Promises and pitfalls of topological data analysis for brain connectivity analysis

Topological state-space estimation of functional human brain networks

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.

Altered topological structure of the brain white matter in maltreated children through topological data analysis

Childhood maltreatment may adversely affect brain development and consequently influence behavioral, emotional, and psychological patterns during adulthood. In this study, we propose an analytical pipeline for modeling the altered topological structure of brain white matter in maltreated and typically developing children. We perform topological data analysis (TDA) to assess the alteration in the global topology of the brain white matter structural covariance network among children. We use persistent homology, an algebraic technique in TDA, to analyze topological features in the brain covariance networks constructed from structural magnetic resonance imaging and diffusion tensor imaging. We develop a novel framework for statistical inference based on the Wasserstein distance to assess the significance of the observed topological differences. Using these methods in comparing maltreated children with a typically developing control group, we find that maltreatment may increase homogeneity in white matter structures and thus induce higher correlations in the structural covariance; this is reflected in the topological profile. Our findings strongly suggest that TDA can be a valuable framework to model altered topological structures of the brain. The MATLAB codes and processed data used in this study can be found at https://github.com/laplcebeltrami/maltreated.

Simplicial complex entropy for time series analysis

The complex behavior of many systems in nature requires the application of robust methodologies capable of identifying changes in their dynamics. In the case of time series (which are sensed values of a system during a time interval), several methods have been proposed to evaluate their irregularity. However, for some types of dynamics such as stochastic and chaotic, new approaches are required that can provide a better characterization of them. In this paper we present the simplicial complex approximate entropy, which is based on the conditional probability of the occurrence of elements of a simplicial complex. Our results show that this entropy measure provides a wide range of values with details not easily identifiable with standard methods. In particular, we show that our method is able to quantify the irregularity in simulated random sequences and those from low-dimensional chaotic dynamics. Furthermore, it is possible to consistently differentiate cardiac interbeat sequences from healthy subjects and from patients with heart failure, as well as to identify changes between dynamical states of coupled chaotic maps. Our results highlight the importance of the structures revealed by the simplicial complexes, which holds promise for applications of this approach in various contexts.

Topological Data Analysis for Multivariate Time Series Data

Over the last two decades, topological data analysis (TDA) has emerged as a very powerful data analytic approach that can deal with various data modalities of varying complexities. One of the most commonly used tools in TDA is persistent homology (PH), which can extract topological properties from data at various scales. The aim of this article is to introduce TDA concepts to a statistical audience and provide an approach to analyzing multivariate time series data. The application's focus will be on multivariate brain signals and brain connectivity networks. Finally, this paper concludes with an overview of some open problems and potential application of TDA to modeling directionality in a brain network, as well as the casting of TDA in the context of mixed effect models to capture variations in the topological properties of data collected from multiple subjects.

Simple topological task-based functional connectivity features predict longitudinal behavioral change of fluid reasoning in the RANN cohort

Recent attention has been given to topological data analysis (TDA), and more specifically persistent homology (PH), to identify the underlying shape of brain network connectivity beyond simple edge pairings by computing connective components across different connectivity thresholds (see Sizemore et al., 2019). In the present study, we applied PH to task-based functional connectivity, computing 0-dimension Betti (B0) curves and calculating the area under these curves (AUC); AUC indicates how quickly a single connected component is formed across correlation filtration thresholds, with lower values interpreted as potentially analogous to lower whole-brain system segregation (e.g., Gracia-Tabuenca et al., 2020). One hundred sixty-three participants from the Reference Ability Neural Network (RANN) longitudinal lifespan cohort (age 20-80 years) were tested in-scanner at baseline and five-year follow-up on a battery of tests comprising four domains of cognition (i.e., Stern et al., 2014). We tested for 1.) age-related change in the AUC of the B0 curve over time, 2.) the predictive utility of AUC in accounting for longitudinal change in behavioral performance and 3.) compared system segregation to the PH approach. Results demonstrated longitudinal age-related decreases in AUC for Fluid Reasoning, with these decreases predicting longitudinal declines in cognition, even after controlling for demographic and brain integrity factors; moreover, change in AUC partially mediated the effect of age on change in cognitive performance. System segregation also significantly decreased with age in three of the four cognitive domains but did not predict change in cognition. These results argue for greater application of TDA to the study of aging.

Epileptic detection in single and multi-lead EEG signals using persistent homology based on bi-directional weighted visibility graphs

Epilepsy is a widespread neurological disorder, and its recurrence and suddenness are making automatic detection of seizure an urgent necessity. For this purpose, this paper performs topological data analysis (TDA) of electroencephalographic (EEG) signals by the medium of graphs to explore the potential brain activity information they contain. Through our innovative method, we first map the time series of epileptic EEGs into bi-directional weighted visibility graphs (BWVGs), which give more comprehensive reflections of the signals compared to previous existing structures. Traditional graph-theoretic measurements are generally partial and mainly consider differences or correlations in vertices or edges, whereas persistent homology (PH), the essential part of TDA, provides an alternative way of thinking by quantifying the topology structure of the graphs and analyzing the evolution of these topological properties with scale changes. Therefore, we analyze the PH for BWVGs and then obtain the two indicators of persistence and birth-death for homology groups to reflect the topology of the mapping graphs of EEG signals and reveal the discrepancies in brain dynamics. Furthermore, we adopt neural networks (NNs) for the automatic detection of epileptic signals and successfully achieve a classification accuracy of 99.67% when distinguishing among three different sets of EEG signals from seizure, seizure-free, and healthy subjects. In addition, to accommodate multi-leads, we propose a classifier that incorporates graph structure to distinguish seizure and seizure-free EEG signals. The classification accuracies of the two subjects used in the classifier are as high as 99.23% and 94.76%, respectively, indicating that our proposed model is useful for the analysis of EEG signals.

