Abstract
Cross frequency coupling (CFC) is emerging as a fundamental feature of brain activity, correlated with brain function and dysfunction. Many different types of CFC have been identified through application of numerous data analysis methods, each developed to characterize a specific CFC type. Choosing an inappropriate method weakens statistical power and introduces opportunities for confounding effects. To address this, we propose a statistical modeling framework to estimate high frequency amplitude as a function of both the low frequency amplitude and low frequency phase; the result is a measure of phase-amplitude coupling that accounts for changes in the low frequency amplitude. We show in simulations that the proposed method successfully detects CFC between the low frequency phase or amplitude and the high frequency amplitude, and outperforms an existing method in biologically-motivated examples. Applying the method to in vivo data, we illustrate how CFC evolves during seizures and is affected by electrical stimuli.
Introduction
Brain rhythms - as recorded in the local field potential (LFP) or scalp electroencephalogram (EEG) - are believed to play a critical role in coordinating brain networks. By modulating neural excitability, these rhythmic fluctuations provide an effective means to control the timing of neuronal firing [20, 8]. Oscillatory rhythms have been categorized into different frequency bands (e.g., theta [4-10 Hz], gamma [30-80 Hz]) and associated with many functions: the theta band with memory, plasticity, and navigation [20]; the gamma band with local coupling and competition [34, 5]. In addition, gamma and high-gamma (80-200 Hz) activity have been identified as surrogate markers of neuronal firing [48, 42, 23, 47, 69, 50], observable in the EEG and LFP.
In general, lower frequency rhythms engage larger brain areas and modulate spatially localized fast activity [6, 11, 64, 38, 37]. For example, the phase of low frequency rhythms has been shown to modulate and coordinate neural spiking [63, 26, 22] via local circuit mechanisms that provide discrete windows of increased excitability. This interaction, in which fast activity is coupled to slower rhythms, is a common type of cross-frequency coupling (CFC). This particular type of CFC has been shown to carry behaviorally relevant information (e.g., related to position [28, 1], memory [53], decision making and coordination [17, 46, 73, 25]). More generally, CFC has been observed in many brain areas [6, 11, 16, 59, 21, 9], and linked to specific circuit and dynamical mechanisms [26]. The degree of CFC in those areas has been linked to working memory, neuronal computation, communication, learning and emotion [58, 30, 10, 18, 33, 39, 32, 31, 54], and clinical disorders [24, 72, 66, 3, 19], including epilepsy [68]. Although the cellular mechanisms giving rise to some neural rhythms are relatively well understood (e.g. gamma [71, 70, 40], and theta [58]), the neuronal substrate of CFC itself remains obscure.
Analysis of CFC focuses on relationships between the amplitude, phase, and frequency of two rhythms from different frequency bands. The notion of CFC, therefore, subsumes more specific types of coupling, including: phase-phase coupling (PPC), phase-amplitude coupling (PAC), and amplitude-amplitude coupling (AAC) [26]. PAC has been observed in rat striatum and hippocampus [59] and human cortex [9], AAC has been observed between the alpha and gamma rhythms in visual cortex [27, 29, 55], and between theta and gamma rhythms during spatial navigation [52], and both PAC and AAC have been observed between alpha and gamma rhythms [44]. Many quantitative measures exist to characterize different types of CFC, including: mean vector length or modulation index [9, 57], phase-locking value [21, 36, 61], envelope-to-signal correlation [7], analysis of amplitude spectra [12], coherence between amplitude and signal [15], coherence between the time course of power and signal [44], and eigendecomposition of multichannel covariance matrices [13]. Overall, these different measures have been developed from different principles and made suitable for different purposes, as shown in comparative studies [57, 12, 45, 43].
Despite the richness of this methodological toolbox, it has limitations. For example, because each method focuses on one type of CFC, the choice of method restricts the type of CFC detectable in data. Applying a method to detect PAC in data with both PAC and AAC may: (i) falsely report no PAC in the data, or (ii) miss the presence of significant AAC in the same data. Changes in the low frequency power can also affect measures of PAC; increases in low frequency power can increase the signal to noise ratio of phase and amplitude variables, increasing the measure of PAC, even when the phase-amplitude coupling remains constant [2, 60, 30]. Furthermore, many experimental or clinical factors (e.g., stimulation parameters, age or sex of subject) can impact CFC in ways that are difficult to characterize with existing methods [14]. These observations suggest that an accurate measure of PAC would control for confounding variables, including the power of low frequency oscillations.
To that end, we propose here a generalized linear model (GLM) framework to assess CFC between the high-frequency amplitude and, simultaneously, the low frequency phase and amplitude. This formal statistical inference framework builds upon previous work [35, 45, 65, 60] to address the limitations of existing CFC measures. In what follows, we show that this framework successfully detects CFC in simulated signals, including cases in which a method lacking the low frequency amplitude predictor fails. We compare this method to the modulation index, and show that in signals with CFC dependent on the low-frequency amplitude, the proposed method detects true CFC that cannot be detected with the modulation index. We apply this framework to in vivo recordings from human and rat cortex and show examples of how accounting for AAC reveals changes in PAC over the course of seizure, and how to incorporate new covariates directly into the model framework.
Methods
Estimation of the phase and amplitude envelope
To study CFC we estimate three quantities: the phase of the low frequency signal, ϕlow; the amplitude envelope of the high frequency signal, Ahigh; and the amplitude envelope of the low frequency signal, Alow. To do so, we first bandpass filter the data into low frequency (4-7 Hz) and high frequency (100-140 Hz) signals, Vlow and Vhigh, respectively, using a least-squares linear-phase FIR filter of order 375 for the high frequency signal, and order 50 for the low frequency signal. Here we choose specific high and low frequency ranges of interest, motivated by previous in vivo observations [9, 59, 51]. However, we note that this method is flexible and not dependent on this choice. We use the Hilbert transform to compute the analytic signals of Vlow and Vhigh, and from these compute the phase and amplitude of the low frequency signal (Alow and ϕlow) and the amplitude of the high frequency signal (Ahigh).
