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Partial entropy decomposition reveals higher-order information structures in human brain activity

Varley, Thomas F., Pope, Maria, Maria Grazia, Joshua, Sporns, Olaf·2023·Proceedings of the National Academy of Sciences
DOI 10.1073/pnas.2300888120PMC PMC10372615
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Abstract

The standard approach to modeling the human brain as a complex system is with a network, where the basic unit of interaction is a pairwise link between two brain regions. While powerful, this approach is limited by the inability to assess higher-order interactions involving three or more elements directly. In this work, we explore a method for capturing higher-order dependencies in multivariate data: the partial entropy decomposition (PED). Our approach decomposes the joint entropy of the whole system into a set of nonnegative atoms that describe the redundant, unique, and synergistic interactions that compose the system’s structure. PED gives insight into the mathematics of functional connectivity and its limitation. When applied to resting-state fMRI data, we find robust evidence of higher-order synergies that are largely invisible to standard functional connectivity analyses. Our approach can also be localized in time, allowing a frame-by-frame analysis of how the distributions of redundancies and synergies change over the course of a recording. We find that different ensembles of regions can transiently change from being redundancy-dominated to synergy-dominated and that the temporal pattern is structured in time. These results provide strong evidence that there exists a large space of unexplored structures in human brain data that have been largely missed by a focus on bivariate network connectivity models. This synergistic structure is dynamic in time and likely will illuminate interesting links between brain and behavior. Beyond brain-specific application, the PED provides a very general approach for understanding higher-order structures in a variety of complex systems.

