. 2019 Jan 15;185:361-378.
Epub 2018 Oct 18.
Measuring Transient Phase-Amplitude Coupling Using Local Mutual Information
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Measuring Transient Phase-Amplitude Coupling Using Local Mutual Information
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Here we demonstrate the suitability of a local mutual information measure for estimating the temporal dynamics of cross-frequency coupling (CFC) in brain electrophysiological signals. In CFC, concurrent activity streams in different frequency ranges interact and transiently couple. A particular form of CFC, phase-amplitude coupling (PAC), has raised interest given the growing amount of evidence of its possible role in healthy and pathological brain information processing. Although several methods have been proposed for PAC estimation, only a few have addressed the estimation of the temporal evolution of PAC, and these typically require a large number of experimental trials to return a reliable estimate. Here we explore the use of mutual information to estimate a PAC measure (MIPAC) in both continuous and event-related multi-trial data. To validate these two applications of the proposed method, we first apply it to a set of simulated phase-amplitude modulated signals and show that MIPAC can successfully recover the temporal dynamics of the simulated coupling in either continuous or multi-trial data. Finally, to explore the use of MIPAC to analyze data from human event-related paradigms, we apply it to an actual event-related human electrocorticographic (ECoG) data set that exhibits strong PAC, demonstrating that the MIPAC estimator can be used to successfully characterize amplitude-modulation dynamics in electrophysiological data.
Cross-frequency coupling; ECoG; Mutual information; Phase-amplitude coupling.
Copyright © 2018 Elsevier Inc. All rights reserved.
Figure 1:. Schematic of Z-space showing the k = 2 nearest-neighbors search for data point z
X and Y are represented in the horizontal and vertical axes respectively. The two nearest samples to z are i A and B. Red dotted lines show the distance in the marginal spaces from the samples A and B to the sample z. Sample i B is the k = 2 neighbor and its maximum marginal distance to the sample z ( i ε/2) defines the width of the marginal neighborhoods for X and Y, represented by the vertical and horizontal blue and orange stripes. Four sample points (1′, 2′, A, B) are within distance ε/2 in the marginal space of X ( n ( x i) = 4); six sample points (1, 2, 3, 4, 5, A) are within distance ε/2 in the marginal space of Y ( n( y i) = 6). Several other samples (unlabeled) are outside the k = 2 neighborhood.
Figure 2:. MIPAC from a simulated PAC signal with on/off boxcar coupling
Here the sampling rate of the simulated signal was
Srate = 500 Hz, the frequency of the carrier was f = 40 Hz and the frequency of the modulator was c f = 5 Hz. m (A): Coupling strength alternated between ‘on’ and ‘off’ across signal segments. (B): In the simulated signal: instantaneous low-frequency phase at 5 Hz (red), narrow-band high-frequency signal at 40 Hz (grey), and instantaneous high-frequency amplitude at 40 Hz (blue). (C): MIPAC estimated from the simulated signal for f = 40 Hz, amp f = 5 Hz. Time courses of estimated MIPAC (red) and local MI (only positive values are shown) before low-pass filtering to obtain the MIPAC (pink) are shown; both have the same units (nats). Overall MI between instantaneous phase and amplitude computed as the average local MI is shown in the legend. Latencies with statistically significant coupling estimates ( phase p < 0.05, uncorrected) appear in light grey.
Figure 3:. MIPAC estimates of simulated PAC signals with linearly increasing coupling.
(A): Linearly increasing ramps of coupling strength in a simulated PAC signal. (B): In the simulated signal, instantaneous phase (red) at 5 Hz, (grey) narrow-band signal at 40 Hz and (blue) instantaneous 40-Hz amplitude. (C): MIPAC estimate for the simulated signal f = 40 Hz, c f = 5 Hz. The (red) MIPAC estimate and (pink) local MI (only positive values are shown) before being low-pass filtered. Overall MI between instantaneous phase and amplitude computed as the average local MI is shown in the legend. Latencies with statistically significant coupling estimates ( m p < 0.05, uncorrected) appear in light grey.
Figure 4:. MIPAC time course estimate for a simulated PAC signal with absolute value of a sinusoid used as coupling.
(A): Absolute value of a sinusoid used as coupling strength in a simulated PAC signal. (B): In the simulated signal, instantaneous phase (red) at 5 Hz, (grey) narrow-band signal at 40 Hz and (blue) instantaneous amplitude at 40 Hz. (C): Time course of MIPAC estimated from the simulated signal with f = 40 Hz, c f = 5 Hz. The (red) MIPAC estimate and (pink) local MI (only positive values are shown) before being low-pass filtered. Overall MI between instantaneous phase and amplitude computed as the average local MI is shown in the legend. Latencies with statistically significant coupling estimates ( m p < 0.05, uncorrected) appear in light grey
Figure 5:. MIPAC estimation using Instantaneous MIPAC on simulated PAC signals with noise added.
In this simulation, the time course of MIPAC was estimated for the same signals simulated in Figs. 2–4 but with noise added (
SNR = 10). Estimated (red) MIPAC, and (pink) local MI before low-pass filtering are shown. Latencies with statistically significant coupling estimates ( p < 0.05, uncorrected) appear in light grey.
Figure 6:. Convergence in MIPAC estimation.
