Fig 1.
Evolution and statistics of phase metrics for two noisy oscillators with (A) identical and (B) different frequencies.
(a, b, f, g) Phase difference, Δθ1,2, (black), and angle of the cPLV, ϕ1,2, (red and magenta for significant PLV and blue otherwise), with the top plots depicting zoomed phases of the shaded area in the lower plots. (c, h) PLV (blue) and its mean value (black), and two levels of significance (magenta and red). (d, e, i, j) PDF (dashed red and full magenta line), histograms of phase differences Δθ1,2 and angle ϕ1,2 during epochs of synchronization, and their circular mean values (red and magenta arrows). Two different surrogates procedures (high surr and low surr) are used for the levels of significance. Parameters: τ = 0.01s, D = 5, K = 30 (A) ω1,2 = 12 ⋅ 2π rad/s; (B) ω1,2 = 12 ⋅ [0.95, 1.05] ⋅ 2π rad/s.
Fig 2.
Evolution and statistics of phase metrics for two NA oscillators being (A) in- and (B) anti-phase synchronized.
(a, b, h, i) Phase differences Δθ1,2(t), (black) and ϕ1,2(t), (red and magenta for significant PLV, and blue otherwise) with top plots zooming the shaded interval of the lower plots. (c, j) PLV (blue) and its mean value (black), and levels of significance (magenta and red). (d, k) Effective coupling strength, Keff (black) compared with the absolute value of the frequency difference, Δωeff, (blue if ω1 < ω2 and red otherwise), and (e, l) effective natural frequencies (red for ω1 and blue for ω2). (f, g, m, n) PDF (dashed red and full magenta line), histograms of phase differences during epochs of synchronization and their means (arrows). Parameters: ω1,2 = 12 ⋅ [1.03, 0.97] ⋅ 2π rad/s, ϵ1,2 = [3.6, 3.6], , D = 5 (A) τ = 0.01s,
, K = 25, ϵK = 5; (B) τ = 0.03s,
, K = 30, ϵK = 6.
Fig 3.
Sketch of spatial distribution of the delays and connectivity matrices.
Bimodal δ distributed delays with τ1 (blue) τ2 (red). (a) Spatially homogeneous (random) delays and (b) heterogeneous, delay-imposed structure.
Fig 4.
(a) Values of the arcsine function and (b) an example for transformation of ΩΔτ and .
(a) Blue for arcsin(x) ∈ [−π/2, π/2] and red for arcsin(x) ∈ [π/2, 3π/2] in the Cartesian plane. (b) Values of Ωτ1,2 (black) and ΩΔτ and (red), and ΩΔτ* = ΩΔτ ± π and
(blue), which appear due to the summation sin(Ωτ1 + ϕi) + sin(Ωτ2 + ϕi + 2π).
Fig 5.
Delay-coupled heterogeneous oscillators with homogeneous bimodal-δ delays.
Synchronization at frequency (A) Ω < μ and (B) Ω > μ. (a, b, e, f) Relative phases ϕi(t) of the synchronized and two unsynchronized oscillators (black) closest to the limits Ω ± Kr cos ΩΔτ. For comparison ±(Ω − μ)t are shown with dashed lines. Oscillators with (a, e) ωi < Ω and (b, f) ωi > Ω. (c, g) Geometric representation of ϕi of the synchronized oscillators (different shades of red diamonds for ωi < Ω and blue circles for ωi > Ω) at the end of the simulations. Limits and
are dashed red. The arrows show the complex order parameter (black), angles Ωτ1,2 (blue), and
and ΩΔτ (red). (d, h) PDF of the natural frequencies, the frequency Ω (black vertical line), and the limits of synchronization Ω ± rK cos ΩΔτ (red). Entrained (blue and red) and the first two un-synchronized (black) oscillators are consistent across the plots, and the rest are green. Parameters: Number of oscillators: N = 300, Lorentzian natural frequencies with μ = 1Hz and γ = 1 (A) τ = [0.02, 0.37]s, K = 6, (B) τ = [0.07, 0.63]s, K = 8.
Fig 6.
Identical noisy phase oscillators coupled with homogeneous bimodal-δ delays and log-normal coupling strengths.
Synchronization at frequency (A, C, D) Ω > μ and (B) Ω < μ. (a, f, k, p) Phases ϕi(t) of the synchronized (colored coded with the in-strength) and unsynchronized (dashed black) oscillators, and the mean phase, ±(Ω − μ)t (dashed lines). (b, g, l, q) Scatter plot of averaged relative phases and nodes in-strengths, showing synchronized (red) and unsynchronized (green) oscillators. Black line is the theoretical prediction Eq (21) and blue is the linear fit of the correlation. (c, h, m, r) Node strengths color-coded with their phases (filled circles are synchronized, and empty squares are unsynchronized), and their PDF. (d, i, n, s) Phases of synchronized oscillators color-coded with their in-strength, and their PDF, ρ(ϕ) and (e, j, o t) geometric representation and the complex order parameter (black arrow). Parameters: N = 300, simulation time (A-C) tfin = 200s and (D) tfin = 2000s. (A, B, D) D = 1, (C) D = 0.1. Log-normally distributed in-strengths with σK = 2μK, (A, C, D) μK = 6 and (B) μK = 8. Delays (A, C, D) τ = [0.02, 0.37]s, (B) τ = [0.07, 0.63]s.
Table 1.
Change of the relative phases ϕi for increasing coupling strengths for spatially random delays.
Fig 7.
Delay-imposed populations of coupled heterogeneous phase oscillators.
