Persistent brain network homology from the perspective of dendrogram

Predicting the Conversion from Mild Cognitive Impairment to Alzheimer's Disease Using Graph Frequency Bands and Functional Connectivity-Based Features

Accurate prediction of the progression from mild cognitive impairment (MCI) to Alzheimer's disease (AD) is crucial for disease management. Machine learning techniques have demonstrated success in classifying AD and MCI cases, particularly with the use of resting-state functional magnetic resonance imaging (rs-fMRI) data.This study utilized three years of rs-fMRI data from the ADNI, involving 142 patients with stable MCI (sMCI) and 136 with progressive MCI (pMCI). Graph signal processing was applied to filter rs-fMRI data into low, middle, and high frequency bands. Connectivity-based features were derived from both filtered and unfiltered data, resulting in a comprehensive set of 100 features, including global graph metrics, minimum spanning tree (MST) metrics, triadic interaction metrics, hub tendency metrics, and the number of links. Feature selection was enhanced using particle swarm optimization (PSO) and simulated annealing (SA). A support vector machine (SVM) with a radial basis function (RBF) kernel and a 10-fold cross-validation setup were employed for classification. The proposed approach demonstrated superior performance, achieving optimal accuracy with minimal feature utilization. When PSO selected five features, SVM exhibited accuracy, specificity, and sensitivity rates of 77%, 70%, and 83%, respectively. The identified features were as follows: (Mean of clustering coefficient, Mean of strength)/Radius/(Mean Eccentricity, and Modularity) from low/middle/high frequency bands of graph. The study highlights the efficacy of the proposed framework in identifying individuals at risk of AD development using a parsimonious feature set. This approach holds promise for advancing the precision of MCI to AD progression prediction, aiding in early diagnosis and intervention strategies.

From pixels to connections: exploring in vitro neuron reconstruction software for network graph generation

Digital reconstruction has been instrumental in deciphering how in vitro neuron architecture shapes information flow. Emerging approaches reconstruct neural systems as networks with the aim of understanding their organization through graph theory. Computational tools dedicated to this objective build models of nodes and edges based on key cellular features such as somata, axons, and dendrites. Fully automatic implementations of these tools are readily available, but they may also be purpose-built from specialized algorithms in the form of multi-step pipelines. Here we review software tools informing the construction of network models, spanning from noise reduction and segmentation to full network reconstruction. The scope and core specifications of each tool are explicitly defined to assist bench scientists in selecting the most suitable option for their microscopy dataset. Existing tools provide a foundation for complete network reconstruction, however more progress is needed in establishing morphological bases for directed/weighted connectivity and in software validation.

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.

Greater white matter degeneration and lower structural connectivity in non-amnestic vs. amnestic Alzheimer's disease

Introduction

Multimodal evidence indicates Alzheimer's disease (AD) is characterized by early white matter (WM) changes that precede overt cognitive impairment. WM changes have overwhelmingly been investigated in typical, amnestic mild cognitive impairment and AD; fewer studies have addressed WM change in atypical, non-amnestic syndromes. We hypothesized each non-amnestic AD syndrome would exhibit WM differences from amnestic and other non-amnestic syndromes.

Materials and methods

Participants included 45 cognitively normal (CN) individuals; 41 amnestic AD patients; and 67 patients with non-amnestic AD syndromes including logopenic-variant primary progressive aphasia (lvPPA, n = 32), posterior cortical atrophy (PCA, n = 17), behavioral variant AD (bvAD, n = 10), and corticobasal syndrome (CBS, n = 8). All had T1-weighted MRI and 30-direction diffusion-weighted imaging (DWI). We performed whole-brain deterministic tractography between 148 cortical and subcortical regions; connection strength was quantified by tractwise mean generalized fractional anisotropy. Regression models assessed effects of group and phenotype as well as associations with grey matter volume. Topological analyses assessed differences in persistent homology (numbers of graph components and cycles). Additionally, we tested associations of topological metrics with global cognition, disease duration, and DWI microstructural metrics.

Results

Both amnestic and non-amnestic patients exhibited lower WM connection strength than CN participants in corpus callosum, cingulum, and inferior and superior longitudinal fasciculi. Overall, non-amnestic patients had more WM disease than amnestic patients. LvPPA patients had left-lateralized WM degeneration; PCA patients had reductions in connections to bilateral posterior parietal, occipital, and temporal areas. Topological analysis showed the non-amnestic but not the amnestic group had more connected components than controls, indicating persistently lower connectivity. Longer disease duration and cognitive impairment were associated with more connected components and fewer cycles in individuals' brain graphs.

Discussion

We have previously reported syndromic differences in GM degeneration and tau accumulation between AD syndromes; here we find corresponding differences in WM tracts connecting syndrome-specific epicenters. Determining the reasons for selective WM degeneration in non-amnestic AD is a research priority that will require integration of knowledge from neuroimaging, biomarker, autopsy, and functional genetic studies. Furthermore, longitudinal studies to determine the chronology of WM vs. GM degeneration will be key to assessing evidence for WM-mediated tau spread.

Unified topological inference for brain networks in temporal lobe epilepsy using the Wasserstein distance

Persistent homology offers a powerful tool for extracting hidden topological signals from brain networks. It captures the evolution of topological structures across multiple scales, known as filtrations, thereby revealing topological features that persist over these scales. These features are summarized in persistence diagrams, and their dissimilarity is quantified using the Wasserstein distance. However, the Wasserstein distance does not follow a known distribution, posing challenges for the application of existing parametric statistical models. To tackle this issue, we introduce a unified topological inference framework centered on the Wasserstein distance. Our approach has no explicit model and distributional assumptions. The inference is performed in a completely data driven fashion. We apply this method to resting-state functional magnetic resonance images (rs-fMRI) of temporal lobe epilepsy patients collected from two different sites: the University of Wisconsin-Madison and the Medical College of Wisconsin. Importantly, our topological method is robust to variations due to sex and image acquisition, obviating the need to account for these variables as nuisance covariates. We successfully localize the brain regions that contribute the most to topological differences. A MATLAB package used for all analyses in this study is available at https://github.com/laplcebeltrami/PH-STAT.