Modeling framework to assess CFC
Generalized linear models (GLMs) provide a principled framework to assess CFC [45, 35, 60]. Here, we present four models to analyze different types of CFC.
The PAC model
The PAC model relates Ahigh, the response variable, to a linear combination of ϕlow, the predictor variable, expressed in a spline basis: where the conditional distribution of Ahigh given ϕlow is modeled as a Gamma random variable with mean parameter μ and shape parameter ν, and βk are undetermined coefficients, which we refer to collectively as βPAC. We choose this distribution as it guarantees real, positive amplitude values. The functions {f1, ···, fn} correspond to spline basis functions, with n control points equally spaced between 0 and 2π, used to approximate ϕlow. We use a tension parameter of 0.5, which controls the smoothness of the splines. Here, we fix n = 10, which is a reasonable choice for smooth PAC with one or two broad peaks [35]. For a more detailed discussion and simulation examples of the PAC model, see [35]. We note that the choices of distribution and link function differ from those in [45, 60], where the normal distribution and identity link are used instead.
The CFC model
The CFC model extends the PAC model in Equation 1 by including three additional predictors in the GLM: Alow, the low frequency amplitude; and interaction terms between the low frequency amplitude and the low frequency phase: Alow sin(ϕlow), and Alow cos(ϕlow). These new terms allow assessment of phase-amplitude coupling while accounting for linear amplitude-amplitude dependence and more complicated phase-dependent relationships on the low frequency amplitude without introducing many more parameters. Compared to the original PAC model in Equation 1, including these new terms increases the number of variables to n + 3, and the length of the coefficient vector βCFC to n + 3. These changes result in the following model:
Here, the conditional distribution of Ahigh given ϕlow and Alow is modeled as a Gamma random variable with mean parameter μ and shape parameter ν, and βk are undetermined coefficients.
The AAC model
The AAC model relates the high frequency amplitude to the low frequency amplitude: where the conditional distribution of Ahigh given Alow is modeled as a Gamma random variable with mean parameter μ and shape parameter ν. The predictor consists of a single variable and a constant, and the length of the coefficient vector βAAC is 2.
The null model
The null model consists of a single, constant predictor variable. This model has a single parameter βnull) to estimate:
Intuitively, the null model in Equation 4 fits the average high frequency amplitude without dependence on the low frequency phase, low frequency amplitude, or any other predictor.
The statistics RPAC, RAAC, and RCFC
We compute three measures of CFC, RPAC, RAAC, and RCFC, which use the four models defined in the previous section. We evaluate each model in the three-dimensional space (ϕlow, Alow, Ahigh) and calculate the statistics RPAC, RAAC, and RCFC as the maximum absolute fractional difference between the null model mean response function and the PAC, AAC, and CFC models’ mean response functions, respectively. We use the MATLAB function fitglm to estimate the models; we note that this procedure estimates the dispersion directly for the gamma distribution.
To compute RPAC, RAAC, and RCFC, we first evaluate the null model. We do so by estimating the coefficient βnull and computing the mean response function Snull of the null model in the three-dimensional (ϕlow, Alow, Ahigh) space, which appears as a flat plane (Figure 1A, red).
To compute RPAC, the statistic characterizing PAC, we first estimate the coefficients βPAC from the PAC model. From this, we then compute estimates for the high frequency amplitude using the PAC model with 100 evenly spaced values of ϕlow between −π and π. Evaluating the PAC model in this way results in a mean response function CPAC. We extend this PAC curve CPAC in the Alow dimension to create a surface SPAC in the three-dimensional (ϕlow, Alow, Ahigh) space (Figure 1A, blue). The surface SPAC has the same structure as the curve CPAC in the (ϕlow, Ahigh) space, and remains constant along the dimension Alow. From this, we compute RPAC as or the maximum absolute fractional difference between the null and PAC mean response functions Snull and SPAC. Intuitively, a low value of RPAC indicates an absence of PAC, while a high value of RPAC indicates the presence of PAC.
Similarly, we compute the RAAC statistic, which characterizes AAC, by first estimating the coefficient vector βAAC from the AAC model. We then find the corresponding estimate for high frequency amplitude for 640 evenly spaced values between the 5th and 95th quantiles of Alow observed; we choose these quantiles to avoid extremely small or large values of Alow. This creates a mean response function which appears as a curve CAAC in the two-dimensional (Alow, Ahigh) space. We extend this two-dimensional curve to a three-dimensional surface SAAC in the (ϕlow, Alow, Ahigh) space which extends CAAC along the ϕlow dimension (Figure 1B). We use this surface to compute RAAC as or the maximum absolute fractional difference between the null and AAC mean response functions Snull and SAAC. Again, a low value of RAAC indicates an absence of AAC, while a high value indicates the presence of AAC.
Finally, we compute the statistic RCFC, which characterizes CFC, using a similar procedure to those developed for RPAC and RAAC. We first compute estimates of the parameters βCFC for the CFC model. Again, we find corresponding estimates for Ahigh by fixing Alow at one of 640 evenly spaced values between the 5th and 95th quantiles of Alow observed and computing the corresponding high frequency amplitude values from the CFC model over 100 evenly spaced values of ϕlow between −π and π. This creates a two-dimensional curve CCFC in the two-dimensional (ϕlow, Ahigh) space with fixed Alow. We repeat this procedure for all 640 values of Alow to create a surface SCFC in the three-dimensional space (ϕlow, Alow, Ahigh); see Figure 1C.
