Lack of evidence for cross-frequency phase-phase coupling between theta and gamma oscillations in the hippocampus

Phase-amplitude coupling between theta and multiple gamma sub-bands hallmarks hippocampal activity and is believed to take part in information routing. More recently, theta and gamma oscillations were also reported to exhibit reliable phase-phase coupling, or n:m phase-locking. The existence of n:m phase-locking suggests an important mechanism of neuronal coding that has long received theoretical support. However, here we show that n:m phase-locking (1) is much lower than previously reported, (2) highly depends on epoch length, (3) does not statistically differ from chance (when employing proper surrogate methods), and that (4) filtered white noise has similar n:m scores as actual data. Moreover, (5) the diagonal stripes in theta-gamma phase-phase histograms of actual data can be explained by theta harmonics. These results point to lack of theta-gamma phase-phase coupling in the hippocampus, and suggest that studies investigating n:m phase-locking should rely on appropriate statistical controls, otherwise they could easily fall into analysis pitfalls.

phase-locking, also relies on assessing the constancy of the difference between two 58 phase time series (Tass et al., 1998). However, in this case the original phase time 59 series are accelerated, so that their instantaneous frequency can match. Formally, 60 n:m phase-locking occurs when ∆φ ( ) = * φ ( ) − * φ ( ) is constant,

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where * φ ( * φ ) denotes the phase of oscillation B (A) accelerated n (m) 62 times (Tass et al., 1998). For example, the instantaneous phase of theta oscillations 63 at 8 Hz needs to be accelerated 5 times to match in frequency a 40-Hz gamma. A 64 1:5 phase-phase coupling is then said to occur if the instantaneous phase of theta 65 accelerated 5 times has constant difference to the instantaneous gamma phase; or, 66 in other words, if 5 gamma cycles have fixed phase relationship to 1 theta cycle. 67 Cross-frequency phase-phase coupling has previously been hypothesized to 68 take part in memory processes (Lisman and Idiart, 1995

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We first certified that we could reliably detect n:m phase-locking when 79 present. To that end, we simulated a 20-s LFP signal containing white noise, high-80 amplitude theta (8 Hz) and low-amplitude gamma (40 Hz) oscillations ( Figure 1A, 81 top). The phase time series for theta and gamma were obtained from the simulated 82 LFP after band-pass filtering ( Figure 1A, bottom). Figure 1B depicts three versions of 83 accelerated theta phases (m=3, 5 and 7) along with the instantaneous gamma phase 84 (n=1). Also shown is the time series of the difference between gamma and 85 accelerated theta phases (∆φ ). The instantaneous phase difference is roughly 86 constant only for m=5; when m=3 or 7, it changes over time, precessing forwards 87 (m=3) or backwards (m=7) at a rate of 16 Hz. Consequently, ∆φ distribution is 88 uniform over 0 and 2π for m=3 or 7, but highly concentrated for m=5 ( Figure 1C). 89 The concentration (or "constancy") of the phase difference distribution is commonly 90 used as a metric of n:m phase-locking. This metric is defined as the length of the 91 mean resultant vector (Rn:m) over unitary vectors whose angle is the instantaneous 92 phase difference ( ∆φ ( ) ), and thereby it varies between 0 and 1. For any pair of 93 filtered LFPs, an Rn:m "curve" can be obtained by varying m for n=1 fixed. When 94 filtered at theta and gamma, the simulated LFP exhibited a prominent peak for n:m = 95 1:5 ( Figure 1D, left), which shows that Rn:m successfully detects n:m phase-locking. 96 Importantly, in this first case gamma frequency was an integer multiple of theta 97 frequency (5 x 8 Hz = 1 x 40 Hz), or, in other words, gamma frequency was a 98 harmonic of theta frequency. We next repeated the analysis above for a similar LFP, 99 but simulated with a slightly different gamma frequency, set to 39.9 Hz. In this 100 second case the Rn:m curve exhibited no prominent peak ( Figure 1D, right). 101 Therefore, Rn:m is highly sensitive to the exact peak frequency of the analyzed 102 oscillations. This fact alone casts some doubt on the existence of n:m phase-locking 103 between actual brain oscillations, as their peak frequencies are unlikely to be perfect 104 integer multiples. 105 We next sought to understand why previous studies found theta-gamma 106 phase-phase coupling in actual LFP recordings despite the observations above 107  In each case, phase-phase coupling was high within the ratio of the analyzed 117 frequency ranges: Rn:m peaked at m=4-6 for S, at m=7-11 for M, and at m=12-118 20 for F. Therefore, the existence of a "bump" in the n:m phase-locking curve may 119 merely reflect the ratio of the filtered bands and should not be considered as 120 evidence for cross-frequency phase-phase coupling; even filtered white-noise 121 signals exhibit such a pattern. 122 Qualitatively similar results were found for 1-and 10-s epochs; however, Rn:m 123 values were considerably lower for the latter ( Figure 2B   No phase-phase coupling should be detected in white noise, and therefore 155 Original Rn:m values should not differ from chance. However, as shown in Figure 2F  We next proceeded to analyze hippocampal CA1 recordings from 7 rats, 172 focusing on periods of prominent theta activity (active waking and REM sleep). We Based on the above, we concluded that there is no evidence for n:m phase-184 locking in actual hippocampal LFPs. We next sought to investigate what causes the 185 diagonal stripes in phase-phase plots that are appealingly suggestive of 1:5 coupling 186 ( Figure 3B). In Figure 4 we analyze a representative LFP with prominent theta 187 oscillations at ~7 Hz recorded during REM sleep. We constructed phase-phase plots 188 using LFP components narrowly filtered at theta and its harmonics: 14, 21, 28 and 189 35 Hz. For each harmonic frequency, the phase-phase plot exhibited diagonal 190 stripes whose number was determined by the harmonic order (i.e., the 1 st harmonic 191 exhibited two stripes, the 2 nd harmonic three stripes, the 3 rd , four stripes and the 4 th , 192 five stripes; Figure 4B i-iv). Interestingly, when the LFP was filtered at a broad  phase-phase plots were due to theta-gamma coupling; no such stripes were 273 apparent in phase-phase plots constructed from the average over surrogate epochs 274 (c.f. their Figure 6). However, here we show that the presence of diagonal stripes in 275 phase-phase plots is not sufficient to conclude the existence of phase-phase 276 coupling; actually, these stripes can be explained by theta harmonics alone. 277 Accordingly, the analysis of synthetic LFPs having no gamma oscillations and theta 278 simulated as a saw-tooth wave leads to similar results ( Figure S7). That is, even 279 though by definition the theta saw-tooth wave has no theta-gamma phase-phase 280 coupling, filtering at the gamma band will reflect the harmonic of theta with largest 281 amplitude within the filtered band and give rise to diagonal stripes in phase-phase 282 plots. These would be spuriously deemed as statistically significant if we were to use 283 the same statistical analysis as in Belluscio et al. (2012), since phase-phase plots 284 obtained from the average of 1000 surrogates with shuffled phase lags exhibit no 285 diagonal stripes ( Figure S7). 286

