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# On cross-frequency phase-phase coupling between theta and gamma oscillations in the hippocampus.

### Author information

- 1
- Brain Institute, Federal University of Rio Grande do Norte, Natal, Brazil.

### Abstract

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.

#### KEYWORDS:

brain rhythms; cross-frequency coupling; electrophysiology; local field potential; neuronal oscillations; neuroscience; rat

- PMID:
- 27925581
- PMCID:
- PMC5199196
- DOI:
- 10.7554/eLife.20515

- [Indexed for MEDLINE]

**A**) Traces show 500 ms of the instantaneous phase time series of two Kuramoto oscillators (see Materials and methods). When uncoupled (top panels), the mean natural frequencies of the ‘theta’ and ‘gamma’ oscillator are 8 Hz (blue) and 43 Hz (red), respectively. When coupled (bottom panels), the oscillators have mean frequencies of 8.5 Hz and 42.5 Hz. (

**B**) Top blue traces show the instantaneous phase of the coupled theta oscillator for the same period as in A but accelerated

*m*times, where m = 3 (left), 5 (middle) and 7 (right). Middle red traces reproduce the instantaneous phase of the coupled gamma oscillator (i.e., n = 1). Bottom black traces show the instantaneous phase difference between gamma and accelerated theta phases (). Notice roughly constant only when theta is accelerated m = 5 times, which indicates 1:5 phase-locking. See for the uncoupled case. (

**C**) distributions for the coupled case (epoch length = 100 s). Notice uniform distributions for n:m = 1:3 and 1:7, and a highly concentrated distribution for n:m = 1:5. The black arrow represents the mean resultant vector for each case (see Materials and methods). The length of this vector (R

_{n:m}) measures the level of n:m phase-locking. See for the uncoupled case. (

**D**) Phase-locking levels for a range of n:m ratios for the uncoupled (left) and coupled (right) oscillators (epoch length = 100 s).

**DOI:**http://dx.doi.org/10.7554/eLife.20515.003

**A,B**) Panels show the same as in , but for the uncoupled oscillators. Notice roughly uniform distributions.

**DOI:**http://dx.doi.org/10.7554/eLife.20515.004

**A**) Example white-noise signal (black) along with its theta- (blue) and gamma- (red) filtered components. The corresponding instantaneous phases are also shown. (

**B**) n:m phase-locking levels for 1- (left) and 10 s (right) epochs, computed for noise filtered at theta (θ; 4–12 Hz) and at three gamma bands: slow gamma (γ

_{S}; 30–50 Hz), middle gamma (γ

_{M}; 50–90 Hz) and fast gamma (γ

_{F}; 90–150 Hz). Notice R

_{n:m}peaks in each case. (

**C**) Boxplot distributions of θ−γ

_{S}R

_{1:5}values for different epoch lengths (n = 2100 simulations per epoch length). The inset shows representative distributions for 0.3- and 100 s epochs. (

**D**) Overview of surrogate techniques. See text for details. (

**E**) Top panels show representative distributions for single surrogate runs (

*Time Shift*; 10 runs of 1 s epochs), along with the corresponding R

_{n:m}values. The bottom panel shows the pooled distribution; the R

_{n:m}of the pooled distribution is lower than the R

_{n:m}of single runs (compare with values for 1- and 10 s epochs in panel

**C**). (

**F**) Top, n:m phase-locking levels computed for 1- (left) or 10 s (right) epochs using either the

*Original*or five surrogate methods (insets are a zoomed view of R

_{n:m}peaks). Bottom, R

_{1:5}values for white noise filtered at θ and γ

_{S}. Original R

_{n:m}values are not different from R

_{n:m}values obtained from single surrogate runs of

*Random Permutation*and

*Time Shift*procedures. Less conservative surrogate techniques provide lower R

_{n:m}values and lead to the spurious detection of θ−γ

_{S}phase-phase coupling in white noise. *p<0.01, n = 2100 per distribution, one-way ANOVA with Bonferroni post-hoc test.

**DOI:**http://dx.doi.org/10.7554/eLife.20515.005

**A**) Distribution of the phase difference between two consecutive samples for white noise band-pass filtered at theta (4–12 Hz, top) and slow gamma (30–50 Hz, bottom). Epoch length = 100 s; sampling rate = 1000 Hz (dt = 0.001 s). Notice that the top histogram peaks at ~0.05, which corresponds to 2*3.14*8*0.001 (i.e., 2*π*f

_{c}*dt, where f

_{c}is the center frequency), and the bottom histogram peaks at ~0.25 =2*3.14*40*0.001. (

**B**) R

_{n:m}curves computed for theta- and slow gamma-filtered white-noise signals. The black curve was obtained using continuous 1 s long time series sampled at 1000 Hz. The red curve was obtained by also analyzing 1000 data points, but which were subsampled at 20 Hz (subsampling was performed after filtering). Notice R

_{n:m}peak at n:m = 1:5 only for the former case. See also for similar results in hippocampal LFPs.