We define RCFC as the maximum fractional distance between the null surface and CFC surface. We compute the statistic RCFC as i.e., the maximum absolute fractional difference between the null and CFC surfaces in the three-dimensional space (ϕlow, Alow, Ahigh); see Figure 1C.
Estimating 95% confidence intervals for RPAC, RAAC, and RCFC
We compute 95% confidence intervals for RPAC, RAAC, and RCFC via a parametric bootstrap method [35]. Given a vector of estimated coefficients βx for x = {PAC, AAC, or CFC}, we use its estimated covariance and estimated mean to generate 10,000 normally distributed coefficient sample vectors , j ∈ {0, …, 10000}, x = {PAC, AAC, or CFC}. For each , we then compute the high frequency amplitude values from the PAC, AAC, or CFC model, . Finally, we compute the Rx statistic for each j as
We note that in this procedure we use the value of Snull estimated from the data. The 95% confidence intervals for the statistic are the values of at the 0.025 and 0.975 quantiles [35].
Assessing significance of AAC, PAC, and CFC with analytic p-values
To assess whether evidence exists for significant CFC - present as PAC, AAC, or both - we perform three statistical tests. First, to test for significant CFC, we perform a chi-squared test between the null and CFC models, with 12 degrees of freedom, corresponding to the 12 additional parameters in the CFC model compared to the null model. We use a chi-squared test to compare nested models and evaluate whether additional parameters provide a significant improvement in model fit. We note that the null hypothesis (H0) is that the additional parameters all equal zero, and the alternative hypothesis (HA) is that at least one of the additional parameters is nonzero. A p-value below 0.05 indicates that at least one of the parameters in the CFC is significantly different from zero, suggesting that coupling of some kind is present. We label this p-value pCFC.
Next, to test for significant AAC, we perform a chi-squared test between the PAC model and CFC model, with three degrees of freedom due to the three additional parameters in the CFC model. We consider a p-value below 0.05 significant, indicating that at least one of the parameters unique to the CFC is significantly different from zero, i.e., the model is improved by including the low frequency amplitude terms, which provides evidence for amplitude-amplitude coupling. We label this p-value pAAC.
Finally, to test for significant PAC, we perform a chi-squared test between the AAC and CFC models, with 11 degrees of freedom. A p-value below 0.05 signifies that at least one of the parameters unique to the CFC is significantly different from zero, i.e. the model is improved by including the low frequency phase, which provides evidence for phase-amplitude coupling. We label this p-value pPAC.
Assessing significance of AAC, PAC, and CFC with bootstrap p-values
The chi-squared test is an analytic approach to assess the significance of model predictors, and performs well when the mismatch between the data and model is small. However, for real data these assumptions might not hold. This could result in inaccurate - under or over estimates - of significance. To address this, we implement an alternative measure to assess significance using bootstrap p-values.
We implement the bootstrap as follows. Given two signals Vlow and Vhigh, and the resulting output values RPAC, RAAC, and RCFC, we apply the Amplitude Adjusted Fourier Transform (AAFT) algorithm [56] on Vhigh to generate a surrogate signal . In the AAFT algorithm, we first reorder the values of Vhigh by creating a random Gaussian signal W and ordering the values of Vhigh to match W. For example, if the highest value of W occurs at index j, then the highest value of Vhigh will be reordered to occur at index j. Next, we apply the Fourier Transform (FT) to the reordered Vhigh and randomize the phase of the frequency domain signal. This signal is then inverse Fourier transformed and rescaled to have the same amplitude distribution as the original signal Vhigh. In this way, the algorithm produces a permutation of Vhigh such that the power spectrum and amplitude distribution of the original signal are preserved.
We create 1000 such surrogate signals , and calculate , and between Vlow and each . We define our p-values pPAC, pAAC, and pCFC as the proportion of values in , and greater than the original values RPAC, RAAC, and RCFC, respectively. If the proportion is zero, we set p = 0.0005. We show in Results that this method performs well in all stimulation scenarios considered.
We calculate p-values for the modulation index (MI) in this same way. The modulation index calculates the distribution of high frequency amplitudes versus low frequency phases and measures the distance from this distribution to a uniform distribution of amplitudes. Given the signals Vlow and Vhigh, and the resulting modulation index value MI between them, we calculate the MI between Vlow and 1000 surrogate permutations of Vhigh using the AAFT algorithm. We set pMI to be the proportion of these resulting values greater than the original value MI.
Summary of the Modeling Framework
We summarize the model estimation framework (Figure 2) as follows: Given a high frequency signal Vhigh and a low frequency signal Vlow, if pCFC > 0.05, we conclude there is no statistically significant evidence for CFC in the data. If pCFC < 0.05, we conclude there is statistically significant evidence for CFC, and examine the values pAAC and pPAC. If pAAC < 0.05 and pPAC > 0.05, we conclude there is statistically significant evidence for AAC but not PAC. If pAAC > 0.05 and pPAC < 0.05, we conclude there is statistically significant evidence for PAC but not AAC. Finally, if pAAC < 0.05 and pPAC < 0.05, we conclude there is statistically significant evidence for both PAC and AAC.
