Spurious cross-frequency amplitude-amplitude coupling in nonstationary, nonlinear signals

Abstract

Recent studies of brain activities show that cross-frequency coupling (CFC) play an important role in memory and learning. Many measures have been proposed to investigate the CFC phenomenon, including the correlation between the amplitude envelopes of two brain waves at different frequencies - cross-frequency amplitude-amplitude coupling (AAC). In this short communication, we describe how nonstationary, nonlinear oscillatory signals may produce spurious cross-frequency AAC. Utilizing the empirical mode decomposition, we also propose a new method for assessment of AAC that can potentially reduce the effects of nonlinearity and nonstatonarity and, thus, help to avoid the detection of artificial AACs. We compare the performances of this new method and the traditional Fourier-based AAC method. We also discuss the strategies to identify potential spurious AACs.

Keywords: amplitude-amplitude coupling; cross-frequency coupling; empirical mode decomposition; modulation index; phase-amplitude coupling.

Diag. 1

The procedures of the two methods for the assessment of cross-frequency amplitude-amplitude coupling. (A) The spectral cross-frequency comodulation analysis (SCFCA). (B) The intrinsic mode amplitude-amplitude coupling (IMAAC). FT: Fourier transform; PS: power spectrum; CFPC: cross-frequency power correlation; AACC: amplitude-amplitude coupling comodulogram; EEMD: ensemble empirical mode decomposition method; IMF: intrinsic mode function; HT: Hilbert transform; PAC: phase-amplitude coupling;

Fig. 1

Demonstration of the intrinsic mode amplitude-amplitude coupling (IMAAC) analysis. (A) A synthetic signal with 10-Hz and 40-Hz oscillations whose amplitudes are simultaneously modulated by a 2-Hz sinusoidal wave. (B) Two intrinsic mode functions (IMFs) of the signal in A and their instantaneous amplitudes (envelopes). IMFs are obtained using EEMD. IMF1 and IMF2 are the 40-Hz and 10-Hz oscillatory components in the raw data, respectively. The envelope of each IMF is obtained using the Hilbert transform, i.e., Envelope 1 for IMF 1 and Envelope 2 for IMF 2. (C) Two IMFs of Envelope 2 in B. The first IMF corresponds to the 2-Hz oscillation. (D) The phase-amplitude distribution (PA distribution) between the first IMF (2Hz) and second IMF of Envelope 2 (phase) and Envelope 1 (amplitude). (E) The modulation indices (MIs) is calculated based on the PA distributions in D, and the MI between the IMF 1 and IMF 2 is weight-averaged according to the standard deviations (SD) of Envelope 2`s IMFs (see C). The MI will be assigned to a frequency-frequency plane at the coordinate 10 Hz x 40 Hz because the frequencies of IMF 1 and IMF 2 in all cycles are 40 Hz and 10 Hz, respectively.

Fig.2

Construction of nonlinear waveforms with different degrees of asymmetry. (A) An example of asymmetric wave (target period (TP) = 0.1 sec) is created using two sinusoidal waves. We use Part I (from [-π/2 0]) and Part ? (from [-π -π/2]) for the one with period 2 (p2) = 0.16 sec, and Part ? in the other one with period 1 (p1) = 0.04 sec. (B) We fix TP as 0.1 sec (or target frequency (TF) = 10 Hz) and change the period ratio (p2/p1) to control the degree of asymmetry. The sawtooth wave is the extreme case of asymmetric wave.

Fig.3

A synthetic signal with nonstationary, nonlinear oscillations. Nonlinear waveforms are created based on the prototype in Fig. 2 with p2/p1 = 4. The amplitude of each cycle is a random number in the range of 0-4. The frequency of all cycles is 10 Hz. The synthetic signal also has a component of Gaussian white noise with the standard deviation equal to 10% of the oscillatory component.

Fig.4

SCFCA results of synthetic signals with asymmetric 10-Hz oscillation and nonstationary envelope. (A-D) Multitaper power spectra (PS) and (E-H) AAC comodulograms (AACCs) for signals with different degrees of asymmetry: (A, E) p2/p1 = 1, (B, F) p2/p1 = 4, (C, G) p2/p1 = 9, and (D, H) sawtooth. AACCs are obtained from cross-frequency correlation of PS in (A-D). For all signals, the nonstationary envelope is the same as shown in Fig. 3. The nonlinearity associated with the asymmetrical waveform leads to powers at harmonics, the nonstationarity causes the smearing of power densities at the fundamental and harmonic frequencies.

Fig.5

IMACC results of the synthetic signals with asymmetric oscillation and nonstationary envelope. (A-D) IMFs (black lines) and their instantaneous amplitudes or envelops (red lines), and (E-H) AACC of the same signals used in Fig. 4. (A, E) p2/p1 = 1, (B, F) p2/p1 = 4, (C, G) p2/p1 = 9, and (D, H) sawtooth.

Fig.6

Comparison of IMAAC results for signals with spurious and true AACs. (A, B) IMFs and(C, D) AACC of (A) a sawtooth wave (10 Hz) without AAC and (B) a synthetic signal with AAC between two sinusoids (10 Hz and 40 Hz). In both signals, the amplitudes of oscillations are modulated by the same arbitrary nonstationary amplitude.