Phase-amplitude coupling between theta and multiple gamma sub-bands is a hallmark of hippocampal activity and believed to take part in information routing. More recently, theta and gamma oscillations were also reported to exhibit phase-phase coupling, or n:m phase-locking, suggesting an important mechanism of neuronal coding that has long received theoretical support. However, by analyzing simulated and actual LFPs, here we question the existence of theta-gamma phase-phase coupling in the rat hippocampus. We show that the quasi-linear phase shifts introduced by filtering lead to spurious coupling levels in both white noise and hippocampal LFPs, which highly depend on epoch length, and that significant coupling may be falsely detected when employing improper surrogate methods. We also show that waveform asymmetry and frequency harmonics may generate artifactual n:m phase-locking. Studies investigating phase-phase coupling should rely on appropriate statistical controls and be aware of confounding factors; otherwise, they could easily fall into analysis pitfalls.

DOI: http://dx.doi.org/10.7554/eLife.20515.001

eLife digest

Neuroscientists have long sought to understand how the brain works by analyzing its electrical activity. Placing electrodes on the scalp or lowering them into the brain itself reveals rhythmic waves of activity known as oscillations. These arise when large numbers of neurons fire in synchrony. Recordings reveal that the frequency of these oscillations – the number of cycles of a wave per second, measured in Hertz – can vary between brain regions, and within a single region over time. Moreover, oscillations with different frequencies can co-exist and interact with one another.

Within the hippocampus, an area of the brain involved in memory, two types of oscillations dominate: theta waves and gamma waves. Theta waves are relatively slow waves, with a frequency between 5 and 10 Hertz. Gamma waves are faster, with a frequency of up to 100 Hertz. Recent work has suggested that gamma waves and theta waves show a phenomenon called phase-phase coupling. Since gamma waves are faster than theta waves, multiple cycles of gamma can occur during a single cycle of theta. Phase-phase coupling is the idea that gamma and theta waves align themselves, such that gamma waves always begin at the same relative position within a theta wave. This was thought to help the hippocampus to encode memories.

Using computer simulations and recordings from the rat hippocampus, Scheffer-Teixeira and Tort have now reexamined the evidence for theta-gamma phase-phase coupling. The new results suggest that previous reports describing the phenomenon may have relied on inadequate statistical techniques. Using stringent control analyses, Scheffer-Teixeira and Tort find no evidence for prominent theta-gamma phase-phase coupling in the hippocampus. Instead, the simulations suggest that what appeared to be statistically significant coupling may in reality be an artifact of the previous analysis.

Phase-phase coupling of theta and gamma waves has also been reported in the human hippocampus. The next step therefore is to apply these more robust analysis techniques to data from the human brain. While revisiting previously accepted findings may not always be popular, it will likely be essential if neuroscientists want to accurately understand how new memories are formed.

DOI: http://dx.doi.org/10.7554/eLife.20515.002

Main text


Local field potentials (LFPs) exhibit oscillations of different frequencies, which may co-occur and also interact with one another (Jensen and Colgin, 2007; Tort et al., 2010; Hyafil et al., 2015). Cross-frequency phase-amplitude coupling between theta and gamma oscillations has been well described in the hippocampus, whereby the instantaneous amplitude of gamma oscillations depends on the instantaneous phase of theta (Scheffer-Teixeira et al., 2012; Schomburg et al., 2014). More recently, hippocampal theta and gamma oscillations were also reported to exhibit n:m phase-phase coupling, in which multiple gamma cycles are consistently entrained within one cycle of theta (Belluscio et al., 2012; Zheng and Zhang, 2013; Xu et al., 2013, 2015; Zheng et al., 2016). The existence of different types of cross-frequency coupling suggests that the brain may use different coding strategies to transfer multiplexed information.

Coherent oscillations are believed to take part in network communication by allowing opportunity windows for the exchange of information (Varela et al., 2001; Fries, 2005). Standard phase coherence measures the constancy of the phase difference between two oscillations of the same frequency (Lachaux et al., 1999; Hurtado et al., 2004), and has been associated with cognitive processes such as decision-making (DeCoteau et al., 2007; Montgomery and Buzsáki, 2007; Nácher et al., 2013). Similarly to coherence, cross-frequency phase–phase coupling, or n:m phase-locking, also relies on assessing the constancy of the difference between two phase time series (Tass et al., 1998). However, in this case the original phase time series are accelerated, so that their instantaneous frequencies can match. Formally, n:m phase-locking occurs when Δφnm(t)=nφB(t)mφA(t) is non-uniform but centered around a preferred value, where n*φB (m*φA) denotes the phase of oscillation B (A) accelerated n (m) times (Tass et al., 1998). For example, the instantaneous phase of theta oscillations at 8 Hz needs to be accelerated five times to match in frequency a 40 Hz gamma. A 1:5 phase-phase coupling is then said to occur if theta accelerated five times has a preferred phase lag (i.e., a non-uniform phase difference) in relation to gamma; or, in other words, if five gamma cycles have a consistent phase relationship to one theta cycle.

Cross-frequency phase-phase coupling has previously been hypothesized to take part in memory processes (Lisman and Idiart, 1995; Jensen and Lisman, 2005; Lisman, 2005; Schack and Weiss, 2005; Sauseng et al., 2008, 2009Holz et al., 2010; Fell and Axmacher, 2011). Recent findings suggest that the hippocampus indeed uses such a mechanism (Belluscio et al., 2012; Zheng and Zhang, 2013; Xu et al., 2013, 2015; Zheng et al., 2016). However, by analyzing simulated and actual hippocampal LFPs, in the present work we question the existence of theta-gamma phase-phase coupling.


Measuring n:m phase-locking

We first certified that we could reliably detect n:m phase-locking when present. To that end, we simulated a system of two Kuramoto oscillators – a ‘theta’ and a ‘gamma’ oscillator – exhibiting variability in instantaneous frequency (see Materials and methods). The mean natural frequency of the theta oscillator was set to 8 Hz, while the mean natural frequency of the gamma oscillator was set to 43 Hz (Figure 1A). When coupled, the mean frequencies aligned to a 1:5 factor by changing to 8.5 Hz and 42.5 Hz, respectively (see Guevara and Glass, 1982; García-Alvarez et al., 2008; Canavier et al., 2009). Figure 1B depicts three versions of accelerated theta phases (m = 3, 5 and 7) along with the instantaneous gamma phase (n = 1) of the coupled oscillators (see Figure 1—figure supplement 1 for the uncoupled case). Also shown are the time series of the difference between gamma and accelerated theta phases (Δφnm). The instantaneous phase difference has a preferred lag only for m = 5; when m = 3 or 7, Δφnmchanges over time, precessing forwards (m = 3) or backwards (m = 7) at an average rate of 17 Hz. Consequently, Δφnm distribution is uniform over 0 and 2π for m = 3 or 7, but highly concentrated for m = 5 (Figure 1C). The concentration (or ‘constancy’) of the phase difference distribution is used as a metric of n:m phase-locking. This metric is defined as the length of the mean resultant vector (Rn:m) over unitary vectors whose angle is the instantaneous phase difference (eiΔφnm(t)), and thereby it varies between 0 and 1. For any pair of phase time series, an Rn:m‘curve’ can be calculated by varying m for n = 1 fixed. As shown in Figure 1D, the coupled – but not uncoupled – oscillators exhibited a prominent peak for n:m = 1:5, which shows that Rn:m successfully detects n:m phase-locking.

