Topological Data Analysis for Multivariate Time Series Data

Methods for Brain Connectivity Analysis with Applications to Rat Local Field Potential Recordings

Modeling the brain dependence network is central to understanding underlying neural mechanisms such as perception, action, and memory. In this study, we present a broad range of statistical methods for analyzing dependence in a brain network. Leveraging a combination of classical and cutting-edge approaches, we analyze multivariate hippocampal local field potential (LFP) time series data concentrating on the encoding of nonspatial olfactory information in rats. We present the strengths and limitations of each method in capturing neural dynamics and connectivity. Our analysis begins with exploratory techniques, including correlation, partial correlation, spectral matrices, and coherence, to establish foundational connectivity insights. We then investigate advanced methods such as Granger causality (GC), robust canonical coherence analysis, spectral transfer entropy (STE), and wavelet coherence to capture dynamic and nonlinear interactions. Additionally, we investigate the utility of topological data analysis (TDA) to extract multi-scale topological features and explore deep learning-based canonical correlation frameworks for connectivity modeling. This comprehensive approach offers an introduction to the state-of-the-art techniques for the analysis of dependence networks, emphasizing the unique strengths of various methodologies, addressing computational challenges, and paving the way for future research.

From Density to Void: Why Brain Networks Fail to Reveal Complex Higher-Order Structures

In brain network analysis using resting-state fMRI, there is growing interest in modeling higher-order interactions beyond simple pairwise connectivity via persistent homology. Despite the promise of these advanced topological tools, robust and consistently observed higher-order interactions over time remain elusive. In this study, we investigate why conventional analyses often fail to reveal complex higher-order structures-such as interactions involving four or more nodes-and explore whether such interactions truly exist in functional brain networks. We utilize a simplicial complex framework often used in persistent homology to address this question.

Topological Data Analysis of Ninjin'yoeito Effects Unraveling Complex Interconnections in Patients With Frailty: A Pilot Study

Background Ninjin'yoeito (NYT), a traditional Japanese Kampo medicine, has shown potential in treating frailty and overactive bladder (OAB) symptoms. However, its effects are multifaceted and vary among individuals. This pilot study explored the use of topological data analysis (TDA) and natural language processing (NLP) to evaluate the effect of NYT on frailty in patients with OAB. Methods Fifteen patients with frailty aged 75 or older underwent pelvic floor muscle training (PFMT) and one month of NYT administration. The eight standardized health questionnaires were simplified into a 28-item format using NLP. Persistent homology analysis via TDA revealed the complex, multidimensional effects of NYT, while network graph clustering using the Louvain method identified key health domains influenced by NYT.  Results TDA revealed multiloop structures in the therapeutic effects of NYT, indicating multiple pathways of improvement across physical and mental health domains. Network graph clustering identified four distinct communities linking OAB symptoms with energy, physical function, mental stress, and sleep quality. No significant adverse effects were noted.  Conclusions This pilot study demonstrated the feasibility of using TDA and NLP to analyze the complex effects of NYT on frailty in patients with OAB. These findings suggest that NYT exerts multifaceted therapeutic benefits and further large-scale studies are warranted to explore its long-term efficacy.

Persistent homology reveals robustness loss in inhaled substance abuse rs-fMRI networks

Analyzing functional brain activity through functional magnetic resonance imaging (fMRI) is commonly done using tools from graph theory for the analysis of the correlation matrices. A drawback of these methods is that the networks must be restricted to values of the weights of the edges within certain thresholds and there is no consensus about the best choice of such thresholds. Topological data analysis (TDA) is a recently-developed tool in algebraic topology which allows us to analyze networks through combinatorial spaces obtained from them, with the advantage that all the possible thresholds can be considered at once. In this paper we applied TDA, in particular persistent homology, to study correlation matrices from rs-fMRI, and through statistical analysis, we detected significant differences between the topological structures of adolescents with inhaled substance abuse disorder (ISAD) and healthy controls. We interpreted the topological differences as indicative of a loss of robustness in the functional brain networks of the ISAD population.

Dynamic topological data analysis: a novel fractal dimension-based testing framework with application to brain signals

Topological data analysis (TDA) is increasingly recognized as a promising tool in the field of neuroscience, unveiling the underlying topological patterns within brain signals. However, most TDA related methods treat brain signals as if they were static, i.e., they ignore potential non-stationarities and irregularities in the statistical properties of the signals. In this study, we develop a novel fractal dimension-based testing approach that takes into account the dynamic topological properties of brain signals. By representing EEG brain signals as a sequence of Vietoris-Rips filtrations, our approach accommodates the inherent non-stationarities and irregularities of the signals. The application of our novel fractal dimension-based testing approach in analyzing dynamic topological patterns in EEG signals during an epileptic seizure episode exposes noteworthy alterations in total persistence across 0, 1, and 2-dimensional homology. These findings imply a more intricate influence of seizures on brain signals, extending beyond mere amplitude changes.

Statistical inference for dependence networks in topological data analysis

Topological data analysis (TDA) provide tools that are becoming increasingly popular for analyzing multivariate time series data. One key aspect in analyzing multivariate time series is dependence between components. One application is on brain signal analysis. In particular, various dependence patterns in brain networks may be linked to specific tasks and cognitive processes. These dependence patterns may be altered by various neurological and cognitive impairments such as Alzheimer's and Parkinson's diseases, as well as attention deficit hyperactivity disorder (ADHD). Because there is no ground-truth with known dependence patterns in real brain signals, testing new TDA methods on multivariate time series is still a challenge. Our goal here is to develop novel statistical inference procedures via simulations. Simulations are useful for generating some null distributions of a test statistic (for hypothesis testing), forming confidence regions, and for evaluating the performance of proposed TDA methods. To the best of our knowledge, there are no methods that simulate multivariate time series data with potentially complex user-specified connectivity patterns. In this paper we present a novel approach to simulate multivariate time series with specific number of cycles/holes in its dependence network. Furthermore, we also provide a procedure for generating higher dimensional topological features.