBP → RR ) respiratory influences, and simultaneous presence of baroreflex and nonbaroreflex respiratory influences ( S RESP , SBP → RR ), respectively. We find that fast (short time scale) HRV oscillations-respiratory sinus arrhythmia-originate from the coexistence of baroreflex and nonbaroreflex (central) mechanisms at rest, with a stronger baroreflex involvement during HUT. Focusing on slower HRV oscillations, the baroreflex origin is dominant and MA leads to its higher involvement. Respiration influences independent on baroreflex are present at long time scales, and are enhanced during HUT.

Information is often described as a reduction of uncertainty associated with a restriction of possible choices. Despite appearing in Hartley\'s foundational work on information theory, there is a surprising lack of a formal treatment of this interpretation in terms of exclusions. This paper addresses the gap by providing an explicit characterisation of information in terms of probability mass exclusions. It then demonstrates that different exclusions can yield the same amount of information and discusses the insight this provides about how information is shared amongst random variables-lack of progress in this area is a key barrier preventing us from understanding how information is distributed in complex systems. The paper closes by deriving a decomposition of the mutual information which can distinguish between differing exclusions; this provides surprising insight into the nature of directed information.

The formulation of the Partial Information Decomposition (PID) framework by Williams and Beer in 2010 attracted a significant amount of attention to the problem of defining redundant (or shared), unique and synergistic (or complementary) components of mutual information that a set of source variables provides about a target. This attention resulted in a number of measures proposed to capture these concepts, theoretical investigations into such measures, and applications to empirical data (in particular to datasets from neuroscience). In this Special Issue on \"Information Decomposition of Target Effects from Multi-Source Interactions\" at Entropy, we have gathered current work on such information decomposition approaches from many of the leading research groups in the field. We begin our editorial by providing the reader with a review of previous information decomposition research, including an overview of the variety of measures proposed, how they have been interpreted and applied to empirical investigations. We then introduce the articles included in the special issue one by one, providing a similar categorisation of these articles into: i. proposals of new measures; ii. theoretical investigations into properties and interpretations of such approaches, and iii. applications of these measures in empirical studies. We finish by providing an outlook on the future of the field.