title: Granger causality revisited.
abstract: This technical paper offers a critical re-evaluation of (spectral) Granger causality measures in the analysis of biological timeseries. Using realistic (neural mass) models of coupled neuronal dynamics, we evaluate the robustness of parametric and nonparametric Granger causality. Starting from a broad class of generative (state-space) models of neuronal dynamics, we show how their Volterra kernels prescribe the second-order statistics of their response to random fluctuations; characterised in terms of cross-spectral density, cross-covariance, autoregressive coefficients and directed transfer functions. These quantities in turn specify Granger causality - providing a direct (analytic) link between the parameters of a generative model and the expected Granger causality. We use this link to show that Granger causality measures based upon autoregressive models can become unreliable when the underlying dynamics is dominated by slow (unstable) modes - as quantified by the principal Lyapunov exponent. However, nonparametric measures based on causal spectral factors are robust to dynamical instability. We then demonstrate how both parametric and nonparametric spectral causality measures can become unreliable in the presence of measurement noise. Finally, we show that this problem can be finessed by deriving spectral causality measures from Volterra kernels, estimated using dynamic causal modelling.
authors.affiliation: The Wellcome Trust Centre for Neuroimaging, University College London, Queen Square, London WC1N 3BG, UK. Electronic address: k.friston@ucl.ac.uk.
authors.forename: Karl J
authors.initials: KJ
authors.lastname: Friston
authors.affiliation: Center for Neuroscience and Center for Mind and Brain, University of California-Davis, Davis, CA 95618, USA; Ernst Strüngmann Institute in Cooperation with Max Planck Society, Deutschordenstraße 46, 60528 Frankfurt, Germany.
authors.forename: André M
authors.initials: AM
authors.lastname: Bastos
authors.affiliation: The Wellcome Trust Centre for Neuroimaging, University College London, Queen Square, London WC1N 3BG, UK.
authors.forename: Ashwini
authors.initials: A
authors.lastname: Oswal
authors.affiliation: The Wellcome Trust Centre for Neuroimaging, University College London, Queen Square, London WC1N 3BG, UK.
authors.forename: Bernadette
authors.initials: B
authors.lastname: van Wijk
authors.affiliation: Ernst Strüngmann Institute in Cooperation with Max Planck Society, Deutschordenstraße 46, 60528 Frankfurt, Germany.
authors.forename: Craig
authors.initials: C
authors.lastname: Richter
authors.affiliation: The Wellcome Trust Centre for Neuroimaging, University College London, Queen Square, London WC1N 3BG, UK.
authors.forename: Vladimir
authors.initials: V
authors.lastname: Litvak
mesh_terms: D063132:Connectome / Q000379:methods*; D003627:Data Interpretation, Statistical; D006801:Humans; D008962:Models, Theoretical
keywords: Cross spectra; Dynamic causal modelling; Dynamics; Effective connectivity; Functional connectivity; Granger causality; Neurophysiology