Percentage variance reduction (Δ
V arr) (blue) and MIPAC variance (red) as a function of the number of k-neighbors. The decreasing of values of Δ V ar below the threshold defined by Δ r V ar defines the convergence criteria for the MIPAC estimate. The three columns in the figure show the percentage variance reduction (Δ thresh V arr) (blue) and MIPAC variance for each of the three types of simulations shown earlier without noise (A-C) and with noise added (D-F)
Figure 7:. Relationship between MIPAC estimates and signal parameters.
Functional relationships between MIPAC estimates and parameters
k, S and rate SNR were assessed by computing the correlation of the estimated coupling time courses and Instantaneous MIPAC.
Figure 8:. Estimating MIPAC in a simulated noisy multi-trial PAC data set.
A set of 200 simulated PAC signal trials like that depicted in Fig. 2 were generated to estimate
Event-related MIPAC. Each trial was circularly shifted a random number of points (between 1 and 100), and Gaussian white noise was added to each trial, setting the SNR = 10 Top: An on/off boxcar waveform was used to vary the modulation strength of the simulated PAC signals. Middle: Estimated MIPAC for each trial and latency Bottom: Estimated ERPAC (blue) and Trial-mean MIPAC (red). Standard deviation of MIPAC across trials is shown in light red.
Figure 9:. Comparing results of MImi with those of other modulation indices
(A): Simulated PAC signal with noise added (
SNR = 10) (blue). An on/off boxcar waveform is used to model the PAC modulation strength (red) in the simulated signal. In this simulation, the amplitude frequency is f = 50 Hz, the phase frequency is c f = 7 Hz, and the sampling rate is m S = 500 Hz. (B): MIPAC estimates for all combinations of phase frequencies from 4 to 14 Hz (1 Hz steps) and amplitude frequencies from 30 to 70 Hz (5-Hz steps). Comodulograms using (C) MImi, (D) GLMmi , (E) MVLmi  and (F) KLmi . Non significant values ( rate p < 0.05, uncorrected) appear shaded in (D-F).
Figure 10:. Electrode locations (lower left) and MImi comodulograms computed for face and house stimulus-locked trials at a subset of ECoG channels electrodes.
The comodulogram for responses to face presentations in Channel 16 is magnified in upper left. The MImi maximum (95-Hz amplitude, 16-Hz phase) is highlighted with a dotted box (upper left). Note that there is no MImi following house image presentations for this channel (dashed box lower right). No significance testing was performed on the comodulograms.
Figure 11:. ERP-images for Channel 16 trials time locked to presentations of face and house image stimuli.
Stimulus onset is at 0 ms and trials have been (vertically) smoothed with a 3-trial moving window. Left: Single ECoG trials time-locked to the presentation of face stimuli. Right: Trials time-locked to presentations of house stimuli. Lower panels plot the trial-mean ERP for each trial subset. At Channel 16, located on the Fusiform Gyrus, the response to face stimuli includes a prominent and expected ‘N170’ negativity that does not appear following presentations of house stimuli.
Event-related spectral perturbations (ERSPs) and inter-trial coherences (ITCs) for the same channel (16) time locked to (
left) face and ( right) house image presentations. The trial-mean ERPs are shown in the traces below the ITC panels. The face-stimulus ERSP exhibits a broadband power increase near 200 ms after stimulus onset. Beta-band frequencies (16–30 Hz) exhibit strong (≥ 0.5) ITC near 200 ms. The ERSP and ITC responses to house stimuli ( right) are weaker and peak later (400 ms). Jointly, these phenomena may suggest the presence of event-related phase-amplitude coupling (PAC) near f = 16 Hz and phase famp = 95 Hz. Attending to Section 2.1.3, these bands are represented with horizontal dotted lines with values between [78, 112] Hz ( upper left) and [15, 17] Hz ( lower left) respectively.
Figure 13:. Two single face image presentation trials displaying high-frequency activity:
The figure shows (blue traces) trials 91 and 113 in Fig. 11 and time courses for the same two trials of power near 95 Hz (band limits, 78 Hz to 112 Hz). The computed power increases are consistent with the appearance of high-frequency oscillations superimposed in the ERPs surrounding the negative (‘N170’) ERP peak
Figure 14:. Trial-mean MIPAC and ERPAC computed for Channel 16 at f
phase 16 Hz and f amp 95 Hz.
Top: Trial-mean MIPAC following face image (red) and house image (blue) presentations. Bottom: ERPAC following face image (red) and house image (blue) presentations. Pink shaded areas indicate intervals of significant PAC ( p < 0.05, FWER corrected over both time and conditions) following face image presentations. Both event-related MIPAC and ERPAC measures give similar results that feature a PAC maximum near 185 ms following face image presentations only.
MIPAC-images showing event-related MIPAC time series for individual face image trials.
(A):MIPAC-image with trials sorted by the mean MIPAC value near 200 ms (between the vertical dotted lines). Points with statistically non-significant MIPAC estimates ( p < 0.05, uncorrected) are shown in shaded cream color. (B): Power near 95 Hz in trials sorted as in A. (C): Phase at 16 Hz, trials again sorted as in A. (D): Trial-mean ERPs. (E-H) as in A-D, trials here sorted by 16-Hz phase in a latency interval centered at 200 ms (see dotted lines in panel G). Images in panels A and E are smoothed with a 3-trial moving window.
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Research Support, N.I.H., Extramural
Research Support, Non-U.S. Gov't
Signal Processing, Computer-Assisted*