Fig 1.
Temporal dynamics of phase interactions.
a) Example narrowband signals from two different brain regions (rPARH and rFUS). b) Probability density function of the phase differences, Pr(Δφ), across all pairs of brain regions and all time steps. c) Temporal evolution of Pr(Δφ), for 3 example scanning sessions from different subjects. The color code indicates the values of Pr(Δφ) at each time step (see the color bar at the bottom).White trace: Time course of the order parameter R(t) (right y-axis). d) Probability density function of the order parameter values for all sessions (black) compared to phase-randomized surrogates (red) and broadband signals (brown). e) Power spectrum of the order parameter. Gray traces: single scanning sessions; black trace: mean across all sessions. f) Comparison of the averaged power spectrum of the order parameter for narrowband (black) and broadband signals (brown). Power spectra were normalized by the corresponding maximum value.
Fig 2.
Spatiotemporal synchronization patterns.
A synchronization matrix Q was built at each time step t (t = 1, .., T) by calculating the phase difference of each pair of empirical analytic signals and imposing a synchronization threshold: Qij(t) = 1 if |φj(t)-φi(t)|<π/6 and Qij(t) = 0 otherwise (1 ≤ i, j ≤ n). The temporal evolution of the synchronization matrix Q is shown for an example scanning session (session #3) and for selected time periods; entries equal to one are represented in black, null entries are represented in white. The synchronization patterns framed in blue and red last several seconds and reoccur during disjointed time periods.
Fig 3.
Community structure of spatiotemporal synchronization patterns.
a) The temporal evolution of the synchronization matrix is represented in a n×n×T tensor T. The tensor can be factorized as a sum of K rank-one tensors, each one being an outer product of three vectors, ak, bk, ck, of dimension equal to n, n, and T, respectively (Equation 8). The network communities are contained in the vectors ak, the elements of which give the participation weight of each node (i.e., brain region) in the community k. The temporal activation sk(t) of each community k is related to ck and to the participation weights as: sk(t) = ck(t)Σjak(j). b) Detected community patterns (ak. akT) for the example session #1. c) Top: temporal activation strength of each community, for the scanning session #1 (same colors as in (b)). Bottom: temporal evolution of both the order parameter R(t) (left y-axis) and the total activation strength [S(t) = Σksk(t)] (right y-axis), for session #1. d-e) same as (b-c) but for session #7. f) Correlation matrix between all detected communities from all scanning sessions (top), re-arranged according to cluster membership. Bottom: corresponding dendrogram based on correlation coefficients.
Fig 4.
Topology of the synchronization communities.
a) Spatial organization of the communities obtained with the first half of the sessions (top) and the second half of the sessions (bottom). For each community, the brain regions with the highest participation weights are presented (yellow: 0.1<ak(i)<0.2; orange: ak(i)>0.2). The community patterns of the second half of the sessions were matched to the ones of the first half of the sessions (below each panel the correlation between the community i from the first half-dataset and the community i from the second half-dataset is presented; rc: correlation coefficient; p: p-value). b) The synchronization communities were compared to the resting-state networks obtained using ICA. As a measure of similarity we used the Jaccard index. The Jaccard similarity matrix between ICA-based components and synchronization communities is shown for the first half of the data (left) and the second half of the data (right). The spatial patterns given by the synchronization communities include: the default mode network (DMN, community 1), the somato-motor network (2), the visual network (3), the auditory/somato-motor network (4), the self-referencial/DMN network (5), the right cognitive control network (6), and other networks (7–14) that are overlaps of the previous ones and other functional networks detected using ICA.
Fig 5.
An anatomically-constrained network of phase oscillators approximates the empirically observed phase statistics.
The empirical statistics (black) were compared to the statistics generated by the anatomically connected heterogeneous model (model 1, red) and three control models: 100 realizations of the heterogeneous model with shuffled connectivity (model 2, gray, mean ± 95% confidence interval across realizations (gray area)), the anatomically connected homogeneous model (model 3, green), and the anatomically connected homogeneous stochastic model (model 4, yellow), with noise amplitude σ = 0.2. a) Averaged value of the order parameter, <R>. b) Similarity (1/DKL) between the phase differences distribution, Pr(Δφ), of the empirical data and of each model. All similarity values were normalized to the maximum similarity for model 1. Inset: Pr(Δφ) from the empirical data (black) and from model 1 (red) generated with the best-fit parameter. c) Similarity (1/DKL) between the distribution of the number N of synchronized pairs, Pr(N), of the empirical data and each model (normalized by the maximum similarity for model 1). Inset: Pr(N) distribution from the empirical data (black) and from model 1 (red) generated with the best-fit parameter. d) Agreement (correlation) between the empirical PLV matrix and the models’ PLV matrices. Red area: 95% confidence interval of the sample Pearson correlation coefficient. Insets: PLV matrix from the empirical data and from model 1 generated with the best-fit parameter. e) Peak of the power spectrum of R. The color code represents the empirical frequency power of R and the white line represents the empirical peak frequency. f) Top: time evolution of R for one single fMRI session. Bottom: time evolution of the R of model 1 for G = 0.2. Note different time-scales (x-axis).
Fig 6.
Dynamical range of the anatomically-constrained phase oscillators’ network.
a) Time evolution of the phase difference between two nodes of the anatomically-connected heterogeneous Kuramoto model, for three values of the global coupling G. Left: in the weakly connected case (G = 0.025) the phases run almost independently; middle: with moderate coupling (G = 0.25) the phases tend to lock for short periods of time, as revealed by the deflections in the trajectory of the relative phase, indicating the presence of metastability; right: with strong coupling (G = 1.25) the phases are locked. b) Corresponding oscillations of the two nodes, for the three dynamical regimes.
Fig 7.
Emergence of transient synchronization patterns.
a) Temporal evolution of activation strengths of the communities of the anatomically connected heterogeneous Kuramoto model (G = 0.2 and K = 10). b) Temporal evolution of the total community activation strength S (top) and the order parameter R (bottom) of the model anatomically connected heterogeneous Kuramoto model (G = 0.2 and K = 10). The correlation coefficient between S(t) and R(t) is 0.82 (p<10–3). c) Top: The largest correlation coefficient rmax (and 95% confidence interval) between the model communities and the empirical communities was calculated for various number of components (K = 2, …, 16) as a function of the global coupling. Middle: rmax was compared to the expected upper 95% confidence bound of the largest correlation coefficient (rmax, perm) between the model communities and 103 random permutations of the n elements of each empirical communities, for each K and each G. Bottom: Probability that rmax> rmax, perm.