Topological insights into the neural basis of flexible behavior.

It is widely accepted that there is an inextricable link between neural computations, biological mechanisms, and behavior, but it is challenging to simultaneously relate all three. Here, we show that topological data analysis (TDA) provides an important bridge between these approaches to studying how brains mediate behavior. We demonstrate that cognitive processes change the topological description of the shared activity of populations of visual neurons. These topological changes constrain and distinguish between competing mechanistic models, are connected to subjects' performance on a visual change detection task, and, via a link with network control theory, reveal a tradeoff between improving sensitivity to subtle visual stimulus changes and increasing the chance that the subject will stray off task. These connections provide a blueprint for using TDA to uncover the biological and computational mechanisms by which cognition affects behavior in health and disease.

Artificial intelligence-aided protein engineering: from topological data analysis to deep protein language models.

Protein engineering is an emerging field in biotechnology that has the potential to revolutionize various areas, such as antibody design, drug discovery, food security, ecology, and more. However, the mutational space involved is too vast to be handled through experimental means alone. Leveraging accumulative protein databases, machine learning (ML) models, particularly those based on natural language processing (NLP), have considerably expedited protein engineering. Moreover, advances in topological data analysis (TDA) and artificial intelligence-based protein structure prediction, such as AlphaFold2, have made more powerful structure-based ML-assisted protein engineering strategies possible. This review aims to offer a comprehensive, systematic, and indispensable set of methodological components, including TDA and NLP, for protein engineering and to facilitate their future development.

Advancing Precision Medicine: Algebraic Topology and Differential Geometry in Radiology and Computational Pathology.

Precision medicine aims to provide personalized care based on individual patient characteristics, rather than guideline-directed therapies for groups of diseases or patient demographics. Images-both radiology- and pathology-derived-are a major source of information on presence, type, and status of disease. Exploring the mathematical relationship of pixels in medical imaging ("radiomics") and cellular-scale structures in digital pathology slides ("pathomics") offers powerful tools for extracting both qualitative and, increasingly, quantitative data. These analytical approaches, however, may be significantly enhanced by applying additional methods arising from fields of mathematics such as differential geometry and algebraic topology that remain underexplored in this context. Geometry's strength lies in its ability to provide precise local measurements, such as curvature, that can be crucial for identifying abnormalities at multiple spatial levels. These measurements can augment the quantitative features extracted in conventional radiomics, leading to more nuanced diagnostics. By contrast, topology serves as a robust shape descriptor, capturing essential features such as connected components and holes. The field of topological data analysis was initially founded to explore the shape of data, with functional network connectivity in the brain being a prominent example. Increasingly, its tools are now being used to explore organizational patterns of physical structures in medical images and digitized pathology slides. By leveraging tools from both differential geometry and algebraic topology, researchers and clinicians may be able to obtain a more comprehensive, multi-layered understanding of medical images and contribute to precision medicine's armamentarium.

Segmentation of biomedical images based on a computational topology framework.

The homology groups of a topological space provide us with information about its connectivity and the number and type of holes in it. This type of information can find practical applications in describing the intrinsic structure of an image, as well as in identifying equivalence classes in collections of images. When computing homological characteristics, the existence and strength of the relationships between each pair of points in the topological space are studied. The practical use of this approach begins by building a topological space from the image, in which the computation of the homology groups can be carried out in a feasible time. Once the homological properties are obtained, what follows is the task of translating such information into operations such as image segmentation. This work presents a technique for denoising persistent diagrams and reconstructing the shape of segmented objects using the remaining classes on the diagram. A case study for the segmentation of cell nuclei in histological images is used for demonstration purposes. With this approach: a) topological denoising is achieved by aggregating trivial classes on the persistence diagram, and b) a growing seed algorithm uses the information obtained during the construction of the persistence diagram for the reconstruction of the segmented cell structures.

Topological clustering of multilayer networks.