Development of the Functional Connectome Topology in Adolescence: Evidence from Topological Data Analysis

Adolescence is a crucial developmental period in terms of behavior and mental health. Therefore, understanding how the brain develops during this stage is a fundamental challenge for neuroscience. Recent studies have modeled the brain as a network or connectome, mainly applying measures from graph theory, showing a change in its functional organization, such as an increase in its segregation and integration. Topological Data Analysis (TDA) complements such modeling by extracting high-dimensional features across the whole range of connectivity values instead of exploring a fixed set of connections. This study inquires into the developmental trajectories of such properties using a longitudinal sample of typically developing human participants (N = 98; 53/45 female/male; 6.7-18.1 years), applying TDA to their functional connectomes. In addition, we explore the effect of puberty on individual developmental trajectories. Results showed that the adolescent brain has a more distributed topology structure compared with random networks but is more densely connected at the local level. Furthermore, developmental effects showed nonlinear trajectories for the topology of the whole brain and fronto-parietal networks, with an inflection point and increasing trajectories after puberty onset. These results add to the insights into the development of the functional organization of the adolescent brain.

Alterations in Social Brain Network Topology at Rest in Children With Autism Spectrum Disorder

OBJECTIVE

Underconnectivity in the resting brain is not consistent in autism spectrum disorder (ASD). However, it is known that the functional connectivity of the default mode network is mainly decreased in childhood ASD. This study investigated the brain network topology as the changes in the connection strength and network efficiency in childhood ASD, including the early developmental stages.

METHODS

In this study, 31 ASD children aged 2-11 years were compared with 31 age and sex-matched children showing typical development. We explored the functional connectivity based on graph filtration by assessing the single linkage distance and global and nodal efficiencies using resting-state functional magnetic resonance imaging. The relationship between functional connectivity and clinical scores was also analyzed.

RESULTS

Underconnectivities within the posterior default mode network subregions and between the inferior parietal lobule and inferior frontal/superior temporal regions were observed in the ASD group. These areas significantly correlated with the clinical phenotypes. The global, local, and nodal network efficiencies were lower in children with ASD than in those with typical development. In the preschool-age children (2-6 years) with ASD, the anterior-posterior connectivity of the default mode network and cerebellar connectivity were reduced.

CONCLUSION

The observed topological reorganization, underconnectivity, and disrupted efficiency in the default mode network subregions and social function-related regions could be significant biomarkers of childhood ASD.

A Multiview Deep Learning Method for Brain Functional Connectivity Classification

The brain functional connectivity classification based on deep learning is a research hotspot nowadays. However, the classification performance is far behind the demand of clinical applications. To alleviate the problem, this paper proposes a multiview deep learning method for brain functional connectivity classification. Firstly, the proposed method adopts multiple brain atlases to identify brain regions and thereby builds different brain functional connectivity of different views. Secondly, it uses a multiview feature selection strategy to select out the most discriminative features of each view with the assistance of other views. Then, it trains a stacked autoencoder to extract deep features of the brain functional connectivity of each view. At last, it utilizes a multiview fusion strategy to take full advantage of complementary information of different views for brain functional connectivity classification. The proposed method has been compared with several deep learning-based brain functional connectivity classification methods on three public datasets of neuropsychiatric disorders. The experimental results have validated the superior performance of the proposed method.

Method for persistent topological features extraction of schizophrenia patients' electroencephalography signal based on persistent homology

With the development of network science and graph theory, brain network research has unique advantages in explaining those mental diseases, the neural mechanism of which is unclear. Additionally, it can provide a new perspective in revealing the pathophysiological mechanism of brain diseases from the system level. The selection of threshold plays an important role in brain networks construction. There are no generally accepted criteria for determining the proper threshold. Therefore, based on the topological data analysis of persistent homology theory, this study developed a multi-scale brain network modeling analysis method, which enables us to quantify various persistent topological features at different scales in a coherent manner. In this method, the Vietoris-Rips filtering algorithm is used to extract dynamic persistent topological features by gradually increasing the threshold in the range of full-scale distances. Subsequently, the persistent topological features are visualized using barcodes and persistence diagrams. Finally, the stability of persistent topological features is analyzed by calculating the Bottleneck distances and Wasserstein distances between the persistence diagrams. Experimental results show that compared with the existing methods, this method can extract the topological features of brain networks more accurately and improves the accuracy of diagnostic and classification. This work not only lays a foundation for exploring the higher-order topology of brain functional networks in schizophrenia patients, but also enhances the modeling ability of complex brain systems to better understand, analyze, and predict their dynamic behaviors.

0-Dimensional Persistent Homology Analysis Implementation in Resource-Scarce Embedded Systems

Persistent Homology (PH) analysis is a powerful tool for understanding many relevant topological features from a given dataset. PH allows finding clusters, noise, and relevant connections in the dataset. Therefore, it can provide a better view of the problem and a way of perceiving if a given dataset is equal to another, if a given sample is relevant, and how the samples occupy the feature space. However, PH involves reducing the problem to its simplicial complex space, which is computationally expensive and implementing PH in such Resource-Scarce Embedded Systems (RSES) is an essential add-on for them. However, due to its complexity, implementing PH in such tiny devices is considerably complicated due to the lack of memory and processing power. The following paper shows the implementation of 0-Dimensional Persistent Homology Analysis in a set of well-known RSES, using a technique that reduces the memory footprint and processing power needs of the 0-Dimensional PH algorithm. The results are positive and show that RSES can be equipped with this real-time data analysis tool.