Fig 1.
The dynamic PPC-PAC hypothesis.
(A) A repertoire of synchronous slow oscillations that interact via the PPC and interact with fast oscillations via the PAC. (B) The resulting PAC states. (C) Transitions between the metastable PAC states. The dynamic PPC-PAC hypothesis states that for the resting brain, dynamic changes in PPC strengths (transitions between synchronous states) can cause dynamic and large-scale changes in PAC strengths because of PPC-PAC connectivity (A), and thereby yield transitions between oscillatory states with multiple peak frequencies (B and C). The oscillations of each state can realize the transition to another state by spontaneous fluctuations in the brain; in other words, the underlying states can show metastability.
Fig 2.
The flow of signal processing in metastable states clustering.
(A) A transition dynamics among three PAC states (d = 2) in the state space of xi(t) = cos(2πf1 t + ψi) + 0.5[1 + bi(t) cos(2πf1 t + ψi)] sin(2πf2 t) + ξi(t), where the first, second, and last terms indicated the slow oscillations, amplitude-modulated fast oscillations, and noise, respectively. (B) Observed signals, the first envelopes (instantaneous amplitudes) around f2, and the second envelopes around f1. (C) The labeled sequences. (D) LDA projections of the state space. (E) The bandpass signal of x1(t) around peak frequency f2 and corresponding labels. The frequencies f1 and f2 here were set to 1 Hz and 10 Hz, respectively; modulation index bi(t) dynamically changed among strengths 0.15, 0.5, and 0.85; and noise ξi(t) followed a normal distribution of mean 0 and standard deviation 0.3. The colors in panel A correspond to those in the right column of panels C to E.
Fig 3.
Dynamic changes in the delta-alpha PAC strength.
(A) A representative raw EEG signal at the FC2 electrode. (B) The corresponding first envelope (instantaneous amplitudes) around an alpha-band peak frequency. (C) The second envelope around a delta-band peak frequency. (D) The alpha-band signals. (E) The mean power spectrum of the raw EEG signals as in panel A. (F) The mean power spectrum of the first envelopes as in panel B. The alpha-band and delta-band peak frequencies were estimated from the peaks of mean power spectra in panels E and F, respectively (see the dotted lines in panels E and F and refer to S3 Fig). In panel D, (i) indicates the data corresponding to panel A, while (ii) indicates another signal of faster transition among more states obtained from an individual with a lower AQ score (the signal at electrode POz). The colors in panels A to D indicate distinct states.
Fig 4.
Transition dynamics among delta-alpha PAC states in a lower-dimensional space.
(A to E) Representative delta-alpha PAC dynamics for an individual with higher attention-related AQ subscores (refer to Fig 3A to 3D(i)). (F to J) Representative dynamics for an individual with lower scores (refer to Fig 3D(ii)). (A, F) The trajectory of labeled signals in a plane. (B, G) The corresponding bivariate histograms. (C, H) Surrogate data testing under a condition of d = 2. (D, I) The resulting delta-alpha PAC states (represented by the mean PAC strengths). (E, J) Transitions between the identified PAC states. Surrogate data testing was applied to the density of points indicated by the red circles in panels B and G and the red lines in panels C and H, and the null hypothesis H0 in condition d = 2 was rejected (C and H). The delta-alpha PAC dynamics tended to stay in a state for a longer time and to visit a lower number of states in individuals with higher AQ subscores for attention to detail and attention switching (compare E with J). The colors in panels A, E, F, and J indicate distinct states, as depicted in D and I.
Fig 5.
The four groups of consistent delta-alpha PAC states across individuals.
(A) PCs of across-individual states. (B) Eigenvectors of the first four PCs. The variance explained by the first four PCs was significant, and accounted for 81.6% of total variance. The dataset used here was a set of the modified Z-scores of mean PAC strengths that were concatenated across states and individuals.
Fig 6.
Correlations between delta-alpha PAC dynamics and attention-related AQ subscores.
(A) Scatter plot of the attention-to-detail score against maximal dwell time. (B) Scatter plot of the attention-switching score against the linear sum of the number of states and the alpha-band peak frequency (i) with corresponding factor loadings (ii). (C, D) Scatter plots of the attention-switching score against the number of states and the alpha-band peak frequency, respectively. In each panel, the circles in magenta and green correspond to the representative individual delta-alpha PAC dynamics, as depicted in Fig 4A to 4E and 4F to 4J, respectively. The dotted line in each panel indicates the fitted regression line.
Fig 7.
Estimation of PPC connectivity and the level of fluctuations from delta-band phase dynamics.
(A) Mean phase differences between every pair of delta-band CSD phases with respect to each delta-alpha PAC state μ. (B) The corresponding phase patterns {θμ} (regarded as the synchronous states in the model). (C) The estimated PPC connectivity Cδ as a complex-valued matrix with absolute part
(i) and argument part
(ii). (D) The estimated fluctuation level
. The synchronous states (B) combined with the Kuramoto model resulted in PPC connectivity (C), and the Kuramoto model with PPC connectivity was used for estimation of the fluctuation level (D). The data used in this Figure correspond to Figs 3A to 3D(i) and 4A to 4E.
Fig 8.
Simulation of delta-alpha PAC dynamics by a coupled oscillator system driven by spontaneous fluctuations.
(A to D) The representative simulated delta-alpha PAC dynamics for an individual with higher attention-related AQ subscores (refer to Fig 4A to 4E). (E to H) Representative dynamics for an individual with lower scores (refer to Fig 4F to 4J). (A, E) Time courses of overlaps (i) and the corresponding labels (ii) among delta-alpha PAC states. (B, F) The trajectory of labeled signals in a plane. (C, G) The corresponding bivariate histograms. (D, H) Surrogate data testing under condition d = 2. Surrogate data testing was applied to the density of points indicated by the red circles in panels C and G and the red lines in panels D and H, and the null hypothesis H0 in condition d = 2 was rejected (D and H). The model showed consistent results with the data analysis, evidence of the dynamic PPC-PAC hypothesis.
Fig 9.
Shrinking of the simulated delta-alpha PAC dynamics with a temporally decreasing fluctuation level in the phase space: The qualitative change from the transition dynamics to the dynamics in a single state.
(A) Time courses of the overlaps with their labeled sequences under different initial conditions in cases where the dynamics can converge into one of three states. (B) The time course of the fluctuation level. (C) The trajectories of overlaps in the phase space with their projections. The contour plots on projections in panel C indicate that the spaces filled by transition dynamics (black lines) can include the three states (red, green, and blue lines) as their subsets. The data used in this Figure correspond to the individual with higher attention-related AQ subscores depicted in Figs 3A to 3D(i), 4A to 4E, 7 and 8A to 8D.