Filtering-induced n:m phase-locking in white noise

We next analyzed white-noise signals, in which by definition there is no structured activity; in particular, the spectrum is flat and there is no true n:m phase-locking. Rn:m values measured from white noise should be regarded as chance levels. We band-pass filtered white-noise signals to extract the instantaneous phase of theta (θ: 4–12 Hz) and of multiple gamma bands (Figure 2A): slow gamma (γS: 30–50 Hz), middle gamma (γM: 50–90 Hz), and fast gamma (γF: 90–150 Hz). For each frequency pair, we constructed n:m phase-locking curves for epochs of 1 and 10 s, with n = 1 fixed and m varying from 1 to 25 (Figure 2B). In each case, phase-phase coupling was high within the ratio of the analyzed frequency ranges: Rn:m peaked at m = 4–6 for θ−γS, at m = 7–11 for θ−γM, and at m = 12–20 for θ−γF. Therefore, the existence of a ‘bump’ in the Rn:m curve may merely reflect the ratio of the filtered bands and should not be considered as evidence for cross-frequency phase-phase coupling: even filtered white-noise signals exhibit such a pattern.

The bump in the Rn:m curve of filtered white noise is explained by the fact that neighboring data points are not independent. In fact, the phase shift between two consecutive data points follows a probability distribution highly concentrated around 2*π*fc*dt, where fc is the filter center frequency and dt the sampling period (Figure 2—figure supplement 1). For instance, for dt = 1 ms (1000 Hz sampling rate), consecutive samples of white noise filtered between 4 and 12 Hz are likely to exhibit phase difference of 0.05 rad (8 Hz center frequency); likewise, signals filtered between 30 and 50 Hz are likely to exhibit phase differences of 0.25 rad (40 Hz center frequency). In turn, the ‘sinusoidality’ imposed by filtering leads to non-zero Rn:m values, which peak at the ratio of the center frequencies, akin to the fact that perfect 8 Hz and 40 Hz sine waves have Rn:m = 1 at n:m = 1:5. In accordance to this explanation, no Rn:m bump occurs when data points of the gamma phase time series are made independent by sub-sampling with a period longer than a gamma cycle (Figure 2—figure supplement 1), or when extracting phase values from different trials (not shown). As expected, the effect of filtering-induced sinusoidality on Rn:m values is stronger for narrower frequency bands (Figure 2—figure supplement 2).

Qualitatively similar results were found for 1- and 10 s epochs; however, Rn:m values were considerably lower for the latter (Figure 2B). In fact, for any fixed n:m ratio and frequency pair, Rn:m decreased as a function of epoch length (see Figure 2C for θ−γS and R1:5): the longer the white-noise epoch the more the phase difference distribution becomes uniform. In other words, as standard phase coherence (Vinck et al., 2010) and phase-amplitude coupling (Tort et al., 2010), phase-phase coupling has positive bias for shorter epochs. As a corollary, notice that false-positive coupling may be detected if control (surrogate) epochs are longer than the original epoch.

Statistical testing of n:m phase-locking

We next investigated the reliability of surrogate methods for detecting n:m phase-locking (Figure 2D). The ‘Original’ Rn:m value uses the same time window for extracting theta and gamma phases (Figure 2D, upper panel). A ‘Time Shift’ procedure for creating surrogate epochs has been previously employed (Belluscio et al., 2012; Zheng et al., 2016), in which the time window for gamma phase is randomly shifted between 1 to 200 ms from the time window for theta phase (Figure 2D, upper middle panel). A variant of this procedure is the ‘Random Permutation’, in which the time window for gamma phase is randomly chosen (Figure 2D, lower middle panel). Finally, in the ‘Phase Scramble’ procedure, the timestamps of the gamma phase time series are shuffled (Figure 2D, lower panel); clearly, the latter is the least conservative. For each surrogate procedure, Rn:m values were obtained by two approaches: ‘Single Run’ and ‘Pooled’ (Figure 2E). In the first approach, each surrogate run (e.g., a time shift or a random selection of time windows) produces one Rn:m value (Figure 2E, top panels). In the second, Δφnm from several surrogate runs are first pooled, then a single Rn:m value is computed from the pooled distribution (Figure 2E, bottom panel). As illustrated in Figure 2E, Rn:m computed from a pool of surrogate runs is much smaller than when computed for each individual run. This is due to the dependence of Rn:m on the epoch length: pooling instantaneous phase differences across 10 runs of 1 s surrogate epochs is equivalent to analyzing a single surrogate epoch of 10 s. And the longer the analyzed epoch, the more the noise is averaged out and the lower the Rn:m. Therefore, pooled surrogate epochs summing up to 10 s of total data have lower Rn:m than any individual 1 s surrogate epoch.

No phase-phase coupling should be detected in white noise, and therefore Original Rn:m values should not differ from properly constructed surrogates. However, as shown in Figure 2F for θ−γS as an illustrative case (similar results hold for any frequency pair), θ−γS phase-phase coupling in white noise was statistically significantly larger than in phase-scrambled surrogates (for either Single Run or Pooled distributions). This was true for surrogate epochs of any length, although the longer the epoch, the lower the actual and the surrogate Rn:m values, as expected (compare right and left panels of Figure 2F). Pooled R1:5 distributions derived from either time-shifted (Figure 2F) or randomly permutated epochs (not shown) also led to the detection of false positive θ−γS phase-phase coupling. On the other hand, Original Rn:m values were not statistically different from chance distributions when these were constructed from Single Run Rn:m values for either Time Shift and Random Permutation surrogate procedures (Figure 2F; see also Figure 2—figure supplement 3). We conclude that neither scrambling phases nor pooling individual surrogate epochs should be employed for statistically evaluating n:m phase-locking. Chance distributions should be derived from surrogate epochs of the same length as the original epoch and which preserve phase continuity.

To check if Single Run surrogate distributions are capable of statistically detecting true n:m phase-locking, we next simulated noisy Kuramoto oscillators as in Figure 1, but of mean natural frequencies set to 8 and 40 Hz. Original R1:5 values were much greater than the surrogate distribution for coupled – but not uncoupled – oscillators (Figure 3A). This result illustrates that variability in the instantaneous frequency leads to low n:m phase-locking levels for independent oscillators even when their mean frequencies are perfect integer multiples. On the other hand, coupled oscillators have high Rn:m because variations of their instantaneous frequencies are mutually dependent. We then proceeded to analyze simulated LFPs from a previously published model network (Kopell et al., 2010). The network has two inhibitory interneurons, called O and I cells, which spike at theta and gamma frequency, respectively (for a motivation of this model, see Tort et al., 2007). Compared to Single Run surrogate distributions, the model LFP exhibited significant n:m phase-locking only when the interneurons were coupled; Rn:m levels did not differ from the surrogate distribution for the uncoupled network (Figure 3B). (Note that the Rn:m curve also exhibited a peak for both the uncoupled network and Single Run surrogate data, which is due to the low variability in the instantaneous spike frequency of the model cells; without this variability, however, all networks would display perfect n:m phase-locking).

Figure 3.
Download figureOpen in new tabFigure 3. True n:m phase-locking leads to significant Rn:m values.

(A) The left panels show mean Rn:m curves and distributions of R1:5 values for original and surrogate (Random Permutation/Single Run) data obtained from the simulation of two coupled Kuramoto oscillators (n = 300; epoch length = 30 s; *p<0.001, t-test). The right panels show the same, but for uncoupled oscillators. In these simulations, each oscillator has instantaneous peak frequency determined by a Gaussian distribution; the mean natural frequencies of the theta and gamma oscillators were set to 8 Hz and 40 Hz, respectively (coupling does not alter the mean frequencies since they already exhibit a 1:5 ratio; compare with Figure 1). (B) Top panels show results from a simulation of a model network composed of two mutually connected interneurons, O and I cells, which emit spikes at theta and gamma frequency, respectively (Tort et al., 2007; Kopell et al., 2010). Original n:m phase-locking levels are significantly higher than chance (n = 300; epoch length = 30 s; *p<0.001, t-test). The bottom panels show the same, but for unconnected interneurons. In this case, n:m phase-locking levels are not greater than chance.

