arXiv subjects

Lionel Barnett

Publications and source records attributed to Lionel Barnett.

16 recordsLinked to original sources

arXiv2026

Integrated information theory: the good, the bad and the misunderstood

The integrated information theory of consciousness (IIT) is uniquely ambitious in proposing a mathematical formula, derived from apparently fundamental properties of conscious experience, to describe the quantity and quality of consciousness for any physical system that possesses it. IIT has generated considerable debate, which has engendered some misunderstandings and misrepresentations. Here we address and hope to remedy this. We begin by concisely summarising the essentials of IIT. Given IIT is supposed to apply universally, we do this with reference to an arbitrary patch of matter, as opposed to the usual system of discrete computational units. Then, after briefly summarising IIT's theoretical and empirical achievements, we focus on five points which we consider especially important for driving forward new theory and increasing understanding. First, a high value of the measure $Φ$ is not synonymous with `more consciousness'. We describe how $Φ$ might be replaced with a suite of quantities to obtain a multi-dimensional characterisation of states of consciousness. Second, we describe with nuance the distinct flavour of panpsychism implied by IIT -- whereby space (and time) are tiled with substrates of (proto-) consciousness -- and find this is not problematic for the theory. Third, $Φ$ is not well-defined for real physical systems, and has not been computed on any real physical system. Fourth, so far only proxies for IIT measures have been computed, and not approximations. Fifth, for IIT to fit with current successful theories in fundamental physics, a reformulation in terms of continuous fields would be needed.

q-bio.NC↗

arXiv2026

Granger Causality Maps for Langevin Systems

Wahl et al. (2016, 2017) introduced the idea of Granger causality (GC) maps for Langevin systems: dynamics are localised linearly at each point in phase space as vector Ornstein-Uhlenbeck (VOU) processes, for which GCs may in principle be calculated, thus constructing a GC map on phase space. Their implementation, however, suffered a significant drawback: GCs were approximated from models based on discrete-time stroboscopic sampling of local VOU processes, which is not only computationally inefficient, but more seriously, unfeasible on regions of phase space where local dynamics are unstable, leaving "holes" in the GC maps. We solve these problems by deriving an analytical expression for GC rates associated with a VOU process which, under quite general conditions, yields a meaningful solution even in the unstable case. Applied to GC maps, this not only "fills in the holes", but also furnishes a computationally efficient method of calculation devolving to solution of continuous-time algebraic Riccati equations which, in the case of a univariate source, become simple quadratic equations. We show, furthermore, that the GC rate for VOU processes is invariant under rescaling of the overall fluctuations intensity, so that GC maps may effectively be calculated for deterministic nonlinear dynamical systems, with a residual "ghost of noise" represented by a variance-covariance map.

math-ph↗

arXiv2023

Minimum-phase property of the hemodynamic response function, and implications for Granger Causality in fMRI

Granger Causality (GC) is widely used in neuroimaging to estimate directed statistical dependence among brain regions using time series of brain activity. An important issue is that functional MRI (fMRI) measures brain activity indirectly via the blood-oxygen-level-dependent (BOLD) signal, which affects the temporal structure of the signals and distorts GC estimates. However, some notable applications of GC are not concerned with the GC magnitude but its statistical significance. This is the case for network inference, which aims to build a statistical model of the system based on directed relationships among its elements. The critical question for the viability of network inference in fMRI is whether the hemodynamic response function (HRF) and its variability across brain regions introduce spurious relationships, i.e., statistically significant GC values between BOLD signals, even if the GC between the neuronal signals is zero. It has been mathematically proven that such spurious statistical relationships are not induced if the HRF is minimum-phase, i.e., if both the HRF and its inverse are stable (producing finite responses to finite inputs). However, whether the HRF is minimum-phase has remained contentious. Here, we address this issue using multiple realistic biophysical models from the literature and studying their transfer functions. We find that these models are minimum-phase for a wide range of physiologically plausible parameter values. Therefore, statistical testing of GC is plausible even if the HRF varies across brain regions, with the following limitations. First, the minimum-phase condition is violated for parameter combinations that generate an initial dip in the HRF, confirming a previous mathematical proof. Second, the slow sampling of the BOLD signal (in seconds) compared to the timescales of neural signal propagation (milliseconds) may still introduce spurious GC.

q-bio.NC↗

arXiv2021

Dynamical independence: discovering emergent macroscopic processes in complex dynamical systems

We introduce a notion of emergence for coarse-grained macroscopic variables associated with highly-multivariate microscopic dynamical processes, in the context of a coupled dynamical environment. 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 emphasise 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 via state-space modelling, in both time and frequency domains, facilitating discovery of emergent phenomena at all spatiotemporal scales. We discuss application of the state-space operationalisation to inference of the emergence portrait for neural systems from neurophysiological time-series data. We also examine 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.

