Using Low-Dimensional Manifolds to Map Relationships Between Dynamic Brain Networks

A mechanism for the emergence of low-dimensional structures in brain dynamics

Recent neuroimaging advancements have led to datasets characterized by an overwhelming number of features. Different dimensionality reduction techniques have been employed to uncover low-dimensional manifold representations underlying cognitive functions, while maintaining the fundamental characteristics of the data. These range from linear algorithms to more intricate non-linear methods for manifold extraction. However, the mechanisms responsible for the emergence of these simplified architectures remain a topic of debate. Motivated by concepts from dynamical systems theory, such as averaging and time-scale separation, our study introduces a novel mechanism for the collapse of high dimension brain dynamics onto lower dimensional manifolds. In our framework, fast neuronal activity oscillations average out over time, leading to the resulting dynamics approximating task-related processes occurring at slower time scales. This leads to the emergence of low-dimensional solutions as complex dynamics collapse into slow invariant manifolds. We test this assumption via neural simulations using a simplified model and then enhance the complexity of our simulations by incorporating a large-scale brain network model to mimic realistic neuroimaging signals. We observe in the different cases the convergence of fast oscillatory fluctuations of neuronal activity across time scales that correspond to simulated behavioral configurations. Specifically, by employing various dimensionality reduction techniques and manifold extraction schemes, we observe the reduction of high-dimensional dynamics onto lower-dimensional spaces, revealing emergent low-dimensional solutions. Our findings shed light on the role of frequency and time-scale separation in neuronal activity, proposing and testing a novel theoretical framework for understanding the inner mechanisms governing low-dimensional pattern formation in brain dynamics.

The relationship between demographic factors and brain hierarchical changes following antidepressant treatment in patients remitted from depression

To investigate the associations between demographic factors and brain hierarchical changes following successful selective serotonin reuptake inhibitor (SSRI) treatment, 57 major depressive disorder (MDD) patients who achieved remission after a 12-week SSRI treatment and 39 healthy controls (HCs) were recruited. MDD patients underwent diffusion tensor imaging (DTI) scans before treatment and after the 12-week SSRI treatment. Depression severity was evaluated with the Hamilton Rating Scale for Depression (HAMD) using the total score and the subscales: retardation, cognitive impairment, anxiety, and sleep disturbance. All HCs also underwent DTI scans after enrollment. Building on gradient mapping techniques, we developed a set of measures to quantify the dispersion within functional communities and also studied demographic-relevant differences in the three-dimensional gradient space of remitted MDD patients. We defined the Z-scores of the gradients in the pre-treatment group relative to the HC group as the disease pattern, post-treatment group relative to the HC group as the recovery pattern. The results showed that the disease pattern of depression is associated with age, as older age groups exhibit more severe impairments in depression. A significant difference was detected in the dispersion of the frontoparietal network (FPN) between pre-treatment and post-treatment patients. With the moderating effect of the age of onset, the dispersion of the FPN was related to the improvement in cognitive impairment, the dorsal attention network (DAN) was related to the improvement in retardation symptoms. Our findings help clinicians be alert to the role of demographic effects on clinical efficacy when treating depressed patients.

An Adaptively Weighted Averaging Method for Regional Time Series Extraction of fMRI-Based Brain Decoding

Brain decoding that classifies cognitive states using the functional fluctuations of the brain can provide insightful information for understanding the brain mechanisms of cognitive functions. Among the common procedures of decoding the brain cognitive states with functional magnetic resonance imaging (fMRI), extracting the time series of each brain region after brain parcellation traditionally averages across the voxels within a brain region. This neglects the spatial information among the voxels and the requirement of extracting information for the downstream tasks. In this study, we propose to use a fully connected neural network that is jointly trained with the brain decoder to perform an adaptively weighted average across the voxels within each brain region. We perform extensive evaluations by cognitive state decoding, manifold learning, and interpretability analysis on the Human Connectome Project (HCP) dataset. The performance comparison of the cognitive state decoding presents an accuracy increase of up to 5% and stable accuracy improvement under different time window sizes, resampling sizes, and training data sizes. The results of manifold learning show that our method presents a considerable separability among cognitive states and basically excludes subject-specific information. The interpretability analysis shows that our method can identify reasonable brain regions corresponding to each cognitive state. Our study would aid the improvement of the basic pipeline of fMRI processing.

