Dissecting Spectral Granger Causality through Partial Information Decomposition

TL;DR

This work introduces Partial Decomposition of Granger Causality (PDGC), a tool eliciting redundant and synergistic causal interactions in the pattern of information flow between the subsystems of physiological networks, based on multivariate state-space models expanded in the frequency domain.

Abstract

Granger causality (GC), a popular statistical method for the inference of directional influences between time series measured from a complex network, is sensitive to high-order (non-pairwise) interactions which fundamentally shape the collective network dynamics. This work introduces Partial Decomposition of Granger Causality (PDGC), a tool eliciting redundant and synergistic causal interactions in the pattern of information flow between the subsystems of physiological networks. The tool exploits the framework of partial information decomposition to dissect the multivariate GC from a set of driver random processes to a target process into unique effects carried exclusively by each driver, redundant effects carried identically by more drivers, and synergistic effects carried jointly by some drivers but not by any of them individually. Computation is based on multivariate state-space models expanded in the frequency domain to assess PDGC both in specific bands of physiological interest and in the time domain after whole-band integration. The spectral PDGC was tested in physiological networks probed by measuring the variability series of arterial pressure, heart period, respiration and cerebral blood velocity in patients prone to neurally-mediated syncope compared to healthy controls. This application revealed unprecedented modes of physiological interaction, related to the sympathetic control of low-frequency cardiovascular and cerebrovascular oscillations, characterizing distinctive patterns of autonomic dysfunction. The extraction of high-order causality patterns from the spectral GC favors dissecting the mechanisms of causal influence underlying multivariate interactions among oscillatory processes in many data-driven applications of network science.

Figures (4)

• Figure 1: Granger Causality redundancy lattices for $N$=2 (a) and $N$=3 (b) source processes. The shaded areas denote how single PID atoms in (a) or groups of atoms in (b) are associated to unique (pink, green, orange), synergistic (blue), and redundant (red) components, providing a coarse-grained decomposition of the full GC. Red circles identify the atoms composing the bivariate GC $F_{X_1\rightarrow Y}$ in the two cases.
• Figure 2: Example of PDGC analyses performed for a representative SYNC patient in the resting condition. Panels depict the analyzed variability series (a) together with their estimated power spectrum (b), and the PDGC measures computed to analyze CV interactions (c-e; target $Y=H$, drivers $X=\{R,S\}$) and CB interactions (f-h; target $Y=V$, drivers $X=\{R,M,H\}$). In the two settings, panels depict the spectral and atomic GCs (c,f), the full spectral GC and the terms of its coarse-grained decomposition (d,g), and the time-domain values of the full and coarse-grained GCs computed along with their surrogate distributions (e,h).
• Figure 3: PDGC analysis of CV interactions in nonSYNC subjects (a) and SYNC patients (b). Panels depict the boxplots and individual values of the of the full GC from $\{$RESP,SAP$\}$ to HP (gray), as well as of its coarse grained decomposition evidencing the unique GCs originating from RESP (violet) and SAP (green), the synergistic GC (blue) and the redundant GC (red), computed at REST and during HUT through whole-band integration (top panels) or through integration within the LF band (middle panels) or the HF band (bottom panels); the number of nonSYNC or SYNC participants (out of 13) for whom each measure was deemed as statistically significant according to surrogate data analysis is also reported in each panel. *, $p<0.05$ REST vs. HUT, Wilcoxon signed-rank test.
• Figure 4: PDGC analysis of CB interactions in nonSYNC subjects (a) and SYNC patients (b). Panels depict the boxplots and individual values of the full GC from $\{$RESP,MAP,HP$\}$ to MCBV (gray), as well as of its coarse grained decomposition evidencing the unique GCs originating from RESP (violet), MAP (green) and HP (orange), the synergistic GC (blue) and the redundant GC (red), computed both at REST and during HUT through whole-band integration (top panels) or through integration within the LF band (middle panels) or the HF band (bottom panels); the number of nonSYNC or SYNC participants (out of 13) for whom each measure was deemed as statistically significant according to surrogate data analysis is also reported in each panel. *, $p<0.05$ REST vs. HUT, Wilcoxon signed-rank test.