Automatic epileptic seizure detection based on persistent homology

Epilepsy is a prevalent brain disease, which is quite difficult-to-treat or cure. This study developed a novel automatic seizure detection method based on the persistent homology method. In this study, a Vietoris-Rips (VR) complex filtration model was constructed based on the EEG data. And the persistent homology method was applied to calculate the VR complex filtration barcodes to describe the topological changes of EEG recordings. Afterward, the barcodes as the topological characteristics of EEG signals were fed into the GoogLeNet for classification. The persistent homology is applicable for multi-channel EEG data analysis, where the global topological information is calculated and the features are extracted by considering the multi-channel EEG data as a whole, without the multiple calculations or the post-stitching. Three databases were used to evaluate the proposed approach and the results showed that the approach had high performances in the epilepsy detection. The results obtained from the CHB-MIT Database recordings revealed that the proposed approach can achieve a segment-based averaged accuracy, sensitivity and specificity values of 97.05%, 96.71% and 97.38%, and achieve an event-based averaged sensitivity value of 100% with 1.22 s average detection latency. In addition, on the Siena Scalp Database, the proposed method yields averaged accuracy, sensitivity and specificity values of 96.42%, 95.23% and 97.6%. Multiple tasks of the Bonn Database also showed achieved accuracy of 99.55%, 98.63%, 98.28% and 97.68%, respectively. The experimental results on these three EEG databases illustrate the efficiency and robustness of our approach for automatic detection of epileptic seizure.

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.

The robustness of persistent homology of brain networks to data acquisition-related non-neural variability in resting state fMRI

There is increasing interest in investigating brain function based on functional connectivity networks (FCN) obtained from resting-state functional magnetic resonance imaging (fMRI). FCNs, typically obtained using measures of time series association such as Pearson's correlation, are sensitive to data acquisition parameters such as sampling period. This introduces non-neural variability in data pooled from different acquisition protocols and MRI scanners, negating the advantages of larger sample sizes in pooled data. To address this, we hypothesize that the topology or shape of brain networks must be preserved irrespective of how densely it is sampled, and metrics which capture this topology may be statistically similar across sampling periods, thereby alleviating this source of non-neural variability. Accordingly, we present an end-to-end pipeline that uses persistent homology (PH), a branch of topological data analysis, to demonstrate similarity across FCNs acquired at different temporal sampling periods. PH, as a technique, extracts topological features by capturing the network organization across all continuous threshold values, as opposed to graph theoretic methods, which fix a discrete network topology by thresholding the connectivity matrix. The extracted topological features are encoded in the form of persistent diagrams that can be compared against one another using the earth-moving metric, also popularly known as the Wasserstein distance. We extract topological features from three data cohorts, each acquired at different temporal sampling periods and demonstrate that these features are statistically the same, hence, empirically showing that PH may be robust to changes in data acquisition parameters such as sampling period.

Topological analysis of functional connectivity in Parkinson's disease

Parkinson's disease (PD) is a clinically heterogeneous disorder, which mainly affects patients' motor and non-motor function. Functional connectivity was preliminary explored and studied through resting state functional magnetic resonance imaging (rsfMRI). Through the topological analysis of 54 PD scans and 31 age-matched normal controls (NC) in the Neurocon dataset, leveraging on rsfMRI data, the brain functional connection and the Vietoris-Rips (VR) complex were constructed. The barcodes of the complex were calculated to reflect the changes of functional connectivity neural circuits (FCNC) in brain network. The 0-dimensional Betti number β0 means the number of connected branches in VR complex. The average number of connected branches in PD group was greater than that in NC group when the threshold δ ≤ 0.7. Two-sample Mann-Whitney U test and false discovery rate (FDR) correction were used for statistical analysis to investigate the FCNC changes between PD and NC groups. In PD group, under threshold of 0.7, the number of FCNC involved was significantly differences and these brain regions include the Cuneus_R, Lingual_R, Fusiform_R and Heschl_R. There are also significant differences in brain regions in the Frontal_Inf_Orb_R and Pallidum_R, when the threshold increased to 0.8 and 0.9 (p < 0.05). In addition, when the length of FCNC was medium, there was a significant statistical difference between the PD group and the NC group in the Neurocon dataset and the Parkinson's Progression Markers Initiative (PPMI) dataset. Topological analysis based on rsfMRI data may provide comprehensive information about the changes of FCNC and may provide an alternative for clinical differential diagnosis.

Persistent homology-based functional connectivity and its association with cognitive ability: Life-span study

Brain-segregation attributes in resting-state functional networks have been widely investigated to understand cognition and cognitive aging using various approaches [e.g., average connectivity within/between networks and brain system segregation (BSS)]. While these approaches have assumed that resting-state functional networks operate in a modular structure, a complementary perspective assumes that a core-periphery or rich club structure accounts for brain functions where the hubs are tightly interconnected to each other to allow for integrated processing. In this article, we apply a novel method, persistent homology (PH), to develop an alternative to standard functional connectivity by quantifying the pattern of information during the integrated processing. We also investigate whether PH-based functional connectivity explains cognitive performance and compare the amount of variability in explaining cognitive performance for three sets of independent variables: (1) PH-based functional connectivity, (2) graph theory-based measures, and (3) BSS. Resting-state functional connectivity data were extracted from 279 healthy participants, and cognitive ability scores were generated in four domains (fluid reasoning, episodic memory, vocabulary, and processing speed). The results first highlight the pattern of brain-information flow over whole brain regions (i.e., integrated processing) accounts for more variance of cognitive abilities than other methods. The results also show that fluid reasoning and vocabulary performance significantly decrease as the strength of the additional information flow on functional connectivity with the shortest path increases. While PH has been applied to functional connectivity analysis in recent studies, our results demonstrate potential utility of PH-based functional connectivity in understanding cognitive function.

Hodge Laplacian of Brain Networks

The closed loops or cycles in a brain network embeds higher order signal transmission paths, which provide fundamental insights into the functioning of the brain. In this work, we propose an efficient algorithm for systematic identification and modeling of cycles using persistent homology and the Hodge Laplacian. Various statistical inference procedures on cycles are developed. We validate the our methods on simulations and apply to brain networks obtained through the resting state functional magnetic resonance imaging. The computer codes for the Hodge Laplacian are given in https://github.com/laplcebeltrami/hodge.

Topological data analysis of human brain networks through order statistics

Understanding the common topological characteristics of the human brain network across a population is central to understanding brain functions. The abstraction of human connectome as a graph has been pivotal in gaining insights on the topological properties of the brain network. The development of group-level statistical inference procedures in brain graphs while accounting for the heterogeneity and randomness still remains a difficult task. In this study, we develop a robust statistical framework based on persistent homology using the order statistics for analyzing brain networks. The use of order statistics greatly simplifies the computation of the persistent barcodes. We validate the proposed methods using comprehensive simulation studies and subsequently apply to the resting-state functional magnetic resonance images. We found a statistically significant topological difference between the male and female brain networks.