Statistical Brain Network Analysis

The recent fusion of network science and neuroscience has catalyzed a paradigm shift in how we study the brain and led to the field of brain network analysis. Brain network analyses hold great potential in helping us understand normal and abnormal brain function by providing profound clinical insight into links between system-level properties and health and behavioral outcomes. Nonetheless, methods for statistically analyzing networks at the group and individual levels have lagged behind. We have attempted to address this need by developing three complementary statistical frameworks-a mixed modeling framework, a distance regression framework, and a hidden semi-Markov modeling framework. These tools serve as synergistic fusions of statistical approaches with network science methods, providing needed analytic foundations for whole-brain network data. Here we delineate these approaches, briefly survey related tools, and discuss potential future avenues of research. We hope this review catalyzes further statistical interest and methodological development in the field.

Manifold learning for fMRI time-varying functional connectivity

Whole-brain functional connectivity (FC) measured with functional MRI (fMRI) evolves over time in meaningful ways at temporal scales going from years (e.g., development) to seconds [e.g., within-scan time-varying FC (tvFC)]. Yet, our ability to explore tvFC is severely constrained by its large dimensionality (several thousands). To overcome this difficulty, researchers often seek to generate low dimensional representations (e.g., 2D and 3D scatter plots) hoping those will retain important aspects of the data (e.g., relationships to behavior and disease progression). Limited prior empirical work suggests that manifold learning techniques (MLTs)-namely those seeking to infer a low dimensional non-linear surface (i.e., the manifold) where most of the data lies-are good candidates for accomplishing this task. Here we explore this possibility in detail. First, we discuss why one should expect tvFC data to lie on a low dimensional manifold. Second, we estimate what is the intrinsic dimension (ID; i.e., minimum number of latent dimensions) of tvFC data manifolds. Third, we describe the inner workings of three state-of-the-art MLTs: Laplacian Eigenmaps (LEs), T-distributed Stochastic Neighbor Embedding (T-SNE), and Uniform Manifold Approximation and Projection (UMAP). For each method, we empirically evaluate its ability to generate neuro-biologically meaningful representations of tvFC data, as well as their robustness against hyper-parameter selection. Our results show that tvFC data has an ID that ranges between 4 and 26, and that ID varies significantly between rest and task states. We also show how all three methods can effectively capture subject identity and task being performed: UMAP and T-SNE can capture these two levels of detail concurrently, but LE could only capture one at a time. We observed substantial variability in embedding quality across MLTs, and within-MLT as a function of hyper-parameter selection. To help alleviate this issue, we provide heuristics that can inform future studies. Finally, we also demonstrate the importance of feature normalization when combining data across subjects and the role that temporal autocorrelation plays in the application of MLTs to tvFC data. Overall, we conclude that while MLTs can be useful to generate summary views of labeled tvFC data, their application to unlabeled data such as resting-state remains challenging.

Manifold Learning for fMRI time-varying FC

Whole-brain functional connectivity ( FC ) measured with functional MRI (fMRI) evolve over time in meaningful ways at temporal scales going from years (e.g., development) to seconds (e.g., within-scan time-varying FC ( tvFC )). Yet, our ability to explore tvFC is severely constrained by its large dimensionality (several thousands). To overcome this difficulty, researchers seek to generate low dimensional representations (e.g., 2D and 3D scatter plots) expected to retain its most informative aspects (e.g., relationships to behavior, disease progression). Limited prior empirical work suggests that manifold learning techniques ( MLTs )-namely those seeking to infer a low dimensional non-linear surface (i.e., the manifold) where most of the data lies-are good candidates for accomplishing this task. Here we explore this possibility in detail. First, we discuss why one should expect tv FC data to lie on a low dimensional manifold. Second, we estimate what is the intrinsic dimension (i.e., minimum number of latent dimensions; ID ) of tvFC data manifolds. Third, we describe the inner workings of three state-of-the-art MLTs : Laplacian Eigenmaps ( LE ), T-distributed Stochastic Neighbor Embedding ( T-SNE ), and Uniform Manifold Approximation and Projection ( UMAP ). For each method, we empirically evaluate its ability to generate neuro-biologically meaningful representations of tvFC data, as well as their robustness against hyper-parameter selection. Our results show that tvFC data has an ID that ranges between 4 and 26, and that ID varies significantly between rest and task states. We also show how all three methods can effectively capture subject identity and task being performed: UMAP and T-SNE can capture these two levels of detail concurrently, but L E could only capture one at a time. We observed substantial variability in embedding quality across MLTs , and within- MLT as a function of hyper-parameter selection. To help alleviate this issue, we provide heuristics that can inform future studies. Finally, we also demonstrate the importance of feature normalization when combining data across subjects and the role that temporal autocorrelation plays in the application of MLTs to tvFC data. Overall, we conclude that while MLTs can be useful to generate summary views of labeled tvFC data, their application to unlabeled data such as resting-state remains challenging.