Lack of evidence vs evidence of non-existence 287
One could argue that we were unable to detect statistically significant theta-288 gamma phase-phase coupling because we did not analyze a proper dataset, or else 289 because phase-phase coupling would only occur during certain behavioral states not 290 investigated here. We disagree with these arguments for the following reasons: (1) 291 we could reproduce our results using data from the same laboratory as Belluscio et  Figure S5). Finally, one could argue that gamma 309 oscillations are not continuous but transient, and that assessing phase-phase 310 coupling between theta and transient gamma bursts would require a different type of 311 analysis than employed here. Regarding this argument, we once again stress that 312 we used the exact same methodology as originally used to detect theta-gamma 313 phase-phase coupling (Belluscio et al., 2012). Nevertheless, we also ran analysis 314 only taking into account periods in which gamma amplitude was >2SD above the 315 mean ("gamma bursts") and found no statistically significant phase-phase coupling 316 ( Figure S6). 317 As showcased in Figure 1D values of m (for n=1 fixed) in order to avoid contamination with harmonics. At any 330 event, we also deem unlikely that two genuine brain oscillations would differ in peak 331 frequency by a perfect rational number. 332 The high sensitivity of Rn:m on exact peak frequencies casts some doubt as to 333 whether theta-gamma n:m phase-locking can ever exist in the brain. Contrary to this

studies. 345
Since it is philosophically impossible to prove absence of an effect, the burden 346 of proof should be placed on demonstrating that a true effect exists. In this sense, 347 and to the best of our knowledge, none of previous research investigating theta-348 gamma phase-phase coupling has properly tested Rn:m against chance. Many 349 studies have focused on comparing changes in n:m phase-locking levels, but we 350 believe these can be influenced by other variables such as changes in power. 351 Interestingly, in their pioneer work, Tass and colleagues used filtered white noise to 352 construct surrogate distributions and did not find significant n:m phase-locking 353 among brain oscillations (Tass et al., 1998;Tass et al., 2003). gamma cycles within theta cycles that is consistent across theta cycles would imply 372 phase-phase coupling; indeed, n:m phase-locking is a main feature of computational 373 models of sequence coding by theta-gamma coupling (Lisman and Idiart, 1995;374 Jensen and Lisman, 1996;Jensen et al., 1996). In contrast to these models, 375 however, the absence of theta-gamma phase-phase coupling reported here shows 376 that the theta phases in which gamma cycles begin/end are not fixed across theta 377 cycles, which is to say that gamma cycles are not precisely timed but rather drift; in 378 other words, gamma is not a clock (Burns et al., 2011). 379 If theta-gamma neural coding exists, our results suggest that the precise 380 location of gamma memory slots within a theta cycle is not required for such a code, 381 and that the ordering of the represented items would be more important than the 382 exact spike timing of the cell assemblies that represent the items (Lisman and 383 Jensen, 2013). 384

Conclusion 385
In summary, while absence of evidence is not evidence of absence, our 386 results strongly suggest that theta-gamma phase-phase coupling does not exist in 387 the hippocampus. We believe that evidence in favor of n:m phase locking in other 388 brain regions and signals should be revisited and, whenever suitable, checked 389 against the more conservative surrogate techniques outlined here. benchmarks such as highest ripple power (see Figure S4 for an example). Similar 403 results were obtained for recordings from other hippocampal layers ( Figure S4). 404 We also analyzed data from 3 additional rats downloaded from the Collaborative 405 Research in Computational Neuroscience data sharing website (www.crcns.org) 406 ( Figure S3) Figure S2). The CSD signals analyzed in Figure S4 were obtained as 427 −A+2B−C for adjacent probe sites. In Figure S5, the independent components were 428 obtained as described in Schomburg et al. (2014); phase-amplitude comodulograms 429 were computed as described in Tort et al. (2010).   Figure 5A); the rightmost panel shows the phase-phase plot computed using the pool of all time-shifted surrogate runs (n=1000).
(B) Same as above, but for the example epoch analyzed in Figure 5B.     (C) Phase-phase plots for theta and LFP band-pass filtered at harmonic frequencies (14, 21, 28 and 35 Hz). Also shown are phase-phase plots for the conventional gamma band (30 -90 Hz) and for pooled surrogate runs.