TOPOLOGICAL LEARNING FOR BRAIN NETWORKS

This paper proposes a novel topological learning framework that integrates networks of different sizes and topology through persistent homology. Such challenging task is made possible through the introduction of a computationally efficient topological loss. The use of the proposed loss bypasses the intrinsic computational bottleneck associated with matching networks. We validate the method in extensive statistical simulations to assess its effectiveness when discriminating networks with different topology. The method is further demonstrated in a twin brain imaging study where we determine if brain networks are genetically heritable. The challenge here is due to the difficulty of overlaying the topologically different functional brain networks obtained from resting-state functional MRI onto the template structural brain network obtained through diffusion MRI.

The persistent homology of genealogical networks

Genealogical networks (i.e. family trees) are of growing interest, with the largest known data sets now including well over one billion individuals. Interest in family history also supports an 8.5 billion dollar industry whose size is projected to double within 7 years [FutureWise report HC-1137]. Yet little mathematical attention has been paid to the complex network properties of genealogical networks, especially at large scales. The structure of genealogical networks is of particular interest due to the practice of forming unions, e.g. marriages, that are typically well outside one's immediate family. In most other networks, including other social networks, no equivalent restriction exists on the distance at which relationships form. To study the effect this has on genealogical networks we use persistent homology to identify and compare the structure of 101 genealogical and 31 other social networks. Specifically, we introduce the notion of a network's persistence curve, which encodes the network's set of persistence intervals. We find that the persistence curves of genealogical networks have a distinct structure when compared to other social networks. This difference in structure also extends to subnetworks of genealogical and social networks suggesting that, even with incomplete data, persistent homology can be used to meaningfully analyze genealogical networks. Here we also describe how concepts from genealogical networks, such as common ancestor cycles, are represented using persistent homology. We expect that persistent homology tools will become increasingly important in genealogical exploration as popular interest in ancestry research continues to expand.

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.

Topological analysis of brain dynamics in autism based on graph and persistent homology

Autism spectrum disorder (ASD) is a heterogeneous disorder with a rapidly growing prevalence. In recent years, the dynamic functional connectivity (DFC) technique has been used to reveal the transient connectivity behavior of ASDs' brains by clustering connectivity matrices in different states. However, the states of DFC have not been yet studied from a topological point of view. In this paper, this study was performed using global metrics of the graph and persistent homology (PH) and resting-state functional magnetic resonance imaging (fMRI) data. The PH has been recently developed in topological data analysis and deals with persistent structures of data. The structural connectivity (SC) and static FC (SFC) were also studied to know which one of the SC, SFC, and DFC could provide more discriminative topological features when comparing ASDs with typical controls (TCs). Significant discriminative features were only found in states of DFC. Moreover, the best classification performance was offered by persistent homology-based metrics and in two out of four states. In these two states, some networks of ASDs compared to TCs were more segregated and isolated (showing the disruption of network integration in ASDs). The results of this study demonstrated that topological analysis of DFC states could offer discriminative features which were not discriminative in SFC and SC. Also, PH metrics can provide a promising perspective for studying ASD and finding candidate biomarkers.

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.

Topological data analysis in biomedicine: A review

Significant technological advances made in recent years have shepherded a dramatic increase in utilization of digital technologies for biomedicine- everything from the widespread use of electronic health records to improved medical imaging capabilities and the rising ubiquity of genomic sequencing contribute to a "digitization" of biomedical research and clinical care. With this shift toward computerized tools comes a dramatic increase in the amount of available data, and current tools for data analysis capable of extracting meaningful knowledge from this wealth of information have yet to catch up. This article seeks to provide an overview of emerging mathematical methods with the potential to improve the abilities of clinicians and researchers to analyze biomedical data, but may be hindered from doing so by a lack of conceptual accessibility and awareness in the life sciences research community. In particular, we focus on topological data analysis (TDA), a set of methods grounded in the mathematical field of algebraic topology that seeks to describe and harness features related to the "shape" of data. We aim to make such techniques more approachable to non-mathematicians by providing a conceptual discussion of their theoretical foundations followed by a survey of their published applications to scientific research. Finally, we discuss the limitations of these methods and suggest potential avenues for future work integrating mathematical tools into clinical care and biomedical informatics.

Positron Emission Tomography in the Neuroimaging of Autism Spectrum Disorder: A Review

Autism spectrum disorder (ASD) is a basket term for neurodevelopmental disorders characterized by marked impairments in social interactions, repetitive and stereotypical behaviors, and restricted interests and activities. Subtypes include (A) disorders with known genetic abnormalities including fragile X syndrome, Rett syndrome, and tuberous sclerosis and (B) idiopathic ASD, conditions with unknown etiologies. Positron emission tomography (PET) is a molecular imaging technology that can be utilized in vivo for dynamic and quantitative research, and is a valuable tool for exploring pathophysiological mechanisms, evaluating therapeutic efficacy, and accelerating drug development in ASD. Recently, several imaging studies on ASD have been published and physiological changes during ASD progression was disclosed by PET. This paper reviews the specific radioligands for PET imaging of critical biomarkers in ASD, and summarizes and discusses the similar and different discoveries in outcomes of previous studies. It is of great importance to identify general physiological changes in cerebral glucose metabolism, cerebral blood flow perfusion, abnormalities in neurotransmitter systems, and inflammation in the central nervous system in ASD, which may provide excellent points for further ASD research.

Intralayer and interlayer synchronization in multiplex network with higher-order interactions

Recent developments in complex systems have witnessed that many real-world scenarios, successfully represented as networks, are not always restricted to binary interactions but often include higher-order interactions among the nodes. These beyond pairwise interactions are preferably modeled by hypergraphs, where hyperedges represent higher-order interactions between a set of nodes. In this work, we consider a multiplex network where the intralayer connections are represented by hypergraphs, called the multiplex hypergraph. The hypergraph is constructed by mapping the maximal cliques of a scale-free network to hyperedges of suitable sizes. We investigate the intralayer and interlayer synchronizations of such multiplex structures. Our study unveils that the intralayer synchronization appreciably enhances when a higher-order structure is taken into consideration in spite of only pairwise connections. We derive the necessary condition for stable synchronization states by the master stability function approach, which perfectly agrees with the numerical results. We also explore the robustness of interlayer synchronization and find that for the multiplex structures with many-body interaction, the interlayer synchronization is more persistent than the multiplex networks with solely pairwise interaction.

