Fig 1.
Electronic circuit of a pair of van der Pol oscillators and recorded electric potential.
(a) Schematic of electronic circuit of two coupled van der Pol oscillators, where xi and yi are positions for recording electric potential, Rk denotes resistors, and Ci denotes condensers. Electronic units U1 and U2 represent the multiplier and operational amplifiers, respectively. Rcoupling is a resistor whose resistance is the parameter of the strength of connectivity. (b) Experimental data of electric potentials x1 and y1 show the limit-cycle oscillator under the same-frequency (129.1 Hz) coupling condition (gray dots and line). The black trajectory shows the theoretical value computed by the van der Pol oscillator Eqs (15–18). Here, the frequency is 142.1 Hz. Blue dots represent the zero-phase reference points on the experimental data, which were determined automatically via Hilbert transformation. Green crosses represent the theoretical zero-phase reference points defined as the peak points of xi. Red dots denote the adjusted zero-phase reference points. (c) x2 and y2 show the oscillators under the same-frequency oscillator condition. The frequency of the experimental data is 132.5 Hz and that of the theoretical trajectory is 146.4 Hz. (d) Recorded electric potentials show the slow limit-cycle oscillator under cross-frequency coupling conditions (experimental frequency, 64.1 Hz; theoretical frequency, 71.1 Hz). (e) x2 and y2 denote the fast oscillator (experimental frequency, 131.1 Hz; theoretical frequency, 146.4 Hz).
Fig 2.
Syllable and prosody rhythms in speech sound.
(a) Example of speech stimulus. The stimulus consisted of noise and a four-syllable Japanese word. The red line represents a speech wave. The blue line represents the presented sound wave, which consists of speech plus noise sounds. (b) Speech envelope was computed as the absolute value of Hilbert-transformed speech sound. (c) Syllabic rhythms were computed from the speech envelope through the bandpass filter within 3–6 Hz. (d) Prosodic rhythms were computed from the speech envelope through the bandpass filter within 1–3 Hz.
Fig 3.
Estimated coupling function for numerical simulation data.
Upper-left diagram shows the network structure. The estimated coupling functions (red lines) were nearly identical to the correct functions (dashed black line). The gray dots represent the phase time-series data. When the interaction did not exist, the estimated coupling function was identically zero. The proposed method estimated all coupling functions correctly for the simulation data.
Fig 4.
Estimated coupling function of electronic circuit.
(a) The diagram shows the coupling direction between oscillators of the same frequency. The first oscillator was coupled to the second oscillator. (b) The red line shows the estimated phase coupling function with the natural frequency in the same-frequency coupling case. The dashed black line shows the theoretical coupling function. The coupling function from the second to first oscillator Γ12 is identically zero. When there is no interaction, the coupling function is nearly zero. The gray dots show the experimental data points. (c) The coupling functions from the first to second oscillator Γ21. (d) The blue line shows the phase difference histogram of the experimental data in the case of 1:1 phase locking (experimental histogram). The red line shows the simulated histogram calculated in the phase oscillator model estimated from the experimental data (estimated histogram). The dashed black line shows the simulated histogram calculated in the phase oscillator model using the theoretical natural frequencies and coupling functions (theoretical histogram). (e) In the cross-frequency coupling case, the slow oscillator was coupled to the fast oscillator. (f) The coupling function from the fast to slow oscillator is identically zero. (g) The coupling function from the slow to fast oscillator. (h) The experimental, estimated, and theoretical histogram in the 1:2 phase-locking case.
Fig 5.
Estimated distribution of phase difference between EEG data and syllable envelope.
(a) Experimental histogram of phase difference between the theta oscillation on the Cz electrode and the syllabic rhythm (the histograms show phase locking). The gray lines represent histograms of individual participants and the blue line represents the histogram averaged over all participants. (b) Histograms obtained from the simulated data in the estimated phase oscillator model. The averaged histogram is similar to the experimental histogram. The gray lines represent the phase difference histograms of individual participants. The red line represents the average of the simulated histograms. (c) Blue lines represent the averaged experimental histogram and the standard error of mean (SEM). Red lines represent the averaged simulated histograms and the SEM. (d) Estimated coupling functions Γθ,s from syllabic rhythm to theta oscillation. The gray and red lines represent the results of individual participants and the average results of all participants, respectively. (e) Estimated coupling functions Γs,θ are considerably smaller than the opposite directional coupling functions. (f) Simulated histograms where the coupling functions Γs,θ are removed. The effect on the original phase-locking state was negligible. (g) Histograms where Γθ,s were removed are nearly flat.
Fig 6.
Estimated distribution of phase difference between EEG data and prosody envelope.
(a) Experimental phase difference histograms for 1:2 phase locking. (b) Simulated histograms based on the estimated phase oscillator model. (c) Blue lines represent the average and SEM of experimental phase difference histograms. Red lines represent the average and SEM of simulated histograms. (d) Estimated coupling functions Γθ,p. (e) Estimated coupling functions Γp,θ. (f) Simulated histograms where coupling functions Γp,θ are removed. (g) Simulated histograms where Γθ,p is removed are uniform.