1 On cross-frequency phase-phase coupling between theta and 2 gamma oscillations in the hippocampus 3 4 Robson Scheffer-Teixeira & Adriano BL Tort 5 6 Brain Institute, Federal University of Rio Grande do Norte, Natal, RN 59056-450, Brazil. 7 8 9 10 11 12 13 14 Correspondence: 15 Adriano B. L. Tort, M.D., Ph.D. 16 Brain Institute, Federal University of Rio Grande do Norte 17 Rua Nascimento de Castro, 2155 18 Natal, RN 59056-450, Brazil 19 Tel: +55 84 32152709 - Fax: +55 84 32153185 20 [email protected]

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21 Abstract

22 Phase-amplitude coupling between theta and multiple gamma sub-bands is a

23 hallmark of hippocampal activity and believed to take part in information

24 routing. More recently, theta and gamma oscillations were also reported to

25 exhibit phase-phase coupling, or n:m phase-locking, suggesting an important

26 mechanism of neuronal coding that has long received theoretical support.

27 However, by analyzing simulated and actual LFPs, here we question the

28 existence of theta-gamma phase-phase coupling in the rat hippocampus. We

29 show that the quasi-linear phase shifts introduced by filtering lead to spurious

30 coupling levels in both white noise and hippocampal LFPs, which highly

31 depend on epoch length, and that significant coupling may be falsely detected

32 when employing improper surrogate methods. We also show that waveform

33 asymmetry and frequency harmonics may generate artifactual n:m phase-

34 locking. Studies investigating phase-phase coupling should rely on

35 appropriate statistical controls and be aware of confounding factors;

36 otherwise, they could easily fall into analysis pitfalls.

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38 Introduction

39 Local field potentials (LFPs) exhibit oscillations of different frequencies, which

40 may co-occur and also interact with one another (Jensen and Colgin, 2007; Tort et

41 al., 2010; Hyafil et al., 2015). Cross-frequency phase-amplitude coupling between

42 theta and gamma oscillations has been well described in the hippocampus, whereby

43 the instantaneous amplitude of gamma oscillations depends on the instantaneous

44 phase of theta (Scheffer-Teixeira et al., 2012; Schomburg et al., 2014). More

45 recently, hippocampal theta and gamma oscillations were also reported to exhibit

46 n:m phase-phase coupling, in which multiple gamma cycles are consistently

47 entrained within one cycle of theta (Belluscio et al., 2012; Zheng et al., 2013; Xu et

48 al., 2013; Xu et al., 2015; Zheng et al., 2016). The existence of different types of

49 cross-frequency coupling suggests that the brain may use different coding strategies

50 to transfer multiplexed information.

51 Coherent oscillations are believed to take part in network communication by

52 allowing opportunity windows for the exchange of information (Varela et al., 2001;

53 Fries, 2005). Standard phase coherence measures the constancy of the phase

54 difference between two oscillations of the same frequency (Lachaux et al., 1999;

55 Hurtado et al., 2004), and has been associated with cognitive processes such as

56 decision-making (DeCoteau et al., 2007; Montgomery and Buzsáki, 2007; Nácher et

57 al., 2013). Similarly to coherence, cross-frequency phase–phase coupling, or n:m

58 phase-locking, also relies on assessing the constancy of the difference between two

59 phase time series (Tass et al., 1998). However, in this case the original phase time

60 series are accelerated, so that their instantaneous frequencies can match. Formally,

61 n:m phase-locking occurs when ∆() = ∗ () −∗() is non-uniform

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62 but centered around a preferred value, where ∗ ∗ denotes the phase

63 of oscillation B (A) accelerated n (m) times (Tass et al., 1998). For example, the

64 instantaneous phase of theta oscillations at 8 Hz needs to be accelerated 5 times to

65 match in frequency a 40-Hz gamma. A 1:5 phase-phase coupling is then said to

66 occur if theta accelerated 5 times has a preferred phase lag (i.e., a non-uniform

67 phase difference) in relation to gamma; or, in other words, if 5 gamma cycles have a

68 consistent phase relationship to 1 theta cycle.

69 Cross-frequency phase-phase coupling has previously been hypothesized to

70 take part in memory processes (Lisman and Idiart, 1995; Jensen and Lisman, 2005;

71 Lisman, 2005; Schack and Weiss, 2005; Sauseng et al., 2008; Sauseng et al., 2009;

72 Holtz et al., 2010; Fell and Axmacher, 2011). Recent findings suggest that the

73 hippocampus indeed uses such a mechanism (Belluscio et al., 2012; Zheng et al.,

74 2013; Xu et al., 2013; Xu et al., 2015; Zheng et al., 2016). However, by analyzing

75 simulated and actual hippocampal LFPs, in the present work we question the

76 existence of theta-gamma phase-phase coupling.