A Spatiotemporal Brain Network Analysis of Alzheimer's Disease Based on Persistent Homology

Current brain network studies based on persistent homology mainly focus on the spatial evolution over multiple spatial scales, and there is little research on the evolution of a spatiotemporal brain network of Alzheimer's disease (AD). This paper proposed a persistent homology-based method by combining multiple temporal windows and spatial scales to study the spatiotemporal evolution of brain functional networks. Specifically, a time-sliding window method was performed to establish a spatiotemporal network, and the persistent homology-based features of such a network were obtained. We evaluated our proposed method using the resting-state functional MRI (rs-fMRI) data set from Alzheimer's Disease Neuroimaging Initiative (ADNI) with 31 patients with AD and 37 normal controls (NCs). In the statistical analysis experiment, most network properties showed a better statistical power in spatiotemporal networks than in spatial networks. Moreover, compared to the standard graph theory properties in spatiotemporal networks, the persistent homology-based features detected more significant differences between the groups. In the clustering experiment, the brain networks on the sliding windows of all subjects were clustered into two highly structured connection states. Compared to the NC group, the AD group showed a longer residence time and a higher window ratio in a weak connection state, which may be because patients with AD have not established a firm connection. In summary, we constructed a spatiotemporal brain network containing more detailed information, and the dynamic spatiotemporal brain network analysis method based on persistent homology provides stronger adaptability and robustness in revealing the abnormalities of the functional organization of patients with AD.

Topological Features of Electroencephalography are Robust to Re-referencing and Preprocessing

Electroencephalography (EEG) is among the most widely diffused, inexpensive, and adopted neuroimaging techniques. Nonetheless, EEG requires measurements against a reference site(s), which is typically chosen by the experimenter, and specific pre-processing steps precede analyses. It is therefore valuable to obtain quantities that are minimally affected by reference and pre-processing choices. Here, we show that the topological structure of embedding spaces, constructed either from multi-channel EEG timeseries or from their temporal structure, are subject-specific and robust to re-referencing and pre-processing pipelines. By contrast, the shape of correlation spaces, that is, discrete spaces where each point represents an electrode and the distance between them that is in turn related to the correlation between the respective timeseries, was neither significantly subject-specific nor robust to changes of reference. Our results suggest that the shape of spaces describing the observed configurations of EEG signals holds information about the individual specificity of the underlying individual's brain dynamics, and that temporal correlations constrain to a large degree the set of possible dynamics. In turn, these encode the differences between subjects' space of resting state EEG signals. Finally, our results and proposed methodology provide tools to explore the individual topographical landscapes and how they are explored dynamically. We propose therefore to augment conventional topographic analyses with an additional-topological-level of analysis, and to consider them jointly. More generally, these results provide a roadmap for the incorporation of topological analyses within EEG pipelines.

Neural evidence for image quality perception based on algebraic topology

In this paper, the algebraic topological characteristics of brain networks composed of electroencephalogram(EEG) signals induced by different quality images were studied, and on that basis, a neurophysiological image quality assessment approach was proposed. Our approach acquired quality perception-related neural information via integrating the EEG collection with conventional image assessment procedures, and the physiologically meaningful brain responses to different distortion-level images were obtained by topological data analysis. According to the validation experiment results, statistically significant discrepancies of the algebraic topological characteristics of EEG data evoked by a clear image compared to that of an unclear image are observed in several frequency bands, especially in the beta band. Furthermore, the phase transition difference of brain network caused by JPEG compression is more significant, indicating that humans are more sensitive to JPEG compression other than Gaussian blur. In general, the algebraic topological characteristics of EEG signals evoked by distorted images were investigated in this paper, which contributes to the study of neurophysiological assessment of image quality.

EVOLUTIONARY DE RHAM-HODGE METHOD

The de Rham-Hodge theory is a landmark of the 20^th Century's mathematics and has had a great impact on mathematics, physics, computer science, and engineering. This work introduces an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from a filtration, which induces a family of evolutionary de Rham complexes. While the present method can be easily applied to close manifolds, the emphasis is given to more challenging compact manifolds with 2-manifold boundaries, which require appropriate analysis and treatment of boundary conditions on differential forms to maintain proper topological properties. Three sets of unique evolutionary Hodge Laplacians are proposed to generate three sets of topology-preserving singular spectra, for which the multiplicities of zero eigenvalues correspond to exactly the persistent Betti numbers of dimensions 0, 1 and 2. Additionally, three sets of non-zero eigenvalues further reveal both topological persistence and geometric progression during the manifold evolution. Extensive numerical experiments are carried out via the discrete exterior calculus to demonstrate the potential of the proposed paradigm for data representation and shape analysis of both point cloud data and density maps. To demonstrate the utility of the proposed method, the application is considered to the protein B-factor predictions of a few challenging cases for which existing biophysical models break down.

Understanding Changes in the Topology and Geometry of Financial Market Correlations during a Market Crash

In econophysics, the achievements of information filtering methods over the past 20 years, such as the minimal spanning tree (MST) by Mantegna and the planar maximally filtered graph (PMFG) by Tumminello et al., should be celebrated. Here, we show how one can systematically improve upon this paradigm along two separate directions. First, we used topological data analysis (TDA) to extend the notions of nodes and links in networks to faces, tetrahedrons, or k-simplices in simplicial complexes. Second, we used the Ollivier-Ricci curvature (ORC) to acquire geometric information that cannot be provided by simple information filtering. In this sense, MSTs and PMFGs are but first steps to revealing the topological backbones of financial networks. This is something that TDA can elucidate more fully, following which the ORC can help us flesh out the geometry of financial networks. We applied these two approaches to a recent stock market crash in Taiwan and found that, beyond fusions and fissions, other non-fusion/fission processes such as cavitation, annihilation, rupture, healing, and puncture might also be important. We also successfully identified neck regions that emerged during the crash, based on their negative ORCs, and performed a case study on one such neck region.

Topological Learning and Its Application to Multimodal Brain Network Integration

A long-standing challenge in multimodal brain network analyses is to integrate topologically different brain networks obtained from diffusion and functional MRI in a coherent statistical framework. Existing multimodal frameworks will inevitably destroy the topological difference of the networks. In this paper, we propose a novel topological learning framework that integrates networks of different topology through persistent homology. Such challenging task is made possible through the introduction of a new topological loss that bypasses intrinsic computational bottlenecks and thus enables us to perform various topological computations and optimizations with ease. We validate the topological loss in extensive statistical simulations with ground truth to assess its effectiveness of discriminating networks. Among many possible applications, we demonstrate the versatility of topological loss in the twin imaging study where we determine the extend to which brain networks are genetically heritable.