Multilayer networks continue to gain significant attention in many areas of study, particularly due to their high utility in modeling interdependent systems such as critical infrastructures, human brain connectome, and socioenvironmental ecosystems. However, clustering of multilayer networks, especially using the information on higher-order interactions of the system entities, still remains in its infancy. In turn, higher-order connectivity is often the key in such multilayer network applications as developing optimal partitioning of critical infrastructures in order to isolate unhealthy system components under cyber-physical threats and simultaneous identification of multiple brain regions affected by trauma or mental illness. In this paper, we introduce the concepts of topological data analysis to studies of complex multilayer networks and propose a topological approach for network clustering. The key rationale is to group nodes based not on pairwise connectivity patterns or relationships between observations recorded at two individual nodes but based on how similar in shape their local neighborhoods are at various resolution scales. Since shapes of local node neighborhoods are quantified using a topological summary in terms of persistence diagrams, we refer to the approach as clustering using persistence diagrams (CPD). CPD systematically accounts for the important heterogeneous higher-order properties of node interactions within and in-between network layers and integrates information from the node neighbors. We illustrate the utility of CPD by applying it to an emerging problem of societal importance: vulnerability zoning of residential properties to weather- and climate-induced risks in the context of house insurance claim dynamics.

Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors.

Highly resolved spatial data of complex systems encode rich and nonlinear information. Quantification of heterogeneous and noisy data-often with outliers, artifacts, and mislabeled points-such as those from tissues, remains a challenge. The mathematical field that extracts information from the shape of data, topological data analysis (TDA), has expanded its capability for analyzing real-world datasets in recent years by extending theory, statistics, and computation. An extension to the standard theory to handle heterogeneous data is multiparameter persistent homology (MPH). Here we provide an application of MPH landscapes, a statistical tool with theoretical underpinnings. MPH landscapes, computed for (noisy) data from agent-based model simulations of immune cells infiltrating into a spheroid, are shown to surpass existing spatial statistics and one-parameter persistent homology. We then apply MPH landscapes to study immune cell location in digital histology images from head and neck cancer. We quantify intratumoral immune cells and find that infiltrating regulatory T cells have more prominent voids in their spatial patterns than macrophages. Finally, we consider how TDA can integrate and interrogate data of different types and scales, e.g., immune cell locations and regions with differing levels of oxygenation. This work highlights the power of MPH landscapes for quantifying, characterizing, and comparing features within the tumor microenvironment in synthetic and real datasets.

Homogenization of word relationships in schizophrenia: Topological analysis of cortical semantic representations.

AIM: Patients with schizophrenia typically exhibit symptoms of disorganized thought and display concreteness and over-inclusion in verbal reports, depending on the level of abstraction. While concreteness and over-inclusion may appear contradictory, the underlying psychopathology that explains these symptoms remains unclear. In the current study, we used functional magnetic resonance imaging with an encoding modeling approach to examine how concepts of various words, represented as brain activity, are anomalously connected at different levels of abstraction in patients with schizophrenia. METHODS: Fourteen individuals diagnosed with schizophrenia and 17 healthy controls underwent functional magnetic resonance imaging to measure brain activity representing concepts of various words. We used a persistent homology (PH) method to analyze the topological structures of word representations in schizophrenia patients, healthy controls, and random data, across different levels of abstraction by varying dissimilarity scales in the representation space. RESULTS: The results revealed that patients with schizophrenia exhibited more homogeneous word relationships across different levels of abstraction compared with healthy controls. Additionally, topological structures exhibited a shift toward a random network structure in patients with schizophrenia compared with controls. The PH method successfully distinguished semantic representations of patients with schizophrenia from those of controls. CONCLUSIONS: The current results provide an explanation for the mechanisms underlying the deficits in abstraction ability observed in schizophrenia. The isotopic connection of individual concepts reflects both the reduction of contextual connections at a semantically fine-grained scale and the absence of clear boundaries between related concepts at a coarse scale, which lead to concreteness and over-inclusion, respectively.

Topological Persistence Guided Knowledge Distillation for Wearable Sensor Data.

