Tensorpac: An open-source Python toolbox for tensor-based phase-amplitude coupling measurement in electrophysiological brain signals.

Estimation process of a corrected 10↔100hz PAC.For illustration, here simulated raw data contains a coupling between a 10 hz phase and a 100 hz amplitude. First, the raw data is respectively filtered with bandpass filters centered on 100hz and 10hz. Then, the complex analytic form of each signal is obtained using the Hilbert transform. The phase is extracted from the 10Hz signal (angle of analytic signal) and power from the 100Hz signal (amplitude of analytic signal). An uncorrected PAC measure is obtained from these two signals. To estimate the null distribution of the measure in the absence of any genuine coupling, the amplitude signal is split into two blocks at a random time point and the temporal order of those two blocks is swapped. Then, the PAC is estimated using this swapped version of amplitude and the originally extracted phase. By repeating this process and cutting at a random point, for example 200 times, we can obtain a distribution of surrogate values for which there is no genuine coupling. Finally, a corrected PAC estimate is obtained through z-score normalization of the uncorrected PAC using this distribution.

Synthetic signals to simulate phase-amplitude coupling.Illustration of synthetic signals that can be generated to simulate phase-amplitude coupling (left column) and associated comodulogram (right column). A: First row shows an example of a signal that contains a 5↔120hz coupling defined as proposed by Tort et al. [46]. B: Bottom signal also contains a 5↔20hz coupling but defined as proposed in Dupré la Tour et al. [41].

Comparison of the main PAC methods currently implemented in Tensorpac.The comodulograms were computed using the A: MVL, B: MI, C: HR, D: ndPac, E: PLV and F: gcPAC, from 20 trials of simulated data containing a 10↔100hz phase-amplitude coupling. The data is available in Tensorpac and can be used to validate and benchmark other methods.

Comparison between corrected and uncorrected PAC.A: PAC comodulogram is computed for several (phase, amplitude) pairs. B: For each of those pairs, we estimate the distribution of surrogates and plot the mean comodulogram of these permutations. Note that both uncorrected and surrogate PAC comodulograms exhibit a spurious peak in the very low frequency phase. C: The true 10↔100hz coupling is finally retrieved by subtracting the mean of the surrogate distribution (panel B) from the uncorrected PAC (panel A).

Example of event-related phase-amplitude coupling (ERPAC).We first generate 300 one-second trials each containing a 10↔100hz coupling. Next, one-second of random noise is appended to these signals. The depicted A: ERPAC and B: gcERPAC represents time-resolved PAC estimation over the two-second window, computed with a phase between [9, 11] hz and for multiple amplitudes.

Estimation of the preferred phase.Illustrative example of the preferred phase estimation based on 100 trials generated with coupling between 6hz↔100hz and where the amplitude is locked to the 6hz phase at 45° (π/4). A: The 100 hz amplitude is first binned according to the phase using 18 slices of 20° each. The sum of the amplitude inside each slice is plotted as a histogram and the preferred phase is identified as the phase for which the amplitude is maximum (45°). B: An alternative polar visualization available in Tensorpac displays the strength across multiple amplitude frequency bands. Phase is binned as before, but now multiple amplitude signals from different bands are calculated for each phase bin. In these polar plots, the angle represents the phase of the low-frequency (here 6Hz), and the radial axis represents different frequencies considered for the amplitude signal. The color depicts the average value of the amplitude of a given frequency inside the corresponding phase bin. The preferred 45° phase for the 6hz↔100hz PAC is clear in this representation.

Investigation of the presence of a phase peak and data-driven exploration of the optimal bandwidth.In this example, we first generated a 6↔70hz phase-amplitude coupling. A: The PSD retrieves the presence of the phase peak around 6hz. B: For a fixed phase filtered in [5, 7] hz, we search for the optimal amplitude band, defined as the bandwidth for which the PAC is maximum. The triangular freq-freq representation depicts coupling strength across many possible combinations of amplitude frequency bounds, where the x-axis corresponds to the starting frequency and the y-axis to the ending frequency. Here, the PAC is maximum for an amplitude range of [61, 79] hz.

Computation time of the vector and tensor-based implementations.We first generate a relatively small dataset composed of 100 trials of 3000 time points each. We then evaluated the comodulogram by extracting 26 phases and 24 amplitudes. The comodulogram is then either computed using one-dimensional time-series (vector-based) or directly using multidimensional arrays (tensor-based). Computing time is compared as a function of PAC method (A) or as a ratio where the computing time using tensors is divided by the one using vectors (B).