From mathematics to medicine: A practical primer on topological data analysis (TDA) and the development of related analytic tools for the functional discovery of latent structure in fMRI data

fMRI is the preeminent method for collecting signals from the human brain in vivo, for using these signals in the service of functional discovery, and relating these discoveries to anatomical structure. Numerous computational and mathematical techniques have been deployed to extract information from the fMRI signal. Yet, the application of Topological Data Analyses (TDA) remain limited to certain sub-areas such as connectomics (that is, with summarized versions of fMRI data). While connectomics is a natural and important area of application of TDA, applications of TDA in the service of extracting structure from the (non-summarized) fMRI data itself are heretofore nonexistent. “Structure” within fMRI data is determined by dynamic fluctuations in spatially distributed signals over time, and TDA is well positioned to help researchers better characterize mass dynamics of the signal by rigorously capturing shape within it. To accurately motivate this idea, we a) survey an established method in TDA (“persistent homology”) to reveal and describe how complex structures can be extracted from data sets generally, and b) describe how persistent homology can be applied specifically to fMRI data. We provide explanations for some of the mathematical underpinnings of TDA (with expository figures), building ideas in the following sequence: a) fMRI researchers can and should use TDA to extract structure from their data; b) this extraction serves an important role in the endeavor of functional discovery, and c) TDA approaches can complement other established approaches toward fMRI analyses (for which we provide examples). We also provide detailed applications of TDA to fMRI data collected using established paradigms, and offer our software pipeline for readers interested in emulating our methods. This working overview is both an inter-disciplinary synthesis of ideas (to draw researchers in TDA and fMRI toward each other) and a detailed description of methods that can motivate collaborative research.

The topology of data: Opportunities for cancer research

Motivation

Topological methods have recently emerged as a reliable and interpretable framework for extracting information from high-dimensional data, leading to the creation of a branch of applied mathematics called Topological Data Analysis (TDA). Since then, TDA has been progressively adopted in biomedical research. Biological data collection can result in enormous datasets, comprising thousands of features and spanning diverse datatypes. This presents a barrier to initial data analysis as the fundamental structure of the dataset becomes hidden, obstructing the discovery of important features and patterns. TDA provides a solution to obtain the underlying shape of datasets over continuous resolutions, corresponding to key topological features independent of noise. TDA has the potential to support future developments in healthcare as biomedical datasets rise in complexity and dimensionality. Previous applications extend across the fields of neuroscience, oncology, immunology and medical image analysis. TDA has been used to reveal hidden subgroups of cancer patients, construct organizational maps of brain activity and classify abnormal patterns in medical images. The utility of TDA is broad and to understand where current achievements lie, we have evaluated the present state of TDA in cancer data analysis.

Results

This article aims to provide an overview of TDA in Cancer Research. A brief introduction to the main concepts of TDA is provided to ensure that the article is accessible to readers who are not familiar with this field. Following this, a focussed literature review on the field is presented, discussing how TDA has been applied across heterogeneous datatypes for cancer research.

Promises and pitfalls of topological data analysis for brain connectivity analysis

Developing sensitive and reliable methods to distinguish normal and abnormal brain states is a key neuroscientific challenge. Topological Data Analysis, despite its relative novelty, already generated many promising applications, including in neuroscience. We conjecture its prominent tool of persistent homology may benefit from going beyond analysing structural and functional connectivity to effective connectivity graphs capturing the direct causal interactions or information flows. Therefore, we assess the potential of persistent homology to directed brain network analysis by testing its discriminatory power in two distinctive examples of disease-related brain connectivity alterations: epilepsy and schizophrenia. We estimate connectivity from functional magnetic resonance imaging and electrophysiology data, employ Persistent Homology and quantify its ability to distinguish healthy from diseased brain states by applying a support vector machine to features quantifying persistent homology structure. We show how this novel approach compares to classification using standard undirected approaches and original connectivity matrices. In the schizophrenia classification, topological data analysis generally performs close to random, while classifications from raw connectivity perform substantially better; potentially due to topographical, rather than topological, specificity of the differences. In the easier task of seizure discrimination from scalp electroencephalography data, classification based on persistent homology features generally reached comparable performance to using raw connectivity, albeit with typically smaller accuracies obtained for the directed (effective) connectivity compared to the undirected (functional) connectivity. Specific applications for topological data analysis may open when direct comparison of connectivity matrices is unsuitable - such as for intracranial electrophysiology with individual number and location of measurements. While standard homology performed overall better than directed homology, this could be due to notorious technical problems of accurate effective connectivity estimation.

Constructing Connectome Atlas by Graph Laplacian Learning

The recent development of neuroimaging technology and network theory allows us to visualize and characterize the whole-brain functional connectivity in vivo. The importance of conventional structural image atlas widely used in population-based neuroimaging studies has been well verified. Similarly, a "common" brain connectivity map (also called connectome atlas) across individuals can open a new pathway to interpreting disorder-related brain cognition and behaviors. However, the main obstacle of applying the classic image atlas construction approaches to the connectome data is that a regular data structure (such as a grid) in such methods breaks down the intrinsic geometry of the network connectivity derived from the irregular data domain (in the setting of a graph). To tackle this hurdle, we first embed the brain network into a set of graph signals in the Euclidean space via the diffusion mapping technique. Furthermore, we cast the problem of connectome atlas construction into a novel learning-based graph inference model. It can be constructed by iterating the following processes: (1) align all individual brain networks to a common space spanned by the graph spectrum bases of the latent common network, and (2) learn graph Laplacian of the common network that is in consensus with all aligned brain networks. We have evaluated our novel method for connectome atlas construction in comparison with non-learning-based counterparts. Based on experiments using network connectivity data from populations with neurodegenerative and neuropediatric disorders, our approach has demonstrated statistically meaningful improvement over existing methods.

Disrupted metabolic connectivity in dopaminergic and cholinergic networks at different stages of dementia from 18F-FDG PET brain persistent homology network

Dementia is related to the cellular accumulation of β-amyloid plaques, tau aggregates, or α-synuclein aggregates, or to neurotransmitter deficiencies in the dopaminergic and cholinergic pathways. Cellular and neurochemical changes are both involved in dementia pathology. However, the role of dopaminergic and cholinergic networks in metabolic connectivity at different stages of dementia remains unclear. The altered network organisation of the human brain characteristic of many neuropsychiatric and neurodegenerative disorders can be detected using persistent homology network (PHN) analysis and algebraic topology. We used 18F-fluorodeoxyglucose positron emission tomography (18F-FDG PET) imaging data to construct dopaminergic and cholinergic metabolism networks, and used PHN analysis to track the evolution of these networks in patients with different stages of dementia. The sums of the network distances revealed significant differences between the network connectivity evident in the Alzheimer's disease and mild cognitive impairment cohorts. A larger distance between brain regions can indicate poorer efficiency in the integration of information. PHN analysis revealed the structural properties of and changes in the dopaminergic and cholinergic metabolism networks in patients with different stages of dementia at a range of thresholds. This method was thus able to identify dysregulation of dopaminergic and cholinergic networks in the pathology of dementia.

