To fully characterize the information that two source variables carry about a third target variable, one must decompose the total information into redundant, unique, and synergistic components, i.e., obtain a partial information decomposition (PID). However, Shannon's theory of information does not provide formulas to fully determine these quantities. Several recent studies have begun addressing this. Some possible definitions for PID quantities have been proposed and some analyses have been carried out on systems composed of discrete variables. Here we present an in-depth analysis of PIDs on Gaussian systems, both static and dynamical. We show that, for a broad class of Gaussian systems, previously proposed PID formulas imply that (i) redundancy reduces to the minimum information provided by either source variable and hence is independent of correlation between sources, and (ii) synergy is the extra information contributed by the weaker source when the stronger source is known and can either increase or decrease with correlation between sources. We find that Gaussian systems frequently exhibit net synergy, i.e., the information carried jointly by both sources is greater than the sum of information carried by each source individually. Drawing from several explicit examples, we discuss the implications of these findings for measures of information transfer and information-based measures of complexity, both generally and within a neuroscience setting. Importantly, by providing independent formulas for synergy and redundancy applicable to continuous time-series data, we provide an approach to characterizing and quantifying information sharing amongst complex system variables.

Granger causality analysis is a popular method for inference on directed interactions in complex systems of many variables. A shortcoming of the standard framework for Granger causality is that it only allows for examination of interactions between single (univariate) variables within a system, perhaps conditioned on other variables. However, interactions do not necessarily take place between single variables but may occur among groups or ``ensembles'' of variables. In this study we establish a principled framework for Granger causality in the context of causal interactions among two or more multivariate sets of variables. Building on Geweke's seminal 1982 work, we offer additional justifications for one particular form of multivariate Granger causality based on the generalized variances of residual errors. Taken together, our results support a comprehensive and theoretically consistent extension of Granger causality to the multivariate case. Treated individually, they highlight several specific advantages of the generalized variance measure, which we illustrate using applications in neuroscience as an example. We further show how the measure can be used to define ``partial'' Granger causality in the multivariate context and we also motivate reformulations of ``causal density'' and ``Granger autonomy.'' Our results are directly applicable to experimental data and promise to reveal new types of functional relations in complex systems, neural and otherwise.

A recent measure of ‘integrated information’, Φ_DM, quantifies the extent to which a system generates more information than the sum of its parts as it transitions between states, possibly reflecting levels of consciousness generated by neural systems. However, Φ_DM is defined only for discrete Markov systems, which are unusual in biology; as a result, Φ_DM can rarely be measured in practice. Here, we describe two new measures, Φ_E and Φ_AR, that overcome these limitations and are easy to apply to time-series data. We use simulations to demonstrate the in-practice applicability of our measures, and to explore their properties. Our results provide new opportunities for examining information integration in real and model systems and carry implications for relations between integrated information, consciousness, and other neurocognitive processes. However, our findings pose challenges for theories that ascribe physical meaning to the measured quantities. A key feature of the human brain is its ability to represent a vast amount of information, and to integrate this information in order to produce specific and selective behaviour, as well as a stream of unified conscious scenes. Attempts have been made to quantify so-called ‘integrated information’ by formalizing in mathematics the extent to which a system as a whole generates more information than the sum of its parts. However, so far, the resulting measures have turned out to be inapplicable to real neural systems. In this paper we introduce two new measures that can be applied to both realistic neural models and to time-series data garnered from a broad range of neuroimaging and electrophysiological methods. Our work provides new opportunities for examining the role of integrated information in cognition and consciousness, and indeed in the function of any complex biological system. However, our results also pose challenges for theories that ascribe a direct physical meaning to any version of integrated information so far described.

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.

Changes in conscious level have been associated with changes in dynamical integration and segregation among distributed brain regions. Recent theoretical developments emphasize changes in directed functional (i.e., causal) connectivity as reflected in quantities such as ‘integrated information’ and ‘causal density’. Here we develop and illustrate a rigorous methodology for assessing causal connectivity from electroencephalographic (EEG) signals using Granger causality (GC). Our method addresses the challenges of non-stationarity and bias by dividing data into short segments and applying permutation analysis. We apply the method to EEG data obtained from subjects undergoing propofol-induced anaesthesia, with signals source-localized to the anterior and posterior cingulate cortices. We found significant increases in bidirectional GC in most subjects during loss-of-consciousness, especially in the beta and gamma frequency ranges. Corroborating a previous analysis we also found increases in synchrony in these ranges; importantly, the Granger causality analysis showed higher inter-subject consistency than the synchrony analysis. Finally, we validate our method using simulated data generated from a model for which GC values can be analytically derived. In summary, our findings advance the methodology of Granger causality analysis of EEG data and carry implications for integrated information and causal density theories of consciousness.

To truly eliminate Cartesian ghosts from the science of consciousness, we must describe consciousness as an aspect of the physical. Integrated Information Theory states that consciousness arises from intrinsic information generated by dynamical systems; however existing formulations of this theory are not applicable to standard models of fundamental physical entities. Modern physics has shown that fields are fundamental entities, and in particular that the electromagnetic field is fundamental. Here I hypothesise that consciousness arises from information intrinsic to fundamental fields. This hypothesis unites fundamental physics with what we know empirically about the neuroscience underlying consciousness, and it bypasses the need to consider quantum effects.