Deep learning methods have achieved a lot of success in various applications involving converting wearable sensor data to actionable health insights. A common application areas is activity recognition, where deep-learning methods still suffer from limitations such as sensitivity to signal quality, sensor characteristic variations, and variability between subjects. To mitigate these issues, robust features obtained by topological data analysis (TDA) have been suggested as a potential solution. However, there are two significant obstacles to using topological features in deep learning: (1) large computational load to extract topological features using TDA, and (2) different signal representations obtained from deep learning and TDA which makes fusion difficult. In this paper, to enable integration of the strengths of topological methods in deep-learning for time-series data, we propose to use two teacher networks - one trained on the raw time-series data, and another trained on persistence images generated by TDA methods. These two teachers are jointly used to distill a single student model, which utilizes only the raw time-series data at test-time. This approach addresses both issues. The use of KD with multiple teachers utilizes complementary information, and results in a compact model with strong supervisory features and an integrated richer representation. To assimilate desirable information from different modalities, we design new constraints, including orthogonality imposed on feature correlation maps for improving feature expressiveness and allowing the student to easily learn from the teacher. Also, we apply an annealing strategy in KD for fast saturation and better accommodation from different features, while the knowledge gap between the teachers and student is reduced. Finally, a robust student model is distilled, which can at test-time uses only the time-series data as an input, while implicitly preserving topological features. The experimental results demonstrate the effectiveness of the proposed method on wearable sensor data. The proposed method shows 71.74% in classification accuracy on GENEActiv with WRN16-1 (1D CNNs) student, which outperforms baselines and takes much less processing time (less than 17 sec) than teachers on 6k testing samples.

Topological Data Analysis in Cardiovascular Signals: An Overview.

Topological data analysis (TDA) is a recent approach for analyzing and interpreting complex data sets based on ideas a branch of mathematics called algebraic topology. TDA has proven useful to disentangle non-trivial data structures in a broad range of data analytics problems including the study of cardiovascular signals. Here, we aim to provide an overview of the application of TDA to cardiovascular signals and its potential to enhance the understanding of cardiovascular diseases and their treatment in the form of a literature or narrative review. We first introduce the concept of TDA and its key techniques, including persistent homology, Mapper, and multidimensional scaling. We then discuss the use of TDA in analyzing various cardiovascular signals, including electrocardiography, photoplethysmography, and arterial stiffness. We also discuss the potential of TDA to improve the diagnosis and prognosis of cardiovascular diseases, as well as its limitations and challenges. Finally, we outline future directions for the use of TDA in cardiovascular signal analysis and its potential impact on clinical practice. Overall, TDA shows great promise as a powerful tool for the analysis of complex cardiovascular signals and may offer significant insights into the understanding and management of cardiovascular diseases.

Singular-Value-Decomposition-Based Matrix Surgery.

This paper is motivated by the need to stabilise the impact of deep learning (DL) training for medical image analysis on the conditioning of convolution filters in relation to model overfitting and robustness. We present a simple strategy to reduce square matrix condition numbers and investigate its effect on the spatial distributions of point clouds of well- and ill-conditioned matrices. For a square matrix, the SVD surgery strategy works by: (1) computing its singular value decomposition (SVD), (2) changing a few of the smaller singular values relative to the largest one, and (3) reconstructing the matrix by reverse SVD. Applying SVD surgery on CNN convolution filters during training acts as spectral regularisation of the DL model without requiring the learning of extra parameters. The fact that the further away a matrix is from the non-invertible matrices, the higher its condition number is suggests that the spatial distributions of square matrices and those of their inverses are correlated to their condition number distributions. We shall examine this assertion empirically by showing that applying various versions of SVD surgery on point clouds of matrices leads to bringing their persistent diagrams (PDs) closer to the matrices of the point clouds of their inverses.

Understanding Higher-Order Interactions in Information Space.