Stability of synchronization in simplicial complexes

Various systems in physics, biology, social sciences and engineering have been successfully modeled as networks of coupled dynamical systems, where the links describe pairwise interactions. This is, however, too strong a limitation, as recent studies have revealed that higher-order many-body interactions are present in social groups, ecosystems and in the human brain, and they actually affect the emergent dynamics of all these systems. Here, we introduce a general framework to study coupled dynamical systems accounting for the precise microscopic structure of their interactions at any possible order. We show that complete synchronization exists as an invariant solution, and give the necessary condition for it to be observed as a stable state. Moreover, in some relevant instances, such a necessary condition takes the form of a Master Stability Function. This generalizes the existing results valid for pairwise interactions to the case of complex systems with the most general possible architecture.

Learning dynamic graph embeddings for accurate detection of cognitive state changes in functional brain networks

Mounting evidence shows that brain functions and cognitive states are dynamically changing even in the resting state rather than remaining at a single constant state. Due to the relatively small changes in BOLD (blood-oxygen-level-dependent) signals across tasks, it is difficult to detect the change of cognitive status without requiring prior knowledge of the experimental design. To address this challenge, we present a dynamic graph learning approach to generate an ensemble of subject-specific dynamic graph embeddings, which allows us to use brain networks to disentangle cognitive events more accurately than using raw BOLD signals. The backbone of our method is essentially a representation learning process for projecting BOLD signals into a latent vertex-temporal domain with the greater biological underpinning of brain activities. Specifically, the learned representation domain is jointly formed by (1) a set of harmonic waves that govern the topology of whole-brain functional connectivities and (2) a set of Fourier bases that characterize the temporal dynamics of functional changes. In this regard our dynamic graph embeddings provide a new methodology to investigate how these self-organized functional fluctuation patterns oscillate along with the evolving cognitive status. We have evaluated our proposed method on both simulated data and working memory task-based fMRI datasets, where our dynamic graph embeddings achieve higher accuracy in detecting multiple cognitive states than other state-of-the-art methods.

Effect of APOE ε4 on multimodal brain connectomic traits: a persistent homology study

BACKGROUND

Although genetic risk factors and network-level neuroimaging abnormalities have shown effects on cognitive performance and brain atrophy in Alzheimer's disease (AD), little is understood about how apolipoprotein E (APOE) ε4 allele, the best-known genetic risk for AD, affect brain connectivity before the onset of symptomatic AD. This study aims to investigate APOE ε4 effects on brain connectivity from the perspective of multimodal connectome.

RESULTS

Here, we propose a novel multimodal brain network modeling framework and a network quantification method based on persistent homology for identifying APOE ε4-related network differences. Specifically, we employ sparse representation to integrate multimodal brain network information derived from both the resting state functional magnetic resonance imaging (rs-fMRI) data and the diffusion-weighted magnetic resonance imaging (dw-MRI) data. Moreover, persistent homology is proposed to avoid the ad hoc selection of a specific regularization parameter and to capture valuable brain connectivity patterns from the topological perspective. The experimental results demonstrate that our method outperforms the competing methods, and reasonably yields connectomic patterns specific to APOE ε4 carriers and non-carriers.

CONCLUSIONS

We have proposed a multimodal framework that integrates structural and functional connectivity information for constructing a fused brain network with greater discriminative power. Using persistent homology to extract topological features from the fused brain network, our method can effectively identify APOE ε4-related brain connectomic biomarkers.

Multi-dimensional persistent feature analysis identifies connectivity patterns of resting-state brain networks in Alzheimer's disease

OBJECTIVE

The characterization of functional brain network is crucial to understanding the neural mechanisms associated with Alzheimer's disease (AD) and mild cognitive impairment (MCI). Some studies have shown that graph theoretical analysis could reveal changes of the disease-related brain networks by thresholding edge weights. But the choice of threshold depends on ambiguous cognitive conditions, which leads to the lack of interpretability. Recently, persistent homology (PH) was proposed to record the persistence of topological features of networks across every possible thresholds, reporting a higher sensitivity than graph theoretical features in detecting network-level biomarkers of AD. However, most research on PH focused on 0-dimensional features (persistence of connected components) reflecting the intrinsic topology of the brain network, rather than 1-dimensional features (persistence of cycles) with an interesting neurobiological communication pattern. Our aim is to explore the multi-dimensional persistent features of brain networks in the AD and MCI patients, and further to capture valuable brain connectivity patterns.

APPROACH

We characterized the change rate of the connected component numbers across graph filtration using the functional derivative curves, and examined the persistence landscapes that vectorize the persistence of cycle structures. After that, the multi-dimensional persistent features were validated in disease identification using a K-nearest neighbor algorithm. Furthermore, a connectivity pattern mining framework was designed to capture the disease-specific brain structures.

MAIN RESULTS

We found that the multi-dimensional persistent features can identify statistical group differences, quantify subject-level distances, and yield disease-specific connectivity patterns. Relatively high classification accuracies were received when compared with graph theoretical features.

SIGNIFICANCE

This work represents a conceptual bridge linking complex brain network analysis and computational topology. Our results can be beneficial for providing a complementary objective opinion to the clinical diagnosis of neurodegenerative diseases.

Persistence codebooks for topological data analysis

Persistent homology is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs) which are 2D multisets of points. Their variable size makes them, however, difficult to combine with typical machine learning workflows. In this paper we introduce persistence codebooks, a novel expressive and discriminative fixed-size vectorized representation of PDs that adapts to the inherent sparsity of persistence diagrams. To this end, we adapt bag-of-words, vectors of locally aggregated descriptors and Fischer vectors for the quantization of PDs. Persistence codebooks represent PDs in a convenient way for machine learning and statistical analysis and have a number of favorable practical and theoretical properties including 1-Wasserstein stability. We evaluate the presented representations on several heterogeneous datasets and show their (high) discriminative power. Our approach yields comparable—and partly even higher—performance in much less time than alternative approaches.

