Re-visiting Riemannian geometry of symmetric positive definite matrices for the analysis of functional connectivity

ℛSCZ: A Riemannian schizophrenia diagnosis framework based on the multiplexity of EEG-based dynamic functional connectivity patterns

Abnormal electrophysiological (EEG) activity has been largely reported in schizophrenia (SCZ). In the last decade, research has focused to the automatic diagnosis of SCZ via the investigation of an EEG aberrant activity and connectivity linked to this mental disorder. These studies followed various preprocessing steps of EEG activity focusing on frequency-dependent functional connectivity brain network (FCBN) construction disregarding the topological dependency among edges. FCBN belongs to a family of symmetric positive definite (SPD) matrices forming the Riemannian manifold. Due to its unique geometric properties, the whole analysis of FCBN can be performed on the Riemannian geometry of the SPD space. The advantage of the analysis of FCBN on the SPD space is that it takes into account all the pairwise interdependencies as a whole. However, only a few studies have adopted a FCBN analysis on the SPD manifold, while no study exists on the analysis of dynamic FCBN (dFCBN) tailored to SCZ. In the present study, I analyzed two open EEG-SCZ datasets under a Riemannian geometry of SPD matrices for the dFCBN analysis proposing also a multiplexity index that quantifies the associations of multi-frequency brainwave patterns. I adopted a machine learning procedure employing a leave-one-subject-out cross-validation (LOSO-CV) using snapshots of dFCBN from (N-1) subjects to train a battery of classifiers. Each classifier operated in the inter-subject dFCBN distances of sample covariance matrices (SCMs) following a rhythm-dependent decision and a multiplex-dependent one. The proposed ℛSCZ decoder supported both the Riemannian geometry of SPD and the multiplexity index DC reaching an absolute accuracy (100 %) in both datasets in the virtual default mode network (DMN) source space.

Manifold Learning-Based Common Spatial Pattern for EEG Signal Classification

EEG signal classification using Riemannian manifolds has shown great potential. However, the huge computational cost associated with Riemannian metrics poses challenges for applying Riemannian methods, particularly in high-dimensional feature data. To address these, we propose an efficient ensemble method called MLCSP-TSE-MLP, which aims to reduce the computational cost while achieving superior performance. MLCSP of the ensemble utilizes a Riemannian graph embedding strategy to learn intrinsic low-dimensional sub-manifolds, enhancing discrimination. TSE uses the Euclidean mean as the reference point for tangent space mapping and reducing computational cost. Finally, the ensemble incorporates the MLP classifier to offer improved classification performance. Classification results conducted on three datasets demonstrate that MLCSP-TSE-MLP achieves significant superior performance compared to various competing methods. Notably, the MLCSP-TSE module achieves a remarkable increase in training speed and exhibits much lower test time compared to traditional Riemannian methods. Based on these results, we believe that the proposed MLCSP-TSE-MLP is a powerful tool for handling high-dimensional data and holds great potential for practical applications.

Riemannian frameworks for the harmonization of resting-state functional MRI scans

Magnetic Resonance Imaging provides unprecedented images of the brain. Unfortunately, scanners and acquisition protocols can significantly impact MRI scans. The development of statistical methods able to reduce this variability without altering the relevant information in the scans, often coined harmonization methods, has been the topic of an increasing research effort supported by the recent growth of publicly available neuroimaging data sets and new possibilities for combining them to achieve greater statistical power. In this work, we focus on the challenges specifically raised by the harmonization of resting-state functional MRI scans. We propose to harmonize resting-state fMRI scans by reducing the impact of covariates such as scanner differences and scanning protocols on their associated functional connectomes and then propagating the changes back to the rs-fMRI time series. We use Riemannian geometric frameworks to preserve the mathematical properties of functional connectomes during their harmonization, and we demonstrate how state-of-the-art harmonization methods can be embedded within these frameworks to reduce covariates effects while preserving the relevant clinical information associated with aging or brain disorders. During our experiments, a large set of synthetic data was generated and processed to compare eighty variants of the proposed approach. The framework achieving the best harmonization was then applied to three low-dimensional data sets made of 712 sets of fMRI time series provided by the ABIDE consortium and two high-dimensional data sets obtained by processing 1527 rs-fMRI scans provided by the Human Connectome Project, the Framingham Heart Study and the Genetics of Brain Structure and Function study. These experiments established that our new framework could successfully harmonize low-dimensional connectomes and voxelwise functional time series and confirmed the need for preserving connectomes properties during their harmonization.

