Fig 1.
Basic properties of oscillatory phase synchronization.
(A) The underlying model of this study was phase synchronization of two coupled phase-oscillators which could correspond to oscillatory signals measured from separate cortical areas. Phase-locking is the amount of consistency of instantaneous phases between the two oscillators. Phase-locking is resulting from synchronization process governed by two principal factors as described by Theory of Weakly Coupled Oscillators (TWCO): The intrinsic (natural) frequency ω and the coupling strength κ. The intrinsic frequency difference (detuning Δω) between oscillators determines the phase precession. The coupling strength κ determines the interaction strength, which is a function of the phase-relation (defined by the phase response curve, PRC). (B) The detuning Δω and the coupling strength κ defined a 2-dimensional space, in which phase-locking (gray shading) occurs within certain ranges. In a noiseless system, full synchrony (phase locking of 1) occurs in a limited area of detuning and coupling strength that appears as inverted triangle (Arnold tongue). The stronger the coupling strength, the more detuning is possible while still reaching full synchrony. Full synchrony (C) occurs if the oscillators converge on a common frequency (no phase precession). The phase-relation distribution exhibits a strong peak at the attractor phase relation. (D) Complete asynchrony is only possible when the oscillators are uncoupled. The phase-precession is smooth and the phase-difference distribution uniform. (E) The state of partial synchrony is characterized by phase-locking between 0 and 1. In most regions in the Δω vs. κ space, phase-locking might be close to 0. Yet, close to the Arnold tongue the phase-locking might still be relevant. The oscillators do not converge to a common frequency, but exhibit phase precession. The phase precession does not have a smooth trajectory, but is modulated depending on the phase-relation. This leads to non-uniform phase-difference distribution with a peak at the phase-relation in which the oscillators have the smallest frequency difference. In noisy phase-oscillatory systems, the partial synchronized regime can be the most dominant regime. (F) Because the phase-precession (= instantaneous frequency difference) is not smooth and changes as a function of phase-relation, it implies phase-relation dependent frequency modulations (PrFM). (G) We also included phase-relation dependent amplitude modulations (PrAM) due to observations in many of our neural network simulations. We assume here that PrAM, in the ranges included here, did not substantially change the phase trajectories and hence TWCO is still an adequate theoretical framework.
Fig 2.
Analytical and numerical results of Coherence estimation of phase-locking with different levels of extrinsic (measurement) noise.
In (A-B) we first show two examples. In (A) oscillator X and Y had a detuning of 3Hz and did not interact. The power spectra (middle panel) show the two power peak of the two oscillators. The coherence spectrum (right panel) was flat as expected. In (B), the oscillators did interact (κ = 1), where oscillator X influenced the phase trajectory of oscillator Y. The power spectra of oscillator Y show two extra power peaks with ± the detuning. These are the so-called modulation sidebands, well described in the cross-frequency coupling literature [62]. Notice that one of the sidebands overlap with the power peak of oscillator X. The coherence spectrum show a strong phase-locking estimate, much higher than expected. This is because the coherence estimate reflected mostly the locking between the sideband of oscillator Y and the main power peak of oscillator X, which can be completely unrelated to the actual phase-locking of oscillators X ad Y. (C) Rendering of the Arnold tongue, shown with a 1/2 cross-section at the level of a 0.75 coupling strength, for which phase locking values are plotted as a function of positive, increasing intrinsic frequency differences between oscillators X and Y (Δω). Here, we did not add PrAM to the oscillatory signal. We compared the numerically (red dot) and analytically derived (black line) Coherence with the analytically derived true phase-locking (purple line) between two oscillators as a function of frequency detuning (Δω) and different levels of SNR. We used trial-number corrected squared coh values to minimize inflation due to a finite number of trials. In the partially synchronized states associated with different Δω values in the selected coupling condition, we observed strong deviations of Coherence from the true locking. The coh2values became more inflated with higher SNR. The numerically computed coh2 matched with the analytically derived coh2. (D) The impact of different levels of PrAM is shown with different level of SNR. The oscillators were uncoupled and hence asynchronous (in the condition indicated by the fat dot at the bottom of the Arnold tongue) and the true locking was therefore 0. The oscillators had a phase precession of 3Hz (chosen condition is located off the midline of the Arnold tongue). We observed strong deviations from the true locking with increasing PrAM and SNR. The numerically and analytically derived values matched.
Fig 3.
Comparison of spectrally and non-spectrally based approaches for the estimation of phase locking.
