Cross-Modulated Amplitudes and Frequencies Characterize Interacting Components in Complex Systems

The dynamics of complex systems is characterized by oscillatory components on many time scales. To study the interactions between these components we analyze the cross modulation of their instantaneous amplitudes and frequencies, separating synchronous and antisynchronous modulation. We apply our novel technique to brain-wave oscillations in the human electroencephalogram and show that interactions between the $\alpha$ wave and the $\delta$ or $\beta$ wave oscillators as well as spatial interactions can be quantified and related with physiological conditions (e.g., sleep stages). Our approach overcomes the limitation to oscillations with similar frequencies and enables us to quantify directly nonlinear effects such as positive or negative frequency modulation.

Figure 1

Analysis of cross modulation in EEG recordings. The raw data (b) is decomposed into $x_k^{\alpha }$ [black in (a)] and $x_k^{{\delta }_1}$ [in (c)], $t_k=k\times 0.005\mathrm{s}$ (sampling rate 200 Hz). Amplitudes $A_k^{\alpha }$ and $A_k^{{\delta }_1}$ [Eq. (1), blue in (a),(c)] are reconstructed using the Hilbert transformed signals ${\tilde{x}}_k^{\alpha }$ and ${\tilde{x}}_k^{{\delta }_1}$ [red]. Next, phases ${\phi }_k^{\alpha }$ and ${\phi }_k^{{\delta }_1}$ [purple in (d),(f)] of amplitude oscillations [blue in (d),(f); shown parts differ from those in (a),(c)] are calculated. Finally, complex exponentials [inset in (e)] of the phase differences ${\phi }_k^{\alpha }-{\phi }_k^{{\delta }_1}$ [green in (e)] are averaged yielding ${\gamma }_{\nu }$ for the 30 s segment.

Figure 2

(a) Instantaneous $\alpha$-band frequency (blue) and filtered version $f_k^{\alpha }$ (red). (b) Results of long-term correlation analysis of $f_k^{\theta }$ (red, representative for other bands except $\alpha$) and $f_k^{\alpha }$ (green) versus physiological state. The bars show the average scaling exponents $H$ calculated with CMA (equivalent with DFA1) and the standard errors of the means.

Figure 3

Results of modulation analysis of instantaneous amplitudes (a) and frequencies (d) as well as amplitude-frequency cross-modulation analysis (b),(c). The four $36\times 36$ matrices show the modulation coefficients ${\gamma }_{+}$ [(c) and upper triangles in (a),(d)] and ${\gamma }_{-}$ [(b) and lower triangles in (a),(d)] with modulation degrees increasing from blue to green, yellow and red. The six first rows and columns refer to the ${\delta }_1$ band of the electrodes O1-M2 (occipital left), O2-M1 (occipital right), C3-M2 (central left), C4-M1 (central right), Fp1-M2 (frontal left), and Fp2-M1 (frontal right); repeated for each band. The color code is modified in (a), increasing the values in the lower triangle by a factor of 8 and decreasing them in the upper triangle by 2 to enhance details.