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Multivariate information theory uncovers synergistic subsystems of the human cerebral cortex

Varley, Thomas F., Pope, Maria, Faskowitz, Joshua, Sporns, Olaf·2023·Communications Biology
DOI 10.1038/s42003-023-04843-wPMC PMC10125999
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Abstract

One of the most well-established tools for modeling the brain is the functional connectivity network, which is constructed from pairs of interacting brain regions. While powerful, the network model is limited by the restriction that only pairwise dependencies are considered and potentially higher-order structures are missed. Here, we explore how multivariate information theory reveals higher-order dependencies in the human brain. We begin with a mathematical analysis of the O-information, showing analytically and numerically how it is related to previously established information theoretic measures of complexity. We then apply the O-information to brain data, showing that synergistic subsystems are widespread in the human brain. Highly synergistic subsystems typically sit between canonical functional networks, and may serve an integrative role. We then use simulated annealing to find maximally synergistic subsystems, finding that such systems typically comprise ≈10 brain regions, recruited from multiple canonical brain systems. Though ubiquitous, highly synergistic subsystems are invisible when considering pairwise functional connectivity, suggesting that higher-order dependencies form a kind of shadow structure that has been unrecognized by established network-based analyses. We assert that higher-order interactions in the brain represent an under-explored space that, accessible with tools of multivariate information theory, may offer novel scientific insights.

