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The authors have declared that no competing interests exist.

Analysis of the coupling between the phases and amplitudes of oscillations within the same continuously sampled signal has provided interesting insights into the physiology of memory and other brain process, and, more recently, the pathophysiology of parkinsonism and other movement disorders. Technical aspects of the analysis have a significant impact on the results. We present an empirical exploration of a variety of analysis design choices that need to be considered when measuring phase-amplitude coupling (PAC). We studied three alternative filtering approaches to the commonly used Kullback–Leibler distance-based method of PAC analysis, including one method that uses wavelets, one that uses constant filter settings, and one in which filtering of the data is optimized for individual frequency bands. Additionally, we introduce a time-dependent PAC analysis technique that takes advantage of the inherent temporality of wavelets. We examined how the duration of the sampled data, the stability of oscillations, or the presence of artifacts affect the value of the “modulation index”, a commonly used parameter describing the degree of PAC. We also studied the computational costs associated with calculating modulation indices by the three techniques. We found that wavelet-based PAC performs better with similar or less computational cost than the two other methods while also allowing to examine temporal changes of PAC. We also show that the reliability of PAC measurements strongly depends on the duration and stability of PAC, and the presence (or absence) of artifacts. The best parameters to be used for PAC analyses of long samples of data may differ, depending on data characteristics and analysis objectives. Prior to settling on a specific PAC analysis approach for a given set of data, it may be useful to conduct an initial analysis of the time-dependence of PAC using our time-resolved PAC analysis.

There is substantial interest in new methods to analyze local field potentials (LFPs) or electroencephalography/electrocorticogram (EEG/ECoG) signals in normal and disease states [

We generated color-coded maps (comodulograms) depicting the strength of the coupling between the amplitude and the phase of frequencies within data that were synthetically generated to represent specific data characteristics such as a defined SNR, or (known) PAC at specific frequencies. MI values were used as a measure of PAC. All analyses were carried out with custom-written MATLAB scripts (MATLAB 9.3; MathWorks, Natick, MA, USA), using parallelized code running on up to four CPU cores. In other analyses, the use of the various methods of PAC analysis in “real” cortical local field potential data was also explored.

We generated synthetic LFP-like signals [

Since sin(

The SNR was calculated as

Spontaneous local field potentials (LFPs) were recorded from an awake nonhuman Rhesus monkey (male, 6kg at the time of recording). The experiments in this animal occurred in the context of another study focused on the pathophysiology of primate parkinsonism and are only described very briefly here. These studies were approved by the Institutional Animal Care and Use Committee of Emory University. The animal originated from the breeding colony at the Yerkes National Primate Research Center. After habituation to the laboratory environment, the animal underwent a surgical procedure under isoflurane anesthesia during which a metal recording chamber was placed on the animal’s skull to give us access to the primary motor cortex (M1). A head fixation bolt was also placed on the skull. Subsequent to the surgery, the animal was chronically treated with weekly injections of the dopaminergic neurotoxin 1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine (MPTP), to induce parkinsonism. Once stable moderate parkinsonism was documented, the animal underwent recording studies aimed at recording M1 LFP signals. These recordings were carried out with a 16-contact linear array electrode (NeuroNexus, Ann Arbor, MI) which was acutely placed into M1, and positioned to allow recordings from deep cortical layers. The resulting signals were band-pass filtered (.1 Hz–700 Hz) [

Phase/amplitude “targets” were randomly selected for the synthetic signals with a generator of uniformly distributed random numbers (MATLAB’s rand function) that guaranteed that the “phase” target (within the range of 0−50 Hz), was smaller than the “amplitude” target (within the range of 0−400 Hz), and that the duration of time during which the PAC target was present exceeded 20 s. These signal ranges were chosen to cover a large portion of the range of frequencies that would typically be found in neural signals. Similar ranges have been used in previous studies [

For the calculation of KL-distance based MI values (using an algorithm described by Tort et al. [

For the CM, we used a method described in Ref. [_{θ} represents phase filtering. Similarly, the amplitude signals, _{s}. The ranges and bandwidths can be adjusted with smaller phase bandwidths, leading to a PAC analysis with slightly higher frequency resolution (

For the VM, we used the same center frequencies as were used for the CM (4−52 Hz in 2 Hz steps for phase and 15−400 Hz in 5 Hz steps for amplitude) with a simpler 2nd order Butterworth filter (4th order after application of the MATLAB “filtfilt” function) [

The WM uses the same center frequencies as the methods above (4−52 Hz in 2 Hz steps for phase and 15−400 Hz in 5 Hz steps for amplitude) but instead of filtering, a convolution with Morlet wavelets is used. This wavelet’s cycles linearly progressed from 3−10 Hz as phase frequency increased to 50 Hz or as amplitude frequency increased to 400 Hz (see also [