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78 Results

79 Measuring n:m phase-locking

80 We first certified that we could reliably detect n:m phase-locking when

81 present. To that end, we simulated a system of two Kuramoto oscillators – a “theta”

82 and a “gamma” oscillator – exhibiting variability in instantaneous frequency (see

83 Material and Methods). The mean natural frequency of the theta oscillator was set to

84 8 Hz, while the mean natural frequency of the gamma oscillator was set to 43 Hz

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85 (Figure 1A). When coupled, the mean frequencies aligned to a 1:5 factor by

86 changing to 8.5 Hz and 42.5 Hz, respectively (see Guevara and Glass, 1982;

87 García-Alvarez et al., 2008; Canavier et al., 2009). Figure 1B depicts three versions

88 of accelerated theta phases (m=3, 5 and 7) along with the instantaneous gamma

89 phase (n=1) of the coupled oscillators (see Figure 1 – figure supplement 1 for the

90 uncoupled case). Also shown are the time series of the difference between gamma

91 and accelerated theta phases (∆φ). The instantaneous phase difference has a

92 preferred lag only for m=5; when m=3 or 7, ∆φ changes over time, precessing

93 forwards (m=3) or backwards (m=7) at an average rate of 17 Hz. Consequently,

94 ∆φ distribution is uniform over 0 and 2π for m=3 or 7, but highly concentrated for

95 m=5 (Figure 1C). The concentration (or “constancy”) of the phase difference

96 distribution is used as a metric of n:m phase-locking. This metric is defined as the

97 length of the mean resultant vector (Rn:m) over unitary vectors whose angle is the

( ) 98 instantaneous phase difference (∆ ), and thereby it varies between 0 and 1.

99 For any pair of phase time series, an Rn:m “curve” can be calculated by varying m for

100 n=1 fixed. As shown in Figure 1D, the coupled – but not uncoupled – oscillators

101 exhibited a prominent peak for n:m = 1:5, which shows that Rn:m successfully detects

102 n:m phase-locking.

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

104 We next analyzed white-noise signals, in which by definition there is no

105 structured activity; in particular, the spectrum is flat and there is no true n:m phase-

106 locking. Rn:m values measured from white noise should be regarded as chance

107 levels. We band-pass filtered white-noise signals to extract the instantaneous phase

108 of theta (θ: 4 – 12 Hz) and of multiple gamma bands (Figure 2A): slow gamma (γS:

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109 30 – 50 Hz), middle gamma (γM: 50 – 90 Hz), and fast gamma (γF: 90 – 150 Hz). For

110 each θ−γ pair, we constructed n:m phase-locking curves for epochs of 1 and 10

111 seconds, with n=1 fixed and m varying from 1 to 25 (Figure 2B). In each case,

112 phase-phase coupling was high within the ratio of the analyzed frequency ranges:

113 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.

114 Therefore, the existence of a “bump” in the Rn:m curve may merely reflect the ratio of

115 the filtered bands and should not be considered as evidence for cross-frequency

116 phase-phase coupling: even filtered white-noise signals exhibit such a pattern.

117 The bump in the Rn:m curve of filtered white noise is explained by the fact that

118 neighboring data points are not independent. In fact, the phase shift between two

119 consecutive data points follows a probability distribution highly concentrated around

120 2*π*fc*dt, where fc is the filter center frequency and dt the sampling period (Figure 2

121 – figure supplement 1). For instance, for dt = 1 ms (1000 Hz sampling rate),

122 consecutive samples of white noise filtered between 4 and 12 Hz are likely to exhibit

123 phase difference of 0.05 rad (8 Hz center frequency); likewise, signals filtered

124 between 30 and 50 Hz are likely to exhibit phase differences of 0.25 rad (40 Hz

125 center frequency). In turn, the “sinusoidality” imposed by filtering leads to non-zero

126 Rn:m values, which peak at the ratio of the center frequencies, akin to the fact that

127 perfect 8-Hz and 40-Hz sine waves have Rn:m = 1 at n:m = 1:5. In accordance to this

128 explanation, no Rn:m bump occurs when data points of the gamma phase time series

129 are made independent by sub-sampling with a period longer than a gamma cycle

130 (Figure 2 – figure supplement 1), or when extracting phase values from different

131 trials (not shown). As expected, the effect of filtering-induced sinusoidality on Rn:m

132 values is stronger for narrower frequency bands (Figure 2 – figure supplement 2).

