冠動脈バイパス術患者における術前アスピリン投与中止時期の検討

研究科: 千葉大学大学院医学薬学府学位:千大院医薬博甲第医1081号博士(医学)千葉大

A dynamical systems approach for estimating phase interactions between rhythms of different frequencies from experimental data

<div><p>Synchronization of neural oscillations as a mechanism of brain function is attracting increasing attention. Neural oscillation is a rhythmic neural activity that can be easily observed by noninvasive electroencephalography (EEG). Neural oscillations show the same frequency and cross-frequency synchronization for various cognitive and perceptual functions. However, it is unclear how this neural synchronization is achieved by a dynamical system. If neural oscillations are weakly coupled oscillators, the dynamics of neural synchronization can be described theoretically using a phase oscillator model. We propose an estimation method to identify the phase oscillator model from real data of cross-frequency synchronized activities. The proposed method can estimate the coupling function governing the properties of synchronization. Furthermore, we examine the reliability of the proposed method using time-series data obtained from numerical simulation and an electronic circuit experiment, and show that our method can estimate the coupling function correctly. Finally, we estimate the coupling function between EEG oscillation and the speech sound envelope, and discuss the validity of these results.</p></div

Electronic circuit of a pair of van der Pol oscillators and recorded electric potential.

<p>(a) Schematic of electronic circuit of two coupled van der Pol oscillators, where <i>x</i>_<i>i</i> and <i>y</i>_<i>i</i> are positions for recording electric potential, <i>R</i>_<i>k</i> denotes resistors, and <i>C</i>_<i>i</i> denotes condensers. Electronic units U₁ and U₂ represent the multiplier and operational amplifiers, respectively. <i>R</i>_{<i>coupling</i>} is a resistor whose resistance is the parameter of the strength of connectivity. (b) Experimental data of electric potentials <i>x</i>₁ and <i>y</i>₁ 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 (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005928#pcbi.1005928.e036" target="_blank">15</a>–<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005928#pcbi.1005928.e039" target="_blank">18</a>). 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 <i>x</i>_<i>i</i>. Red dots denote the adjusted zero-phase reference points. (c) <i>x</i>₂ and <i>y</i>₂ 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) <i>x</i>₂ and <i>y</i>₂ denote the fast oscillator (experimental frequency, 131.1 Hz; theoretical frequency, 146.4 Hz).</p

Estimated coupling function of electronic circuit.

<p>(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.</p

Estimated distribution of phase difference between EEG data and syllable envelope.

<p>(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 Γ_{<i>θ</i>,<i>s</i>} 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 Γ_{<i>s</i>,<i>θ</i>} are considerably smaller than the opposite directional coupling functions. (f) Simulated histograms where the coupling functions Γ_{<i>s</i>,<i>θ</i>} are removed. The effect on the original phase-locking state was negligible. (g) Histograms where Γ_{<i>θ</i>,<i>s</i>} were removed are nearly flat.</p

Syllable and prosody rhythms in speech sound.

<p>(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.</p

Estimated distribution of phase difference between EEG data and prosody envelope.

<p>(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 Γ_{<i>θ</i>,<i>p</i>}. (e) Estimated coupling functions Γ_{<i>p</i>,<i>θ</i>}. (f) Simulated histograms where coupling functions Γ_{<i>p</i>,<i>θ</i>} are removed. (g) Simulated histograms where Γ_{<i>θ</i>,<i>p</i>} is removed are uniform.</p

Measured data of acceleration perturbation for Impulse method

Measured data of the expriment with intermittent acceleration perturbations. Data contain the speed of treadmill and the leg motion of human subjects

Measured data of deceleration perturbation for Impulse method

Measured data of the expriment with intermittent deceleration perturbation. Data contain the speed of treadmill and the leg motion of human subjects

The ratio of the length of the double support phase to the length of the whole walking cycle.

<p>The bars show the average and standard deviation for each subject.</p