Default Mode Network Analysis of APOE Genotype in Cognitively Unimpaired Subjects Based on Persistent Homology

Current researches on default mode network (DMN) in normal elderly have mainly focused on finding some dysfunctional areas with decreased or increased connectivity. The global network dynamics of apolipoprotein E (APOE) e4 allele group is rarely studied. In our previous brain network study, we have demonstrated the advantage of persistent homology. It can distinguish robust and noisy topological features over multiscale nested networks, and the derived properties are more stable. In this study, for the first time we applied persistent homology to analyze APOE-related effects on whole-brain functional network. In our experiments, the risk allele group exhibited lower network radius and modularity in whole brain DMN based on graph theory, suggesting the abnormal organization structure. Moreover, two suggested measures from persistent homology detected significant differences between groups within the left hemisphere and in the whole brain in two datasets. They were more statistically sensitive to APOE genotypic differences than standard graph-based measures. In summary, we provide evidence that the e4 genotype leads to distinct DMN functional alterations in the early phases of Alzheimer's disease using persistent homology approach. Our study offers a novel insight to explore potential biomarkers in healthy elderly populations carrying APOE e4 allele.

Weighted persistent homology for osmolyte molecular aggregation and hydrogen-bonding network analysis

It has long been observed that trimethylamine N-oxide (TMAO) and urea demonstrate dramatically different properties in a protein folding process. Even with the enormous theoretical and experimental research work on these two osmolytes, various aspects of their underlying mechanisms still remain largely elusive. In this paper, we propose to use the weighted persistent homology to systematically study the osmolytes molecular aggregation and their hydrogen-bonding network from a local topological perspective. We consider two weighted models, i.e., localized persistent homology (LPH) and interactive persistent homology (IPH). Boltzmann persistent entropy (BPE) is proposed to quantitatively characterize the topological features from LPH and IPH, together with persistent Betti number (PBN). More specifically, from the localized persistent homology models, we have found that TMAO and urea have very different local topology. TMAO is found to exhibit a local network structure. With the concentration increase, the circle elements in these networks show a clear increase in their total numbers and a decrease in their relative sizes. In contrast, urea shows two types of local topological patterns, i.e., local clusters around 6 Å and a few global circle elements at around 12 Å. From the interactive persistent homology models, it has been found that our persistent radial distribution function (PRDF) from the global-scale IPH has same physical properties as the traditional radial distribution function. Moreover, PRDFs from the local-scale IPH can also be generated and used to characterize the local interaction information. Other than the clear difference of the first peak value of PRDFs at filtration size 4 Å, TMAO and urea also shows very different behaviors at the second peak region from filtration size 5 Å to 10 Å. These differences are also reflected in the PBNs and BPEs of the local-scale IPH. These localized topological information has never been revealed before. Since graphs can be transferred into simplicial complexes by the clique complex, our weighted persistent homology models can be used in the analysis of various networks and graphs from any molecular structures and aggregation systems.

White Matter Brain Network Research in Alzheimer's Disease Using Persistent Features

Despite the severe social burden caused by Alzheimer’s disease (AD), no drug than can change the disease progression has been identified yet. The structural brain network research provides an opportunity to understand physiological deterioration caused by AD and its precursor, mild cognitive impairment (MCI). Recently, persistent homology has been used to study brain network dynamics and characterize the global network organization. However, it is unclear how these parameters reflect changes in structural brain networks of patients with AD or MCI. In this study, our previously proposed persistent features and various traditional graph-theoretical measures are used to quantify the topological property of white matter (WM) network in 150 subjects with diffusion tensor imaging (DTI). We found significant differences in these measures among AD, MCI, and normal controls (NC) under different brain parcellation schemes. The decreased network integration and increased network segregation are presented in AD and MCI. Moreover, the persistent homology-based measures demonstrated stronger statistical capability and robustness than traditional graph-theoretic measures, suggesting that they represent a more sensitive approach to detect altered brain structures and to better understand AD symptomology at the network level. These findings contribute to an increased understanding of structural connectome in AD and provide a novel approach to potentially track the progression of AD.

Topological Data Analysis Reveals Robust Alterations in the Whole-Brain and Frontal Lobe Functional Connectomes in Attention-Deficit/Hyperactivity Disorder

Attention-deficit/hyperactivity disorder (ADHD) is a developmental disorder characterized by difficulty to control the own behavior. Neuroimaging studies have related ADHD with the interplay of fronto-parietal attention systems with the default mode network (DMN; Castellanos and Aoki, 2016). However, some results have been inconsistent, potentially due to methodological differences in the analytical strategies when defining the brain functional network, i.e., the functional connectivity threshold and/or the brain parcellation scheme. Here, we make use of topological data analysis (TDA) to explore the brain connectome as a function of the filtration value (i.e., the connectivity threshold), instead of using a static connectivity threshold. Specifically, we characterized the transition from all nodes being isolated to being connected into a single component as a function of the filtration value. We explored the utility of such a method to identify differences between 81 children with ADHD (45 male, age: 7.26–17.61 years old) and 96 typically developing children (TDC; 59 male, age: 7.17–17.96 years old), using a public dataset of resting state (rs)fMRI in human subjects. Results were highly congruent when using four different brain segmentations (atlases), and exhibited significant differences for the brain topology of children with ADHD, both at the whole-brain network and the functional subnetwork levels, particularly involving the frontal lobe and the DMN. Therefore, this is a solid approach that complements connectomics-related methods and may contribute to identify the neurophysio-pathology of ADHD.

The shape of things to come: Topological data analysis and biology, from molecules to organisms

Shape is data and data is shape. Biologists are accustomed to thinking about how the shape of biomolecules, cells, tissues, and organisms arise from the effects of genetics, development, and the environment. Less often do we consider that data itself has shape and structure, or that it is possible to measure the shape of data and analyze it. Here, we review applications of topological data analysis (TDA) to biology in a way accessible to biologists and applied mathematicians alike. TDA uses principles from algebraic topology to comprehensively measure shape in data sets. Using a function that relates the similarity of data points to each other, we can monitor the evolution of topological features-connected components, loops, and voids. This evolution, a topological signature, concisely summarizes large, complex data sets. We first provide a TDA primer for biologists before exploring the use of TDA across biological sub-disciplines, spanning structural biology, molecular biology, evolution, and development. We end by comparing and contrasting different TDA approaches and the potential for their use in biology. The vision of TDA, that data are shape and shape is data, will be relevant as biology transitions into a data-driven era where the meaningful interpretation of large data sets is a limiting factor.