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133 Qualitatively similar results were found for 1- and 10-s epochs; however, Rn:m

134 values were considerably lower for the latter (Figure 2B). In fact, for any fixed n:m

135 ratio and frequency pair, Rn:m decreased as a function of epoch length (see Figure

136 2C for θ−γS and R1:5): the longer the white-noise epoch the more the phase

137 difference distribution becomes uniform. In other words, as standard phase

138 coherence (Vinck et al., 2010) and phase-amplitude coupling (Tort et al., 2010),

139 phase-phase coupling has positive bias for shorter epochs. As a corollary, notice that

140 false-positive coupling may be detected if control (surrogate) epochs are longer than

141 the original epoch.

142 Statistical testing of n:m phase-locking

143 We next investigated the reliability of surrogate methods for detecting n:m

144 phase-locking (Figure 2D). The “Original” Rn:m value uses the same time window for

145 extracting theta and gamma phases (Figure 2D, upper panel). A “Time Shift”

146 procedure for creating surrogate epochs has been previously employed (Belluscio et

147 al., 2012; Zheng et al., 2016), in which the time window for gamma phase is

148 randomly shifted between 1 to 200 ms from the time window for theta phase (Figure

149 2D, upper middle panel). A variant of this procedure is the “Random Permutation”, in

150 which the time window for gamma phase is randomly chosen (Figure 2D, lower

151 middle panel). Finally, in the “Phase Scramble” procedure, the timestamps of the

152 gamma phase time series are shuffled (Figure 2D, lower panel); clearly, the latter is

153 the least conservative. For each surrogate procedure, Rn:m values were obtained by

154 two approaches: “Single Run” and “Pooled” (Figure 2E). In the first approach, each

155 surrogate run (e.g., a time shift or a random selection of time windows) produces one

156 Rn:m value (Figure 2E, top panels). In the second, ∆φ from several surrogate runs

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157 are first pooled, then a single Rn:m value is computed from the pooled distribution

158 (Figure 2E, bottom panel). As illustrated in Figure 2E, Rn:m computed from a pool of

159 surrogate runs is much smaller than when computed for each individual run. This is

160 due to the dependence of Rn:m on the epoch length: pooling instantaneous phase

161 differences across 10 runs of 1-s surrogate epochs is equivalent to analyzing a

162 single surrogate epoch of 10 seconds. And the longer the analyzed epoch, the more

163 the noise is averaged out and the lower the Rn:m. Therefore, pooled surrogate

164 epochs summing up to 10 seconds of total data have lower Rn:m than any individual

165 1-s surrogate epoch.

166 No phase-phase coupling should be detected in white noise, and therefore

167 Original Rn:m values should not differ from properly constructed surrogates. However,

168 as shown in Figure 2F for θ−γS as an illustrative case (similar results hold for any

169 frequency pair), θ−γS phase-phase coupling in white noise was statistically

170 significantly larger than in phase-scrambled surrogates (for either Single Run or

171 Pooled distributions). This was true for surrogate epochs of any length, although the

172 longer the epoch, the lower the actual and the surrogate Rn:m values, as expected

173 (compare right and left panels of Figure 2F). Pooled R1:5 distributions derived from

174 either time-shifted (Figure 2F) or randomly permutated epochs (not shown) also led

175 to the detection of false positive θ−γS phase-phase coupling. On the other hand,

176 Original Rn:m values were not statistically different from chance distributions when

177 these were constructed from Single Run Rn:m values for either Time Shift and

178 Random Permutation surrogate procedures (Figure 2F; see also Figure 2 – figure

179 supplement 3). We conclude that neither scrambling phases nor pooling individual

180 surrogate epochs should be employed for statistically evaluating n:m phase-locking.

181 Chance distributions should be derived from surrogate epochs of the same length as

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182 the original epoch and which preserve phase continuity.

183 To check if Single Run surrogate distributions are capable of statistically

184 detecting true n:m phase-locking, we next simulated noisy Kuramoto oscillators as in

185 Figure 1, but of mean natural frequencies set to 8 and 40 Hz. Original R1:5 values

186 were much greater than the surrogate distribution for coupled – but not uncoupled –

187 oscillators (Figure 3A). This result illustrates that variability in instantaneous

188 frequency leads to low n:m phase-locking levels for independent oscillators even

189 when their mean frequencies are perfect integer multiples. On the other hand,

190 coupled oscillators have high Rn:m because variations of their instantaneous

191 frequencies are mutually dependent. We then proceeded to analyze simulated LFPs

192 from a previously published model network (Kopell et al., 2010). The network has

193 two inhibitory interneurons, called O and I cells, which spike at theta and gamma

194 frequency, respectively (see Tort et al., 2007 for a motivation of this model).