DOI: http://dx.doi.org/10.7554/eLife.20515.009

Spurious n:m phase-locking due to non-sinusoidal waveforms

The simulations above show that Single Run surrogates can properly detect n:m phase-locking for oscillators exhibiting variable instantaneous frequency, which is the case of hippocampal theta and gamma oscillations. However, it should be noted that high asymmetry of the theta waveform may also lead to statistically significant Rn:m values per se. As illustrated in Figure 4A, a non-sinusoidal oscillation such as a theta sawtooth wave can be decomposed into a sum of sine waves at the fundamental and harmonic frequencies, which have decreasing amplitude (i.e., the higher the harmonic frequency, the lower the amplitude). Importantly, the harmonic frequency components are n:m phase-locked to each other: the first harmonic exhibits a fixed 1:2 phase relationship to the fundamental frequency, the second harmonic a 1:3 relationship, and so on (Figure 4B). Of note, the higher frequency harmonics not only exhibit cross-frequency phase-phase coupling to the fundamental theta frequency but also phase-amplitude coupling, since they have higher amplitude at the theta phases where the sharp deflection occurs (Figure 4C left and Figure 4—figure supplement 1; see also Kramer et al., 2008 and Tort et al., 2013).

The gamma-filtered component of a theta sawtooth wave of variable peak frequency thus displays spurious gamma oscillations (i.e., theta harmonics) that have a consistent phase relationship to the theta cycle irrespective of variations in cycle length. In randomly permutated data, however, the theta phases associated with spurious gamma differ from cycle to cycle due to the variability in instantaneous theta frequency. As a result, the spurious n:m phase-coupling induced by sharp signal deflections is significantly higher than the Random Permutation/Single Run surrogate distribution (Figure 4C right and Figure 4—figure supplement 2 top row). Interestingly, the significance of this spurious effect is much lower when using the Time Shift procedure (Figure 4—figure supplement 2 bottom row), probably due to the proximity between the original and the time-shifted time series (200 ms maximum distance).

Assessing n:m phase-locking in actual LFPs

We next proceeded to analyze hippocampal CA1 recordings from seven rats, focusing on the periods of prominent theta activity (active waking and REM sleep). We found similar results between white noise and actual LFP data. Namely, Rn:m curves peaked at n:m ratios according to the filtered bands, and Rn:m values were lower for longer epochs (Figure 5A; compare with Figure 2B). As shown in Figure 5B, Original Rn:m values were not statistically different from a proper surrogate distribution (Random Permutation/Single Run) in epochs of up to 100 s (but see Figure 10). Noteworthy, as with white-noise data (Figure 2F), false positive phase-phase coupling would be inferred if an inadequate surrogate method were employed (Time Shift/Pooled) (Figure 5B).

We also found no difference between original and surrogate n:m phase-locking levels when employing the metric described in Sauseng et al. (2009) (Figure 5—figure supplement 1), and when estimating theta phase by interpolating phase values between 4 points of the theta cycle (trough, ascending, peak and descending points) as performed in Belluscio et al. (2012) (Figure 5—figure supplement 2). The latter was somewhat expected since the phase-phase coupling results in Belluscio et al. (2012) did not depend on this particular method of phase estimation (see their Figure 6Ce). Moreover, coupling levels did not statistically differ from zero when using the pairwise phase consistency metric described in Vinck et al. (2010) (Figure 5—figure supplement 1).

We further confirmed our results by analyzing data from three additional rats recorded in an independent laboratory (Figure 5—figure supplement 3; see Materials and methods). In addition, we also found similar results in LFPs from other hippocampal layers than s. pyramidale (Figure 5—figure supplement 4), in neocortical LFPs (not shown), in current-source density (CSD) signals (Figure 5—figure supplement 4), in independent components that isolate activity of specific gamma sub-bands (Schomburg et al., 2014) (Figure 5—figure supplement 5), and in transient gamma bursts (Figure 5—figure supplement 6).

On diagonal stripes in phase-phase plots

Since Original Rn:m values were not greater than Single Run surrogate distributions, we concluded that there is lack of convincing evidence for n:m phase-locking in the hippocampal LFPs analyzed here. However, as in previous reports (Belluscio et al., 2012; Zheng et al., 2016), phase-phase plots (2D histograms of theta phase vs gamma phase) of actual LFPs displayed diagonal stripes (Figure 6), which seem to suggest phase-phase coupling. We next sought to investigate what causes the diagonal stripes in phase-phase plots.

Figure 6.
Download figureOpen in new tabFigure 6. Phase-phase plots of hippocampal LFPs display diagonal stripes.

Phase-phase plot for theta and slow gamma (average over animals; n = 7 rats). Notice diagonal stripes suggesting phase-phase coupling.

DOI: http://dx.doi.org/10.7554/eLife.20515.022

In Figure 7 we analyze a representative LFP with prominent theta oscillations at ~7 Hz recorded during REM sleep. Due to the non-sinusoidal shape of theta (Belluscio et al., 2012; Sheremet et al., 2016), the LFP also exhibited spectral peaks at harmonic frequencies (Figure 7A). We constructed phase–phase plots using LFP components narrowly filtered at theta and its harmonics: 14, 21, 28 and 35 Hz. Similarly to the sawtooth wave (Figure 4B), the phase-phase plots exhibited diagonal stripes whose number was determined by the harmonic order (i.e., the 1st harmonic exhibited two stripes, the second harmonic three stripes, the third, four stripes and the fourth, five stripes; Figure 7Bi–iv). Interestingly, when the LFP was filtered at a broad gamma band (30–90 Hz), we observed five diagonal stripes, the same number as when narrowly filtering at 35 Hz; moreover, both gamma and 35 Hz filtered signals exhibited the exact same phase lag (Figure 7Biv–v). Therefore, these results indicate that the diagonal stripes in phase-phase plots may be influenced by theta harmonics. Under this interpretation, signals filtered at the gamma band would be likely to exhibit as many stripes as expected for the first theta harmonic falling within the filtered band. Consistent with this possibility, we found that the peak frequency of theta relates to the number of stripes (Figure 8).

As in previous studies (Belluscio et al., 2012; Zheng et al., 2016), phase-phase plots constructed using data averaged from individual time-shifted epochs exhibited no diagonal stripes (Figure 7Bvi and Figure 8). This is because different time shifts lead to different phase lags; the diagonal stripes of individual surrogate runs that could otherwise be apparent cancel each other out when combining data across multiple runs of different lags (Figure 8—figure supplement 1). Moreover, as in Belluscio et al. (2012), the histogram counts that give rise to the diagonal stripes were deemed statistically significant when compared to the mean and standard deviation over individual counts from time-shifted surrogates (Figure 7—figure supplement 1 and Figure 8).

To gain further insight into what generates the diagonal stripes, we next analyzed white-noise signals. As shown in Figure 9A, phase-phase plots constructed from filtered white-noise signals also displayed diagonal stripes. Since white noise has no harmonics, these results show that the sinusoidality induced by the filter can by itself lead to diagonal stripes in phase-phase plots, in the same way that it leads to a bump in the Rn:m curve (Figure 2 and Figure 2—figure supplement 1). Importantly, as in actual LFPs, bin counts in phase-phase plots of white-noise signals were also deemed statistically significant when compared to the distribution of bin counts from time-shifted surrogates (Figure 9A). Since by definition white noise has no n:m phase-locking, we concluded that the statistical analysis of phase-phase plots as originally introduced in Belluscio et al. (2012) is too liberal. Nevertheless, we found that phase-phase plots of white noise were no longer statistically significant when using the same approach as in Belluscio et al. (2012) but corrected for multiple comparisons (i.e., the number of bins) by the Holm-Bonferroni method (the FDR correction still led to significant bins; not shown). This result was true for different epoch lengths and also when computing surrogate phase-phase plots using the Random Permutation procedure (Figure 9A). Consistently, for all epoch lengths, Original Rn:m values fell inside the distribution of Single Run surrogate Rn:m values computed using either Time Shift and Random Permutation procedures (Figure 9B).

Figure 9.
Download figureOpen in new tabFigure 9. Phase-phase plots of white-noise signals display diagonal stripes.