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Fig. 1. The limits of bivariate functional connectivity. (A) In triads, bivariate functional connectivity is largely independent of synergistic structure (all correlations computed with the Pearson correlation coefficient) and (B) is very positively correlated with redundant structure. (C) In tetrads, bivariate functional connectivity is strongly negatively correlated with synergistic structure and (D) is strongly correlated with redundant structure. (E and F) Triads that have all elements within one FC community have significantly less synergistic structure than those that have elements with two communities, while for redundant structure, there was a clear pattern that the more FC communities a triad straddled, the lower its overall redundant structure. (G and H) The same pattern was even more pronounced in tetrads: As the number of FC communities a tetrad straddled increased, the expected synergistic structure climbed, while expected redundant structure fell.
Fig. 1. The limits of bivariate functional connectivity. (A) In triads, bivariate functional connectivity is largely independent of synergistic structure (all correlations computed with the Pearson correlation coefficient) and (B) is very positively correlated with redundant structure. (C) In tetrads, bivariate functional connectivity is strongly negatively correlated with synergistic structure and (D) is strongly correlated with redundant structure. (E and F) Triads that have all elements within one FC community have significantly less synergistic structure than those that have elements with two communities, while for redundant structure, there was a clear pattern that the more FC communities a triad straddled, the lower its overall redundant structure. (G and H) The same pattern was even more pronounced in tetrads: As the number of FC communities a tetrad straddled increased, the expected synergistic structure climbed, while expected redundant structure fell.CC BY-NC-ND · PubMed Central
Fig. 2. Redundant and synergistic hypergraph community structure. (A and B) Surface plots of the two communities structures: On the Left is the redundant structure and on the Right is the synergistic structure. We can see that both patterns are largely symmetrical for both information-sharing modes, although the synergistic structure has two large, lateralized communities. (C and D) The coclassification matrices for redundant structure (Left) and the synergistic structure (Right). The higher the value of a pair, the more frequently the hypergraph modularity maximization (31) assigns those two regions to the same hypercommunity. The yellow squares indicate the seven canonical Yeo functional networks (32), and we can see that the higher-order redundant structure matches the bivariate Yeo systems well (despite consisting of information shared redundantly across three nodes). In contrast, the synergistic structure largely fails to match the canonical network structure at all. (E) For each of the 95 subjects and for each of the 1,000 permutation nulls used to significance test the NMI between subject-level community structure and the master level structure, we computed the log-ratio of the empirical NMI to the null NMI. For redundancy, there was not a single null, over any subject, that was greater than the associated empirical NMI. For the case of the synergy, only 0.6% of nulls were greater than their associated empirical NMI.
Fig. 2. Redundant and synergistic hypergraph community structure. (A and B) Surface plots of the two communities structures: On the Left is the redundant structure and on the Right is the synergistic structure. We can see that both patterns are largely symmetrical for both information-sharing modes, although the synergistic structure has two large, lateralized communities. (C and D) The coclassification matrices for redundant structure (Left) and the synergistic structure (Right). The higher the value of a pair, the more frequently the hypergraph modularity maximization (31) assigns those two regions to the same hypercommunity. The yellow squares indicate the seven canonical Yeo functional networks (32), and we can see that the higher-order redundant structure matches the bivariate Yeo systems well (despite consisting of information shared redundantly across three nodes). In contrast, the synergistic structure largely fails to match the canonical network structure at all. (E) For each of the 95 subjects and for each of the 1,000 permutation nulls used to significance test the NMI between subject-level community structure and the master level structure, we computed the log-ratio of the empirical NMI to the null NMI. For redundancy, there was not a single null, over any subject, that was greater than the associated empirical NMI. For the case of the synergy, only 0.6% of nulls were greater than their associated empirical NMI.CC BY-NC-ND · PubMed Central
Fig. 3. Time-resolved analysis. (A) Surface plots for the distributions of relative synergies and relative redundancies across the human brain. These results match prior work by Luppi et al. (33), with the primary sensory and motor cortex being relatively redundant, while multimodal association areas being relatively synergistic. (B) Over the course of one subject’s scan (1100 TRs), the total redundant and synergistic structure varies over time, although never so much that the curves cross (i.e., there is never more redundant structure than synergistic structure present). (C) Instantaneous redundant and synergistic structures are anticorrelated (r = −0.83, P < 10−50). (D) Redundancy is positively correlated with the amplitude of bivariate cofluctuations (Pearson’s Pearson’s r = 0.6, P < 10−50), and (E) synergy is negatively correlated with cofluctuation amplitude (Pearson’s r = −0.43, P < 10−50). (F) For each TR, we show the difference in the rank-redundancy and rank-synergy for each node (red indicates a higher rank-redundancy than rank-synergy and vice versa for blue). When nodes are stratified by Yeo system (32) (gray, horizontal lines), it is clear that different systems alternate between high-redundancy and high-synergy configurations in different ways. (G) For every pair of columns in Panel F. we compute the Pearson correlation between them to construct a time × time similarity matrix, which we then clustered using the MRCC algorithm (30). Note that rows and columns are not in time order, but rather, reordered to reveal the state-structure of the time series. (H) Five example states (centroids of each community shown in Panel G) projected onto the cortical surface. It is clear that the instantaneous pattern of relative synergies and redundancies varies from the average structure presented in Panel A. For example, in states 3 and 4, the visual system is highly redundant (as in the average); however, in state 5, the visual system is synergistic.
Fig. 3. Time-resolved analysis. (A) Surface plots for the distributions of relative synergies and relative redundancies across the human brain. These results match prior work by Luppi et al. (33), with the primary sensory and motor cortex being relatively redundant, while multimodal association areas being relatively synergistic. (B) Over the course of one subject’s scan (1100 TRs), the total redundant and synergistic structure varies over time, although never so much that the curves cross (i.e., there is never more redundant structure than synergistic structure present). (C) Instantaneous redundant and synergistic structures are anticorrelated (r = −0.83, P < 10−50). (D) Redundancy is positively correlated with the amplitude of bivariate cofluctuations (Pearson’s Pearson’s r = 0.6, P < 10−50), and (E) synergy is negatively correlated with cofluctuation amplitude (Pearson’s r = −0.43, P < 10−50). (F) For each TR, we show the difference in the rank-redundancy and rank-synergy for each node (red indicates a higher rank-redundancy than rank-synergy and vice versa for blue). When nodes are stratified by Yeo system (32) (gray, horizontal lines), it is clear that different systems alternate between high-redundancy and high-synergy configurations in different ways. (G) For every pair of columns in Panel F. we compute the Pearson correlation between them to construct a time × time similarity matrix, which we then clustered using the MRCC algorithm (30). Note that rows and columns are not in time order, but rather, reordered to reveal the state-structure of the time series. (H) Five example states (centroids of each community shown in Panel G) projected onto the cortical surface. It is clear that the instantaneous pattern of relative synergies and redundancies varies from the average structure presented in Panel A. For example, in states 3 and 4, the visual system is highly redundant (as in the average); however, in state 5, the visual system is synergistic.CC BY-NC-ND · PubMed Central

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Metadata

Authors
Varley, T. F., Pope, M., Grazia, M., Joshua, & Sporns, O.
Affiliations
  • School of Informatics, Computing and Engineering, Indiana University, Bloomington, IN 47405
  • Department of Psychological and Brain Sciences, Indiana University, Bloomington, IN 47405
ORCID
Venue
Proceedings of the National Academy of Sciences
Publisher
National Academy of Sciences
Published
2023-07-19
Volume
120
Issue
30
ISSN
  • 0027-8424
  • 1091-6490
DOI
10.1073/pnas.2300888120
PMC
PMC10372615
Cited by
103
References
3
Language
en
Licence
creativecommons.org/licenses/by-nc-nd/4.0/ (vor)
Funders
National Science Foundation (1735095)

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