195 Compared to Single Run surrogate distributions, the model LFP exhibited significant

196 n:m phase-locking only when the interneurons were coupled; Rn:m levels did not

197 differ from the surrogate distribution for the uncoupled network (Figure 3B). (Note

198 that the Rn:m curve also exhibited a peak for both the uncoupled network and Single

199 Run surrogate data, which is due to the low variability in the instantaneous spike

200 frequency of the model cells; without this variability, however, all networks would

201 display perfect n:m phase-locking).

202

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

204 The simulations above show that Single Run surrogates can properly detect

205 n:m phase-locking for coupled oscillators exhibiting variable instantaneous

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206 frequency, which is the case of hippocampal theta and gamma oscillations. However,

207 it should be noted that high asymmetry of the theta waveform may also lead to

208 statistically significant Rn:m values per se. As illustrated in Figure 4A, a non-

209 sinusoidal oscillation such as a theta sawtooth wave can be decomposed into a sum

210 of sine waves at the fundamental and harmonic frequencies, which have decreasing

211 amplitude (i.e., the higher the harmonic frequency, the lower the amplitude).

212 Importantly, the harmonic frequency components are n:m phase-locked to each

213 other: the first harmonic exhibits a fixed 1:2 phase relationship to the fundamental

214 frequency, the second harmonic a 1:3 relationship, and so on (Figure 4B). Of note,

215 the higher frequency harmonics not only exhibit cross-frequency phase-phase

216 coupling to the fundamental theta frequency but also phase-amplitude coupling,

217 since they have higher amplitude at the theta phases where the sharp deflection

218 occurs (Figure 4C left and Figure 4 – figure supplement 1; see also Kramer et al.,

219 2008 and Tort et al., 2013).

220 The gamma-filtered component of a theta sawtooth wave of variable peak

221 frequency thus displays spurious gamma oscillations (i.e., theta harmonics) that

222 have consistent phase relationship to the theta cycle irrespective of variations in

223 cycle length. In randomly permutated data, however, the theta phases associated to

224 spurious gamma differ from cycle to cycle due to the variability in instantaneous

225 theta frequency. As a result, the spurious n:m phase-coupling induced by sharp

226 signal deflections is significantly higher than the Random Permutation/Single Run

227 surrogate distribution (Figure 4C right and Figure 4 – figure supplement 2 top row).

228 Interestingly, the significance of this spurious effect is much lower when using the

229 Time Shift procedure (Figure 4 – figure supplement 2 bottom row), probably due to

230 the proximity between the original and the time-shifted time series (200 ms maximum

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231 distance).

232

233 Assessing n:m phase-locking in actual LFPs

234 We next proceeded to analyze hippocampal CA1 recordings from 7 rats,

235 focusing on periods of prominent theta activity (active waking and REM sleep). We

236 found similar results between white noise and actual LFP data. Namely, Rn:m curves

237 peaked at n:m ratios according to the filtered bands, and Rn:m values were lower for

238 longer epochs (Figure 5A; compare with Figure 2B). As shown in Figure 5B, Original

239 Rn:m values were not statistically different from a proper surrogate distribution

240 (Random Permutation/Single Run) in epochs of up to 100 seconds (but see Figure

241 10). Noteworthy, as with white-noise data (Figure 2F), false positive phase-phase

242 coupling would be inferred if an inadequate surrogate method were employed (Time

243 Shift/Pooled) (Figure 5B).

244 We also found no difference between original and surrogate n:m phase-

245 locking levels when employing the metric described in Sauseng et al. (2009) (Figure

246 5 – figure supplement 1), and when estimating theta phase by interpolating phase

247 values between 4 points of the theta cycle (trough, ascending, peak and descending

248 points) as performed in Belluscio et al. (2012) (Figure 5 – figure supplement 2). The

249 latter was somewhat expected since the phase-phase coupling results in Belluscio et

250 al. (2012) did not depend on this particular method of phase estimation (see their

251 Figure 6Ce). Moreover, coupling levels did not statistically differ from zero when

252 using the pairwise phase consistency metric described in Vinck et al. (2010) (Figure

253 5 – figure supplement 1).

254 We further confirmed our results by analyzing data from 3 additional rats

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255 recorded in an independent laboratory (Figure 5 – figure supplement 3; see Material

256 and Methods). In addition, we also found similar results in LFPs from other

257 hippocampal layers than s. pyramidale (Figure 5 – figure supplement 4), in

258 neocortical LFPs (not shown), in current-source density (CSD) signals (Figure 5 –

259 figure supplement 4), in independent components that isolate activity of specific

260 gamma sub-bands (Schomburg et al., 2014) (Figure 5 – figure supplement 5), and in

261 transient gamma bursts (Figure 5 – figure supplement 6).