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Fig. 1 Understanding O-information in the context of the TSE complexity. The TSE complexity provides a system-wide summary statistic of how integrated the system is at every scale. The O-information can be understood as measuring how sensitive the global integration is to the removal of single elements (on average). a The panel shows a TSE curve for a low-synergy system: TC(X)/N > DTC(X)/N, so Ω(X) > 0. The erasure of any single element, on average, does not change the overall integration of the remaining (N − 1) elements much more than would be expected in the null case. b The panel shows the case where TC(X)/N < DTC(X)/N, so Ω(X) < 0 and the system is synergy dominated. Intuitively, this can be understood by recognizing that, on average, the removal of any of the N elements causes a large decrease in the integration among the remaining (N − 1) elements.
Fig. 1 Understanding O-information in the context of the TSE complexity. The TSE complexity provides a system-wide summary statistic of how integrated the system is at every scale. The O-information can be understood as measuring how sensitive the global integration is to the removal of single elements (on average). a The panel shows a TSE curve for a low-synergy system: TC(X)/N > DTC(X)/N, so Ω(X) > 0. The erasure of any single element, on average, does not change the overall integration of the remaining (N − 1) elements much more than would be expected in the null case. b The panel shows the case where TC(X)/N < DTC(X)/N, so Ω(X) < 0 and the system is synergy dominated. Intuitively, this can be understood by recognizing that, on average, the removal of any of the N elements causes a large decrease in the integration among the remaining (N − 1) elements.CC BY · PubMed Central
Fig. 2 Approximating TSE complexity with total correlation and dual total correlation. Data are from 100,000 randomly selected subsets of 10 nodes (blue: HCP data; red: MICA data), with TSE complexity computed exactly, by sampling all subsystems. a Sum of TC + DTC versus TSE complexity; HCP: R = 0.998, p = 0; MICA: R = 0.999, p = 0. b DTC versus TSE; HCP: R = 0.982, p = 0; MICA: R = 0.992, p = 0.
Fig. 2 Approximating TSE complexity with total correlation and dual total correlation. Data are from 100,000 randomly selected subsets of 10 nodes (blue: HCP data; red: MICA data), with TSE complexity computed exactly, by sampling all subsystems. a Sum of TC + DTC versus TSE complexity; HCP: R = 0.998, p = 0; MICA: R = 0.999, p = 0. b DTC versus TSE; HCP: R = 0.982, p = 0; MICA: R = 0.992, p = 0.CC BY · PubMed Central
Fig. 3 Information measures computed from randomly sampled 10-node subsets. a Fraction (left) and number (right) of subsets with negative O-information, obtained by randomly sampling subsets from the HCP (blue) and MICA (red) FC matrix. Fraction and number estimated from samples of 5000 (3–12 nodes) or 200 (13–15 nodes) subsets with negative O-information. As the size of the subset grows, the fraction expressing an overall synergy-dominated structure (negative O-information) drops precipitously, while their absolute number continues to climb due to combinatorics. b The relation between O-information and TSE complexity in 100,000 randomly sampled 10-node subsystems (HCP data). While very few randomly sampled sets have negative O-information (see panel a), TSE complexity generally increases with the strength of the dependencies visible to the O-information (R = 0.642, p = 0). c The participation quantifies, for each node pair, how often they are encountered as part of a subset with negative o-information (10 nodes, 5000 random samples, HCP data), The plot shows the relation of the participation against the FC, with each data point representing one of the 19,900 unique node pairs. Node pairs with strong mutual FC (positive or negative) are rarely encountered as part of the same synergistic subset, while node pairs that are more frequently encountered tend to show weak FC. Spearman’s rho between absolute FC and participation: ρ = − 0.504, p = 0.
Fig. 3 Information measures computed from randomly sampled 10-node subsets. a Fraction (left) and number (right) of subsets with negative O-information, obtained by randomly sampling subsets from the HCP (blue) and MICA (red) FC matrix. Fraction and number estimated from samples of 5000 (3–12 nodes) or 200 (13–15 nodes) subsets with negative O-information. As the size of the subset grows, the fraction expressing an overall synergy-dominated structure (negative O-information) drops precipitously, while their absolute number continues to climb due to combinatorics. b The relation between O-information and TSE complexity in 100,000 randomly sampled 10-node subsystems (HCP data). While very few randomly sampled sets have negative O-information (see panel a), TSE complexity generally increases with the strength of the dependencies visible to the O-information (R = 0.642, p = 0). c The participation quantifies, for each node pair, how often they are encountered as part of a subset with negative o-information (10 nodes, 5000 random samples, HCP data), The plot shows the relation of the participation against the FC, with each data point representing one of the 19,900 unique node pairs. Node pairs with strong mutual FC (positive or negative) are rarely encountered as part of the same synergistic subset, while node pairs that are more frequently encountered tend to show weak FC. Spearman’s rho between absolute FC and participation: ρ = − 0.504, p = 0.CC BY · PubMed Central
Fig. 4 Topography and functional specialization of randomly sampled synergistic subsets in the brain. Data in panels a and b was derived from a random sample of 100,000 synergistic 10-node subsets (HCP data). a Drawing random sub-samples of 500 subsets, we computed their Jaccard similarity, capturing the number of nodes in common between each subset pair. The similarity matrix was clustered using the kmeans algorithm, iterating between 2 and 30 clusters, with 10,000 repetitions. Optimal cluster quality was determined using the ‘silhouette’ criterion on the resulting cluster assignments. Random samples consistently yielded around 9–11 optimal clusters, with one example (10 clusters) shown in this panel. A Jaccard similarity of 0.25 corresponds to two subsets having 4 out of 10 nodes in common. b Frequency of individual node participation across 100,000 synergistic subsets, displayed on a surface rendering of the cerebral cortex indicating the boundaries of the 200 nodes used for constructing the FC matrix. c Each of the 200 nodes is affiliated with one of 7 canonical functional systems60. Frequency of participation of individual nodes in synergistic subsets (negative O-information, subset size ranging from 3 to 15 nodes) is aggregated (averaged) for each functional system. The plot displays the ratio of empirical frequency over the expected frequency if nodes were selected by chance. A ratio > 1 or < 1 indicates that the system is over-represented or under-represented, respectively, in synergistic subsets. Sample sizes identical to those used in Fig. 4a.