Uncovering shape signatures of resting-state functional connectivity by geometric deep learning on Riemannian manifold

Functional neural activities manifest geometric patterns, as evidenced by the evolving network topology of functional connectivities (FC) even in the resting state. In this work, we propose a novel manifold‐based geometric neural network for functional brain networks (called “Geo‐Net4Net” for short) to learn the intrinsic low‐dimensional feature representations of resting‐state brain networks on the Riemannian manifold. This tool allows us to answer the scientific question of how the spontaneous fluctuation of FC supports behavior and cognition. We deploy a set of positive maps and rectified linear unit (ReLU) layers to uncover the intrinsic low‐dimensional feature representations of functional brain networks on the Riemannian manifold taking advantage of the symmetric positive‐definite (SPD) form of the correlation matrices. Due to the lack of well‐defined ground truth in the resting state, existing learning‐based methods are limited to unsupervised methodologies. To go beyond this boundary, we propose to self‐supervise the feature representation learning of resting‐state functional networks by leveraging the task‐based counterparts occurring before and after the underlying resting state. With this extra heuristic, our Geo‐Net4Net allows us to establish a more reasonable understanding of resting‐state FCs by capturing the geometric patterns (aka. spectral/shape signature) associated with resting states on the Riemannian manifold. We have conducted extensive experiments on both simulated data and task‐based functional resonance magnetic imaging (fMRI) data from the Human Connectome Project (HCP) database, where our Geo‐Net4Net not only achieves more accurate change detection results than other state‐of‐the‐art counterpart methods but also yields ubiquitous geometric patterns that manifest putative insights into brain function.

Mixed Modeling Frameworks for Analyzing Whole-Brain Network Data

Brain network analyses have exploded in recent years and hold great potential in helping us understand normal and abnormal brain function. Network science approaches have facilitated these analyses and our understanding of how the brain is structurally and functionally organized. However, the development of statistical methods that allow relating this organization to health outcomes has lagged behind. We have attempted to address this need by developing mixed modeling frameworks that allow relating system-level properties of brain networks to outcomes of interest. These frameworks serve as a synergistic fusion of multivariate statistical approaches with network science, providing a needed analytic (modeling and inferential) foundation for whole-brain network data. In this chapter we delineate these approaches that have been developed for single-task and multitask (longitudinal) brain network data, illustrate their utility with data applications, detail their implementation with a user-friendly Matlab toolbox, and discuss ongoing work to adapt the methods to (within-task) dynamic network analysis.

Embedding Functional Brain Networks in Low Dimensional Spaces Using Manifold Learning Techniques

Background: fMRI data is inherently high-dimensional and difficult to visualize. A recent trend has been to find spaces of lower dimensionality where functional brain networks can be projected onto manifolds as individual data points, leading to new ways to analyze and interpret the data. Here, we investigate the potential of two powerful non-linear manifold learning techniques for functional brain networks representation: (1) T-stochastic neighbor embedding (t-SNE) and (2) Uniform Manifold Approximation Projection (UMAP) a recent breakthrough in manifold learning. Methods: fMRI data from the Human Connectome Project (HCP) and an independent study of aging were used to generate functional brain networks. We used fMRI data collected during resting state data and during a working memory task. The relative performance of t-SNE and UMAP were investigated by projecting the networks from each study onto 2D manifolds. The levels of discrimination between different tasks and the preservation of the topology were evaluated using different metrics. Results: Both methods effectively discriminated the resting state from the memory task in the embedding space. UMAP discriminated with a higher classification accuracy. However, t-SNE appeared to better preserve the topology of the high-dimensional space. When networks from the HCP and aging studies were combined, the resting state and memory networks in general aligned correctly. Discussion: Our results suggest that UMAP, a more recent development in manifold learning, is an excellent tool to visualize functional brain networks. Despite dramatic differences in data collection and protocols, networks from different studies aligned correctly in the embedding space.

