Synergy and redundancy in the Granger causal analysis of dynamical networks

Abstract

We analyze, by means of Granger causality (GC), the effect of synergy and redundancy in the inference (from time series data) of the information flow between subsystems of a complex network. While we show that fully conditioned GC (CGC) is not affected by synergy, the pairwise analysis fails to prove synergetic effects. In cases when the number of samples is low, thus making the fully conditioned approach unfeasible, we show that partially conditioned GC (PCGC) is an effective approach if the set of conditioning variables is properly chosen. Here we consider two different strategies (based either on informational content for the candidate driver or on selecting the variables with highest pairwise influences) for PCGC and show that, depending on the data structure, either one or the other might be equally valid. On the other hand, we observe that fully conditioned approaches do not work well in the presence of redundancy, thus suggesting the strategy of separating the pairwise links in two subsets: those corresponding to indirect connections of the CGC (which should thus be excluded) and links that can be ascribed to redundancy effects and, together with the results from the fully connected approach, provide a better description of the causality pattern in the presence of redundancy. Finally we apply these methods Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.to two different real datasets. First, analyzing electrophysiological data from an epileptic brain, we show that synergetic effects are dominant just before seizure occurrences. Second, our analysis applied to gene expression time series from HeLa culture shows that the underlying regulatory networks are characterized by both redundancy and synergy.

Quantifying Information Modification in Developing Neural Networks via Partial Information Decomposition

“…We also note that there are earlier applications of the concept of synergy (meant as synergistic mutual information) to neural data (e.g., [46][47][48][49]) that relied on the computation of interaction information. However, when interpreting these studies, it should be kept in mind that these report the difference between shared information and synergistic information-as detailed by Williams and Beer [10].…”

Information Decomposition in Multivariate Systems: Definitions, Implementation and Application to Cardiovascular Networks

“…Operational definitions of these concepts have been proposed in recent years, which allow to quantify predictive information through measures of prediction entropy or full-predictability [11,12], information storage through the self-entropy or self-predictability [11,13], information transfer through transfer entropy or Granger causality [14], and information modification through entropy and prediction measures of net redundancy/synergy [11,15] or separate measures derived from partial information decomposition [16,17]. All these measures have been successfully applied in diverse fields of science ranging from cybernetics to econometrics, climatology, neuroscience and others [6,7,[18][19][20][21][22][23][24][25][26][27][28]. In particular, recent studies have implemented these measures in cardiovascular physiology to study the short-term dynamics of the cardiac, vascular and respiratory systems in terms of information storage, transfer and modification [12,13,29].…”

Synergetic and Redundant Information Flow Detected by Unnormalized Granger Causality: Application to Resting State fMRI

“…The following examples show that conditioned GC tends to be reduced in presence of redundancy and increased in presence of synergy, the latter occurrence being a problem for pairwise GC, see [15,29].…”