Fig 1.
Example of a preprocessed signal.
The driver x is extracted from the raw signal z with a bandpass filter, and then subtracted from z to give y. A PAC effect is present as we see stronger high frequency oscillations at the peaks of the driver. The signal y is presented here as it looks before the temporal whitening step.
Fig 2.
Example on simulated signals: Pipeline, signal, conditional PSD, and comodulogram.
(a) Pipeline of the method. We applied it with (p, m) = (10, 1) on two simulated signals: (b) Simulated signal with PAC and (c) Simulated signal without PAC. (d) From a fitted model, we computed the PSD conditionally to the driver’s phase ϕx; each line is centered independently to show amplitude modulation. PAC can be identified in the fluctuation of the PSD as the driver’s phase varies: around 50 Hz, the PSD has more power for one phase than for another. This figure corresponds to a single driver’s frequency fx = 3.0 Hz. (e) Applying this method to a range of driver’s frequency, we build a comodulogram, which quantifies the PAC between each pair of frequencies.
Fig 3.
Preferred phase and temporal delay are different.
The temporal delay τ is distinct from the preferred phase ϕ0. (a) When both are equal to zero, the high frequency bursts happen in the driver’s peaks. (b) When τ = 0 and ϕ ≠ 0, the bursts are shifted in time with respect to the driver’s peaks, and this shift varies depending on the instantaneous frequency of the driver. (c) When τ ≠ 0 and ϕ = 0, the bursts are shifted in time with respect to the driver’s peaks, and this shift is constant over the signal. In this case, note how the driver’s phase corresponding to the bursts varies depending on the instantaneous frequency of the driver. (d) τ and ϕ0 can also be both non-zero. (e-h) Negative log-likelihood of DAR models, fitted with different delays between the driver and the high frequencies. The method correctly estimates the delay even when ϕ0 ≠ 0. (i-l) PSD conditional to the driver’s phase, estimated through a DAR model with the best estimated delay. The maximum amplitude occurs at the phase ϕ0.
Fig 4.
Driver’s frequency and bandwidth selection (simulations).
(a,b) Two examples of grid-search over both fx and Δfx, over simulated signals with (a) Δfx(simu) = 0.4 Hz and (b) Δfx(simu) = 1.6 Hz. For all bandwidths (except 6.4 Hz), the negative log-likelihood is minimal at the correct frequency fx = 4 Hz. (c) To see more precisely the bandwidth estimation, we plot the negative log-likelihood (relative to each minimum for readability), for several ground-truth bandwidth Δfx(simu), with fx = 4 Hz. For each line, the minimum correctly estimates the ground-truth bandwidth (depicted as a diamond), showing empirically that the parameter selection method gives satisfying results.
Fig 5.
Driver’s frequency and bandwidth selection (real signals).
(a,c,d) Negative log-likelihood of the fitted model for a grid of filtering parameters fx and Δfx: (a) rodent striatum, (c) rodent hippocampus, (d) human auditory cortex. The optimal bandwidth was very large (3.2 Hz), and the optimal center frequency changed as the bandwidth increased, suggesting that the optimal driver had a wide asymmetrical spectrum. (b) This portion of the rodent striatal signal shows two examples of driver with different bandwidths: The wide-band driver better follows the raw signal, and independently also leads to a better fit in DAR models.
Fig 6.
PSD conditional to the driver’s phase.
Dataset: Rodent striatum (a,d), rodent hippocampus (b,e), human auditory cortex (c,f). We derive the conditional PSD from the fitted DAR models. In one plot, each line shows at a given frequency the amplitude modulation with respect to the driver’s phase. The driver bandwidth Δfx is 0.4 Hz (top row) and 3.2 Hz (bottom row). Note that the maximum amplitude is not always at a phase of 0 or π (i.e. respectively the peaks or the troughs of the slow oscillation). In figure (d), we can also observe that the peak frequency is slightly modulated by the phase of the driver (phase-frequency coupling).
Fig 7.
Comodulograms with parameters optimizing the likelihood.
Dataset: rodent striatum (top), rodent hippocampus (middle) and human auditory cortex (down). Methods, from left to right: [Ozkurt et al. 2011] [26], [Penny et al. 2008] [22], [Tort et al. 2010] [23], DAR models (ours). The DAR model parameters and the driver bandwidth are chosen to be optimal with respect to the likelihood. White lines outline the regions with p < 0.01.
Fig 8.
Comodulograms with parameters optimizing the comodulgoram sharpness.
Dataset: rodent striatum (top), rodent hippocampus (middle) and human auditory cortex (down). Methods, from left to right: [Ozkurt et al. 2011] [26], [Penny et al. 2008] [22], [Tort et al. 2010] [23], DAR models (our). The driver bandwidth is chosen to have a well defined maximum in driver frequency: Δfx = 0.4 Hz. The DAR model parameters are chosen to give similar results than the other methods: (p, m) = (10, 2). White lines outline the regions with p < 0.01.
Fig 9.
Frequencies of the maximum PAC value, with four methods: a DAR model with (p, m) = (10, 1), the GLM-based model from Penny et al. [22], and two non-parametric models from Tort et al. [23] and Ozkurt et al. [26]. Each point corresponds to one signal out of 200. Kernel density estimates are represented above and at the right of each scatter plot. From left to right, the simulated signals last T = 2, 4 and 8 seconds. The signals are simulated with a PAC between 3 Hz and 50 Hz. The DAR models correctly estimate this pair of frequency even with a short signal length, as well as the GLM-based metric [22], while the two other metrics [23, 26] are strongly affected by the small length of the signals, and do not estimate the correct pair of frequency.
Fig 10.
Estimating delays over 20 simulations for each delay.
The delays are estimated on both direct time (a) and reverted time (b). The delays are normalized: τ = 1 corresponds to one driver oscillation, i.e. 1/fx sec. The modified CFD shows a bias and only serves for qualitative comparison. The delay estimation based on DAR models correctly estimates the delays, with no visible bias.
Fig 11.
Estimated delays on rodent and human datasets.
(a) Negative log likelihood for multiple delays, with the human cortical signal. (b) Optimal delay for the three signals, computed with model selection on DAR models, and with a CFD method modified to provide a delay. The error bars indicate the standard deviation obtained with a block-bootstrap strategy. For the rodent hippocampal data, the delay is positive: the low frequency oscillation precedes the high frequency oscillation. For the human cortical data and the rodent striatal data, the delay is negative: The high frequency oscillation precedes the low frequency oscillation.
Fig 12.
Comodulograms obtained on a simulated signal with spurious PAC.
Spurious PAC was generated using a spike train at 10 Hz, as described in [94]. All four methods, including the proposed one, detect some significant PAC, even though there is no nested oscillations.