(A) Representative phase-phase plots computed for white-noise signals. Notice the presence of diagonal stripes for both 100 s (left) and 1200 s (right) epochs. The colormaps underneath show the p-values of the original bin counts when compared to the mean and standard deviation over bin counts of single time-shifted surrogate runs. Also shown are significance maps after correcting for multiple comparisons (Holm-Bonferroni) using either time-shifted (top) or randomly permutated (bottom) surrogate runs. No bin count was considered statistically significant after the correction. (B) The top panels show original Rn:m curves (green) plotted along with Single Run distributions of Rn:m curves of time-shifted (orange) and randomly permutated (red) surrogates for different epoch lengths (shades denote the 2.5th–97.5th percentile interval; n = 2100 per distribution). The bottom panels show the same in a zoomed scale.

DOI: http://dx.doi.org/10.7554/eLife.20515.027

The observations above suggest that the diagonal stripes in phase-phase plots of hippocampal LFPs may actually be caused by filtering-induced sinusoidality, as opposed to being an effect of theta harmonics as we first interpreted. To test this possibility, we next revisited the significance of phase-phase plots of actual LFPs. For epochs of up to 100 s, we found similar results as in white noise, namely, bin counts were no longer statistically significant after correcting for multiple comparisons (Holm-Bonferroni method); this was true when using either the Time Shift or Random Permutation procedures (Figure 10A). Surprisingly, however, when analyzing much longer time series (10 or 20 min of concatenated periods of REM sleep), several bin counts became statistically significant when compared to randomly permutated, but not time-shifted, surrogates (Figure 10A). Moreover, this result reflected in the Rn:m curves: the Original Rn:m curve fell within the distribution of Time Shift/Single Run surrogate Rn:m values for all analyzed lengths, but outside the distribution of Random Permutation/Single Run surrogates for the longer time series (Figure 10B). We believe such a finding relates to what we observed for synthetic sawtooth waves, in which Random Permutation was more sensitive than Time Shift to detect the significance of the artifactual coupling caused by waveform asymmetry (Figure 4—figure supplement 2). In this sense, the n:m phase-locking between fundamental and harmonic frequencies would persist for small time shifts (±200 ms), albeit in different phase relations, while it would not resist the much larger time shifts obtained through random permutations. However, irrespective of this explanation, it should be noted that since the n:m phase-locking metrics cannot separate artifactual from true coupling, the possibility of the latter cannot be discarded. But if this is the case, we consider unlikely that the very low coupling level (~0.03) would have any physiological significance.

We conclude that the diagonal stripes in phase-phase plots of both white noise and actual LFPs are mainly caused by a temporary n:m alignment of the phase time-series secondary to the filtering-induced sinusoidality, and as such they are also apparent in surrogate data (Figure 8—figure supplement 1 and Figure 10—figure supplement 1). However, for actual LFPs there is a second influence, which can only be detected when analyzing very long epoch lengths, and which we believe is due to theta harmonics.


Theta and gamma oscillations are hallmarks of hippocampal activity during active exploration and REM sleep (Buzsáki et al., 2003; Csicsvari et al., 2003; Montgomery et al., 2008). Theta and gamma are well known to interact by means of phase-amplitude coupling, in which the instantaneous gamma amplitude waxes and wanes as a function of theta phase (Bragin et al., 1995; Scheffer-Teixeira et al., 2012; Caixeta et al., 2013). This particular type of cross-frequency coupling has been receiving large attention and related to functional roles (Canolty and Knight, 2010; Hyafil et al., 2015). In addition to phase-amplitude coupling, theta and gamma oscillations can potentially interact in many other ways (Jensen and Colgin, 2007; Hyafil et al., 2015). For example, the power of slow gamma oscillations may be inversely related to theta power (Tort et al., 2008), suggesting amplitude-amplitude coupling. Recently, it has been reported that theta and gamma in hippocampal LFPs would also couple by means of n:m phase-locking (Belluscio et al., 2012; Zheng and Zhang, 2013; Xu et al., 2013, 2015; Zheng et al., 2016). Among other implications, this finding was taken as evidence for network models of working memory (Lisman and Idiart, 1995; Jensen and Lisman, 2005; Lisman, 2005) and for a role of basket cells in generating cross-frequency coupling (Belluscio et al., 2012; Buzsáki and Wang, 2012). However, our results show a lack of convincing evidence for n:m phase-locking in the two hippocampal datasets analyzed here, and further suggest that previous work may have spuriously detected phase-phase coupling due to an improper use of surrogate methods, a concern also raised for phase-amplitude coupling (Aru et al., 2015).

Statistical inference of phase-phase coupling

When searching for phase-phase coupling between theta and gamma, we noticed that our Rn:m values differed from those reported in previous studies (Belluscio et al., 2012; Xu et al., 2013, 2015; Zheng et al., 2016). We suspected that this could be due to differences in the duration of the analyzed epochs. We then investigated the dependence of Rn:m on epoch length, and found a strong positive bias for shorter epochs. In addition, Rn:m values exhibit greater variability across samples as epoch length decreases for both white noise and actual data (e.g., compare in Figure 5B the data dispersion in Original R1:5 or R1:8 boxplots for different epoch lengths). Since theta and gamma peak frequencies are not constant in these signals, the longer the epoch, the more the theta and gamma peak frequencies are allowed to fluctuate and the more apparent the lack of coupling. On the other hand, Δφnm distribution becomes less uniform for shorter epochs. The dependence of n:m phase-coupling metrics on epoch length has important implications in designing surrogate epochs for testing the statistical significance of actual Rn:m values. Of note, methodological studies on 1:1 phase-synchrony have properly used single surrogate runs of the same length as the original signal (Le Van Quyen et al., 2001; Hurtado et al., 2004). As demonstrated here, spurious detection of phase-phase coupling may occur if surrogate epochs are longer than the original epoch. This is the case when one lumps together several surrogate epochs before computing Rn:m. When employing proper controls, our results show that Rn:m values of real data do not differ from surrogate values in theta epochs of up to 100 s. Moreover, the prominent bump in the Rn:m curve disappears when subsampling data at a lower frequency than gamma for both white noise and hippocampal LFPs (see Figure 2—figure supplement 1 and Figure 5—figure supplement 7), which suggests that it is due to the statistical dependence among contiguous data points introduced by the filter (which we referred to as ‘filtering-induced sinusoidality’).

Therefore, even though the n:m phase-locking metric Rn:m is theoretically well-defined and varies between 0 and 1, an estimated Rn:m value in isolation does not inform if two oscillations exhibit true phase-coupling or not. This can only be inferred after testing the statistical significance of the estimated Rn:m value against a proper surrogate distribution (but notice that false-positive cases may occur due to waveform asymmetry; Figure 4C). While constructing surrogate data renders the metric computationally more expensive, such an issue is not specific for measuring n:m phase-locking but also happens for other metrics commonly used in the analysis of neurophysiological data, such as coherence, spike-field coupling, phase-amplitude coupling, mutual information and directionality measures, among many others (Le Van Quyen et al., 2001; Hurtado et al., 2004; Pereda et al., 2005; Tort et al., 2010).

