Human brain is a complex yet economically organized system, where a small portion of critical hub regions support the majority of brain functions. The identification of common hub nodes in a population of networks is often simplified as a voting procedure on the set of identified hub nodes across individual brain networks, which ignores the intrinsic data geometry and partially lacks the reproducible findings in neuroscience. Hence, we propose a first-ever group-wise hub identification method to identify hub nodes that are common across a population of individual brain networks. Specifically, the backbone of our method is to learn common graph embedding that can represent the majority of local topological profiles. By requiring orthogonality among the graph embedding vectors, each graph embedding as a data element is residing on the Grassmannian manifold. We present a novel Grassmannian manifold optimization scheme that allows us to find the common graph embeddings, which not only identify the most reliable hub nodes in each network but also yield a population-based common hub node map. Results of the accuracy and replicability on both synthetic and real network data show that the proposed manifold learning approach outperforms all hub identification methods employed in this evaluation.

Functional connectivities (FC) of brain network manifest remarkable geometric patterns, which is the gateway to understanding brain dynamics. In this work, we present a novel geometric-attention neural network to characterize the time-evolving brain state change from the functional neuroimages by tracking the trajectory of functional dynamics on high-dimension Riemannian manifold of symmetric positive definite (SPD) matrices. Specifically, we put the spotlight on learning the common state-specific manifold signatures that represent the underlying cognition. In this context, the driving force of our neural network is tied up with the learning of the evolution functionals on the Riemannian manifold of SPD matrix that underlies the known evolving brain states. To do so, we train a convolution neural network (CNN) on the Riemannian manifold of SPD matrices to seek for the putative low-dimension feature representations, followed by an end-to-end recurrent neural network (RNN) to yield the time-varying mapping function of SPD matrices which fits the evolutionary trajectories of the underlying states. Furthermore, we devise a geometric attention mechanism in CNN, allowing us to discover the latent geometric patterns in SPD matrices that are associated with the underlying states. Notably, our work has the potential to understand how brain function emerges behavior by investigating the geometrical patterns from functional brain networks, which is essentially a correlation matrix of neuronal activity signals. Our proposed manifold-based neural network achieves promising results in predicting brain state changes on both simulated data and task functional neuroimaging data from Human Connectome Project, which implies great applicability in neuroscience studies.

Converging evidence shows that disease-relevant brain alterations do not appear in random brain locations, instead, their spatial patterns follow large-scale brain networks. In this context, a powerful network analysis approach with a mathematical foundation is indispensable to understand the mechanisms of neuropathological events as they spread through the brain. Indeed, the topology of each brain network is governed by its native harmonic waves, which are a set of orthogonal bases derived from the Eigen-system of the underlying Laplacian matrix. To that end, we propose a novel connectome harmonic analysis framework that provides enhanced mathematical insights by detecting frequency-based alterations relevant to brain disorders. The backbone of our framework is a novel manifold algebra appropriate for inference across harmonic waves. This algebra overcomes the limitations of using classic Euclidean operations on irregular data structures. The individual harmonic differences are measured by a set of common harmonic waves learned from a population of individual Eigen-systems, where each native Eigen-system is regarded as a sample drawn from the Stiefel manifold. Specifically, a manifold optimization scheme is tailored to find the common harmonic waves, which reside at the center of the Stiefel manifold. To that end, the common harmonic waves constitute a new set of neurobiological bases to understand disease progression. Each harmonic wave exhibits a unique propagation pattern of neuropathological burden spreading across brain networks. The statistical power of our novel connectome harmonic analysis approach is evaluated by identifying frequency-based alterations relevant to Alzheimer's disease, where our learning-based manifold approach discovers more significant and reproducible network dysfunction patterns than Euclidean methods.

In recent years, the field of network science has enabled researchers to represent the highly complex interactions in the brain in an approachable yet quantitative manner. One exciting finding since the advent of brain network research was that the brain network can withstand extensive damage, even to highly connected regions. However, these highly connected nodes may not be the most critical regions of the brain network, and it is unclear how the network dynamics are impacted by removal of these key nodes. This work seeks to further investigate the resilience of the human functional brain network. Network attack experiments were conducted on voxel-wise functional brain networks and region-of-interest (ROI) networks of 5 healthy volunteers. Networks were attacked at key nodes using several criteria for assessing node importance, and the impact on network structure and dynamics was evaluated. The findings presented here echo previous findings that the functional human brain network is highly resilient to targeted attacks, both in terms of network structure and dynamics.

This paper presents registration via embedded maps (REM), a deformable registration algorithm for images with varying topology. The algorithm represents 3-D images as 4-D manifolds in a Riemannian space (referred to as embedded maps). Registration is performed as a surface evolution matching one embedded map to another using a diffusion process. The approach differs from those existing in that it takes an a priori estimation of image regions where topological changes are present, for example lesions, and generates a dense vector field representing both the shape and intensity changes necessary to match the images. The algorithm outputs both a diffeomorphic deformation field and an intensity displacement which corrects the intensity difference caused by topological changes. Multiple sets of experiments are conducted on magnetic resonance imaging (MRI) with lesions from OASIS and ADNI datasets. These images are registered to either a brain template or images of healthy individuals. An exemplar case registering a template to an MRI with tumor is also given. The resulting deformation fields were compared with those obtained using diffeomorphic demons, where topological changes are not modeled. These sets of experiments demonstrate the efficacy of our proposed REM method for registration of brain MRI with severe topological differences.