Maturational delay and asymmetric information flow of brain connectivity in SHR model of ADHD revealed by topological analysis of metabolic networks

Attention-deficit hyperactivity disorder (ADHD) is a complex brain development disorder characterized by hyperactivity/impulsivity and inattention. A major hypothesis of ADHD is a lag of maturation, which is supported mainly by anatomical studies evaluating cortical thickness. Here, we analyzed changes of topological characteristics of whole-brain metabolic connectivity in twelve SHR rats selected as ADHD-model rats by confirming behavior abnormalities using the marble burying test, open field test, and delay discounting task and 12 Wistar Kyoto rats as the control group, across development from 4 weeks old (childhood) and 6 weeks old (entry of puberty). A topological approach based on graph filtrations revealed a lag in the strengthening of limbic-cortical/subcortical connections in ADHD-model rats. This in turn related to impaired modularization of memory and reward-motivation associated regions. Using mathematical network analysis techniques such as single linkage hierarchical clustering and volume entropy, we observed left-lateralized connectivity in the ADHD-model rats at 6 weeks old. Our findings supported the maturational delay of metabolic connectivity in the SHR model of ADHD, and also suggested the possibility of impaired and compensative reconfiguration of information flow over the brain network.

Weighted persistent homology for biomolecular data analysis

In this paper, we systematically review weighted persistent homology (WPH) models and their applications in biomolecular data analysis. Essentially, the weight value, which reflects physical, chemical and biological properties, can be assigned to vertices (atom centers), edges (bonds), or higher order simplexes (cluster of atoms), depending on the biomolecular structure, function, and dynamics properties. Further, we propose the first localized weighted persistent homology (LWPH). Inspired by the great success of element specific persistent homology (ESPH), we do not treat biomolecules as an inseparable system like all previous weighted models, instead we decompose them into a series of local domains, which may be overlapped with each other. The general persistent homology or weighted persistent homology analysis is then applied on each of these local domains. In this way, functional properties, that are embedded in local structures, can be revealed. Our model has been applied to systematically study DNA structures. It has been found that our LWPH based features can be used to successfully discriminate the A-, B-, and Z-types of DNA. More importantly, our LWPH based principal component analysis (PCA) model can identify two configurational states of DNA structures in ion liquid environment, which can be revealed only by the complicated helical coordinate system. The great consistence with the helical-coordinate model demonstrates that our model captures local structure variations so well that it is comparable with geometric models. Moreover, geometric measurements are usually defined in local regions. For instance, the helical-coordinate system is limited to one or two basepairs. However, our LWPH can quantitatively characterize structure information in regions or domains with arbitrary sizes and shapes, where traditional geometrical measurements fail.

Propofol Anesthesia Alters Spatial and Topologic Organization of Rat Brain Metabolism

BACKGROUND

Loss of consciousness during anesthesia reduces local and global rate of cerebral glucose metabolism. Despite this, the influence of gradual anesthetic-induced changes on consciousness across the entire brain metabolic network has barely been studied. The purpose of the present study was to identify specific cerebral metabolic patterns characteristic of different consciousness/anesthesia states induced by intravenous anesthetic propofol.

METHODS

At various times, 20 Sprague-Dawley adult rats were intravenously administered three different dosages of propofol to induce different anesthetic states: mild sedation (20 mg · kg · h), deep sedation (40 mg · kg · h), and deep anesthesia (80 mg · kg · h). Using [F]fluorodeoxyglucose positron emission tomography brain imaging, alterations in the spatial pattern of metabolic distribution and metabolic topography were investigated by applying voxel-based spatial covariance analysis and graph-theory analysis.

RESULTS

Evident reductions were found in baseline metabolism along with altered metabolic spatial distribution during propofol-induced anesthesia. Moreover, graph-theory analysis revealed a disruption in global and local efficiency of the metabolic brain network characterized by decreases in metabolic connectivity and energy efficiency during propofol-induced deep anesthesia (mild sedation global efficiency/local efficiency = 0.6985/0.7190, deep sedation global efficiency/local efficiency = 0.7444/0.7875, deep anesthesia global efficiency/local efficiency = 0.4498/0.6481; mild sedation vs. deep sedation, global efficiency: P = 0.356, local efficiency: P = 0.079; mild sedation vs. deep anesthesia, global efficiency: P < 0.0001, local efficiency: P < 0.0001; deep sedation vs. deep anesthesia, global efficiency: P < 0.0001, local efficiency: P < 0.0001). A strong spatial correlation was also found between cerebral metabolism and metabolic connectivity strength, which decreased significantly with deepening anesthesia level (correlation coefficients: mild sedation, r = 0.55, deep sedation, r = 0.47; deep anesthesia, r = 0.23; P < 0.0001 between the sedation and deep anesthesia groups).

CONCLUSIONS

The data revealed anesthesia-related alterations in spatial and topologic organization of metabolic brain network, as well as a close relationship between metabolic connectivity and cerebral metabolism during propofol anesthesia. These findings may provide novel insights into the metabolic mechanism of anesthetic-induced loss of consciousness.

A UNIVARIATE PERSISTENT BRAIN NETWORK FEATURE BASED ON THE AGGREGATED COST OF CYCLES FROM THE NESTED FILTRATION NETWORKS

A threshold-free feature in brain network analysis can help circumvent the curse of arbitrary network thresholding for binary network conversions. Here, Persistent Homology is the inspiration for defining a new aggregation cost based on the number of cycles, or for tracking the first Betti number in a nested filtration network within the graph. Our theoretical analysis shows that the proposed aggregated cost of cycles (ACC) is monotonically increasing and thus we define a univariate persistent feature based on the shape of ACC. The proposed statistic has advantages compared to the First Betti Number Plot (BNP1), which only tracks the total number of cycles at each filtration. We show that our method is sensitive to both the topology of modular networks and the difference in the number of cycles in a network. Our method outperforms its counterparts in a synthetic dataset, while in a real-world one it achieves results comparable with the BNP1. Our proposed framework enriches univariate measures for discovering brain network dissimilarities for better categorization of distinct stages in Alzheimer's Disease (AD).

Persistent Feature Analysis of Multimodal Brain Networks Using Generalized Fused Lasso for EMCI Identification

Early Mild Cognitive Impairment (EMCI) involves very subtle changes in brain pathological process, and thus identification of EMCI can be challenging. By jointly analyzing cross-information among different neuroimaging data, an increased interest recently emerges in multimodal fusion to better understand clinical measurements with respect to both structural and functional connectivity. In this paper, we propose a novel multimodal brain network modeling method for EMCI identification. Specifically, we employ the structural connectivity based on diffusion tensor imaging (DTI), as a constraint, to guide the regression of BOLD time series from resting state functional magnetic resonance imaging (rs-fMRI). In addition, we introduce multiscale persistent homology features to avoid the uncertainty of regularization parameter selection. An empirical study on the Alzheimer's Disease Neuroimaging Initiative (ADNI) database demonstrates that the proposed method effectively improves classification performance compared with several competing approaches, and reasonably yields connectivity patterns specific to different diagnostic groups.