After the signals were appropriately filtered and binned (as described above), the resulting binned filtered signals (_{i}. The amplitude of each signal, _{j}, was calculated as _{i} and _{j}, we binned _{i} into 18 non-overlapping 20-degree bins, covering the entire 0−360° range, and calculated the mean amplitude in each bin k as _{ij} scores resulting from this procedure are high if the distribution of mean amplitudes differs substantially from a uniform distribution. Doing this process for every combination of _{ij}) is color-coded. To further quantify our data, we summed up specific rectangular regions of comodulograms (referred to as SumMI, see below). To further reduce noise, a surrogate analysis can be performed. Each surrogate consists of the original amplitude signal and a “shuffled” phase signal for which a PAC is computed [

Measurements of computational costs of calculating comodulogram were carried out using a series of synthetic signals with varying lengths. The measurements were done with MATLAB’s “tic” and “toc” routines. We report the median of measurements using five randomly chosen PAC targets for each of 199 different data lengths from 10 s to 1000 s for each variant of MI calculations (using the CM, VM or WM). For these calculations, we did not generate plots of comodulograms, in order to avoid inclusion of plotting runtime in our measurements.

We developed an algorithm to automatically detect PAC peaks in comodulograms. We focused this analysis on the accuracy of capturing multiple or changing targets during PAC capture, as we feel that this was one of the most important components of PAC analysis that could not be addressed through other technical solutions, such as increasing signal duration or lowering the SNR. For this analysis, the comodulogram matrix, ^{−T} values, rounded down, with a value of _{u}, and _{ℓ} define the empirically chosen thresholds of the curve, N is the current number of confidence regions, and 0 ≤ ^{−3}) are assigned a confidence value greater than _{u} (chosen to be 0.9) while MI values lower than ^{−5}), are assigned a confidence value below _{ℓ} (chosen to be 0.1) We then added each adjacent pixel in the comodulogram to the confidence region if its value too was higher than the matching threshold. This process was then repeated for several rounds “growing” the region in each direction until adjacent pixels were either below the threshold or a maximum distance, _{max} = 24 (found through optimization below), of pixels away from the center. These added pixels were given a confidence value based on their respective distance to the center, ^{−d/δ} _{i}, were given a confidence value of 0.1 × _{i}, ^{2} multiplier penalizes confidence values below 25% and favorably weights any confidence value over 50%. The values for the maximum distance, _{max} = 24, confidence decay,

We measured the computer runtime needed to generate comodulograms with each of the three methods over a series of 199 epochs with varying lengths, each containing five randomly placed PAC targets (

Comparison of the computational costs of three filtering methods when calculating MI values for construction of comodulograms. Each data point represents the median of the amounts of time needed for PAC analysis of five random synthetic targets at a given signal length. Bars denote standard deviations. Signals were generated to present a sampling rate of 1000 Hz. Blue symbols, CM; orange symbols, VM; gray symbols, WM.

Our analysis shows that the VM takes much longer than the other two methods. Even shorter time scales (< 20

The length of the signal (number of samples) also influences the reliability of PAC target detection. Tort et al. found that longer signal lengths generally reduced MI variation but did not recommend a minimal signal length [

A-I: comodulograms of signals with “target” PAC at 20 Hz (phase) and 130 Hz (amplitude), using the CM (A, D, G), VM (B, E, H), or WM (C, F, I), respectively, and signal lengths of 5 s (A-C), 10 s (D-F), and 20 s (G-I). J shows the sum of the modulation indices around the target (phase: 14−26 Hz; amplitude: 100−160 Hz; see red rectangle in A) and their logarithmic trend lines. K shows the percentage of PAC found in the red rectangle around the target over that of the entire comodulogram. In each comodulogram, the area below the white dashed line represents an area of low confidence (as frequencies for phase and amplitude are close together). All comodulograms are scaled to the same color range. The axis titles ‘Phase’ and ‘amplitude’ in A-I refer to frequencies, measured in Hz.

To examine the effect of different levels of noise on PAC measurements, we generated PAC measures on the same 100 s synthetic signal using a target combination of 20 Hz (phase) and 130 Hz (amplitude), with different SNR values (see

The plots in A-I show example comodulograms computed for SNR = 0.1 (A-C), SNR = 2 (D-F), and SNR = 10 (G-I), with each column representing a different method (CM (A, D, G), VM (B, E, H), or WM (C, F, I), respectively). J shows a comparison of the SumMIs around the target (phase: 14−26 Hz range; amplitude: 100−160 Hz range; see red rectangle in A) and their logarithmic trendlines. K shows the proportion of the sum of PAC values in the rectangle around the target and the sum of all PAC values in the comodulogram.