An Explainable Geometric-Weighted Graph Attention Network for Identifying Functional Networks Associated with Gait Impairment

One of the hallmark symptoms of Parkinson's Disease (PD) is the progressive loss of postural reflexes, which eventually leads to gait difficulties and balance problems. Identifying disruptions in brain function associated with gait impairment could be crucial in better understanding PD motor progression, thus advancing the development of more effective and personalized therapeutics. In this work, we present an explainable, geometric, weighted-graph attention neural network (xGW-GAT) to identify functional networks predictive of the progression of gait difficulties in individuals with PD. xGW-GAT predicts the multi-class gait impairment on the MDS-Unified PD Rating Scale (MDS-UPDRS). Our computational- and data-efficient model represents functional connectomes as symmetric positive definite (SPD) matrices on a Riemannian manifold to explicitly encode pairwise interactions of entire connectomes, based on which we learn an attention mask yielding individual- and group-level explainability. Applied to our resting-state functional MRI (rs-fMRI) dataset of individuals with PD, xGW-GAT identifies functional connectivity patterns associated with gait impairment in PD and offers interpretable explanations of functional subnetworks associated with motor impairment. Our model successfully outperforms several existing methods while simultaneously revealing clinically-relevant connectivity patterns. The source code is available at https://github.com/favour-nerrise/xGW-GAT.

A joint subspace mapping between structural and functional brain connectomes

Understanding the connection between the brain's structural connectivity and its functional connectivity is of immense interest in computational neuroscience. Although some studies have suggested that whole brain functional connectivity is shaped by the underlying structure, the rule by which anatomy constraints brain dynamics remains an open question. In this work, we introduce a computational framework that identifies a joint subspace of eigenmodes for both functional and structural connectomes. We found that a small number of those eigenmodes are sufficient to reconstruct functional connectivity from the structural connectome, thus serving as low-dimensional basis function set. We then develop an algorithm that can estimate the functional eigen spectrum in this joint space from the structural eigen spectrum. By concurrently estimating the joint eigenmodes and the functional eigen spectrum, we can reconstruct a given subject's functional connectivity from their structural connectome. We perform elaborate experiments and demonstrate that the proposed algorithm for estimating functional connectivity from the structural connectome using joint space eigenmodes gives competitive performance as compared to the existing benchmark methods with better interpretability.

Functional connectivity learning via Siamese-based SPD matrix representation of brain imaging data

Measurement of brain functional connectivity has become a dominant approach to explore the interaction dynamics between brain regions of subjects under examination. Conventional functional connectivity measures largely originate from deterministic models on empirical analysis, usually demanding application-specific settings (e.g., Pearson's Correlation and Mutual Information). To bridge the technical gap, this study proposes a Siamese-based Symmetric Positive Definite (SPD) Matrix Representation framework (SiameseSPD-MR) to derive the functional connectivity of brain imaging data (BID) such as Electroencephalography (EEG), thus the alternative application-independent measure (in the form of SPD matrix) can be automatically learnt: (1) SiameseSPD-MR first exploits graph convolution to extract the representative features of BID with the adjacency matrix computed considering the anatomical structure; (2) Adaptive Gaussian kernel function then applies to obtain the functional connectivity representations from the deep features followed by SPD matrix transformation to address the intrinsic functional characteristics; and (3) Two-branch (Siamese) networks are combined via an element-wise product followed by a dense layer to derive the similarity between the pairwise inputs. Experimental results on two EEG datasets (autism spectrum disorder, emotion) indicate that (1) SiameseSPD-MR can capture more significant differences in functional connectivity between neural states than the state-of-the-art counterparts do, and these findings properly highlight the typical EEG characteristics of ASD subjects, and (2) the obtained functional connectivity representations conforming to the proposed measure can act as meaningful markers for brain network analysis and ASD discrimination.

