The Nash embedding theorem demonstrates that any compact manifold can be isometrically embedded in a Euclidean space. Assuming the complex brain states form a high-dimensional manifold in a topological space, we propose a manifold learning framework, termed Thought Chart, to reconstruct and visualize the manifold in a low-dimensional space. Furthermore, it serves as a data-driven approach to discover the underlying dynamics when the brain is engaged in a series of emotion and cognitive regulation tasks. EEG-based temporal dynamic functional connectomes are created based on 20 psychiatrically healthy participants EEG recordings during resting state and an emotion regulation task. Graph dissimilarity space embedding was applied to all the dynamic EEG connectomes. In order to visualize the learned manifold in a lower dimensional space, local neighborhood information is reconstructed via k-nearest neighbor-based nonlinear dimensionality reduction (NDR) and epsilon distance-based NDR. We showed that two neighborhood constructing approaches of NDR embed the manifold in a two-dimensional space, which we named Thought Chart. In Thought Chart, different task conditions represent distinct trajectories. Properties such as the distribution or average length in the 2-D space may serve as useful parameters to explore the underlying cognitive load and emotion processing during the complex task. In sum, this framework is a novel data-driven approach to the learning and visualization of underlying neurophysiological dynamics of complex functional brain data.

Fragile X premutation carriers (fXPC) are characterized by 55-200 CGG trinucleotide repeats in the 5' untranslated region on the Xq27.3 site of the X chromosome. Clinically, they are associated with the fragile X-Associated Tremor/Ataxia Syndrome, a late-onset neurodegenerative disorder with diffuse white matter neuropathology. Here, we conducted first-ever graph theoretical network analyses in fXPCs using 30-direction diffusion-weighted magnetic resonance images acquired from 42 healthy controls aged 18-44 years (HC; 22 male and 20 female) and 46 fXPCs (16 male and 30 female). Globally, we found no differences between the fXPCs and HCs within each gender for all global graph theoretical measures. In male fXPCs, global efficiency was significantly negatively associated with the number of CGG repeats. For nodal measures, significant group differences were found between male fXPCs and male HCs in the right fusiform and the right ventral diencephalon (for nodal efficiency), and in the left hippocampus [for nodal clustering coefficient (CC)]. In female fXPCs, CC in the left superior parietal cortex correlated with counting performance in an enumeration task.

Background

Many recent studies have separately investigated functional and white matter (WM) based structural connectivity, yet their relationship remains less understood. In this paper, we proposed the functional-by-structural hierarchical (FSH) mapping to integrate multimodal connectome data from resting state fMRI (rsfMRI) and the whole brain tractography-derived connectome.

Methods

FSH first observes that the level of resting-state functional correlation between any two regions in general decreases as the graph distance of the corresponding structural connectivity matrix between them increases. As not all white matter tracts are actively in use (i.e., "utilized") during resting state, FSH thus models the rsfMRI correlation as an exponential decay function of the graph distance of the rsfMRI-informed structural connectivity or rsSC. rsSC is mathematically computed by multiplying entry-by-entry the tractography-derived structural connectivity matrix with a binary white matter "utilization matrix" U. U thus encodes whether any specific WM tract is being utilized during rsFMRI, and is estimated using simulated annealing. We applied this technique and investigated the hierarchical modular structure of rsSC from 7 depressed subjects and 7 age/gender matched controls.

Results

No significant group differences were detected in the modular structures of either the resting state functional connectome or the whole brain tractography-derived connectome. By contrast, FSH results revealed significantly different patterns of association in the bilateral posterior cingulate cortex and right precuneus, with the depressed group exhibiting stronger associations among regions instrumental in self-referential operations.

Discussion

The results of this study support that enhanced sensitivity can be obtained by integrating multimodal imaging data using FSH, a novel computational technique that may increase power to detect group differences in brain connectomes.

This article presents a novel approach for understanding information exchange efficiency and its decay across hierarchies of modularity, from local to global, of the structural human brain connectome. Magnetic resonance imaging techniques have allowed us to study the human brain connectivity as a graph, which can then be analyzed using a graph-theoretical approach. Collectively termed brain connectomics, these sophisticated mathematical techniques have revealed that the brain connectome, like many networks, is highly modular and brain regions can thus be organized into communities or modules. Here, using tractography-informed structural connectomes from 46 normal healthy human subjects, we constructed the hierarchical modularity of the structural connectome using bifurcating dendrograms. Moving from fine to coarse (i.e., local to global) up the connectome's hierarchy, we computed the rate of decay of a new metric that hierarchically preferentially weighs the information exchange between two nodes in the same module. By computing "embeddedness"-the ratio between nodal efficiency and this decay rate, one could thus probe the relative scale-invariant information exchange efficiency of the human brain. Results suggest that regions that exhibit high embeddedness are those that comprise the limbic system, the default mode network, and the subcortical nuclei. This supports the presence of near-decomposability overall yet relative embeddedness in select areas of the brain. The areas we identified as highly embedded are varied in function but are arguably linked in the evolutionary role they play in memory, emotion and behavior.

This paper describes novel methods for constructing the intrinsic geometry of the human brain connectome using dimensionality-reduction techniques. We posit that the high-dimensional, complex geometry that represents this intrinsic topology can be mathematically embedded into lower dimensions using coupling patterns encoded in the corresponding brain connectivity graphs. We tested both linear and nonlinear dimensionality-reduction techniques using the diffusion-weighted structural connectome data acquired from a sample of healthy subjects. Results supported the nonlinearity of brain connectivity data, as linear reduction techniques such as the multidimensional scaling yielded inferior lower-dimensional embeddings. To further validate our results, we demonstrated that for tractography-derived structural connectome more influential regions such as rich-club members of the brain are more centrally mapped or embedded. Further, abnormal brain connectivity can be visually understood by inspecting the altered geometry of these three-dimensional (3D) embeddings that represent the topology of the human brain, as illustrated using simulated lesion studies of both targeted and random removal. Last, in order to visualize brains intrinsic topology we have developed software that is compatible with virtual reality technologies, thus allowing researchers to collaboratively and interactively explore and manipulate brain connectome data.