The recent studies assessing theta-gamma phase-phase coupling in hippocampal LFPs have not tested the significance of individual Rn:m values against chance (Belluscio et al., 2012; Zheng and Zhang, 2013; Xu et al., 2013, 2015; Zheng et al., 2016). Two studies (Belluscio et al., 2012; Zheng et al., 2016) statistically inferred the existence of n:m phase-locking by comparing empirical phase-phase plots with those obtained from the average of 1000 time-shifted surrogate runs. Specifically, Belluscio et al. (2012) established a significance threshold for each phase-phase bin based on the mean and standard deviation of individual surrogate counts in that bin, and showed that the bin counts leading to diagonal stripes were statistically significant. Here we were able to replicate these results (Figure 7—figure supplement 1 and Figure 8). However, we note that a phase-phase bin count is not a metric of n:m phase-locking; it does not inform coupling strength and even coupled oscillators have bins with non-significant counts. A bin count would be analogous to a phase difference vector (eiΔφnm(t)), which is also not a metric of n:m phase-locking per se, but used to compute one. That is, in the same way that the Rn:m considers all phase difference vectors, n:m phase-locking can only be inferred when considering all bin counts in a phase-phase plot. In this sense, by analyzing the phase-phase plot as a whole, it was assumed that the appearance diagonal stripes was due to theta-gamma coupling; no such stripes were apparent in phase-phase plots constructed from the average over all surrogate runs (see Figure 6A in Belluscio et al., 2012). However, here we showed that single time-shifted surrogate runs do exhibit diagonal stripes (Figure 8—figure supplement 1 and Figure 10—figure supplement 1), that is, similar stripes exist at the level of a Single Run surrogate analysis, in the same way that Single Run surrogates also exhibit a bump in the Rn:m curve. Averaging 1000 surrogate phase-phase plots destroys the diagonal stripes since different time shifts lead to different phase lags. Moreover, since the average is the sum divided by a scaling factor (the sample size), computing the average phase-phase plot is equivalent to computing a single phase-phase plot using the pool of all surrogate runs, which is akin to the issue of computing a single Rn:m value from a pooled surrogate distribution (Figure 2). Note that even bin counts in phase-phase plots of white noise are considered significant under the statistical analysis introduced in Belluscio et al. (2012) (Figure 9A). Nevertheless, this was no longer the case when adapting their original framework to include a Holm-Bonferroni correction for multiple comparisons (Figure 9A).

Here we showed that the presence of diagonal stripes in phase-phase plots is not sufficient to conclude the existence of phase-phase coupling. The diagonal stripes are simply a visual manifestation of a maintained phase relationship, and as such they essentially reflect what Rn:m measures: that is, the ‘clearer’ the stripes, the higher the Rn:m. Therefore, in addition to true coupling, the same confounding factors that influence Rn:m also influence phase-phase plots, such as filtering-induced sinusoidality and frequency harmonics. Our results suggest that the former is a main factor, because white-noise signals have no harmonics but nevertheless display stripes in phase-phase plots (Figure 9A). In accordance, no stripes are observed in phase-phase plots of white noise when subsampling the time series (Figure 10—figure supplement 2; see also Figure 2—figure supplement 1). However, in actual LFPs filtering is not the only influence: (1) for the same filtered gamma band (30–50 Hz), the number of stripes relates to theta frequency (Figure 8); (2) for very long time series (i.e., 10–20 min of concatenated data), the stripes in phase-phase plots of actual data – but not of white noise – persist after correcting for multiple comparisons when employing Random Permutation/Single Run surrogates (Figure 10A); (3) a striped-like pattern remains in phase-phase plots of actual LFPs after subsampling the time series (Figure 10—figure supplement 2). Consistently, Rn:m values of actual LFPs are greater than those of white noise in 1200 s epochs (~0.03 vs ~0.005, compare the bottom right panels of Figures 9B and 10B). Interestingly, Original Rn:m values of actual LFPs are not statistically different from the distribution of Time Shift/Single Run surrogates even for the very long epochs (Figure 10B), which suggests that Random Permutation is more powerful than Time Shift and should therefore be preferred. Though a very weak but true coupling effect cannot be discarded, based on our analysis of sawtooth waves (Figure 4 and Figure 4—figure supplement 2), we believe these results can be explained by theta harmonics, which would remain phase-locked to the fundamental frequency under small time shifts. Sharp signal deflections have been previously recognized to generate artifactual phase-amplitude coupling (Kramer et al., 2008; Scheffer-Teixeira et al., 2013; Tort et al., 2013; Aru et al., 2015; Lozano-Soldevilla et al., 2016). Interestingly, Hyafil (2015) recently suggested that the non-sinusoidality of alpha waves could underlie the 1:2 phase-locking between alpha and beta observed in human EEG (Nikulin and Brismar, 2006; see also Palva et al., 2005). To the best of our knowledge, there is currently no metric capable of automatically distinguishing true cross-frequency coupling from waveform-induced artifacts in collective signals such as LFP, EEG and MEG signals. Ideally, learning how the signal is generated from the activity of different neuronal populations would answer whether true cross-frequency coupling exists or not (Hyafil et al., 2015), but unfortunately this is methodologically challenging.

Lack of evidence vs evidence of non-existence

One could argue that we did not analyze a proper dataset, or else that prominent phase-phase coupling would only occur during certain behavioral states not investigated here. We disagree with these arguments for the following reasons: (1) we could reproduce our results using a second dataset from an independent laboratory (Figure 5—figure supplement 3), and (2) we examined the same behavioral states in which n:m phase-locking was reported to occur (active waking and REM sleep). One could also argue that there exists multiple gammas, and that different gamma types are most prominent in different hippocampal layers (Colgin et al., 2009; Scheffer-Teixeira et al., 2012; Tort et al., 2013; Schomburg et al., 2014; Lasztóczi and Klausberger, 2014); therefore, prominent theta-gamma phase-phase coupling could exist in other hippocampal layers not investigated here. We also disagree with this possibility because: (1) we examined the same hippocampal layer in which theta-gamma phase-phase coupling was reported to occur (Belluscio et al., 2012); moreover, (2) we found similar results in all hippocampal layers (we recorded LFPs using 16-channel silicon probes, see Materials and methods) (Figure 5—figure supplement 4) and (3) in parietal and entorhinal cortex recordings (not shown). Furthermore, similar results hold when (4) filtering LFPs within any gamma sub-band (Figure 5 and Figure 5—figure supplement 1 to 6), (5) analyzing CSD signals (Figure 5—figure supplement 4), or (6) analyzing independent components that maximize activity within particular gamma sub-bands (Schomburg et al., 2014) (Figure 5—figure supplement 5). Finally, one could argue that gamma oscillations are not continuous but transient, and that assessing phase-phase coupling between theta and transient gamma bursts would require a different type of analysis than employed here. Regarding this argument, we once again stress that we used the exact same methodology as originally used to detect theta-gamma phase-phase coupling (Belluscio et al., 2012). Nevertheless, we also ran analysis only taking into account periods in which gamma amplitude was >2 SD above the mean (‘gamma bursts’) and found no statistically significant phase-phase coupling (Figure 5—figure supplement 6).

Following Belluscio et al. (2012), other studies also reported theta-gamma phase-phase coupling in the rodent hippocampus (Zheng and Zhang, 2013; Xu et al., 2013, 2015; Zheng et al., 2016) and amygdala (Stujenske et al., 2014). In addition, human studies had previously reported theta-gamma phase-phase coupling in scalp EEG (Sauseng et al., 2008, 2009; Holz et al., 2010). Most of these studies, however, have not tested the statistical significance of coupling levels against chance (Sauseng et al., 2008, 2009; Holz et al., 2010; Zheng and Zhang, 2013; Xu et al., 2013, 2015; Stujenske et al., 2014), while Zheng et al. (2016) based their statistical inferences on the inspection of diagonal stripes in phase-phase plots as originally introduced in Belluscio et al. (2012). We further note that epoch length was often not informed in the animal studies. Based on our results, we believe that differences in analyzed epoch length are likely to explain the high variability of Rn:m values across different studies, from ~0.4 (Zheng et al., 2016) down to 0.02 (Xu et al., 2013).

Since it is philosophically impossible to prove the absence of an effect, the burden of proof should be placed on demonstrating that a true effect exists. In this sense, and to the best of our knowledge, none of previous research investigating theta-gamma phase-phase coupling has properly tested Rn:m against chance. Many studies have focused on comparing changes in n:m phase-locking levels, but we believe these can be influenced by other variables such as changes in power, which affect the signal-to-noise ratio and consequently also the estimation of the phase time series. Interestingly, in their pioneer work, Tass and colleagues used filtered white noise to construct surrogate distributions and did not find significant n:m phase-locking among brain oscillations (Tass et al., 1998, 2003). On the other hand, it is theoretically possible that n:m phase-locking exists but can only be detected by other types of metrics yet to be devised. In any case, our work shows that there is currently no convincing evidence for genuine theta-gamma phase-phase coupling using the same phase-locking metric (Rn:m) as employed in previous studies (Belluscio et al., 2012; Zheng and Zhang, 2013Xu et al., 2013, 2015; Stujenske et al., 2014; Zheng et al., 2016), at least when examining LFP epochs of up to 100 s of prominent theta activity. For longer epoch lengths, though, we did find that Rn:m values of hippocampal LFPs may actually differ from those of randomly permuted, but not time-shifted, surrogates (Figure 10B). While we tend to ascribe such result to the effect of theta harmonics, we note that the possibility of true coupling cannot be discarded. But we are particularly skeptical that the very low levels of coupling strength observed in long LFP epochs would be physiologically meaningful.