In biological signals, PAC may be inconstant, or shift during the recording period. For instance, in cortical signals, factors such as ongoing behaviors, perception of stimuli, or the state of arousal may alter PAC [

The three comodulograms (A-C) show the same signal (D) presenting a randomly chosen target present for a random duration, followed by a gap of random length, and another randomly chosen target present for a random duration. E shows the match score of the three different methods over 500 trials. * denotes significance (

The three comodulograms (A-C) show examples of an analysis of a signal with five successively appearing random targets of varying lengths (signal shown in the spectrogram in (D)). E shows a Match score analysis (see methods) for 500 random trials. *,

Since short signal durations reduce the sensitivity and accuracy of PAC detection (

Part A of the figure shows the time-resolved PAC for the signal in 5D, calculated by using the WM. B and C show the same 3D representation, projected onto the Phase-Time plane (B) or the Amplitude-Time plane (C). The red lines denote the synthetic targets. All comodulograms are scaled to the same color range.

Red lines indicate 1 s “burst” periods of 20 Hz (phase)/130 Hz (amplitude) PAC. Note that the phase axis is only shown for 15 − 30 Hz to better visualize the 3D structure.

The KL distance MI is considered to be the best current method for calculating PAC because it is noise-tolerant and amplitude-independent. Our studies show that several issues limit the interpretation of PAC analyses, including the duration of the underlying data stream, the SNR of the signal (which influences the detectability and shape of the detected PAC), and the method used to detect PAC. Given the sensitivity of the analysis to these factors, it is important to communicate not only the results of comodulogram analyses, but also to be explicit regarding the specifics of the data used, and the algorithm used for filtering the data. Additionally, initial time-resolved PAC analyses may help to determine some of the parameters to be used in subsequent conventional PAC analyses, specifically the most appropriate signal length which may differ between studies, dependent on signal stability. We found that the WM had overall the most favorable performance profile among the methods tested. This method has the additional advantage of providing the temporal features of the input signal (Figs

This figure shows an example of the appearance of a false peak in PAC analysis. The result was seen when the target phase frequency was within 12 the amplitude frequency. For this example, the target phase frequency was set to 40 Hz and the target amplitude frequency to 60 Hz. The target is marked as a red x in these plots. The false peak was detected in comodulograms generated with the CM, WM, and the VM (top to bottom).

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A, B: Effect of a 4 Hz, 2 Hz, 1 Hz phase bandwidth on an example synthetic LFP with low phase and low amplitude (marked as a red x) using both the CM (A) and VM (B). C: The figure shows comodulograms of the same synthetic signal, using the WM with cycle length progressing linearly from 3 − 10 Hz (top) and from 1.5 − 20 Hz (second from top), as well as with stationary cycle length of 3, 6.5, and 10 Hz (bottom three plots). D, E: Effect of a 4 Hz, 2 Hz, 1 Hz phase bandwidth on an example synthetic LFP with high phase and high amplitude (marked as a red x) using both the CM (A) and VM (B). C: PAC on the same synthetic LFP is shown using the WM with cycle lengths progressing linearly from 3 − 10 Hz and from 1.5 − 20 Hz (top plots), as well as with stationary cycle lengths of 3, 6.5, and 10 Hz (bottom three plots). Dotted lines are marked around the PAC computed with the 4Hz bandwidth (A, B, D, E) or the 3-10 Hz cycle length (C, F) for comparison.

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A 60-second section of data with a constant synthetic target at 20 Hz (phase) and 130 Hz (amplitude) was examined using the CM, WM, and the VM. The left column (colorbars indicate range of modulation indices) of plots shows the analysis of the original data. The right column shows the analysis of the data, z-scored to 100 phase-shuffled surrogates (colorbars indicate range of z-score ranges).

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A 3-second section of data with a constant synthetic target at 20 Hz (phase) and 130 Hz (amplitude) was examined using the CM, WM, and the VM. The left column (colorbars indicate ranges of modulation indices) of plots show the analysis of the original data. The right column shows the analysis of the data, z-scored to 100 phase-shuffled surrogates (colorbars indicate z-score ranges).

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A 60 s signal of M1 LFP activity recorded in a nonhuman primate was examined with the CM, WM, and VM (A). The same signal was then examined again after being z-scored to 100 phase-shuffled surrogates (B).

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Supported by NIH/NINDS grant P50-NS098685 (Udall Center grant), NIH/ORIP grant P51-OD011132 (Yerkes National Primate Research Center), and NIH/NCATS grant UL1TR002378.