nlin.AO↗

arXiv2021

Information Flow in First-Order Potts Model Phase Transition

Phase transitions abound in nature and society, and, from species extinction to stock market collapse, their prediction is of widespread importance. In earlier work we showed that Global Transfer Entropy, a general measure of information flow, was found to peak away from the transition on the disordered side for the Ising model, a canonical second-order transition. Here we show that (a) global transfer entropy also peaks on the disordered side of the transition of finite first-order transitions, i.e., those which have finite latent heat and no correlation length divergence, such as ecology dynamics on coral reefs, and (b) analysis of information flow across state boundaries unifies both transition orders. We obtain the first information-theoretic result for the high-order Potts model and the first demonstration of early warning of a first-order transition. The unexpected earlier finding that global transfer entropy peaks on the disordered side of a transition is also found for finite first-order systems, albeit not in the thermodynamic limit. By noting that the interface length of clusters in each phase is the dominant region of information flow, we unify the information theoretic behaviour of first and second-order transitions.

cond-mat.stat-mech↗

arXiv2019

Inferring the temporal structure of directed functional connectivity in neural systems: some extensions to Granger causality

Neural processes in the brain operate at a range of temporal scales. Granger causality, the most widely-used neuroscientific tool for inference of directed functional connectivity from neurophsyiological data, is traditionally deployed in the form of one-step-ahead prediction regardless of the data sampling rate, and as such yields only limited insight into the temporal structure of the underlying neural processes. We introduce Granger causality variants based on multi-step, infinite-future and single-lag prediction, which facilitate a more detailed and systematic temporal analysis of information flow in the brain.

stat.AP↗

arXiv2018

Information Flow in Finite Flocks

We simulate the canonical Vicsek model and estimate the flow of information as a function of noise (the variability in the extent to which each animal aligns with its neighbours). We show that the global transfer entropy for finite flocks not only fails to peak near the phase transition, as demonstrated for the canonical 2D Ising model, but remains constant from the transition to very low noise values. This provides a foundation for future study regarding information flow in more complex models and real-world flocking data.

cond-mat.stat-mech↗

arXiv2018

Information Flow in Finite Flocks with Topological Interactions

We simulate the Vicsek model utilising topological neighbour interactions and estimate information theoretic quantities as a function of noise, the variability in the extent to which each animal aligns with its neighbours, and the flock direction. We show that these quantities, mutual information and global transfer entropy, are in fact dependent on observation time, and in comparison to the canonical Vicsek model which utilises range-based interactions, the topological variant converges to the long-term limiting behaviour with smaller observation windows. Finally, we show that in contrast to the metric model, which exhibits maximal information flow for the ordered regime, the topological model maintains this maximal information flow beyond the phase transition and into the disordered regime.

cond-mat.stat-mech↗

arXiv2018

Solved problems and remaining challenges for Granger causality analysis in neuroscience: A response to Stokes and Purdon (2017)

Granger-Geweke causality (GGC) is a powerful and popular method for identifying directed functional (`causal') connectivity in neuroscience. In a recent paper, Stokes and Purdon [1] raise several concerns about its use. They make two primary claims: (1) that GGC estimates may be severely biased or of high variance, and (2) that GGC fails to reveal the full structural/causal mechanisms of a system. However, these claims rest, respectively, on an incomplete evaluation of the literature, and a misconception about what GGC can be said to measure. Here we explain how existing approaches (as implemented, for example, in our popular MVGC software [2,3]) resolve the first issue, and discuss the frequently-misunderstood distinction between functional and effective neural connectivity which underlies Stokes and Purdon's second claim. [1] Patrick A. Stokes and Patrick. L. Purdon (2017), A study of problems encountered in Granger causality analysis from a neuroscience perspective, Proc. Natl. Acad. Sci. USA 114(34):7063-7072. [2] Lionel Barnett and Anil K. Seth (2012), The MVGC Multivariate Granger Causality Matlab toolbox, http://users.sussex.ac.uk/~lionelb/MVGC/ [3] Lionel Barnett and Anil K. Seth (2014), The MVGC multivariate Granger causality toolbox: A new approach to Granger-causal inference, J. Neurosci. Methods 223:50-68

stat.ME↗

arXiv2017

Review of Data Structures for Computationally Efficient Nearest-Neighbour Entropy Estimators for Large Systems with Periodic Boundary Conditions

Information theoretic quantities are extremely useful in discovering relationships between two or more data sets. One popular method---particularly for continuous systems---for estimating these quantities is the nearest neighbour estimators. When system sizes are very large or the systems have periodic boundary conditions issues with performance and correctness surface, however solutions are known for each problem. Here we show that these solutions are inappropriate in systems that simultaneously contain both features and discuss a lesser known alternative solution involving Vantage Point trees that is capable of addressing both issues.

