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Multimodal Higher-Order Brain Networks: A Topological Signal Processing Perspective

Breno C. Bispo, Stefania Sardellitti, Juliano B. Lima, Fernando A. N. Santos

Abstract

Brain connectomics is still largely dominated by pairwise-based models, such as graphs, which cannot represent circulatory or higher-order functional interactions. In this paper, we propose a multimodal framework based on Topological Signal Processing (TSP) that models the brain as a higher-order topological domain and treats functional interactions as discrete vector fields. We integrate diffusion MRI and resting-state fMRI to learn subject-specific brain cell complexes, where statistically validated structural connectivity defines a sparse scaffold and phase-coupling functional edge signals drive the inference of higher-order interactions (HOIs). Using Hodge-theoretic tools, spectral filtering, and sparse signal representations, our framework disentangles brain connectivity into divergence (source-sink organization), gradient (potential-driven coordination), and curl (circulatory HOIs), enabling the characterization of temporal dynamics through the lens of discrete vector calculus. Across 100 healthy young adults from Human Connectome Project, node-based HOIs are highly individualized, yet robust mesoscale structure emerges under functional-system aggregation. We identify a distributed default mode network-centered gradient backbone and limbic-centered rotational flows; divergence polarization and curl profiles defining circulation regimes with insightful occupancy and dwell-time statistics. These topological signatures yield significant brain-behavior associations, revealing a relevant higher-order organization intrinsic to edge-based models. By making divergence, circulation, and recurrent mesoscale coordination directly measurable, this work enables a principled and interpretable topological phenotyping of brain function.

Multimodal Higher-Order Brain Networks: A Topological Signal Processing Perspective

Abstract

Brain connectomics is still largely dominated by pairwise-based models, such as graphs, which cannot represent circulatory or higher-order functional interactions. In this paper, we propose a multimodal framework based on Topological Signal Processing (TSP) that models the brain as a higher-order topological domain and treats functional interactions as discrete vector fields. We integrate diffusion MRI and resting-state fMRI to learn subject-specific brain cell complexes, where statistically validated structural connectivity defines a sparse scaffold and phase-coupling functional edge signals drive the inference of higher-order interactions (HOIs). Using Hodge-theoretic tools, spectral filtering, and sparse signal representations, our framework disentangles brain connectivity into divergence (source-sink organization), gradient (potential-driven coordination), and curl (circulatory HOIs), enabling the characterization of temporal dynamics through the lens of discrete vector calculus. Across 100 healthy young adults from Human Connectome Project, node-based HOIs are highly individualized, yet robust mesoscale structure emerges under functional-system aggregation. We identify a distributed default mode network-centered gradient backbone and limbic-centered rotational flows; divergence polarization and curl profiles defining circulation regimes with insightful occupancy and dwell-time statistics. These topological signatures yield significant brain-behavior associations, revealing a relevant higher-order organization intrinsic to edge-based models. By making divergence, circulation, and recurrent mesoscale coordination directly measurable, this work enables a principled and interpretable topological phenotyping of brain function.

Paper Structure

This paper contains 11 sections, 7 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Topological Signal Processing Framework
  3. Topological signal processing background
  4. Joint learning of topology and sparse signals
  5. Learning brain topology from data
  6. Results and discussion
  7. Structural-functional organization of brain topologies
  8. Dominant gradient and rotational flow organization
  9. Source/sink organization and behavioral variability
  10. Circulation organization and behavioral variability
  11. Conclusion

Figures (5)

  • Figure 1: An example of a CC of order $2$.
  • Figure 2: Hierarchical organization and inter-subject consistency of learned brain 2-CCs. \ref{['subfig:cell_counts']} Subject-specific counts of edges ($E$) and HOCs (triangles $P_3$, quadrilaterals $P_4$, and pentagons $P_5$). \ref{['subfig:prevalence']} Cell- and subnet-level prevalence distributions across subjects; 3D brain visualizations with ROIs at atlas Euclidean centroids are provided in the Supplementary Material. \ref{['subfig:top_cell_prev']} Five most prevalent node-level triangles and squares. \ref{['subfig:top_subnet_prev']} Five most prevalent subnet combinations across subjects.
  • Figure 3: Dominant gradient and rotational flows: \ref{['subfig:mean_significant_gradient_flow']}-\ref{['subfig:mean_significant_curl_flow']} show the group-averaged dominant gradient and rotational flows, respectively. Node color denotes FS membership, node size scales with degree, and edge color intensity represents mean flow magnitude across subjects (red: gradient; blue: rotational); 3D visualizations are available in the Supplementary Materials. \ref{['subfig:heatmap_dominant_gradient_curl_flows']} summarizes both components in a matrix view (gradient/rotational in above/below diagonal), with hemispheric boundaries marked on the diagonal (left/right: olive/dark green color).
  • Figure 4: Mesoscale source-sink organization and behavioral associations: \ref{['subfig:div_distribution']} Empirical and surrogate z-scored divergence distributions across subjects; \ref{['subfig:divergence_patterns']} time-averaged divergence by FS, shown as boxplots (median, IQR; whiskers $1.5\times$IQR) with overlaid data points; brain-behavior association between \ref{['subfig:correlation_plot_div_lim_pmat24']} LIM divergence and fluid intelligence; and \ref{['subfig:correlation_plot_div_sc_cardsort']} SC divergence and executive flexibility (Card Sort). Outliers were removed from association analyses using the $1.5\times$IQR rule.
  • Figure 5: Mesoscale rotational regimes and behavioral associations. \ref{['subfig:curl_distribution']} Empirical and surrogate z-scored curl distributions across subjects, highlighting conservative, predominant, and strong regimes. \ref{['subfig:occupancy_curl_patterns']} Fractional occupancy and \ref{['subfig:dwell_time_curl_patterns']} dwell time across functional-system pairings. Brain–behavior associations between \ref{['subfig:correlation_plot_curl_outliers_dmn_lim_language_task_story']} DMN–LIM strong-regime dwell time and Language Task Story performance, \ref{['subfig:correlation_plot_curl_popular_lim_fp_dexterity']} FP–LIM predominant-regime occupancy and motor dexterity, and \ref{['subfig:correlation_plot_curl_zero_dmn_lim_vocab']} DMN–LIM conservative-regime dwell time and receptive vocabulary.