Synthetic Time Series with PAC
We construct synthetic time series to examine the performance of the CFC model as follows. First, we simulate 20 s of pink noise data such that the power spectrum scales as 1/f. We then filter these data into low (4-7 Hz) and high (100-140 Hz) frequency bands, as described in Methods: Estimation of the phase and amplitude envelope, creating signals Vlow and Vhigh. Next, we couple the amplitude of the high frequency signal to the phase of the low frequency signal. To do so, we first locate the peaks of Vlow and determine the times tk, k = {1,2,3,…, K}, of the K relative extrema. We note that these times correspond to ϕlow = 0. We then create a smooth modulation signal M which consists of a 42 ms Hanning window of height 1 + IPAC centered at each tk, and a value of 1 at all other times (Figure 3A). The intensity parameter IPAC in the modulation signal corresponds to the strength of PAC. IPAC = 0.0 corresponds to no PAC, while IPAC = 1.0 results in a 100% increase in the high frequency amplitude at each tk, creating strong PAC. We create a new signal with the same phase as Vhigh, but with amplitude dependent on the phase of Vlow by setting
We create the final voltage trace V as where Vpink is a new instance of pink noise multiplied by a small constant c = 0.01. In the signal V, brief increases of the high frequency activity occur at a specific phase (0 radians) of the low frequency signal.
Synthetic Time Series with AAC
To generate synthetic time series with dependence on the low frequency amplitude, we follow the procedure in the preceding section to generate Vlow, Vhigh, and Alow. We then induce amplitude-amplitude coupling between the low and high frequency components by creating a new signal such that where IAAC is the intensity parameter corresponding to the strength of amplitude-amplitude coupling. We define the final voltage trace V as where Vpink is a new instance of pink noise multiplied by a small constant c = 0.01 (Figure 3B).
Code Availability
The code to perform this analysis is available for reuse and further development at https://github.com/Eden-Kramer-Lab/GLM-CFC.
Results
We first examine the performance of the CFC measure through simulation examples, using analytic p-values. In doing so, we show that the statistics RCFC, RPAC, and RAAC accurately detect different types of cross-frequency coupling, increase with the intensity of coupling, and detect weak PAC coupled to the low frequency amplitude. We show that the proposed method is less sensitive to changes in low frequency power, and outperforms an existing PAC measure that lacks dependence on the low frequency amplitude. We conclude with an example application to human microelectrode array recordings during seizure, and propose that the RCFC measure identifies CFC not detected in a standard PAC measure.
The absence of CFC produces no significant detections of coupling
We first consider simulated signals without CFC. To create these signals, we follow the procedure in Methods: Synthetic Time Series with PAC with the modulation intensity set to zero (IPAC = 0). In the resulting signals, Ahigh is approximately constant and does not depend on ϕlow or Alow. We estimate the null, PAC, AAC, and CFC models from these data; we show example fits of the model surfaces in Figures 4B. We observe that the PAC, AAC, and CFC models exhibit small modulations in the estimated high frequency amplitude envelope as a function of the low frequency phase and amplitude.
To assess the distribution of significant R values in the case of no cross-frequency coupling, we simulate 1000 instances of the pink noise signals (each of 20 s) and apply the null, PAC, AAC, and CFC models to each instance, plotting significant R values in Figure 4C. We find that for all 1000 instances, pCFC, pPAC, and pAAC are greater than 0.05, indicating no evidence of significant AAC or PAC in these simulated data, as expected.
The proposed method accurately detects PAC
We next consider signals that possess phase-amplitude coupling, but lack amplitude-amplitude coupling. To do so, we simulate a 20 s signal with Ahigh modulated by ϕlow (Figure 4D); more specifically, Ahigh increases when ϕlow is near 0 radians (see Methods, IPAC = 1). We then estimate the null, PAC, AAC, and CFC models from these data; we show example fits in Figure 4E. We find that the PAC and CFC models deviate from the null model at the preferred phase ϕlow = 0. We note that, because the data do not depend on the low frequency amplitude (Alow), the PAC and CFC models have very similar shapes in the (ϕlow, Alow, Ahigh) space, and the AAC model is nearly flat.
Simulating 1000 instances of these 20 s signals with induced phase-amplitude coupling, we find pAAC < 0.05 for only 0.5% of the simulations, while pPAC, pPCF < 0.05 for 100% of the simulations, indicating that the inclusion of the Alow parameter in our model seldom improves the fit. We find that the significant values of RPAC and RCFC lie well above 0 (Figure 4F), and that as the intensity of the simulated phase-amplitude coupling increases (see Methods), so do the statistics RPAC and RCFC (Figure 4G), while the few values of spurious significant RAAC remain close to 0. We conclude that the proposed method accurately detects the presence of phase-amplitude coupling in these simulated data.
The proposed method accurately detects AAC
We next consider signals with amplitude-amplitude coupling, but without phase-amplitude coupling. We simulate a 20 s signal such that Ahigh is modulated by Alow (see Methods, IAAC = 1); when Alow is large, so is AMgh (Figure 4H). We then estimate the null, PAC, AAC, and CFC models (example fits in Figure 4I). We find that the AAC model deviates from the null model, increasing along the Alow axis, and that the CFC model closely follows this trend, while the PAC model remains mostly flat, as expected.
Simulating 1000 instances of these signals we find that pAAC, pCFC < 0.05 for 100% of simulations, while pPAC < 0.05 for 0% of simulations, indicating that inclusion of the ϕlow terms in the model do not improve the model fit. The significant values of RCFC and RAAC are similar, lying above 0 (Figure 4J), and increases in the intensity of AAC produce increases in both RAAC and RCFC (Figure 4K). We conclude that the proposed method accurately detects the presence of amplitude-amplitude coupling.
