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.

Integrated Information Theory (IIT) is a prominent theory of consciousness that has at its centre measures that quantify the extent to which a system generates more information than the sum of its parts. While several candidate measures of integrated information (“ $\mathsf{\Phi }$ ”) now exist, little is known about how they compare, especially in terms of their behaviour on non-trivial network models. In this article, we provide clear and intuitive descriptions of six distinct candidate measures. We then explore the properties of each of these measures in simulation on networks consisting of eight interacting nodes, animated with Gaussian linear autoregressive dynamics. We find a striking diversity in the behaviour of these measures—no two measures show consistent agreement across all analyses. A subset of the measures appears to reflect some form of dynamical complexity, in the sense of simultaneous segregation and integration between system components. Our results help guide the operationalisation of IIT and advance the development of measures of integrated information and dynamical complexity that may have more general applicability.

Background: Brain interdependencies can be studied from either a structural/anatomical perspective (“structural connectivity”) or by considering statistical interdependencies (“functional connectivity” [FC]). Interestingly, while structural connectivity is by definition pairwise (white-matter fibers project from one region to another), FC is not. However, most FC analyses only focus on pairwise statistics and they neglect higher order interactions. A promising tool to study high-order interdependencies is the recently proposed O-Information, which can quantify the intrinsic statistical synergy and the redundancy in groups of three or more interacting variables. Methods: We analyzed functional magnetic resonance imaging (fMRI) data obtained at rest from 164 healthy subjects with ages ranging in 10 to 80 years and used O-Information to investigate how high-order statistical interdependencies are affected by age. Results: Older participants (from 60 to 80 years old) exhibited a higher predominance of redundant dependencies compared with younger participants, an effect that seems to be pervasive as it is evident for all orders of interaction. In addition, while there is strong heterogeneity across brain regions, we found a “redundancy core” constituted by the prefrontal and motor cortices in which redundancy was evident at all the interaction orders studied. Discussion: High-order interdependencies in fMRI data reveal a dominant redundancy in functions such as working memory, executive, and motor functions. Our methodology can be used for a broad range of applications, and the corresponding code is freely available. Impact statement Past research has showcased multiple changes to the brain's structural and functional properties caused by aging. Here we expand prior work through recent advancements in multivariate information theory, which provide richer and more theoretically principled analyses than existing alternatives. We show that the brains of older participants contain more redundant information at multiple spatial scales—that is, activation in different brain regions is less diverse, compared with younger participants—and identify a “redundancy core” constituted by prefrontal and motor cortices, which might explained impaired performance in the old population in functions such as working memory and executive control.

Network inference algorithms are valuable tools for the study of large-scale neuroimaging datasets. Multivariate transfer entropy is well suited for this task, being a model-free measure that captures nonlinear and lagged dependencies between time series to infer a minimal directed network model. Greedy algorithms have been proposed to efficiently deal with high-dimensional datasets while avoiding redundant inferences and capturing synergistic effects. However, multiple statistical comparisons may inflate the false positive rate and are computationally demanding, which limited the size of previous validation studies. The algorithm we present—as implemented in the IDTxl open-source software—addresses these challenges by employing hierarchical statistical tests to control the family-wise error rate and to allow for efficient parallelization. The method was validated on synthetic datasets involving random networks of increasing size (up to 100 nodes), for both linear and nonlinear dynamics. The performance increased with the length of the time series, reaching consistently high precision, recall, and specificity (>98% on average) for 10,000 time samples. Varying the statistical significance threshold showed a more favorable precision-recall trade-off for longer time series. Both the network size and the sample size are one order of magnitude larger than previously demonstrated, showing feasibility for typical EEG and magnetoencephalography experiments.