Implications for models of neural coding by theta-gamma coupling

Lisman and Idiart (1995) proposed an influential model in which theta and gamma oscillations would interact to produce a neural code. The theta-gamma coding model has since been improved (Jensen and Lisman, 2005; Lisman, 2005; Lisman and Buzsáki, 2008), but its essence remains the same (Lisman and Jensen, 2013): nested gamma cycles would constitute memory slots, which are parsed at each theta cycle. Accordingly, Lisman and Idiart (1995) hypothesized that working memory capacity (7 ± 2) is determined by the number of gamma cycles per theta cycle.

Both phase-amplitude and phase-phase coupling between theta and gamma have been considered experimental evidence for such coding scheme (Lisman and Buzsáki, 2008; Sauseng et al., 2009; Axmacher et al., 2010; Belluscio et al., 2012; Lisman and Jensen, 2013; Hyafil et al., 2015; Rajji et al., 2016). In the case of phase-amplitude coupling, the modulation of gamma amplitude within theta cycles would instruct a reader network when the string of items represented in different gamma cycles starts and terminates. On the other hand, the precise ordering of gamma cycles within theta cycles that is consistent across theta cycles would imply phase-phase coupling; indeed, n:m phase-locking is a main feature of computational models of sequence coding by theta-gamma coupling (Lisman and Idiart, 1995; Jensen and Lisman, 1996; Jensen et al., 1996). In contrast to these models, however, our results show that the theta phases in which gamma cycles begin/end are not fixed across theta cycles, which is to say that gamma cycles are not precisely timed but rather drift; in other words, gamma is not a clock (Burns et al., 2011).

If theta-gamma neural coding exists, our results suggest that the precise location of gamma memory slots within a theta cycle is not required for such a code, and that the ordering of the represented items would be more important than the exact spike timing of the cell assemblies that represent the items (Lisman and Jensen, 2013).


In summary, while absence of evidence is not evidence of absence, our results challenge the hypothesis that theta-gamma phase-phase coupling exists in the hippocampus. At best, we only found significant Rn:m values when examining long LFP epochs (>100 s), but these had very low magnitude (and we particularly attribute their statistical significance to the effects of harmonics). We believe that the evidence in favor of n:m phase-locking in other brain regions and signals could potentially also be explained by simpler effects (e.g., filtering-induced sinusoidality, asymmetrical waveform, and improper statistical tests). While no current technique can differentiate spurious from true phase-phase coupling, previous findings should be revisited and, whenever suitable, checked against the confounding factors and the more conservative surrogate procedures outlined here.

Materials and methods

Animals and surgery

All procedures were approved by our local institutional ethics committee (Comissão de Ética no Uso de Animais - CEUA/UFRN, protocol number 060/2011) and were in accordance with the National Institutes of Health guidelines. We used seven male Wistar rats (2–3 months; 300–400 g) from our breeding colony, kept under 12 hr/12 hr dark-light cycle. We recorded from the dorsal hippocampus through either multi-site linear probes (n = 6 animals; 4 probes had 16 4320 μm2 contacts spaced by 100 μm; 1 probe had 16 703 μm2 contacts spaced by 100 μm; 1 probe had 16 177 μm2 contacts spaced by 50 μm; all probes from NeuroNexus) or single wires (n = 1 animal; 50 μm diameter) inserted at AP −3.6 mm and ML 2.5 mm. Results shown in the main figures were obtained for LFP recordings from the CA1 pyramidal cell layer, identified by depth coordinate and characteristic electrophysiological benchmarks such as highest ripple power (see Figure 5—figure supplement 4 for an example). Similar results were obtained for recordings from other hippocampal layers (Figure 5—figure supplement 4).

We also analyzed data from three additional rats downloaded from the Collaborative Research in Computational Neuroscience data sharing website (www.crcns.org) (Figure 5—figure supplement 3). These recordings are a generous contribution by György Buzsáki’s laboratory (HC3 dataset, Mizuseki et al., 20132014).

Data collection

Recording sessions were performed in an open field (1 m x 1 m) and lasted 4–5 hr. Raw signals were amplified (200x), filtered between 1 Hz and 7.5 kHz (third order Butterworth filter), and digitized at 25 kHz (RHA2116, IntanTech). The LFP was obtained by further filtering between 1–500 Hz and downsampling to 1000 Hz.

Data analysis

Active waking and REM sleep periods were identified from spectral content (high theta/delta power ratio) and video recordings (movements during active waking; clear sleep posture and preceding slow-wave sleep for REM). The results were identical for active waking and REM epochs; throughout this work we only show the latter. The analyzed REM sleep dataset is available at http://dx.doi.org/10.5061/dryad.12t21. MATLAB codes for reproducing our analyses are available at https://github.com/tortlab/phase_phase .

We used built-in and custom-written MATLAB routines. Band-pass filtering was obtained using a least squares finite impulse response (FIR) filter by means of the ‘eegfilt’ function from the EEGLAB Toolbox (Delorme and Makeig, 2004). The filter order was three times the sampling rate divided by the low cutoff frequency. The eegfilt function calls the MATLAB ‘filtfilt’ function, which applies the filter forward and then again backwards to ensure no distortion of phase values. Similar results were obtained when employing other types of filters (Figure 5—figure supplement 8).

The phase time series was estimated through the Hilbert transform. To estimate the instantaneous theta phase of actual data, we filtered the LFP between 4–20 Hz, a bandwidth large enough to capture theta wave asymmetry (Belluscio et al., 2012). Estimating theta phase by the interpolation method described in Belluscio et al. (2012) led to similar results (Figure 5—figure supplement 2).

The CSD signals analyzed in Figure 5—figure supplement 4 were obtained as −A +2B −C, where A, B and C denote LFP signals recorded from adjacent probe sites. In Figure 5—figure supplement 5, the independent components were obtained as described in Schomburg et al. (2014); phase-amplitude comodulograms were computed as described in Tort et al. (2010).

n:m phase-locking

We measured the consistency of the phase difference between accelerated time series (Δφnm(tj)=nφγ(tj)mφθ(tj)). To that end, we created unitary vectors whose angle is the instantaneous phase difference (eiΔφnm(tj)), where j indexes the time sample, and then computed the length of the mean vector: Rn:m=1Nj=1NeiΔφnm(tj), where N is the total number of time samples (epoch length in seconds x sampling frequency in Hz). Rn:m equals 1 when Δφnm is constant for all time samples tj, and 0 when Δφnm is uniformly distributed. This metric is also commonly referred to as ‘mean resultant length’ or ‘mean radial distance’ (Belluscio et al., 2012; Stujenske et al., 2014; Zheng et al., 2016). Qualitatively similar results were obtained when employing the framework introduced in Sauseng et al. (2009), which computes the mean radial distance using gamma phases in separated theta phase bins, or the pairwise phase consistency metric described in Vinck et al. (2010) (Figure 5—figure supplement 1). Phase-phase plots were obtained by first binning theta and gamma phases into 120 bins and next constructing 2D histograms of phase counts, which were smoothed using a Gaussian kernel of σ = 10 bins.


In all cases, theta phase was kept intact while gamma phase was mocked in three different ways: (1) Time Shift: the gamma phase time series is randomly shifted between 1 and 200 ms; (2) Random Permutation: a contiguous gamma phase time series of the same length as the original is randomly extracted from the same session. (3) Phase Scrambling: the timestamps of the gamma phase time series are randomly shuffled (thus not preserving phase continuity). For each case, Rn:m values were computed using either Δφnm distribution for single surrogate runs (Single Run Distribution) or the pooled distribution of Δφnm over 100 surrogate runs (Pooled Distribution).