The proposed method accurately detects the simultaneous occurrence of PAC and AAC
We now consider signals that possess both phase-amplitude coupling and amplitude-amplitude coupling. To do so, we simulate time series data with both AAC and PAC (Figure 4L). In this case, Ahigh increases when ϕlow is near 0 radians and when Alow is large (see Methods, IPAC = 1 and IAAC = 1). We then estimate the null, PAC, AAC, and CFC models from the data and visualize the results (Figure 4M). We find that the PAC model deviates from the null model near ϕlow = 0, and that the AAC model increases linearly with Alow. The CFC model exhibits both of these behaviors, increasing at ϕlow = 0 and as Alow increases.
Simulating 1000 instances of signals with both AAC and PAC present, we find that pAAC, pPAC, and pCFC are all less than 0.05 in 100% of simulations. The distributions of significant RPAC, RAAC, and RCFC values lie above 0, consistent with the presence of both PAC and AAC (Figure 4N), and as the intensity of PAC and AAC increases, so do the values of RPAC, RAAC, and RCFC (Figure 4O). We conclude that the model successfully detects the concurrent presence of PAC and AAC.
The proposed method is less affected by fluctuations in low-frequency amplitude
Increases in low frequency power can increase measures of cross-frequency coupling, although the underlying CFC remains unchanged [2, 14]. Characterizing the impact of this confounding effect is important both to understand measure performance and to produce accurate interpretations of analyzed data. To examine this phenomenon, we perform the following simulation. First, we simulate a signal V with fixed PAC (intensity IPAC = 1, see Methods). Second, we filter V into its low and high frequency components Vlow and Vhigh, respectively. Then, we create a new signal Vk as follows: where k is a value between 1.0 and 2.2, and Vnoise is a pink noise tern (see Methods). We note that the value of k only modifies the amplitude of the low frequency component of V and does not alter the PAC. To analyze the CFC in this new signal we compute RCFC and a measure of PAC in common use: the modulation index MI [57].
We show in Figure 5 population results (1000 realizations of the simulated signal Vk for each k) for the z-scored R and MI values versus the scale factor k; to compute the z-score we use the mean and standard deviation from the k = 1 simulations. We observe that increases in the amplitude of Vlow produce increases in MI and RCFC. However, this increase is more dramatic for MI than for RCFC. We conclude that the statistic RCFC determined from the CFC model — which includes the low frequency amplitude as a predictor in the GLM — is more robust to increases in low frequency power than a method that only includes the low frequency phase.
Sparse PAC is detected when coupled to the low frequency amplitude
While the modulation index has been successfully applied in many contexts [10, 26], instances may exist where this measure is not optimal. For example, because the modulation index was not designed to account for the low frequency amplitude, it may fail to detect CFC when Ahigh depends not only on ϕlow, but also on Alow. In what follows, we consider two such instances. In the first, we hypothesize that since the modulation index considers the distribution of Ahigh at all observed values of ϕlow, it may fail to detect coupling events that occur sparsely at only a subset of appropriate ϕlow occurrences. The CFC model, on the other hand, may detect these sparse events if these events are coupled to Alow. To test this hypothesis, we consider two simulation scenarios in which PAC occurs sparsely in time.
First, we consider a signal V with PAC, and corresponding modulation signal M with intensity value IPAC = 1.0 (see Methods, Figure 6A-B). We then modify this signal to reduce the number of PAC events in a way that depends on Alow. To do so, we preserve PAC at the peaks of Vlow (i.e., when ϕlow = 0), but now only when these peaks are large, more specifically in the top 5% of peak values.
We define a threshold value T to be the 95th quantile of the peak Vlow values, and modify the modulation signal M as follows. When M exceeds 1 (i.e., when ϕlow = 0) and the low frequency amplitude exceeds T (i.e., Alow ≥ T), we make no change to M. Alternatively, when M exceeds 1 and the low frequency amplitude lies below T (i.e., Alow < T), we decrease M to 1 (Figure 6C). In this way, we create a modified modulation signal M1 such that in the resulting signal V1, when ϕlow = 0 and Alow is large enough, Ahigh is increased; and when ϕlow = 0 and Alow is not large enough, there is no change to Ahigh. This signal V1 hence has some number (N) of phase-amplitude coupling events, which is much less than the number of times ϕlow = 0. We label this first scenario the dependent case.
Second, we create a signal V2 with the same number of PAC events (N) as V1, but without dependence on Alow. To do so, we create a new modulation signal M2 such that M2 has the same number of peaks (N) as M1, but these peaks are chosen to occur at ϕlow = 0 and random low frequency amplitude values (Figure 6D). The resulting signal V2 then has the same number of CFC events as V1, but these events occur at times independent of Alow. We call this second scenario the independent case.
We generate 1000 realizations of the simulated signals V1 and V2, and from these signals compute RCFC and MI. We find that in the independent case, i.e., when Ahigh is independent of Alow, MI detects CFC in 18% of simulations and RCFC detects CFC in 25% of simulations. In this scenario, although PAC occurs in the signal V2, this PAC occurs infrequently, and both methods rarely detect it. In the dependent case, i.e., when Ahigh depends on Alow, MI detects PAC in only 16% of simulations, while RCFC detects CFC in 45% of simulations. In this case, although the PAC still occurs infrequently, these occurrences are coupled to Alow, and the CFC model — which includes both Alow and ϕlow as predictors — successfully detects these events much more frequently. We conclude that when the PAC is dependent on Alow, RCFC more accurately detects these sparse coupling events due to the inclusion of the Alow predictor in the model.
The CFC model detects simultaneous PAC and AAC missed in an existing method
As a second example to illustrate the utility of the proposed method, we consider another scenario in which Alow impacts the occurrence of PAC. More specifically, we consider a case in which Ahigh increases at a fixed low frequency phase for high values of Alow, and Ahigh decreases at the same phase for small values of Alow. In this case, we expect that the modulation index may fail to detect the coupling because the distribution of Ahigh over ϕlow would appear uniform when averaged over all values of Alow; the dependence of Ahigh on ϕlow would only become apparent after accounting for Alow.