Post-Stroke Cognitive Impairment: Pathophysiological Insights into Brain Disconnectome from Advanced Neuroimaging Analysis Techniques

The neurological symptoms of stroke have traditionally provided the foundation for functional mapping of the brain. However, there are many unresolved aspects in our understanding of cerebral activity, especially regarding high-level cognitive functions. This review provides a comprehensive look at the pathophysiology of post-stroke cognitive impairment in light of recent findings from advanced imaging techniques. Combining network neuroscience and clinical neurology, our research focuses on how changes in brain networks correlate with post-stroke cognitive prognosis. More specifically, we first discuss the general consequences of stroke lesions due to damage of canonical resting-state large-scale networks or changes in the composition of the entire brain. We also review emerging methods, such as lesion-network mapping and gradient analysis, used to study the aforementioned events caused by stroke lesions. Lastly, we examine other patient vulnerabilities, such as superimposed amyloid pathology and blood-brain barrier leakage, which potentially lead to different outcomes for the brain network compositions even in the presence of similar stroke lesions. This knowledge will allow a better understanding of the pathophysiology of post-stroke cognitive impairment and provide a theoretical basis for the development of new treatments, such as neuromodulation.

A covariate-constraint method to map brain feature space into lower dimensional manifolds

Human brain connectome studies aim to both explore healthy brains, and extract and analyze relevant features associated with pathologies of interest. Usually this consists of modeling the brain connectome as a graph and using graph metrics as features. A fine brain description requires graph metrics computation at the node level. Given the relatively reduced number of patients in standard cohorts, such data analysis problems fall in the high-dimension, low-sample-size framework. In this context, our goal is to provide a machine learning technique that exhibits flexibility, gives the investigator an understanding of the features and covariates, allows visualization and exploration, and yields insight into the data and the biological phenomena at stake. The retained approach is dimension reduction in a manifold learning methodology; the originality is that the investigator chooses one (or several) reduced variables. The proposed method is illustrated in two studies. The first one addresses comatose patients; the second one compares young and elderly populations. The method sheds light on the differences between brain connectivity graphs using graph metrics and potential clinical interpretations of these differences.

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.

Dispersion of functional gradients across the adult lifespan

Ageing is commonly associated with changes to segregation and integration of functional brain networks, but, in isolation, current network-based approaches struggle to elucidate changes across the many axes of functional organisation. However, the advent of gradient mapping techniques in neuroimaging provides a new means of studying functional organisation in a multi-dimensional connectivity space. Here, we studied ageing and behaviourally-relevant differences in a three-dimensional connectivity space using the Cambridge Centre for Ageing Neuroscience cohort (n = 643). Building on gradient mapping techniques, we developed a set of measures to quantify the dispersion within and between functional communities. We detected a strong shift of the visual network across the adult lifespan from an extreme to a more central position in the 3D gradient space. In contrast, the dispersion distance between transmodal communities (dorsal attention, ventral attention, frontoparietal and default mode) did not change. However, these communities themselves were increasingly dispersed with increasing age, reflecting more dissimilar functional connectivity profiles within each community. Increasing dispersion of frontoparietal, attention and default mode networks, in particular, were associated negatively with cognition, measured by fluid intelligence. By using a technique that explicitly captures the ordering of functional systems in a multi-dimensional hierarchical framework, we identified behaviorally-relevant age-related differences of within and between network organisation. We propose that the study of functional gradients across the adult lifespan could provide insights that may facilitate the development of new strategies to maintain cognitive ability across the lifespan in health and disease.