For each animal, behavioral state (active waking or REM sleep) and epoch length, we computed 300 Original Rn:m values using different time windows along with 300 mock Rn:m values per surrogate method. Therefore, in all figures each boxplot was constructed using the same number of samples (=300 x number of animals). For instance, in Figure 5B we used n = 7 animals x 300 samples per animal = 2100 samples (but see Statistics below). In Figure 2, boxplot distributions for the white-noise data were constructed using n = 2100.


Kuramoto oscillators displaying n:m phase-locking were modeled as described in Osipov et al. (2007):φθ˙= ωθ+ εsin(nφγmφθ)φγ˙= ωγ+ εsin(mφθnφγ),

where ε is the coupling strength and ωθ and ωγ are the natural frequencies of theta and gamma, respectively, which followed a Gaussian probability (σ = 5 Hz) at each time step. We used ε = 10, n = 1, m = 5, and dt = 0.001 s. The mean theta and gamma frequencies of each simulation are stated in the main text. For uncoupled oscillators, we set ε = 0.

For implementing the O-I cell network (Figure 3B), we simulated the model previously described in Kopell et al. (2010). We used the same parameters as in Figure 3A of Kopell et al. (2010), with white noise (σ = 0.001) added to the I cell drive to create variations in spike frequency. NEURON (https://www.neuron.yale.edu/) codes for the model are available at ModelDB (https://senselab.med.yale.edu/).

The sawtooth wave in Figure 4C was simulated using dt = 0.001 s. Its instantaneous frequency followed a Gaussian distribution with mean = 8 Hz and σ = 5 Hz; white noise (σ = 0.1) was added to the signal.

In Figures 3 and 4C, boxplot distributions for simulated data were constructed using n = 300.


For white noise data (Figure 2F), given the large sample size (n = 2100) and independence among samples, we used one-way ANOVA with Bonferroni post-hoc test. For statistical analysis of real data (Figure 5B), we avoided nested design and inflation of power and used the mean Rn:m value per animal. In this case, due to the reduced sample size (n = 7) and lack of evidence of normal distribution (Shapiro-Wilk normality test), we used the Friedman’s test and Nemenyi post-hoc test. In Figures 3 and 4C, we tested if Rn:m values of simulated data were greater than the distribution of surrogate values using one-tailed t-tests.



The authors are indebted to the reviewers for many constructive comments and helpful suggestions. The authors are grateful to Jurij Brankačk and Andreas Draguhn for donation of NeuroNexus probes. Supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES). The authors declare no competing financial interests.

Decision letter

Frances K Skinner, University Health Network, Canada

In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.

[Editors’ note: a previous version of this study was rejected after peer review, but the authors submitted for reconsideration. The first decision letter after peer review is shown below.]

Thank you for submitting your work entitled "Lack of evidence for cross-frequency phase-phase coupling between theta and gamma oscillations in the hippocampus" for consideration by eLife. Your article has been favorably evaluated by Timothy Behrens (Senior Editor) and four reviewers, one of whom is a member of our Board of Reviewing Editors.

Our decision has been reached after consultation between the reviewers. Based on these discussions and the individual reviews, we regret to inform you that your work will not be considered further for publication in eLife at this time. Below we provide detailed feedback for your consideration if you choose to re-submit, and we encourage this possibility.

While all of the reviewers felt that this was an important and interesting paper, they also thought that it was presented in a confusing manner, was limited, and could lead to confusion rather than clarification for the field. Given the contrarian nature of the manuscript as presented, this would not be helpful. The reviewers' overall comments are provided here:

1) The paper presentation should be changed to be less confrontational and completely clear on what is being said and done. A well-crafted manuscript with clarity and sufficient explanations is lacking in the present submission.

A) Specifically, a more detailed review of the literature, a more careful presentation and discussion of the simulation results, and a more careful comparison with the procedures utilized in [Belluscio et al. 2012] is needed.

B) We note that phase-coupling is well defined theoretically from the tools of dynamical systems, but what is not clear is a selective measure of cross-frequency phase coupling. Defining an improved phase-phase coupling measure that detects true coupling and ignores artifactual coupling might be possible. Such a measure – even if it's not the optimal measure – might serve as a "patch" to existing approaches.

C) Finding the optimal phase-phase coupling measure (and assessing its statistical properties) would be a challenge. That is, a consistent and operational way to define how cross-frequency phase coupling can be measured (such that white noise or asymmetrical oscillation would not qualify). Clearly stating the lack of a clear measure as an open problem formally would be of value. In essence, the manuscript would need to make these aspects clear (i.e., present limitations of their work and others, possible fixes/challenges associated with having phase-phase coupling measures), and in this way can identify a methodological issue that's driving incorrect conclusions in the literature, and so be of service to the field.

2) Given that a clear measure of cross-frequency phase-coupling does not exist, a change of title should be considered.

3) A repository (of data and algorithms) should complement the work, and would be beneficial for the community moving forth as analyses, clear understanding and interpretation of artificial and biological data would be available.

[Editors’ note: what now follows is the decision letter after the authors submitted for further consideration.]

Thank you for submitting your article "On cross-frequency phase-phase coupling between theta and gamma oscillations in the hippocampus" for consideration by eLife. Your article has been favorably evaluated by Timothy Behrens (Senior Editor) and four reviewers, one of whom is a member of our Board of Reviewing Editors.

The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.

In this resubmitted paper on phase-phase coupling, all of the reviewers thought that several concerns had been addressed and suggestions implemented (such as the title change). The reviewers agreed that this is an important article that highlights a methodological issue impacting our understanding of the brain's activity and function. The authors are commended for sharing their analysis methods (on GitHub) and data.

The authors address three points in their paper: a statistical issue (using appropriates surrogate to detect n:m coupling), a conceptual issue (whether n:m coupling can be dissociated from asymmetrical oscillations), and an experimental one (whether n:m coupling is present in hippocampus). The first and last points are well made. However, this is not the case for the second, and other aspects remain unclear as presented. The authors are encouraged to revise and edit their paper to clarify all of the following issues.

1) A clear distinction between true phase versus estimated phase needs to be made as it is confusing as presented. Specifically:

A) As the authors' state, the theoretical quantity of n:m phase locking is well-defined. However, the estimated quantity of n:m phase locking as deduced from noisy neural data is fraught with difficulty. The authors show here that simply applying the theoretical n:m phase locking equation to data produces estimates that – without appropriate statistical tests – lead to spurious results. This is an important conclusion. Future work might seek to improve or correct (somehow) the estimation of the theoretical quantity, or propose a new theoretical quantity that is more easily estimated.

B) The point in the Discussion (subsection “Statistical inference of phase-phase coupling”, last paragraph) that true n:m coupling can be disentangled from asymmetrical oscillation by visual inspection of the LFP or EEG trace, precisely because they will look very much alike does not seem appropriate. This distinction may be hard to make at the macroscopic level (LFP/EEG) and could rather be searched for in how this signal is generated from the activity of distinct neural subpopulations. If the same gamma oscillations that are shown to phase-lock to theta oscillations can be dissociated neurally from theta oscillations (and indeed the authors did a tremendous job in previous publications at showing that hippocampal theta and gamma rely on distinct interconnected networks), then the evidence is rather of favor of genuine n:m locking. In the converse case, if no population selectively engaged in the faster oscillation can be identified, I would say that the asymmetrical oscillation is the most plausible explanation.

2) More detail of why the stripes in phase-phase plot can be produced by asymmetry of the theta waves needs to be provided.

It is not clear how the authors can conclude for the dataset at hand that stripes in phase-phase plot are due to the asymmetry of the theta waves and not to genuine gamma activity (Results, eighth paragraph). The authors seem to somehow tie this possible caveat to the stripes in phase-phase plots. However, stripes are just a visual inspection for maintained relationship between the two phases, just as (properly controlled) Rnm should measure.