To implement this scenario, we consider the modulation signal M (see Methods) with an intensity value IPAC = 1. We consider all peaks of Alow and set the threshold T to be the 50th quantile (Figure 7A). We then modify the modulation signal M as follows. When M exceeds 1 (i.e., when ϕlow = 0) and the low frequency amplitude exceeds T (i.e., Alow ≥ T), we make no change to M. Alternatively, when M exceeds 1 and the low frequency amplitude lies below T (i.e. Alow < T), we decrease M to 0 (Figure 7B). In this way, we create a modified modulation signal M such that when ϕlow = 0 and Alow is large enough, Ahigh is increased;and when ϕlow = 0 and Alow is small enough, Ahigh is decreased (Figure 7C).
Using this method, we simulate 1000 realizations of this signal at each of 10 values of I between 0 and 1, and calculate MI and RCFC for each signal (Figure 7D). We find that as the intensity increases, the percentage of significant detections for RCFC (i.e. where pCFC < 0.05) approaches 100%, while the percentage of significant detections for MI, (i.e., where pMI < 0.05) remains less than 20%. We conclude that the CFC method more accurately detects the combined PAC and AAC in this simulation compared to the modulation index.
An alternative method to assess significance produces consistent results
In the previous sections, we applied an analytic approach to assess the significance of model predictors. This approach is computationally efficient, and performs well in the simulations above in which the mismatch between the data and model is small. Here, we consider an alternative approach of computing p-values - a bootstrap procedure - that is more expensive to compute but requires fewer assumptions about the data. We apply this bootstrap procedure (see Methods) to four different scenarios: (i) where no CFC is present, (ii) where PAC but not AAC is present, (iii) where AAC but not PAC is present, and (iv) where PAC and AAC are both present. In the first case, to create a signal with no CFC, we simulate 20 s of pink noise data such that the power spectrum scales as 1/f. We choose these simulated data to introduce additional variability in the amplitude of the high frequency signal;unlike the simulation in Figure 4A, in which the value of Ahigh was approximately constant, the value of Ahigh here is more variable. Computing pPAC, pAAC, and pCFC using the bootstrap method, we find that pCFC < 0.05 for 0.4% of simulations, pPAC < 0.05 for 5.2% of simulations, and pAAC < 0.05 for 0.9% of simulations (Figure 8A). We detect no evidence of CFC, consistent with the lack of CFC in these data.
Next, to simulate a signal with PAC but no AAC, we first follow the procedures in Methods to create Vlow, Vhigh, and the modulation signal M. Then, we multiply Vhigh by the modulation signal M to create the voltage trace V = Vlow + M Vhigh + c Vpink, where again Vpink is a new instance of pink noise multiplied by a small constant c = 0.01. We note that by modulating Vhigh directly by M, instead of modulating Ahigh by M, we introduce additional variability in the high frequency amplitude that is not dependent on the low frequency signal. We simulate 1000 instances of V, each time measuring pPAC, pAAC, and pCFC using the bootstrap method, and find that pCFC < 0.05 for 99.3% of simulations, pPAC < 0.05 for 100% of simulations, and pAAC < 0.05 for 0.6% of simulations (Figure 8B). Consistent with the constructed signal, we conclude there is PAC, but no AAC, in these simulated data.
We next simulate a signal with AAC, but not PAC. Following the procedures in Methods: Synthetic Time Series with AAC, we simulate 1000 instances of V, and in each instance measure pPAC, pAAC, and pCFC using the bootstrap method. We find that pCFC < 0.05 for 86% of simulations, pPAC < 0.05 for 3.9% of simulations, and pAAC < 0.05 for 99.5% of simulations (Figure 8C), consistent with the presence of AAC, but not PAC, in the simulated data.
Finally, to simulate a signal with both PAC and AAC, we multiply Vhigh by the modulation signal M to introduce PAC, and then multiply the result by Alow to introduce AAC. We create the final voltage trace V as the sum of Vlow, the PAC and AAC modulated Vhigh, and a new instance of pink noise multiplied by a small constant (c = 0.0į). We find that pCFC < 0.05 for 100% of simulations, pPAC < 0.05 for 100% of simulations, and pAAC < 0.05 for 98.8% of simulations (Figure 8D). We conclude that the bootstrap p-values accurately detects the presence of both PAC and AAC in the simulated data.
Application to in vivo human seizure data
To evaluate the performance of the proposed method on in vivo data, we consider example recordings from human cortex. Two patients (males ages 45, 32 years) with medically intractable focal epilepsy underwent clinically indicated intracranial cortical recordings for epilepsy monitoring. In addition to clinical electrode implantation, these patients were also implanted with a 10×10 (4 mm × 4 mm) NeuroPort microelectrode array (MEA; Blackrock Microsystems, Utah) in a neocortical area expected to be resected with high probability, in either the middle or superior temporal gyrus. The MEA consist of 96 recording platinum-tipped silicon probes, with a length of either 1-mm or 1.5-mm, corresponding to neocortical layer III as confirmed by histology after resection. The reference electrode was either subdural or epidural, chosen dynamically based on recording quality. Signals from the MEA were acquired continuously at 30 kHz per channel. Seizure onset times were determined by an experienced encephalographer (S.S.C.) through inspection of the macroelectrode recordings, referral to the clinical report, and clinical manifestations recorded on video. For a detailed clinical summary of each patient, see patients P1 and P2 of [67]. This research was approved by local Institutional Review Boards at Massachusetts General Hospital and Brigham Women’s Hospitals (Partners Human Research Committee), and at Boston University according to National Institutes of Health guidelines.