The figure shows numerical results of Coherence and PLV estimates of phase-locking between phase-oscillators with both dynamical noise (more broadband) and different level of extrinsic (measurement) noise (uncorrelated between oscillators). First two examples are shown. (A) Oscillators X and Y are interacting (κ = 1 X-> Y) with a detuning of 2Hz (X = 38Hz, Y = 40Hz). The power spectra (here in the gamma range, although exact range is irrelevant here) are shown in the middle-panel. The power spectra do largely overlap and the modulation sidebands cannot be easily observed (but are present in the data). The coherence spectrum (right panel) gave a phase-locking estimate larger than expected with a peak at 38Hz (where the left modulation sideband of oscillator Y overlaps with the power peak of oscillator X). In (B) the detuning was increased to 6Hz (X = 34Hz, Y = 40Hz) with same coupling conditions. Now the left modulation sideband of oscillator Y can be observed as a small peak at 34Hz. The coherence spectrum gave a phase-locking estimate much larger than expected reflecting the influence of the modulation sideband. (C) A 1/2 cross-section of the Arnold tongue, similar to Fig 3A, is shown. The continuous lines represent simulations without PrAM and the dashed lines represent simulation with a PrAM of 20%. We compared the coherence to the expected phase-locking. We used the noise-free instantaneous phases of the phase-oscillators to compute the PL2, which was a good estimator of the analytically derived true phase-locking. We observed that the coh2 values deviated strongly from the expected phase-locking. The exact deviation depended on the detuning frequency and SNR. Including a PrAM of 20% led to a further inflation of the coh2 values. Note also the deviations of coh2 from the PL2 at a zero detuning frequency. (D) The same analysis as in (C) but with PLV values estimated by the SSD-HT method. We observed that for higher SNR the estimate behaved better and remained close to the expected phase locking. At lower SNR the PLV2 showed lower than expected values due to the effect of (uncorrelated) noise. Including a PrAM of 20% led to an inflation of PLV2 values in the lower SNR only.
Fig 4.
Testing performance of coherence and PLV in the estimation of phase-locking performance and information flow of two interacting gamma-generating spiking networks receiving different detuning levels.
(A) The network architectures. Two interconnected excitatory-inhibitory networks consisting of 100 inhibitory cells (fast-spiking type) and 400 excitatory cells (regular spiking) were simulated using Izhikevich formalism [39]. Neurons were interconnected with AMPA (excitatory) and GABAA (inhibitory) connections. The networks generated so called ‘pyramidal-interneuron network gamma’ (PING). The two networks where weakly interconnected by E-I and EE interconnections. The detuning was manipulated by altering the difference in excitatory input drive (to E-cells) between the networks. We used the experimentally and theoretically established observation that the frequency of gamma oscillations is tightly linked to input drive. (B) Generation of test-signals. From each of the 300 1sec trials the ‘LFP’ (population signal) was extracted from each network by summing and smoothing (pseudo Gaussian function of 3ms width) the E-cell spikes. Then we added 1/f noise (exponent = 1.5) to manipulate SNR. Compared to the phase-oscillator model, phase-relation dependent amplitude modulations (PrAM) were generated intrinsically in the model. (C) By changing the relative excitatory input drive to E-cells between the networks, we could manipulate the detuning (frequency difference). (D) Coh2 (in the gamma frequency range 30-50Hz) and PLV2 as a function of input drive difference (ΔE-drive) between networks. Different line colors represent difference SNR (relative power). The black line represents the PLV2 with no noise added (PL). (E) Information flow (combined directions), as measured by transfer entropy (TE), as a function of input drive difference (ΔE-drive) between networks. (F) Variance in information flow explained by coh2 (black) and PLV2 (red) as a function of SNR derived by computing pearson correlation.
Fig 5.
Testing performance of coherence and PLV in estimation of phase-locking performance and information flow of two interacting gamma-generating spiking networks among which the coupling is manipulated by changing the strength of cross-network synaptic connections.
(A) The network architecture as shown in Fig 4. Here, the coupling strength κ between the networks was manipulated changing the E→I and E→E values. The input drive difference was kept the same (Δ1.5mV). B) Example (SNR = ~10) of EE- strength manipulation (0 to 0.02mV) with a fixed IE strength of 0.02mV. The top-plot shows the information flow as measured by transfer entropy (TE), combined for both directions as a function of EE-strength. Middle- and lower panels show the same for coherence2 and PLV estimate respectively. For this particular combination the coh2 estimate behaved oppositely to PLV2 and information flow. (C-F) Surface plots representing effects of all combinations of E→I and E→E strengths on information flow (C), coh2 (D), PLV2 (E) and PrAM strength (F). A comparison of surface plots reveals that coherence not only reflects changes in PLV or information flow with coupling manipulations, but also changes in PrAM. (G-H) Variance in coh2 and PLV2 explained by PrAM (G) and by information flow (TE) (H) plotted as a function of SNR. Changes in coh2 values predominantly reflected changes in PrAM (G), yet explained little variance in information flow between networks with coupling manipulations (H). The opposite was true for PLV2.