Multiscale functional connectivity patterns of the aging brain learned from harmonized rsfMRI data of the multi-cohort iSTAGING study

To learn multiscale functional connectivity patterns of the aging brain, we built a brain age prediction model of functional connectivity measures at seven scales on a large fMRI dataset, consisting of resting-state fMRI scans of 4186 individuals with a wide age range (22 to 97 years, with an average of 63) from five cohorts. We computed multiscale functional connectivity measures of individual subjects using a personalized functional network computational method, harmonized the functional connectivity measures of subjects from multiple datasets in order to build a functional brain age model, and finally evaluated how functional brain age gap correlated with cognitive measures of individual subjects. Our study has revealed that functional connectivity measures at multiple scales were more informative than those at any single scale for the brain age prediction, the data harmonization significantly improved the brain age prediction performance, and the data harmonization in the functional connectivity measures' tangent space worked better than in their original space. Moreover, brain age gap scores of individual subjects derived from the brain age prediction model were significantly correlated with clinical and cognitive measures. Overall, these results demonstrated that multiscale functional connectivity patterns learned from a large-scale multi-site rsfMRI dataset were informative for characterizing the aging brain and the derived brain age gap was associated with cognitive and clinical measures.

Revisiting Riemannian geometry-based EEG decoding through approximate joint diagonalization

Objective.The wider adoption of Riemannian geometry in electroencephalography (EEG) processing is hindered by two factors: (a) it involves the manipulation of complex mathematical formulations and, (b) it leads to computationally demanding tasks. The main scope of this work is to simplify particular notions of Riemannian geometry and provide an efficient and comprehensible scheme for neuroscientific explorations.Approach.To overcome the aforementioned shortcomings, we exploit the concept of approximate joint diagonalization in order to reconstruct the spatial covariance matrices assuming the existence of (and identifying) a common eigenspace in which the application of Riemannian geometry is significantly simplified.Main results.The employed reconstruction process abides to physiologically plausible assumptions, reduces the computational complexity in Riemannian geometry schemes and bridges the gap between rigorous mathematical procedures and computational neuroscience. Our approach is both formally established and experimentally validated by employing real and synthetic EEG data.Significance.The implications of the introduced reconstruction process are highlighted by reformulating and re-introducing two signal processing methodologies, namely the 'Symmetric Positive Definite (SPD) Matrix Quantization' and the 'Coding over SPD Atoms'. The presented approach paves the way for robust and efficient neuroscientific explorations that exploit Riemannian geometry schemes.

Geometric learning of functional brain network on the correlation manifold

The correlation matrix is a typical representation of node interactions in functional brain network analysis. The analysis of the correlation matrix to characterize brain networks observed in several neuroimaging modalities has been conducted predominantly in the Euclidean space by assuming that pairwise interactions are mutually independent. One way to take account of all interactions in the network as a whole is to analyze the correlation matrix under some geometric structure. Recent studies have focused on the space of correlation matrices as a strict subset of symmetric positive definite (SPD) matrices, which form a unique mathematical structure known as the Riemannian manifold. However, mathematical operations of the correlation matrix under the SPD geometry may not necessarily be coherent (i.e., the structure of the correlation matrix may not be preserved), necessitating a post-hoc normalization. The contribution of the current paper is twofold: (1) to devise a set of inferential methods on the correlation manifold and (2) to demonstrate its applicability in functional network analysis. We present several algorithms on the correlation manifold, including measures of central tendency, cluster analysis, hypothesis testing, and low-dimensional embedding. Simulation and real data analysis support the application of the proposed framework for brain network analysis.

Harmonizing functional connectivity reduces scanner effects in community detection

Community detection on graphs constructed from functional magnetic resonance imaging (fMRI) data has led to important insights into brain functional organization. Large studies of brain community structure often include images acquired on multiple scanners across different studies. Differences in scanner can introduce variability into the downstream results, and these differences are often referred to as scanner effects. Such effects have been previously shown to significantly impact common network metrics. In this study, we identify scanner effects in data-driven community detection results and related network metrics. We assess a commonly employed harmonization method and propose new methodology for harmonizing functional connectivity that leverage existing knowledge about network structure as well as patterns of covariance in the data. Finally, we demonstrate that our new methods reduce scanner effects in community structure and network metrics. Our results highlight scanner effects in studies of brain functional organization and provide additional tools to address these unwanted effects. These findings and methods can be incorporated into future functional connectivity studies, potentially preventing spurious findings and improving reliability of results.

