Wavelet bispectral analysis for the study of interactions among oscillators whose basic frequencies are significantly time variable

Bispectral analysis, recently introduced as a technique for revealing time-phase relationships, is extended to make use of wavelets rather than Fourier analysis. It is thus able to encompass instantaneous phase-time dependence for the case of two or more coupled nonlinear oscillators. The method is demonstrated and evaluated by use of test signals from a pair of coupled Poincaré oscillators. It promises to be useful in a wide range of scientific contexts for studies of interacting oscillators whose basic frequencies are significantly time variable.

Figure 1

Variation of the lengths $N_W$ of Morlet-like wavelets with frequency (scale): Morlet wavelet (solid line); adapted Morlet wavelet (dotted line); and fixed wavelet length for high frequencies (dash-dotted line)

Figure 2

Simulation results for a pair of quadratically coupled Poincaré oscillators in the presence of additive Gaussian noise. (a) The test signal $x_{1A}\left(t\right)$ from the first oscillator, after normalization and subtraction of its mean value. The first oscillator has a characteristic frequency $f_1=1.1\text{Hz}$. That of the second oscillator is $f_2=0.25\text{Hz}$. The oscillators are unidirectionally and quadratically coupled with three different coupling strengths: in column (1) ${\eta }_2=0.2$, (2) 0, and (3) 0.2. In column (3) the characteristic frequency of the second oscillator is linearly increased from $f_2=0.25$ to $0.35$ while being at the same time modulated with $0.0013\text{sin}\left(2\pi 0.01t\right)$. Each coupling lasts for 400 s. The sampling frequency $f_s=10\text{Hz}$. Only the first 15 s are shown in each case. (b) The corresponding power spectra of $x_{1A}\left(t\right)$.

Figure 3

Instantaneous frequencies of (a) the first oscillator $f_1$ and (b) the second oscillator $f_2$, used for generating the test signal $x_{1A}$.

Figure 4

(a) The modulus of the wavelet bispectrum $\left|B_W\right|$ calculated for $K=34$ segments, 67% overlapping, with $T_m=8\text{s}$, $G_e=0.001$, using a fixed Morlet wavelet length of $T_{HF}=40\text{s}$ for calculation of the high frequencies. (b) A contour plot of the WB. Above $f_2>0.9\text{Hz}$, the wavelet bispectrum is removed because the triplet (1.1 Hz, 1.1 Hz, 1.1 Hz) produces a high peak that is not physically significant. (c) The biphase ${\varphi }_1$ and (d) the biamplitude $A_1$ for the bifrequency (1 Hz, 0.25 Hz) peak 1, calculated using a 0.1 s time step. These calculations make no allowance for variations in instantaneous frequencies.

Figure 5

Wavelet bispectral estimates based on the instantaneous frequencies $f_1\left(t\right)$ and $f_2\left(t\right)$, which were obtained using the marked events method. (a) The instantaneous biphase ${\varphi }_1$ and (b) the instantaneous biamplitude $A_1$ for the bifrequency (1.1 Hz, 0.25 Hz) peak 1, calculated for $K=34$ segments, 67% overlapping, with $T_m=8\text{s}$, $G_e=0.001$, using a fixed Morlet wavelet length of $T_{HF}=40\text{s}$ for the calculation of high frequencies and a 0.1 s time step.

Figure 6

Simulation results for time-intermittent quadratic couplings in the presence of additive Gaussian noise. (a) The test signal $x_{1B}\left(t\right)$ from the first oscillator, with characteristic frequency $f_1=1.1\text{Hz}$. The characteristic frequency of the second oscillator was $f_2=0.24\text{Hz}$. The oscillators are unidirectionally and quadratically coupled with two different coupling strengths: column (1) ${\eta }_2=0$; and (2) ${\eta }_2=0.2$. The coupling in column (2) is present every 20 s and lasts for 20 s. The signal is 1200 s long and sampled with sampling frequency $f_s=10\text{Hz}$. Only the first 15 s are shown in each case. (b) The corresponding power spectrum of $x_{1B}\left(t\right)$.

Figure 7

Analyses of the test signal with time-intermittent quadratic couplings, in the presence of additive Gaussian noise using the Fourier-based bispectral method. (a) The Fourier-based bispectrum $\left|B_F\right|$ for signal $x_{1B}$ calculated with $K=33$ segments, 66% overlapping and using the Blackman window to reduce leakage and (b) its contour view. The part of the bispectrum above $f_2>1.0\text{Hz}$ has been cut because the triplet (1.1 Hz, 1.1 Hz, 1.1 Hz) produces a high peak that is physically meaningless. (c) The biphase ${\varphi }_1$. (d) The biamplitude $A_1$ for the bifrequency (1.1 Hz, 0.24 Hz), calculated with a 100-s-long window for estimating DFTs. (e) The biphase and (f) the biamplitude calculated with a 130 s window for estimating DFTs. In both cases, a 0.3 s time step was used and the Blackman window was applied.

Figure 8

Analyses of a test signal for time-intermittent quadratic couplings in the presence of additive Gaussian noise using the wavelet bispectral method. (a) The wavelet bispectrum $\left|B_W\right|$ calculated with $K=33$ segments and 66% overlapping. The part of the bispectrum above $f_2>1.0\text{Hz}$ is removed, because the triplet (1.1 Hz, 1.1 Hz, 1.1 Hz) produces a high peak that is physically meaningless. (c) The biphase $\varphi$ and (d) biamplitude $A_1$ for bifrequency (1.1 Hz, 0.24 Hz), calculated with $G_e=0.01$. (e) The biphase $\varphi$ and (f) biamplitude $A_1$, calculated with $G_e=0.0001$. In both cases, the time step was 0.1 s, $T_m=8\text{s}$, and a $T_{HF}=20\text{s}$ fixed Morlet wavelet length was used for estimating high frequencies.