Methods used in topological data analysis naturally capture higher-order interactions in point cloud data embedded in a metric space. This methodology was recently extended to data living in an information space, by which we mean a space measured with an information theoretical distance. One such setting is a finite collection of discrete probability distributions embedded in the probability simplex measured with the relative entropy (Kullback-Leibler divergence). More generally, one can work with a Bregman divergence parameterized by a different notion of entropy. While theoretical algorithms exist for this setup, there is a paucity of implementations for exploring and comparing geometric-topological properties of various information spaces. The interest of this work is therefore twofold. First, we propose the first robust algorithms and software for geometric and topological data analysis in information space. Perhaps surprisingly, despite working with Bregman divergences, our design reuses robust libraries for the Euclidean case. Second, using the new software, we take the first steps towards understanding the geometric-topological structure of these spaces. In particular, we compare them with the more familiar spaces equipped with the Euclidean and Fisher metrics.

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.

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.

Statistical and topological summaries aid disease detection for segmented retinal vascular images.

OBJECTIVE: Disease complications can alter vascular network morphology and disrupt tissue functioning. Microvascular diseases of the retina are assessed by visual inspection of retinal images, but this can be challenging when diseases exhibit silent symptoms or patients cannot attend in-person meetings. We examine the performance of machine learning algorithms in detecting microvascular disease when trained on statistical and topological summaries of segmented retinal vascular images. METHODS: We compute 13 separate descriptor vectors (5 statistical, 8 topological) to summarize the morphology of retinal vessel segmentation images and train support vector machines to predict each image's disease classification from the summary vectors. We assess the performance of each descriptor vector, using five-fold cross validation to estimate their accuracy. We apply these methods to four datasets that were assembled from four existing data repositories; three datasets contain segmented retinal vascular images from one of the repositories, whereas the fourth "All" dataset combines images from four repositories. RESULTS: Among the 13 total descriptor vectors considered, either a statistical Box-counting descriptor vector or a topological Flooding descriptor vector achieves the highest accuracy levels. On the combined "All" dataset, the Box-counting vector outperforms all other descriptors, including the topological Flooding vector which is sensitive to differences in the annotation styles between the different datasets. CONCLUSION: Our work represents a first step to establishing which computational methods are most suitable for identifying microvascular disease and assessing their current limitations. These methods could be incorporated into automated disease assessment tools.

Topological data analysis of zebrafish patterns.

Self-organized pattern behavior is ubiquitous throughout nature, from fish schooling to collective cell dynamics during organism development. Qualitatively these patterns display impressive consistency, yet variability inevitably exists within pattern-forming systems on both microscopic and macroscopic scales. Quantifying variability and measuring pattern features can inform the underlying agent interactions and allow for predictive analyses. Nevertheless, current methods for analyzing patterns that arise from collective behavior capture only macroscopic features or rely on either manual inspection or smoothing algorithms that lose the underlying agent-based nature of the data. Here we introduce methods based on topological data analysis and interpretable machine learning for quantifying both agent-level features and global pattern attributes on a large scale. Because the zebrafish is a model organism for skin pattern formation, we focus specifically on analyzing its skin patterns as a means of illustrating our approach. Using a recent agent-based model, we simulate thousands of wild-type and mutant zebrafish patterns and apply our methodology to better understand pattern variability in zebrafish. Our methodology is able to quantify the differential impact of stochasticity in cell interactions on wild-type and mutant patterns, and we use our methods to predict stripe and spot statistics as a function of varying cellular communication. Our work provides an approach to automatically quantifying biological patterns and analyzing agent-based dynamics so that we can now answer critical questions in pattern formation at a much larger scale.

Constrained Adaptive Distillation Based on Topological Persistence for Wearable Sensor Data.

Wearable sensor data analysis with persistence features generated by topological data analysis (TDA) has achieved great successes in various applications, however, it suffers from large computational and time resources for extracting topological features. In this paper, our approach utilizes knowledge distillation (KD) that involves the use of multiple teacher networks trained with the raw time-series and persistence images generated by TDA, respectively. However, direct transfer of knowledge from the teacher models utilizing different characteristics as inputs to the student model results in a knowledge gap and limited performance. To address this problem, we introduce a robust framework that integrates multimodal features from two different teachers and enables a student to learn desirable knowledge effectively. To account for statistical differences in multimodalities, entropy based constrained adaptive weighting mechanism is leveraged to automatically balance the effects of teachers and encourage the student model to adequately adopt the knowledge from two teachers. To assimilate dissimilar structural information generated by different style models for distillation, batch and channel similarities within a mini-batch are used. We demonstrate the effectiveness of the proposed method on wearable sensor data.