**DOI:**http://dx.doi.org/10.7554/eLife.20515.006

**A**) Mean R

_{n:m}curves computed for 1 s long white-noise signals filtered into different bands (same color labels as in B; n = 2100). Notice that the narrower the filter bandwidth, the higher the R

_{n:m}peak. (

**B**) Mean R

_{n:m}peak values for different filter bandwidths and epoch lengths (n = 2100 simulations per filter setting and epoch length).

**DOI:**http://dx.doi.org/10.7554/eLife.20515.007

*Original*R

_{n:m}values against

*Single Run*R

_{n:m}surrogates.

*Original*R

_{n:m}vs

*Single Run*surrogate values (n = 30 samples per group; epoch length = 1 s). The red dashed line marks p=0.05. The p-value distributions do not statistically differ from the uniform distribution (Kolmogorov-Smirnov test).

**DOI:**http://dx.doi.org/10.7554/eLife.20515.008

_{n:m}values.

**A**) The left panels show mean R

_{n:m}curves and distributions of R

_{1:5}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 ). (

**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 (; ). 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

**A**) The top traces show a theta sawtooth wave along with its decomposition into a sum of sinusoids at the fundamental (7 Hz) and harmonic (14 Hz, 21 Hz, 28 Hz, 35 Hz, etc) frequencies. The bottom panel shows the power spectrum of the sawtooth wave. Notice power peaks at the fundamental and harmonic frequencies. (

**B**) Phase-phase plots (2D histograms of phase counts) for the sawooth wave in A filtered at theta (7 Hz; x-axis phases) and harmonic frequencies (14, 21, 28 and 35 Hz; y-axis phases). (

**C**) The left traces show 500 ms of a sawtooth wave along with its theta- and gamma-filtered components and corresponding phase time series. The sawtooth wave was set to have a variable peak frequency, with mean = 8 Hz; no gamma oscillation was added to the signal. Notice that the sharp deflections of the sawtooth wave give rise to artifactual gamma oscillations in the filtered signal (), which have a consistent phase relationship to the theta cycle. The right panels show that artifactual n:m phase-coupling levels induced by the sharp deflections are significantly higher than the chance distribution (n = 300; epoch length = 30 s; *p<0.001, t-test).

**DOI:**http://dx.doi.org/10.7554/eLife.20515.010

**A**) A theta sawtooth wave along with its theta- (7 Hz) and gamma-filtered (35 Hz) components. Notice that no gamma oscillations exist in the original sawtooth wave, but they spuriously appear when filtering sharp deflections (). The amplitude of the spurious gamma waxes and wanes within theta cycles. (

**B**) Mean gamma amplitude as a function of theta phase.

**DOI:**http://dx.doi.org/10.7554/eLife.20515.011

_{1:5}computed between theta and slow gamma for sawtooth waves simulated as in , but of different epoch lengths and peak frequency variability. Dashed area corresponds to the interquartile range (n = 300). Surrogate data were obtained either by

*Radom Permutation*(top row) or

*Time Shift*(bottom row) procedures. Notice that the longer the epoch or the peak frequency variability, the larger the difference between original and surrogate data, and that this difference is greater for randomly permutated than time-shifted surrogates.

**DOI:**http://dx.doi.org/10.7554/eLife.20515.012

**A**) n:m phase-locking levels for actual hippocampal LFPs. Compare with . (

**B**) Original and surrogate distributions of R

_{n:m}values for slow (R

_{1:5}; left) and middle gamma (R

_{1:8}; right) for different epoch lengths. The original data is significantly higher than the pooled surrogate distribution, but indistinguishable from the distribution of surrogate values computed using single runs. Similar results hold for fast gamma. *p<0.01, n = 7 animals, Friedman’s test with Nemenyi post-hoc test.

**DOI:**http://dx.doi.org/10.7554/eLife.20515.013

**A**) The left plots show the mean radial distance (R) computed for gamma phases in different theta phase bins, as described in . The lines denote the mean ± SD over all channels across animals (n = 16 channels per rat x seven rats); 300 1 s long epochs were analyzed for each channel. Note that original and surrogate R values overlap. The variations of R values within a theta cycle are explained by the different number of theta phase bins (right bar plot), which leads to different number of analyzed samples; the higher the number of analyzed samples, the lower the R (see also ). (

**B**) The first column shows the mean pairwise phase consistency (PPC) between gamma and accelerated theta phases as a function of the number of samples (dashed lines denote SD over individual PPC estimates; n = 112 channels x 1000 PPC estimates per channel). Since PPC requires independent observations (), was randomly sampled to avoid the statistical dependence among neighboring data points imposed by the filter (; see also ). The second column shows mean PPC as function of n:m ratio (individual PPC estimates were computed using 1000 samples); the boxplot distributions show PPC values at selected n:m ratios, as labeled. PPC values are very low for all analyzed frequency pairs and not statistically different from zero.