Fig. 4 Topography and functional specialization of randomly sampled synergistic subsets in the brain. Data in panels a and b was derived from a random sample of 100,000 synergistic 10-node subsets (HCP data). a Drawing random sub-samples of 500 subsets, we computed their Jaccard similarity, capturing the number of nodes in common between each subset pair. The similarity matrix was clustered using the kmeans algorithm, iterating between 2 and 30 clusters, with 10,000 repetitions. Optimal cluster quality was determined using the ‘silhouette’ criterion on the resulting cluster assignments. Random samples consistently yielded around 9–11 optimal clusters, with one example (10 clusters) shown in this panel. A Jaccard similarity of 0.25 corresponds to two subsets having 4 out of 10 nodes in common. b Frequency of individual node participation across 100,000 synergistic subsets, displayed on a surface rendering of the cerebral cortex indicating the boundaries of the 200 nodes used for constructing the FC matrix. c Each of the 200 nodes is affiliated with one of 7 canonical functional systems60. Frequency of participation of individual nodes in synergistic subsets (negative O-information, subset size ranging from 3 to 15 nodes) is aggregated (averaged) for each functional system. The plot displays the ratio of empirical frequency over the expected frequency if nodes were selected by chance. A ratio > 1 or < 1 indicates that the system is over-represented or under-represented, respectively, in synergistic subsets. Sample sizes identical to those used in Fig. 4a.CC BY · PubMed Central
Fig. 5 O-Information, brain topography, and functional specialization of optimally synergistic subsets identified by simulated annealing All panels show data from the HCP sample. a Annealing was carried out 5000 times for each subset size. This plot shows O-information for each optimized subset (gray dots) and their mean (blue line). Note that annealing fails to converge onto any synergistic subsets for subsets containing more than 24 nodes. Optimally negative O-information is achieved for subsets between 8 and 12 nodes. For each subset of size k each node was removed individually and the O-information of the remaining k − 1 nodes was calculated. If the O-information was lower than given by the original k node subset, the contribution of that node was considered redundant and the synergy in the subset attributable to its k − 1 node counterpart. For the vast majority of subsets smaller than 12 nodes, no nodes could be removed in a way that increased synergy, indicating that these subsets consisted of nodes with irreducible synergy (red in Fig. 5a). b Frequency of individual node participation across optimally synergistic 10-node subsets (4021 unique subsets out of 5,000 annealing runs), displayed on a surface rendering of the cerebral cortex (cf. Fig. 4b). c Mean O-information (left) and fraction of synergistic subsets (right) encountered in samples of 20,000 subsets that contained nodes belonging to between 1 and 7 canonical FC systems (HCP data). The mean O-information for samples obtained exclusively from each of the 7 FC systems is indicated (red dots).
Fig. 5 O-Information, brain topography, and functional specialization of optimally synergistic subsets identified by simulated annealing All panels show data from the HCP sample. a Annealing was carried out 5000 times for each subset size. This plot shows O-information for each optimized subset (gray dots) and their mean (blue line). Note that annealing fails to converge onto any synergistic subsets for subsets containing more than 24 nodes. Optimally negative O-information is achieved for subsets between 8 and 12 nodes. For each subset of size k each node was removed individually and the O-information of the remaining k − 1 nodes was calculated. If the O-information was lower than given by the original k node subset, the contribution of that node was considered redundant and the synergy in the subset attributable to its k − 1 node counterpart. For the vast majority of subsets smaller than 12 nodes, no nodes could be removed in a way that increased synergy, indicating that these subsets consisted of nodes with irreducible synergy (red in Fig. 5a). b Frequency of individual node participation across optimally synergistic 10-node subsets (4021 unique subsets out of 5,000 annealing runs), displayed on a surface rendering of the cerebral cortex (cf. Fig. 4b). c Mean O-information (left) and fraction of synergistic subsets (right) encountered in samples of 20,000 subsets that contained nodes belonging to between 1 and 7 canonical FC systems (HCP data). The mean O-information for samples obtained exclusively from each of the 7 FC systems is indicated (red dots).CC BY · PubMed Central

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Authors
Varley, T. F., Pope, M., Faskowitz, J., & Sporns, O.
ORCID
Varley, Thomas F.
Venue
Communications Biology
Publisher
Springer Science and Business Media LLC
Published
2023-04-24
Volume
6
Issue
1
ISSN
2399-3642
DOI
10.1038/s42003-023-04843-w
PMC
PMC10125999
Cited by
101
References
4
Language
en
Update notices
correction (2023-06-07)
Licence
Funders
National Science Foundation (1735095)

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

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How to cite

Cite CoChat for the checks on this page, and the paper for its own findings.

This page
CoChat. (2026). Citation check: Multivariate information theory uncovers synergistic subsystems of the human cerebral cortex. https://app.cochat.ai/papers/10.1038/s42003-023-04843-w
The paper
Varley, T. F., Pope, M., Faskowitz, J., & Sporns, O. (2023). Multivariate information theory uncovers synergistic subsystems of the human cerebral cortex. Communications Biology. https://doi.org/10.1038/s42003-023-04843-w