Synergetic and redundant information flow detected by unnormalized Granger causality: application to resting state fMRI

Objectives: We develop a framework for the analysis of synergy and redundancy in the pattern of information flow between subsystems of a complex network. Methods: The presence of redundancy and/or synergy in multivariate time series data renders difficult to estimate the neat flow of information from each driver variable to a given target. We show that adopting an unnormalized definition of Granger causality one may put in evidence redundant multiplets of variables influencing the target by maximizing the total Granger causality to a given target, over all the possible partitions of the set of driving variables. Consequently we introduce a pairwise index of synergy which is zero when two independent sources additively influence the future state of the system, differently from previous definitions of synergy. Results: We report the application of the proposed approach to resting state fMRI data from the Human Connectome Project, showing that redundant pairs of regions arise mainly due to space contiguity and interhemispheric symmetry, whilst synergy occurs mainly between non-homologous pairs of regions in opposite hemispheres. Conclusions: Redundancy and synergy, in healthy resting brains, display characteristic patterns, revealed by the proposed approach. Significance: The pairwise synergy index, here introduced, maps the informational character of the system at hand into a weighted complex network: the same approach can be applied to other complex systems whose normal state corresponds to a balance between redundant and synergetic circuits.

💡 Research Summary

The paper introduces a novel framework for detecting and quantifying synergy and redundancy in the information flow of complex multivariate systems, based on an unnormalized version of Granger causality (GC). Traditional GC measures the logarithmic ratio of prediction errors, which obscures additive contributions of independent drivers because of the logarithm’s non‑linearity. Consequently, in the presence of redundant or synergistic variables, standard GC either under‑estimates or over‑estimates causal influence, making it unsuitable for dissecting the informational role of variable groups.

To overcome this, the authors define unnormalized Granger causality (δ_u) as the simple difference between the residual variance of a target when a set of drivers B is excluded and when it is included: δ_u(B→α)=ε(x_α|X\B)−ε(x_α|X). This definition restores additivity: if the drivers in B are statistically independent and their effects on the target are additive, then δ_u(B→α)=∑_{β∈B}δ_u(β→α). Leveraging this property, the authors propose quantitative criteria for synergy and redundancy:

• Synergy occurs when the combined influence of a set is less than the sum of individual influences (δ_u(B→α) < Σδ_u(β→α)).
• Redundancy occurs when the combined influence is greater than the sum (δ_u(B→α) > Σδ_u(β→α)).

These definitions are applicable to continuous variables and avoid the pitfalls of interaction information or partial information decomposition, which can yield counter‑intuitive results for Gaussian systems.

The framework also includes a pairwise synergy index (PSI). For any two candidate drivers X_i and X_j influencing a target α, PSI = δ_u({i,j}→α) – δ_u(i→α) – δ_u(j→α). By construction, PSI equals zero when the two sources act independently and additively, providing a clean baseline that previous synergy measures lack.

A key methodological step is the maximization of total unnormalized GC across all possible partitions of the driver set for a given target. By partitioning the n‑1 potential drivers into groups {A_ℓ} that maximize Σ_ℓ δ_u(A_ℓ→α), the algorithm automatically identifies clusters of redundant variables. The authors illustrate this with synthetic examples: (1) a hidden common source creates redundancy among observed variables, leading to a partition where the redundant pair is grouped together; (2) a nonlinear interaction term creates synergy that is missed by pairwise GC but captured by the unnormalized measure.

The authors apply the method to resting‑state functional MRI data from the Human Connectome Project (HCP). After preprocessing, 90 cortical and subcortical regions (based on a standard atlas) serve as nodes. For each region, the BOLD time series is treated as the target, and the remaining regions are candidate drivers. Unnormalized GC and PSI are computed using linear kernels for additive effects and polynomial kernels of degree two for detecting nonlinear interactions.

Results reveal two distinct spatial patterns:

1. Redundancy is dominant among spatially contiguous regions and across hemispheric homologues (e.g., left and right primary visual cortices). This likely reflects shared vascular dynamics, structural connectivity, or common functional modules.
2. Synergy is prominent between non‑homologous regions located in opposite hemispheres (e.g., left prefrontal and right posterior parietal areas). These pairs exhibit PSI > 0, indicating that their joint influence on a target exceeds the sum of their individual contributions, suggesting cooperative integration across functional networks.

These findings support the hypothesis that a healthy resting brain maintains a balance between redundant circuits (providing robustness) and synergistic circuits (enabling flexible integration). The pairwise synergy index also yields a weighted network that can be analyzed with graph‑theoretic tools, opening avenues for comparing healthy versus pathological states (e.g., schizophrenia, epilepsy) where this balance may be disrupted.

In summary, the paper makes three major contributions:

• Introduces unnormalized Granger causality, restoring additivity and enabling clear detection of redundancy and synergy.
• Proposes a pairwise synergy index that is zero for independent additive drivers, offering an intuitive baseline.
• Demonstrates the practical utility of the framework on large‑scale resting‑state fMRI, uncovering biologically plausible patterns of redundant and synergistic information flow.

The methodology is general and can be transferred to other domains with multivariate time series, such as climate science, finance, or genomics, wherever the interplay of redundant and synergistic influences shapes system dynamics.