Predicting cognitive scores with graph neural networks through sample selection learning

Analyzing the relation between intelligence and neural activity is of the utmost importance in understanding the working principles of the human brain in health and disease. In existing literature, functional brain connectomes have been used successfully to predict cognitive measures such as intelligence quotient (IQ) scores in both healthy and disordered cohorts using machine learning models. However, existing methods resort to flattening the brain connectome (i.e., graph) through vectorization which overlooks its topological properties. To address this limitation and inspired from the emerging graph neural networks (GNNs), we design a novel regression GNN model (namely RegGNN) for predicting IQ scores from brain connectivity. On top of that, we introduce a novel, fully modular sample selection method to select the best samples to learn from for our target prediction task. However, since such deep learning architectures are computationally expensive to train, we further propose a learning-based sample selection method that learns how to choose the training samples with the highest expected predictive power on unseen samples. For this, we capitalize on the fact that connectomes (i.e., their adjacency matrices) lie in the symmetric positive definite (SPD) matrix cone. Our results on full-scale and verbal IQ prediction outperforms comparison methods in autism spectrum disorder cohorts and achieves a competitive performance for neurotypical subjects using 3-fold cross-validation. Furthermore, we show that our sample selection approach generalizes to other learning-based methods, which shows its usefulness beyond our GNN architecture.

A Riemannian Revisiting of Structure-Function Mapping Based on Eigenmodes

Understanding the link between brain structure and function may not only improve our knowledge of brain organization, but also lead to better quantification of pathology. To quantify this link, recent studies have attempted to predict the brain's functional connectivity from its structural connectivity. However, functional connectivity matrices live in the Riemannian manifold of the symmetric positive definite space and a specific attention must be paid to operate on this appropriate space. In this work we investigated the implications of using a distance based on an affine invariant Riemannian metric in the context of structure-function mapping. Specifically, we revisit previously proposed structure-function mappings based on eigendecomposition and test them on 100 healthy subjects from the Human Connectome Project using this adapted notion of distance. First, we show that using this Riemannian distance significantly alters the notion of similarity between subjects from a functional point of view. We also show that using this distance improves the correlation between the structural and functional similarity of different subjects. Finally, by using a distance appropriate to this manifold, we demonstrate the importance of mapping function from structure under the Riemannian manifold and show in particular that it is possible to outperform the group average and the so-called glass ceiling on the performance of mappings based on eigenmodes.

Riemannian Geometry of Functional Connectivity Matrices for Multi-Site Attention-Deficit/Hyperactivity Disorder Data Harmonization

The use of multi-site datasets in neuroimaging provides neuroscientists with more statistical power to perform their analyses. However, it has been shown that the imaging-site introduces variability in the data that cannot be attributed to biological sources. In this work, we show that functional connectivity matrices derived from resting-state multi-site data contain a significant imaging-site bias. To this aim, we exploited the fact that functional connectivity matrices belong to the manifold of symmetric positive-definite (SPD) matrices, making it possible to operate on them with Riemannian geometry. We hereby propose a geometry-aware harmonization approach, Rigid Log-Euclidean Translation, that accounts for this site bias. Moreover, we adapted other Riemannian-geometric methods designed for other domain adaptation tasks and compared them to our proposal. Based on our results, Rigid Log-Euclidean Translation of multi-site functional connectivity matrices seems to be among the studied methods the most suitable in a clinical setting. This represents an advance with respect to previous functional connectivity data harmonization approaches, which do not respect the geometric constraints imposed by the underlying structure of the manifold. In particular, when applying our proposed method to data from the ADHD-200 dataset, a multi-site dataset built for the study of attention-deficit/hyperactivity disorder, we obtained results that display a remarkable correlation with established pathophysiological findings and, therefore, represent a substantial improvement when compared to the non-harmonization analysis. Thus, we present evidence supporting that harmonization should be extended to other functional neuroimaging datasets and provide a simple geometric method to address it.

Neural excursions from manifold structure explain patterns of learning during human sensorimotor adaptation

Humans vary greatly in their motor learning abilities, yet little is known about the neural mechanisms that underlie this variability. Recent neuroimaging and electrophysiological studies demonstrate that large-scale neural dynamics inhabit a low-dimensional subspace or manifold, and that learning is constrained by this intrinsic manifold architecture. Here, we asked, using functional MRI, whether subject-level differences in neural excursion from manifold structure can explain differences in learning across participants. We had subjects perform a sensorimotor adaptation task in the MRI scanner on 2 consecutive days, allowing us to assess their learning performance across days, as well as continuously measure brain activity. We find that the overall neural excursion from manifold activity in both cognitive and sensorimotor brain networks is associated with differences in subjects' patterns of learning and relearning across days. These findings suggest that off-manifold activity provides an index of the relative engagement of different neural systems during learning, and that subject differences in patterns of learning and relearning are related to reconfiguration processes occurring in cognitive and sensorimotor networks.