stat.CO↗

arXiv2017

Anomalous Behaviour of Mutual Information in Finite Flocks

The existing consensus is that flocks are poised at criticality, entailing long correlation lengths and a maximal value of Shannon mutual information in the large-system limit. We show, by contrast, that for finite flocks in the long observation time limit, mutual information may not only fail to peak at criticality---as observed for other critical systems---but also diverge as noise tends to zero. This result carries implications for other finite-size, out-of-equilibrium systems, where observation times may vary widely compared to time scales of internal system dynamics; thus it may not be assumed that mutual information locates the phase transition.

cond-mat.stat-mech↗

arXiv2016

Detectability of Granger causality for subsampled continuous-time neurophysiological processes

Granger causality is well established within the neurosciences for inference of directed functional connectivity from neurophysiological data. These data usually consist of time series which subsample a continuous-time biophysiological process. While it is well-known that subsampling can lead to imputation of spurious causal connections where none exist, here we address the equally important issue of the effects of subsampling on the ability to reliably detect causal connections which do exist. Neurophysiological processes typically feature signal propagation delays on multiple time scales; accordingly, we base our analysis on a distributed-lag, continuous-time stochastic model, and consider Granger causality in continuous time at finite prediction horizons. Via exact analytical solutions, we identify relationships among sampling frequency, underlying causal time scales and detectability of causalities. Our analysis reveals complex interactions between the time scale(s) of neural signal propagation and sampling frequency: we demonstrate that Granger causality decays exponentially as the sample time interval increases beyond causal delay times, identify detectability "black spots" and "sweet spots", and show that subsampling may sometimes improve detectability. We also demonstrate that the invariance of Granger causality under causal, invertible filtering fails at finite prediction horizons. We discuss the implications of our results for inference of Granger causality at the neural level from various neurophysiological recording modes, and emphasise that sampling rates for causal analysis of neurophysiological time series should be informed by domain-specific time scales.

stat.AP↗

arXiv2015

Granger causality for state space models

Granger causality, a popular method for determining causal influence between stochastic processes, is most commonly estimated via linear autoregressive modeling. However, this approach has a serious drawback: if the process being modeled has a moving average component, then the autoregressive model order is theoretically infinite, and in finite sample large empirical model orders may be necessary, resulting in weak Granger-causal inference. This is particularly relevant when the process has been filtered, downsampled, or observed with (additive) noise - all of which induce a moving average component and are commonplace in application domains as diverse as econometrics and the neurosciences. By contrast, the class of autoregressive moving average models - or, equivalently, linear state space models - is closed under digital filtering, downsampling (and other forms of aggregation) as well as additive observational noise. Here, we show how Granger causality, conditional and unconditional, in both time and frequency domains, may be calculated simply and directly from state space model parameters, via solution of a discrete algebraic Riccati equation. Numerical simulations demonstrate that Granger causality estimators thus derived have greater statistical power and smaller bias than pure autoregressive estimators. We conclude that the state space approach should be the default for (linear) Granger causality estimation.

math.ST↗

arXiv2012

Transfer Entropy as a Log-likelihood Ratio

Transfer entropy, an information-theoretic measure of time-directed information transfer between joint processes, has steadily gained popularity in the analysis of complex stochastic dynamics in diverse fields, including the neurosciences, ecology, climatology and econometrics. We show that for a broad class of predictive models, the log-likelihood ratio test statistic for the null hypothesis of zero transfer entropy is a consistent estimator for the transfer entropy itself. For finite Markov chains, furthermore, no explicit model is required. In the general case, an asymptotic chi-squared distribution is established for the transfer entropy estimator. The result generalises the equivalence in the Gaussian case of transfer entropy and Granger causality, a statistical notion of causal influence based on prediction via vector autoregression, and establishes a fundamental connection between directed information transfer and causality in the Wiener-Granger sense.

stat.AP↗

arXiv2010

Multivariate Granger Causality and Generalized Variance

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 new 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.

q-bio.NC↗

arXiv2009

Granger causality and transfer entropy are equivalent for Gaussian variables

Granger causality is a statistical notion of causal influence based on prediction via vector autoregression. Developed originally in the field of econometrics, it has since found application in a broader arena, particularly in neuroscience. More recently transfer entropy, an information-theoretic measure of time-directed information transfer between jointly dependent processes, has gained traction in a similarly wide field. While it has been recognized that the two concepts must be related, the exact relationship has until now not been formally described. Here we show that for Gaussian variables, Granger causality and transfer entropy are entirely equivalent, thus bridging autoregressive and information-theoretic approaches to data-driven causal inference.

math-ph↗