a–d A classic example that illustrates manifold embedding; i.e., manifold learning applied to the swiss roll data, which is intrinsically a two-dimensional dataset yet represented in a higher (three-dimensional) space. To estimate the low dimensional embedding of the sampled dataset (b), we first create a graph representation (c), where the nodes represent the data points shown in (b), which are sampled from the underlying manifold illustrated in (a), and the edges indicate the relations (distances and/or similarities) between data points. d The manifold is embedded into the low-dimensional representation that matches its intrinsic dimensionality using the Laplacian eigenmaps manifold learning. e–g Our framework applying the same manifold learning approach, i.e. Laplacian eigenmaps, to extract the manifold underlying brain dynamics measured in fMRI data. e For each time point, the fMRI BOLD signal is parcellated into the 90 brain areas defined by the AAL template and pre-processed as explained in the “Methods“ section. f Using the parcellated fMRI data, the instantaneous phase is computed via Hilbert transform and the phase coherence among brain areas is estimated. This phase coherency matrix characterizes the pairwise synchrony relations between each pair of brain areas at any given time point. g The intrinsic manifold (here illustrated as two-dimensional) underlying the set of all instantaneous phase coherence states is estimated using the Laplacian eigenmaps method. To visualize the changes in phase coherency throughout the intrinsic manifold, for illustration purposes we defined 2-dimensional (2D) bins using the two manifold dimensions, and computed the average phase coherency of data points in those bins. Different colors indicate different sleep stages and wakefulness as defined by polysomnography.
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