Harmonization of multi-site functional connectivity measures in tangent space improves brain age prediction

Brain age prediction based on functional magnetic resonance imaging (fMRI) data has the potential to serve as a biomarker for quantifying brain health. To predict the brain age based on fMRI data robustly and accurately, we curated a large dataset (n = 4259) of fMRI scans from seven different data acquisition sites and computed personalized functional connectivity measures at multiple scales from each subject's fMRI scan. Particularly, we computed personalized large-scale functional networks and generated functional connectivity measures at multiple scales to characterize each fMRI scan. To account for inter-site effects on the functional connectivity measures, we harmonized the functional connectivity measures in their tangent space and then built brain age prediction models on the harmonized functional connectivity measures. We compared the brain age prediction models with alternatives that were built on the functional connectivity measures computed at a single scale and harmonized using different strategies. Comparison results have demonstrated that the best brain age prediction performance was achieved by the prediction model built on the multi-scale functional connectivity measures that were harmonized in tangent space, indicating that multi-scale functional connectivity measures provided richer information than those computed at any single scales and the harmonization of functional connectivity measures in tangent space improved the brain age prediction.

Using distance on the Riemannian manifold to compare representations in brain and in models

Representational similarity analysis (RSA) summarizes activity patterns for a set of experimental conditions into a matrix composed of pairwise comparisons between activity patterns. Two examples of such matrices are the condition-by-condition inner product and correlation matrix. These representational matrices reside on the manifold of positive semidefinite matrices, called the Riemannian manifold. We hypothesize that representational similarities would be more accurately quantified by considering the underlying manifold of the representational matrices. Thus, we introduce the distance on the Riemannian manifold as a metric for comparing representations. Analyzing simulated and real fMRI data and considering a wide range of metrics, we show that the Riemannian distance is least susceptible to sampling bias, results in larger intra-subject reliability, and affords searchlight mapping with high sensitivity and specificity. Furthermore, we show that the Riemannian distance can be used for measuring multi-dimensional connectivity. This measure captures both univariate and multivariate connectivity and is also more sensitive to nonlinear regional interactions compared to the state-of-the-art measures. Applying our proposed metric to neural network representations of natural images, we demonstrate that it also possesses outstanding performance in quantifying similarity in models. Taken together, our results lend credence to the proposition that RSA should consider the manifold of the representational matrices to summarize response patterns in the brain and in models.

Geodesic Distance on Optimally Regularized Functional Connectomes Uncovers Individual Fingerprints

Background: Functional connectomes (FCs) have been shown to provide a reproducible individual fingerprint, which has opened the possibility of personalized medicine for neuro/psychiatric disorders. Thus, developing accurate ways to compare FCs is essential to establish associations with behavior and/or cognition at the individual level. Methods: Canonically, FCs are compared using Pearson's correlation coefficient of the entire functional connectivity profiles. Recently, it has been proposed that the use of geodesic distance is a more accurate way of comparing FCs, one which reflects the underlying non-Euclidean geometry of the data. Computing geodesic distance requires FCs to be positive-definite and hence invertible matrices. As this requirement depends on the functional magnetic resonance imaging scanning length and the parcellation used, it is not always attainable and sometimes a regularization procedure is required. Results: In the present work, we show that regularization is not only an algebraic operation for making FCs invertible, but also that an optimal magnitude of regularization leads to systematically higher fingerprints. We also show evidence that optimal regularization is data set-dependent and varies as a function of condition, parcellation, scanning length, and the number of frames used to compute the FCs. Discussion: We demonstrate that a universally fixed regularization does not fully uncover the potential of geodesic distance on individual fingerprinting and indeed could severely diminish it. Thus, an optimal regularization must be estimated on each data set to uncover the most differentiable across-subject and reproducible within-subject geodesic distances between FCs. The resulting pairwise geodesic distances at the optimal regularization level constitute a very reliable quantification of differences between subjects.