(A, B) In- and (C, D) anti- phase synchronized clusters, at frequency (A, C) Ω < μ and (B, D) Ω > μ. (a, b, e, f, i, j, m, n) Phases ϕi(t) of the synchronized (red and blue) and two unsynchronized oscillators (black) closest to the limits of synchronization, and ±(Ω − μ)t (dashed). (c, g, k, o) Geometric representation of ϕi of the synchronized oscillators (different shades of red diamonds for ωi < Ω and blue circles for ωi > Ω). Limits and
are dashed red, the arrows are for the complex order parameter (black) of each subpopulation (they overlap for in-phase), angles Ωτ1,2 (blue), and
and ΩΔτ (red). (d, h, l, p) PDF of the natural frequencies, frequency of synchronization Ω (black vertical line), and limits of synchronization (red). Entrained oscillators are blue and red, the first two un-synchronized on both sides are black, and the rest are green. The colors of each oscillator are consistent across the plots. Parameters: N = 300, K = 7, Lorentzian natural frequencies with μ = 1Hz and γ = 1. (A) τ = [0.05, 0.2]s, (B) τ = [0.7, 0.95]s, (C) τ = [0.22, 0.47]s, and (D) τ = [0.04, 0.27]s.
Fig 8.
Delay-imposed clusters of identical noisy phase oscillators.
(A, B) In- and (C, D) anti- phase synchronization, with log-normally distributed coupling strengths Kij. (a, f, k, p) Phases ϕi(t) of synchronized oscillators (color-coded with their in-strength) and the mean phase, ±(Ω − μ)t (dashed). (b, g, l, k) Scatter plot of averaged relative phases and nodes strength. Oscillators of the different populations are with opposite pointing triangles. Black line is the theoretical prediction, Eq (33), blue is the linear fit for each population. (c, h, m, r) Node strengths color-coded with their relative phases and their PDF. (d, i, n, s) Phases of the synchronized oscillators (color-coded with their in-strength) and their PDF, and (e, j, o, t) their geometric representation and complex order parameter (black arrow). Parameters: N = 300, K = 7, D = 1. (A) τ = [0.05, 0.2]s, (B) τ = [0.7, 0.95]s, (C) τ = [0.22, 0.47]s, and (D) τ = [0.04, 0.27]s.
Table 2.
Change of the relative phases ϕi for increasing coupling strengths for delay-imposed structure.
Fig 9.
Connectome of a healthy subject.
(a) Normalized weights, (b) logarithmic weights and (c) lengths of the tracks. (d) Joint distribution of weights and lengths, and histograms of weighted lengths for (e) intra- and (f) inter-hemisphere links.
Fig 10.
Simulated dynamics over a healthy human connectome.
In-phase (A, D) anti-phase (B) and intermittent synchronization (C). Top left plot of each panel are relative phases for the synchronized and two unsynchronized oscillators (black) closest to the limits, and ±(Ω − μ)t (dashed). Top middle are scatter plots of nodes averaged phases versus their in-strengths. Nodes of left/right hemisphere are up/down pointing triangles, black line is a theoretical prediction, blue is the linear fit. Top right are the PDF of in-strengths color-coded with nodes’s phases. Middle left are phases of the synchronized oscillators (color-coded with in-strength) and their PDF; and middle right is their geometric representation and complex order parameters (black arrows). Bottom left and right are evolutions of order parameter and mean field frequencies, for whole brain (blue) and for each hemisphere (red and magenta). Order parameters for uncoupled case are green for whole brain and cyan for one hemisphere. Parameters: N = 68 oscillators, noise intensity D = 2. Coupling strengths (A, B) K = 0.8, (C) K = 1.1, (D) K = 1.6, natural frequency (A) f = 10Hz and (B–D) f = 20Hz.
Fig 11.
Evolution and statistics of PLV and phase lags for one (A) intra and (B) inter-hemispheric link.
(a, b, f, g) Phase lag, Δθ, (black), and angle of the cPLV, ϕ, (red and magenta for significant PLV, blue otherwise). (c, h) PLV (blue) and its mean value (black), and two levels of significance (magenta and red). (d, e, i, j) Estimated PDF (magenta line and dashed red) and histograms of phase lags, and their means (red and magenta arrows). Parameters: f = 20Hz, D = 2, (A) K = 1.1, (B) K = 1.16.
Fig 12.
Statistics of PLV and phases for 68 brain regions.
Nodes are as given by Desikan Kiliany parcelation (top) and ordered within hemispheres by the in-strength (bottom). For each link the mean and 1–standard deviation are shown, while white are links with no periods of significant coherence. Parameters: K = 1.18, f = 20Hz, D = 2.
Fig 13.
Statistics of PLV metrics for two significance levels.
(A) Standard deviation of phase lags. (B) Histogram of mean phase lags for all the links, calculated over periods of significance. (C) Evolutions of PLV (top) and phase lags (middle), and their histograms (bottom) for both levels of significance (magenta and red in all plots) for one inter-hemispheric link. Parameters: K = 0.55, f = 20Hz, D = 4.
Fig 14.
PLV and phase lags during different regimes of brain dynamics.
(left to right) Coherence over the whole time-series, mean PLV, mean phase lags ϕ for low and high coherence (they are identical for the bottom row), and histograms and PDF of the mean phase lags of coherent links. White are the links with no significant PLV. Nodes in all matrices are sorted by their in-strength. The level of the noise is D = 0.2f.
Fig 15.
Intra- and inter- hemispheric subnetworks of the 10 strongest nodes.
In- and anti-phase synchronization. Parameters: K = 0.5, D = 0.5.