**DOI:**http://dx.doi.org/10.7554/eLife.20515.014

**A**) n:m phase-locking levels for actual hippocampal LFPs (same dataset as in ). Theta phase was estimated by the interpolation method described in . (

**B**) Original and surrogate distributions of R

_{n:m}values. The original data are significantly higher than surrogate values obtained from pooled , but indistinguishable from single run surrogates. *p<0.01, n = 7 animals, Friedman’s test with Nemenyi post-hoc test.

**DOI:**http://dx.doi.org/10.7554/eLife.20515.015

**A**) n:m phase-locking levels for actual hippocampal LFPs. (

**B**) Original and surrogate distributions of R

_{n:m}values. Results obtained for three rats recorded in an independent laboratory (see Materials and methods).

**DOI:**http://dx.doi.org/10.7554/eLife.20515.016

*Random Permutation/Single Run*) distributions of R

_{n:m}values computed between theta phase and the phase of three gamma sub-bands (1 s long epochs). Different rows show results for different layers. (Right) Distribution of original and surrogate R

_{n:m}values computed for current-source density (CSD) signals (1 s long epochs) in three hippocampal layers:

*s. pyramidale*(top)

*, s. radiatum*(middle), and

*s. lacunosum-moleculare*(bottom). Notice no difference between original and surrogate values. Similar results were found in all animals.

**DOI:**http://dx.doi.org/10.7554/eLife.20515.017

_{n:m}values. R

_{n:m}values were computed for 1 s long epochs (n = 4 animals); surrogate gamma phases were obtained by

*Random Permutation/Single Run*.

**DOI:**http://dx.doi.org/10.7554/eLife.20515.018

**A**) Examples of slow-gamma bursts. Top panels show raw LFPs, along with theta- (thick blue line) and slow gamma-filtered (thin red line) signals. The amplitude envelope of slow gamma is also shown (thick red line). The bottom rows show gamma and accelerated theta phases (m = 5), along with their instantaneous phase difference (). For each gamma sub-band, a ‘gamma burst’ was defined to occur when the gamma amplitude envelope was 2SD above the mean. In these examples, periods identified as slow-gamma bursts are marked with yellow in the amplitude envelope and phase difference time series. Notice variable across different burst events. (

**B**) The left panel shows n:m phase-locking levels for theta phase and the phase of different gamma sub-bands (1 s epochs); for each gamma sub-band, R

_{n:m}values were computed using only theta and gamma phases during periods of gamma bursts. The right panels show original and surrogate (

*Random Permutation/Single Run*) distributions of R

_{n:m}values (n = 4 animals).

**DOI:**http://dx.doi.org/10.7554/eLife.20515.019

_{n:m}curve of hippocampal LFPs highly depends on analyzing contiguous phase time series data.

_{n:m}curves computed for theta- and gamma-filtered hippocampal LFPs. The green curves were obtained using 1 s (top) or 10 s (bottom) continuous epochs of the phase time series, sampled at 1000 Hz (same analysis as in ). The blue curves were obtained by analyzing 1000 data points subsampled from the phase time series at 20 Hz (i.e., 50 ms sampling period, longer than a gamma cycle). The red curves were obtained by analyzing 1000 (top) or 10000 (bottom) data points randomly sampled from the phase time series. These plots show that the prominent bump in the R

_{n:m}curve of actual LFPs only occurs for continuously sampled data (1000 Hz sampling rate), and therefore probably reflects the ‘sinusoidality’ imposed by the filter (see also ). But notice that a small R

_{n:m}bump remains for θ−γ

_{S}(see ). Due to limitation of total epoch length, we could not perform the 20 Hz subsampling analysis for 10000 points, but notice that the blue and red curves coincide for 1000 points.

**DOI:**http://dx.doi.org/10.7554/eLife.20515.020

**A**) Original (green) and surrogate (red) n:m phase-locking levels for actual hippocampal LFPs (same dataset as in ) filtered at theta and slow gamma (1 s epochs). Different rows show results for different types of filters. FIR corresponds to the same finite impulse response filter employed in all other figures. For the infinite impulse response filters (Butterworth and Bessel), the digit on the right denotes the filter order. Wavelet filtering was achieved by convolution with a complex Morlet wavelet with a center frequency of 7 Hz. (

**B**) Original and surrogate distributions of R

_{1:5}values. For each filter type, the original data is significantly higher than surrogate values obtained from pooled , but indistinguishable from single run surrogates. *p<0.01, n = 7 animals, Friedman’s test with Nemenyi post-hoc test.

**DOI:**http://dx.doi.org/10.7554/eLife.20515.021

### Conflict of interest statement

The authors declare that no competing interests exist.