Cluster Persistence for Weighted Graphs.

Persistent homology is a natural tool for probing the topological characteristics of weighted graphs, essentially focusing on their 0-dimensional homology. While this area has been thoroughly studied, we present a new approach to constructing a filtration for cluster analysis via persistent homology. The key advantages of the new filtration is that (a) it provides richer signatures for connected components by introducing non-trivial birth times, and (b) it is robust to outliers. The key idea is that nodes are ignored until they belong to sufficiently large clusters. We demonstrate the computational efficiency of our filtration, its practical effectiveness, and explore into its properties when applied to random graphs.

TREPH: A Plug-In Topological Layer for Graph Neural Networks.

Topological Data Analysis (TDA) is an approach to analyzing the shape of data using techniques from algebraic topology. The staple of TDA is Persistent Homology (PH). Recent years have seen a trend of combining PH and Graph Neural Networks (GNNs) in an end-to-end manner to capture topological features from graph data. Though effective, these methods are limited by the shortcomings of PH: incomplete topological information and irregular output format. Extended Persistent Homology (EPH), as a variant of PH, addresses these problems elegantly. In this paper, we propose a plug-in topological layer for GNNs, termed Topological Representation with Extended Persistent Homology (TREPH). Taking advantage of the uniformity of EPH, a novel aggregation mechanism is designed to collate topological features of different dimensions to the local positions determining their living processes. The proposed layer is provably differentiable and more expressive than PH-based representations, which in turn is strictly stronger than message-passing GNNs in expressive power. Experiments on real-world graph classification tasks demonstrate the competitiveness of TREPH compared with the state-of-the-art approaches.

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.

[Molecular mechanisms of the effect of standardized placental hydrolysate peptides on mitochondria functioning].

BACKGROUND: Human placenta hydrolysates (HPH), the study of which was initiated by the scientific school of Vladimir P. Filatov, are currently being investigated using modern proteomic technologies. HPH is a promising tool for maintaining the function of mitochondria and regenerating tissues and organs with a high content of mitochondria (liver, heart muscle, skeletal muscles, etc.). The molecular mechanisms of action of HPH are practically not studied. AIM: Identification of mitochondrial support mitochondrial function-supporting peptides in HPH (Laennec, produced by Japan Bioproducts). MATERIALS AND METHODS: Data on the chemical structure of the peptides were collected through a mass spectrometric experiment. Then, to establish the amino acid sequences of the peptides, de novo peptide sequencing algorithms based on the mathematical theory of topological and metric analysis of chemographs were applied. Bioinformatic analysis of the peptide composition of HPH was carried out using the integral protein annotation method. RESULTS: The biological functions of 41 peptides in the composition of HPH have been identified and described. Among the target proteins, the activity of which is regulated by the identified peptides and significantly affects the function of mitochondria, are caspases (CASP1, CASP3, CASP4) and other proteins regulating apoptosis (BCL2, CANPL1, PPARA), MAP kinases (MAPK1, MAPK3, MAPK4, MAPK8, MAPK9 , MAPK10, MAPK14), AKT1/GSK3B/MTOR cascade kinases, and a number of other target proteins (ADGRG6 receptor, inhibitor of NF-êB kinase IKKE, pyruvate dehydrogenase 2/3/4, SIRT1 sirtuin deacetylase, ULK1 kinase). CONCLUSION: HPH peptides have been identified that promote inhibition of mitochondrial pore formation, apoptosis, and excessive mitochondrial autophagy under conditions of oxidative/toxic stress, chronic inflammation, and/or hyperinsulinemia.