From Nodes to Edges: Edge-Based Laplacians for Brain Signal Processing

Traditional graph signal processing (GSP) methods applied to brain networks focus on signals defined on the nodes. Thus, they are unable to capture potentially important dynamics occurring on the edges. In this work, we adopt an edge-centric GSP approach to analyze edge signals constructed from 100 unrelated subjects of the Human Connectome Project. Specifically, we describe structural connectivity through the lens of the 1-dimensional Hodge Laplacian, processing signals defined on edges to capture co-fluctuation information between brain regions. We demonstrate that edge-based approaches achieve superior task decoding accuracy in static and dynamic scenarios compared to conventional node-based techniques, thereby unveiling unique aspects of brain functional organization. These findings underscore the promise of edge-focused GSP strategies for deepening our understanding of brain connectivity and functional dynamics.

💡 Research Summary

The manuscript introduces a novel edge‑centric framework for brain signal processing that leverages the 1‑dimensional Hodge Laplacian within the broader context of Topological Signal Processing (TSP). Traditional Graph Signal Processing (GSP) treats each brain region as a node and the corresponding fMRI BOLD time series as node‑level signals, using the graph Laplacian (or its incidence‑matrix formulation) to perform graph Fourier transforms, filtering, and spectral analysis. While effective for many applications, node‑centric GSP cannot directly capture dynamics that inherently reside on connections between regions, such as co‑fluctuations or phase synchrony.

To address this limitation, the authors model the structural connectome as a simplicial complex: 0‑simplices are brain regions, 1‑simplices are white‑matter fiber tracts (edges), and 2‑simplices are triangles formed by three mutually connected regions (cliques). The 1‑dimensional Hodge Laplacian L₁ is defined as the sum of two terms, L↓₁ = B₁ᵀB₁ (edge‑node interactions) and L↑₁ = B₂B₂ᵀ (edge‑triangle interactions), where B₁ is the node‑edge incidence matrix and B₂ encodes edge‑triangle incidences. This operator generalizes the classic graph Laplacian to higher‑order topology and admits an eigendecomposition L₁ = U₁Λ₁U₁ᵀ. The eigenvectors U₁ constitute a basis for the Topological Fourier Transform (TFT) of edge‑level signals.

Two families of edge signals are constructed from the fMRI data of 100 unrelated HCP subjects (119 ROIs: 100 cortical parcels plus 19 subcortical regions). The first family, called amplitude‑based edge signals, multiplies the BOLD time series of each node pair (eᵢⱼ(t) = xᵢ(t)·xⱼ(t)), directly measuring co‑activation strength. The second family extracts instantaneous phases via the Hilbert transform, then combines them using either sine or cosine of the phase difference (eᵢⱼ(t) = sin