Abstract
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.
Highlights
The Nash embedding theorems [1, 2] showed that any Riemannian n-manifold with a C1 positive metric has an isometric embedding in a Euclidean space of dimension 2n+1, even in any small portion of this space
To quantify the dynamic properties and thought trajectory of emotion regulation, we summarize the Euclidean distance along the trajectory in this 2-D space and the average distance from the centroid for each subject
3.1 Thought Chart construction After averaging across theta frequencies (4–7 Hz) and combining both resting and emotion regulation task (ERT) theta connectomes for all time points, 20 healthy subjects contributed a total of 10400 connectomes (1 30 × 20 × 4 )
Summary
The Nash embedding theorems [1, 2] showed that any Riemannian n-manifold with a C1 positive metric has an isometric embedding in a Euclidean space of dimension 2n+1, even in any small portion of this space. Since the Gaussian curvature of a surface is invariant under local isometry based on the Theorema Egregium [3], the manifold properties in a low-dimensional space can provide an insight into the topological structure of the brain. We assume brain states compose a high-dimensional space [4, 7], which can be reconstructed and visualized in a low-dimensional space via dimensionality reduction. A linear approach cannot recognize nonlinear structures in a high-dimensional space [9], making it unsuitable for preserving the global intrinsic geometry for complex dynamic brain data. Nonlinear dimensionality reduction approach isomap yields global coordinates which provide a simple way to analyze and manipulate high-dimensional observations in terms of their intrinsic nonlinear degrees of freedom and produce a globally optimal low-dimensional Euclidean representation [10]
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