Persistent Homological State-Space Estimation of Functional Human Brain Networks at Rest

TL;DR

This work introduces a dynamic topological framework for estimating state spaces of resting state brain networks by leveraging persistent homology and the Wasserstein distance to compare time varying networks. The method combines birth–death decompositions on graph filtrations with topological clustering, yielding interpretable state spaces that outperform standard k means in capturing dynamic topology. A weighted Fourier series based approach replaces sliding windows for robust time varying correlations, enabling scalable computation on large fMRI datasets. In a twin study, the authors demonstrate substantial heritability of dynamic topological states, offering a new topological phenotype for genetic investigations and neuropsychiatric risk assessment.

Abstract

We introduce an innovative, data-driven topological data analysis (TDA) technique for estimating the state spaces of dynamically changing functional human brain networks at rest. Our method utilizes the Wasserstein distance to measure topological differences, enabling the clustering of brain networks into distinct topological states. This technique outperforms the commonly used k-means clustering in identifying brain network state spaces by effectively incorporating the temporal dynamics of the data without the need for explicit model specification. We further investigate the genetic underpinnings of these topological features using a twin study design, examining the heritability of such state changes. Our findings suggest that the topology of brain networks, particularly in their dynamic state changes, may hold significant hidden genetic information. MATLAB code for the method is available at https://github.com/laplcebeltrami/PH-STAT.

Figures (14)

• Figure 1: Proposed topological clustering pipeline used in estimating the state space. Given two weighted graphs $G_1, G_2$, we first perform the birth-death decomposition and partition the edges into sorted birth and death sets (section \ref{['sec:BDD']}). The 0D topological distance $D_{W0}$ between birth values quantifies discrepancies in connected components (section \ref{['sec:dist']}). The 1D topological distance $D_{W1}$ between death values quantifies discrepancies in cycles (section \ref{['sec:dist']}). The combined distance $\mathcal{D}=D_{W0}^2 + D_{W1}^2$ is used in computing the within-cluster distance $l_W$ between graphs. Topological clustering is performed by minimizing $l_W$ over all possible cluster labels $C_1, \cdots, C_k$ (section \ref{['sec:clustering']}).
• Figure 2: Dynamically changing correlation matrices computed from rs-fMRI using the sliding window of size 60 for a subject. The constructed correlation matrices are superimposed on top of the white matter fibers in the MNI space and color coded based on correlation values.
• Figure 3: The birth-death decomposition partitions the edge set into the birth and death edge sets. The birth set forms the maximum spanning tree (MST) and contains edges that create connected components (0D topology). The death set contains edges that do not belong to the maximum spanning tree (MST) and destroys loops (1D topology).
• Figure 4: The corresponding birth and death sets of dynamically changing correlation matrix shown in Figure \ref{['fig:dynamicTDA']}. The horizontal axis is the time point. Columns are the sorted birth and death edge values at the time point.
• Figure 5: Comparison between geometric distance $d_{geo}$ and topological distance $d_{top}$. We used the shortest Euclidean distance between objects as the geometric distance. The left (two circles) and middle (circle and arc) objects are topologically different while the left and right (square and circle) objects are topologically equivalent. The geometric distance cannot discriminate topologically different objects (left and middle) and produces false negatives. The geometric distance incorrectly discriminates topologically equivalent objects (left and right) and produces false positive.

Theorems & Definitions (9)

• theorem 1: Birth-death decomposition
• theorem 2
• definition 1
• definition 2
• theorem 3
• theorem 4
• theorem 5
• theorem 6
• theorem 7