3) The discussion about the caveat of asymmetrical oscillations is unsatisfying for the following reasons:

A) This point is made in the middle of the analysis of experimental data. Since this has nothing to do with the data at hand, in which no evidence for phase coupling was found, this obscures the message. A separate section should be devoted to the caveat of the asymmetrical oscillations, possibly before the analysis of experimental data.

B) The point should be clarified, by not only focussing on the presence of stripes for asymmetrical oscillations, but by explaining conceptually why this dissociation emerges. Possibly it could be illustrated by showing how an asymmetrical wave resembles the sum of multiples sinusoids at the fundamental and harmonics frequencies n:m locked to each other (and possibly with also some phase-amplitude coupling, see Kramer et al.).

4) Any differences between the measure of the authors and that of Belluscio's needs to be made clear. As specified by one of the reviewers:

It is not quite clear whether the methods used are exactly the same. In Belluscio et al., I think the counts were performed first for each surrogate (corresponding to a specific time-shift), and then the distribution of counts over the distinct time-shift was computed for each bin (i.e. Single Run analysis, which should be the correct method). By contrast, it seems that here – not quite clear in the text – surrogates for distinct time shifts were merged before counts were done in Figure 5 and Figure 4—figure supplement 1 (i.e. Pooled analysis, which of course averages the count and creates many false positives). [Note that I use Single Run vs. Pooled terminology to refer to whether the metrics was computed for each surrogate independently (Single Run) or for merged data, the metrics applying yo either Rnm or counts.] I hope that I am being wrong here. Otherwise that would seriously limit the conclusions of the author about invalidating Belluscio's findings.

Note that there is still a statistical problem with the Single Run analysis of phase-phase counts, which relates to the absence of a correction for the number of tested bins (either for a given theta phase or across all theta phases). In any case, the p-value in Belluscio et al. reaches very low value (some p<10^-10), which would probably resist proper corrections. If this is the case, that would provide statistically sound evidence for n:m coupling (asymmetrical theta being still a possible confound). Do the authors observe such low values in their own dataset? Could they remove statistical significance in the stripes in white noise using an appropriate correction? Or does this deceiving statistical significance come from yet another explanation?

[Editors' note: further revisions were requested prior to acceptance, as described below.]

Thank you for resubmitting your work entitled "On cross-frequency phase-phase coupling between theta and gamma oscillations in the hippocampus" for further consideration at eLife. Your revised article has been favorably evaluated by Timothy Behrens (Senior editor), a Reviewing editor, and three reviewers.

The manuscript has been improved but there are remaining issues that need to be addressed. All of the reviewers appreciated the authors' revisions and thought that this work could serve as a useful cautionary tale for neuroscientists as well as a useful starting point for continuing work. Even though it was felt that the results were not completely satisfying, it was also felt that it would be of great benefit in preventing researchers from wandering down the wrong analysis path. It was further noted that if this paper were presented as a statistics paper it could be improved more but then it would probably be less read and absorbed by the neuroscience community for some time.

An overall suggestion is that the authors should be more cautious in their interpretation and clearer about there being room for further progress in editing their manuscript.

Specific issues to address are:

1) The Abstract would benefit from significant revisions considering the various changes already done, and ones to be done. Sentences that should be targeted are: "filtered white noise has similar n:m phase-locking levels as actual data", and "the diagonal stripes in theta-gamma phase-phase histograms of actual data can be explained by theta harmonics".

2) Validity of Belluscio's analysis – overall conclusion:

A) In the rebuttal, the authors consider that the analysis by Belluscio is intrinsically flawed rather than being simply not well controlled statistically. I tend to defend the alternative option (clearly here I do not refer to the analysis of Mean Phase-Phase Plot, which the authors have convincingly showed that is flawed, but to the tests on phase counts; I also want to reassure the authors that I really do not have any personal motivation to go one way or the other). My intuition is that the presence of one single significant count in the phase-phase plot, if appropriately controlled for multiple comparisons, would provide a valid statistical measure. In essence it would not simply detect the presence of stripes (which surrogate run also feature) but measure whether their amplitude is larger than those of surrogates. This could be shown in a quite straightforward way by looking whether significant points in phase-phase plots obtained from white noise persist when a correction for multiple comparisons is applied. The authors seem to agree that they would not. (As for the type of correction, the Bonferroni method that the authors refer to looks too conservative as the bin counts clearly are non-independent; less stringent correction such as FDR or Holm-Bonferroni may be preferred). A count is not a metric for n:m coupling for sure, but it can inform of the particular concentration of the high frequency phase at one specific phase of the high frequency phase. More importantly, what matters here is not a metric of the coupling strength, but a reliable statistical test that selectively detects coupling, and my intuition is that counts in the phase-phase plot may provide one. If a simple correction can be applied to give a sound statistical test, this could allow Belluscio and colleagues to look back at their data and see whether the significant counts indeed resist correction for multiple comparisons.

B) About the very low p-values in Belluscio's data, the authors suggest that "it is possible that, by analyzing a longer epoch length, the influence of theta harmonics becomes more apparent and would lead to lower p-values, while the effect of the filtering-induced sinusoidality is washed out for both actual and surrogate epochs (of the same length)."

Well the manuscript previously demonstrates that n:m coupling measures as well as phase-phase plots cannot tear apart asymmetrical waves from true cross-frequency coupling. Thus, if indeed p-values remain lower than threshold when controlled appropriately, it could equally be due to asymmetrical theta or true theta-gamma phase coupling, but the latter could not be dismissed.

3) The reason for stripes in the phase-phase plot of hippocampal data:

A) In the manuscript authors state that "the diagonal stripes in phase-phase plots are due to theta harmonics and not to genuine gamma activity." If it were true harmonics, that would imply a consistent phase relationship between theta and gamma over sustained periods (not just over small periods as for white noise), i.e. would test positive for n:m coupling. So in my opinion here the effect is rather due to the temporary n:m alignment of phases, just as for white noise. In other words, stripes emerge also when analyzing white noise despite there being no harmonics in the signal.

B) In the rebuttal, the authors defend that "Believing in two different and genuine gamma activities – one coupled at 5 cycles per theta, the other at 4 cycles per theta – would go against the parsimonious principle." (this is when theta frequency evolves). However, as established by theoretical studies of coupled oscillators, n:m coupling is not a fixed property of the network but emerges as a combination of oscillator intrinsic dynamics and the characteristics of the connectivity pattern. See for example Arnold tongue: if we assume there is just one fixed gamma, a fluctuating theta and fixed connectivity, it is perfectly normal that n:m coupling will shift from 1:5 to 1:4 if the lower oscillator accelerates and the ratio of frequency goes from around 5 to around 4.

4) Asymmetrical theta vs. phase-coupled theta-gamma in general

I am coming back to the authors' comment on this in the rebuttal, although this is no longer present in the manuscript. This is a comment for the authors benefit and speaks to the overall comment mentioned above. Sharp edges in LFP/EEG are not by itself an indication that we are measuring a single asymmetrical oscillation as the superposition of a slow and a fast n:m coupled oscillation can also give rise to sharp edges (Figure 4C; note that this can even be obtained with just two oscillations). Thus, sharp edges is no more a selective feature of asymmetrical oscillations than n:m coherence is selective of phase-coupled oscillators. In other words, visual inspection cannot tell us more than statistics.

5) It looks like "Random Perm" is more powerful test than "Time Shift", as mentioned in the first paragraph of the subsection “Lack of evidence for n:m phase-locking in actual LFPs”. If this is the case then it could be stated as a conclusion of the work that "Random perm" should be preferred over "Time Shift" (and of course of the "Scrambling" procedure), as it is less likely to miss existing effects (lower False Rejections rate).

6) Conclusion: the very new paper by Lozano-Soldevilla and colleagues (Frontiers Comp Neuro) provides another example of spurious cross-frequency coupling measures due to asymmetrical oscillations, could be worth referencing.

DOI: http://dx.doi.org/10.7554/eLife.20515.036

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