Detection of quadratic phase coupling by cross-bicoherence and spectral Granger causality in bifrequencies interactions

Quadratic Phase Coupling (QPC) serves as an essential statistical instrument for evaluating nonlinear synchronization within multivariate time series data, especially in signal processing and neuroscience fields. This study explores the precision of QPC detection using numerical estimates derived from cross-bicoherence and bivariate Granger causality within a straightforward, yet noisy, instantaneous multiplier model. It further assesses the impact of accidental statistically significant bifrequency interactions, introducing new metrics such as the ratio of bispectral quadratic phase coupling and the ratio of bivariate Granger causality quadratic phase coupling. Ratios nearing 1 signify a high degree of accuracy in detecting QPC. The coupling strength between interacting channels is identified as a key element that introduces nonlinearities, influencing the signal-to-noise ratio in the output channel. The model is tested across 59 experimental conditions of simulated recordings, with each condition evaluated against six coupling strength values, covering a wide range of carrier frequencies to examine a broad spectrum of scenarios. The findings demonstrate that the bispectral method outperforms bivariate Granger causality, particularly in identifying specific QPC under conditions of very weak couplings and in the presence of noise. The detection of specific QPC is crucial for neuroscience applications aimed at better understanding the temporal and spatial coordination between different brain regions.


S1. Signal-to-noise ratio : SNR
For a signal   (), the average (normalized) power is given by (Grami, 2015): Let  represent a sample i.e. realization of stochastic process   ( = 1, 2, 3) generating time series of our three-channel model of bifrequencies interaction.For any independent white noise signal   (, ) following a standard Gaussian distribution (mean  = 0, standard deviation  = 1), the average power is    = 1.For each channel   (, ), the power of the signal (without noise) is  (  −  ) and the signal-to-noise ratio (SNR) values can be computed as .
The initial phase   =   () of each channel's driving signal is sampled from an independent uniform distribution on period [0, 2).A signal's power and its SNR are positive real-valued random variables, depending on   in general.
The signal recorded by channel 1 is defined by: (1) The average power of channel 1 is The signal recorded by channel 2 is defined by: (3) The average power of channel 2 is  ( 2 − 2 ) = 1, because  2 (, ) 2 = 1 for any  2 > 0 and  2 ∈ [0, 2).Hence, The signal recorded by channel 3 is defined by: The average power of channel 3 is computed as follows where Because | 2 | = 1, we have The white noise signals   () follow a standard Gaussian distribution (mean  = 0, standard deviation  = 1), such that Then, equation ( 9) can be rewritten as Notice that   1 =  2 3 according to equation (1) and 1 and  2 2 are mutually independent.Hence, we obtain For each  3 > 0 , () = 0 with probability 1 as   () is uniformly distributed in [0, 2).According to equation ( 5),   3 = 1 2 .Hence, equation ( 6) can be rewritten as which leads to the signal-to-noise ratio for channel 3 equal to

S2. A simpler three-channel model of bifrequencies interaction
The main body of the paper describes a model with three channels that incorporates both triangular and rectangular waveforms.This design aims to achieve a complex power distribution in the model, resembling the patterns commonly observed in natural phenomena.Both triangular and rectangular waveforms consist exclusively of odd harmonics.Specifically, rectangular waves are characterized by odd harmonics of the form 2 + 1, where the amplitude is proportional to to 1∕(2 + 1).Triangular waves also consist of odd harmonics, but their amplitude is proportional to 1∕(2 + 1) 2 .
A simplified three-channel model of bifrequency interactions, represented by sinusoidal waves  ′ 1 (, ) and  ′ 2 (, ), possesses the same power as the functions  1 (, ) and  2 (, ) used in the main body of the paper.This model maintains the same signal-to-noise ratio (SNR) for each channel and enables investigation into the extent to which the harmonics of the carrier waves in the input channels can affect the detection of quadratic phase coupling, according to the metrics presented in the main text.Therefore, the simplified three-channel model of bifrequency interactions is defined by the following functions: We have conducted an experiment identical to that described in the main text, by merging 128 independent epochs, with each epoch corresponding to a 5-second interval sampled at 100 Hz.This was tested under the same conditions, using the same set of frequencies and the same coupling strengths  (1,2) between interacting channels  ′ 1 and  ′ 2 .Following the procedure of the main study, for all 354 experimental conditions, we computed the  BQPC (i.e., the ratio of bispectral quadratic phase coupling) and the  GQPC (i.e., the ratio of bivariate Granger causality quadratic phase coupling).
We observed that the coupling strength  (1,2) influenced the metrics used to detect QPC (Figure S1, Figure S2).The results were very similar to those obtained using the original triangular and rectangular periodic functions.This comparison is evident when contrasting Figure 4 of the main text with Figure S1 in the supplementary materials.Moreover, even with the oversimplified model signals, the impact of the coupling strength between the interacting channels mirrors the observations made with the original signals discussed in the main paper.We noted a similar discrepancy between the experimental and analytical values of signal-to-noise ratios in channel  ′ 3 .This discrepancy became more pronounced at lower values of  (1,2) , accompanied by a broader dispersion in the ratio between experimental and corresponding analytical values, as seen when comparing Figure S2A with Figure 5A in the main text.Additionally, with the simplified model signals, the bispectral method proved to be superior to bivariate Granger causality for detecting QPC, as demonstrated by the comparison of Figure S2B to compare with Figure 5B in the main text.
In summary, the results obtained with simplified sinusoidal periodic functions are highly consistent with those obtained using triangular and rectangular waveforms as presented in the main text.This observation indicates that the waveform of the input signals plays a minimal role in the measurement of QPC based on the metrics newly described in the main text.These findings suggest that the assessment of QPC by  BQPC and  GQPC is largely unaffected by the waveforms of the input signals.This could represent a significant advantage for applications in time series analysis obtained from biological contexts, particularly in the study of neural signals.