Visual inspection of the LFP data (Figure 9A) reveals the emergence of large amplitude voltage fluctuations during the approximately 80 s seizure. To analyze the CFC in these data, we separate this signal into 20 s segments with 10 s overlap, and analyze each segment using the CFC model framework. While little evidence of CFC appears before the seizure (Figure 9A), during the seizure we find significant (using bootstrap p-values) RPAC, RAAC, and RCFC values.
To further investigate these results, we select a 20 s segment with significant and large RPAC, RAAC, and RCFC values to examine (Figure 9B). Visual inspection reveals the occurrence of large amplitude, low frequency oscillations and small amplitude, high frequency oscillations. To examine the detected coupling in even more detail, we isolate a 2 s segment (Figure 9C), and display the signal V, the high frequency signal Vhigh, the low frequency phase ϕlow, and the low frequency amplitude Alow. We observe that when ϕlow is near π (gray bars in Figure 9C), Ahigh increases, consistent with the presence of PAC and a significant value of RPAC. Examining the low frequency amplitude Alow and high frequency amplitude Ahigh over the same 20 s segment (Figure 9D), we find that Alow and Ahigh increase together, consistent the presence of AAC and a significant value of RAAC.
Repeating this analysis for a seizure recorded from a second patient (Figure 10A), we again find significant RPAC, RAAC, and RCFC values during the seizure (Figure 10A). Selecting a 20 s segment with large and significant RPAC, RAAC, and RCFC values (Figure 10B), and isolating a 1 s interval from this segment (Figure 10C), we observe evidence of PAC and AAC. We find that near ϕlow = 0, there is an increase in Ahigh at two time points (grey bars in Figure 10C) consistent with the presence of PAC. However, at other times when the phase is near ϕlow = 0, we observe no increase in Ahigh, consistent with a lack of PAC. We note that the value of Alow distinguishes these scenarios; when we observe PAC, Alow is large, whereas when we do not observe PAC, Alow is small. Hence, the increase in Ahigh is related not only to ϕlow, but also Alow, i.e., there is also AAC in these data, consistent with the detection of significant PAC and AAC values.
Application to in vivo rodent data during electrical stimulation
As a second example to illustrate the performance of the new method, we consider LFP recordings from from the infralimbic cortex (IL) and basolateral amygdala (BLA) of an outbred Long-Evans rat before and after the delivery of an experimental electrical stimulation intervention described in [4]. 8 microwires in each region, referenced as bipolar pairs, sampled each region’s LFP at 30 kHz. The electrical stimulation was explicitly expected to change inter-regional coupling (see [4] for a detailed description of the experiment). Here we examine how cross-frequency coupling between low frequency (5-8 Hz) IL signals and high frequency (70-110 Hz) BLA signals changes from the pre-stimulus (pre) to the post-stimulus (post) condition. To do so, we filter the data V into low and high frequency signals using the techniques from Methods, and compute the MI and RCFC between each possible BLA-IL pairing, sixteen in total (Figure 11).
For the MI, we find both increases and decreases in the PAC from the pre to post condition, and that the average change in MI across all 16 pairs is negligible (p = 0.72) (Figure 11A). However, using RCFC, we detect a notable increase in CFC from pre to post (p < 0.001) (Figure 11 B). To determine which aspects of coupling impact the CFC, we examine RPAC and RAAC (Figure 11C). We find that the average RPAC remains nearly constant across the pre and post conditions, similar to the MI p = 0.68. However, the average RAAC increases significantly (p < 0.001). Hence, we conclude that the increase in RCFC from the pre to post condition results specifically from an increase in AAC.
We next consider an example of intra-electrode coupling, i.e. for a single electrode we measure the coupling between its low frequency (5-8 Hz) and high frequency (70-110 Hz) component. As both RPAC and MI measure phase-amplitude coupling, we expect a similar change in both measures from the pre to post condition. However, while RPAC decreases, MI increases (Figure 11 D-E).
While a difference in RPAC exists, the magnitude of this difference is small (Figure 11E). To test whether RPAC changes across conditions, we implement a natural extension of the proposed framework. To do so we first concatenate the recordings for a given electrode from the pre condition Vpre and post condition Vpost to create a new signal V*:
From V*, we obtain the corresponding high frequency signal and low frequency signal , and subsequently the high frequency amplitude , low frequency phase , and low frequency amplitude . We then use these data to generate a new model: where P is an indicator vector that signifies whether the signal is in the post condition (Pj = į) or the pre condition (Pj = 0). The final term models the effect of the post condition on PAC: if each Pj = 0, then there is no effect; if Pj > 0 then the impact of phase on Ahigh increases in the post condition; if Pj < 0 then the impact of phase on Ahigh decreases in the post condition.
Fitting this new model to the rodent in vivo data, we then perform a χ2 test to compare Model 2 and Model 11 with 10 degrees of freedom, corresponding to new indicator terms. We obtain p ≈ 0, and therefore reject the null hypothesis that Pj = 0 for all j; we conclude that the condition - pre or post - impacts the PAC. Investigating each value in the indicator vector, we find Pj < 0 for all j (p < 0.001). We therefore conclude that, when in the post condition, there is a decrease in PAC. This analysis supports our initial conclusion based on visual inspection of RPAC (Figure 11 E). We note that this change is not consistent with the MI result (Figure 11 D). While the true change in Rpac remains unknown in these in vivo data, this example illustrates an advantage of the proposed framework: the generalized linear model is easily extended to test the significance of additional covariates motivated by the experiment. While here we applied an indicator function for condition on PAC, many other covariates exist (e.g., sex, stimulus parameters) whose effect on PAC, AAC, or both can be explored in this principled statistical framework.
Discussion
In this paper, we proposed a new method for measuring cross-frequency coupling that accounts for both phase-amplitude coupling and amplitude-amplitude coupling, along with a principled statistical modeling framework to assess the significance of this coupling. We have shown that this method effectively detects CFC, both as PAC and AAC, and is more sensitive to weak PAC obscured by or coupled to low-frequency amplitude fluctuations. Compared to an existing method, the modulation index [57], the newly proposed method more accurately detects scenarios in which PAC is coupled to the low-frequency amplitude. Finally, we applied this method to in vivo data to illustrate examples of PAC and AAC in real systems, and show how to extend the modeling framework to include a new covariate.
One of the most important features of the new method is an increased ability to detect weak PAC coupled to AAC. For example, when sparse PAC events occur only when the low frequency amplitude (Alow) is large, the proposed method detects this coupling while other methods not accounting for Alow miss it. While PAC often occurs in neural data, and has been associated with numerous neurological functions [10, 26], the simultaneous occurrence of PAC and AAC is less well studied [44]. Here, we showed examples of simultaneous PAC and AAC recorded from human cortex during seizure, and we note that this phenomena has been simulated in other works [41].
While the exact mechanisms that support CFC are not well understood [26], the general mechanisms of low and high frequency rhythms have been proposed. Low frequency rhythms are associated with the aggregate activity of large neural populations and modulations of neuronal excitability [20, 62, 8], while high frequency rhythms provided a surrogate measure of neuronal spiking [48, 42, 23, 47, 69, 50, 49]. These two observations provide a physical interpretation for PAC: when a low frequency rhythm modulates the excitability of a neural population, we expect spiking to occur (i.e., an increase in Ahigh) at a particular phase of the low frequency rhythm (ϕlow) when excitation is maximal. These notions also provide a physical interpretation for AAC: increases in Alow produce larger modulations in neural excitability, and therefore increased intervals of neuronal spiking (i.e., increases in Ahigh). Alternatively, decreases in Alow reduce excitability and neuronal spiking (i.e., decreases in Ahigh).
The function of concurrent PAC and AAC, both for healthy brain function and over the course of seizure as illustrated here, is not well understood. However, we note that over the course of seizure, particularly at termination, PAC and AAC are both present. As PAC occurs normally in healthy brain signals, for example during working memory, neuronal computation, communication, learning and emotion [58, 30, 10, 18, 33, 39, 32, 31, 54], these preliminary results may suggest a pathological aspect of strong AAC occurring concurrently with PAC.
Proposed functions of PAC include multi-item encoding, long-distance communication, and sensory parsing [26]. Each of these functions takes advantage of the low frequency phase, encoding different objects or pieces of information in distinct phase intervals of ϕlow. PAC can be interpreted as a type of focused attention; Ahigh modulation occurring only in a particular interval of ϕlow organizes neural activity - and presumably information - into discrete packets of time. Similarly, a proposed function of AAC is to encode the number of represented items, or the amount of information encoded in the modulated signal [26]. A pathological increase in AAC may support the transmission of more information than is needed, overloading the communication of relevant information with irrelevant noise. The attention-based function of PAC, i.e. having reduced high frequency amplitude at phases not containing the targeted information, may be lost if the amplitude of the high frequency oscillation is increased across wide intervals of low frequency phase.
Like all measures of CFC, the proposed method possesses specific limitations. We discuss three limitations here. First, the choice of spline basis to represent the low frequency phase may be inaccurate, for example if the PAC changes rapidly with ϕlow. Second, the value of RAAC depends on the range of Alow observed. This is due to the linear relationship between Alow and Ahigh in the AAC model, which causes the maximum distance between the surfaces SAAC and SCFC to occur at the largest or smallest value of Alow. To mitigate the impact of extreme Alow values on RAAC, we evaluate the surfaces SAAC and SCFC over the 5th to 95th quantiles of Alow. We note that an alternative metric of AAC could instead evaluate the slope of the SAAC surface; to maintain consistency of the PAC and AAC measures, we chose not to implement this alternative measure here. Third, the frequency bands for Vhigh and Vlow must be established before R values are calculated. Hence, if the wrong frequency bands are chosen, coupling may be missed. It is possible, though computationally expensive, to scan over all reasonable frequency bands for both Vhigh and Vlow, calculating R values for each frequency band pair.
The proposed method can easily be extended by inclusion of additional predictors in the GLM. Polynomial Alow predictors, rather than the current linear Alow predictors, may better capture the relationship between Alow and Ahigh. One could also include different types of covariates, for example classes of drugs administered to a patient, or time since an administered stimulus during an experiment. The code developed to implement the method is flexible and modular, which facilitates the addition and removal of predictors, while utilizing the same statistical modeling framework to assess the significance of each predictor. This modular code, available at https://github.com/Eden-Kramer-Lab/GLM-CFC, also allows the user to change latent assumptions, such as choice of frequency bands and filtering method. The code is freely available for reuse and further development.
Rhythms, and particularly the interactions of different frequency rhythms, are an important component for a complete understanding of neural activity. While the mechanisms and functions of some rhythms are well understood, how and why rhythms interact remains uncertain. A first step in addressing these uncertainties is the application of appropriate data analysis tools. Here we provide a new tool to measure coupling between different brain rhythms: the method utilizes a statistical modeling framework that is flexible and captures subtle differences in cross-frequency coupling. We hope that this method will better enable practicing neuroscientists to measure and relate brain rhythms, and ultimately better understand brain function and interactions.
Acknowledgements
This work was supported in part by the National Science Foundation Award #1451384.
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