262

263 On diagonal stripes in phase-phase plots

264 Since Original Rn:m values were not greater than Single Run surrogate

265 distributions, we concluded that there is lack of convincing evidence for n:m phase-

266 locking in the hippocampal LFPs analyzed here. However, as in previous reports

267 (Belluscio et al., 2012; Zheng et al., 2016), phase-phase plots (2D histograms of

268 theta phase vs gamma phase) of actual LFPs displayed diagonal stripes (Figure 6),

269 which seem to suggest phase-phase coupling. We next sought to investigate what

270 causes the diagonal stripes in phase-phase plots.

271 In Figure 7 we analyze a representative LFP with prominent theta oscillations

272 at ~7 Hz recorded during REM sleep. Due to the non-sinusoidal shape of theta

273 (Belluscio et al., 2012; Sheremet et al., 2016), the LFP also exhibited spectral peaks

274 at harmonic frequencies (Figure 7A). We constructed phase–phase plots using LFP

275 components narrowly filtered at theta and its harmonics: 14, 21, 28 and 35 Hz.

276 Similarly to the sawtooth wave (Figure 4B), the phase-phase plots exhibited diagonal

277 stripes whose number was determined by the harmonic order (i.e., the 1st harmonic

278 exhibited two stripes, the 2nd harmonic three stripes, the 3rd, four stripes and the 4th,

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279 five stripes; Figure 7B i-iv). Interestingly, when the LFP was filtered at a broad

280 gamma band (30-90 Hz), we observed 5 diagonal stripes, the same number as when

281 narrowly filtering at 35 Hz; moreover, both gamma and 35-Hz filtered signals

282 exhibited the exact same phase lag (Figure 7B iv-v). Therefore, these results

283 indicate that the diagonal stripes in phase-phase plots may be influenced by theta

284 harmonics. Under this interpretation, signals filtered at the gamma band would be

285 likely to exhibit as many stripes as expected for the first theta harmonic falling within

286 the filtered band. Consistent with this possibility, we found that the peak frequency of

287 theta relates to the number of stripes (Figure 8).

288 As in previous studies (Belluscio et al., 2012; Zheng et al., 2016), phase-

289 phase plots constructed using data averaged from individual time-shifted epochs

290 exhibited no diagonal stripes (Figure 7B vi and Figure 8). This is because different

291 time shifts lead to different phase lags; the diagonal stripes of individual surrogate

292 runs that could otherwise be apparent cancel each other out when combining data

293 across multiple runs of different lags (Figure 8 – figure supplement 1). Moreover, as

294 in Belluscio et al. (2012), the histogram counts that give rise to the diagonal stripes

295 were deemed statistically significant when compared to the mean and standard

296 deviation over individual counts from time-shifted surrogates (Figure 7 – figure

297 supplement 1 and Figure 8).

298 To gain further insight into what generates the diagonal stripes, we next

299 analyzed white-noise signals. As shown in Figure 9A, phase-phase plots constructed

300 from filtered white-noise signals also displayed diagonal stripes. Since white noise

301 has no harmonics, these results show that the sinusoidality induced by the filter can

302 by itself lead to diagonal stripes in phase-phase plots, in the same way that it leads

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303 to a bump in the Rn:m curve (Figure 2 and Figure 2 – figure supplement 1).

304 Importantly, as in actual LFPs, bin counts in phase-phase plots of white-noise signals

305 were also deemed statistically significant when compared to the distribution of bin

306 counts from time-shifted surrogates (Figure 9A). Since by definition white noise has

307 no n:m phase-locking, we concluded that the statistical analysis of phase-phase

308 plots as originally introduced in Belluscio et al. (2012) is too liberal. Nevertheless, we

309 found that phase-phase plots of white noise were no longer statistically significant

310 when using the same approach as in Belluscio et al. (2012) but corrected for multiple

311 comparisons (i.e., the number of bins) by the Holm-Bonferroni method (the FDR

312 correction still led to significant bins; not shown). This result was true for different

313 epoch lengths and also when computing surrogate phase-phase plots using the

314 Random Permutation procedure (Figure 9A). Consistently, for all epoch lengths,

315 Original Rn:m values fell inside the distribution of Single Run surrogate Rn:m values

316 computed using either Time Shift and Random Permutation procedures (Figure 9B).

317 The observations above suggest that the diagonal stripes in phase-phase

318 plots of hippocampal LFPs may actually be caused by filtering-induced sinusoidality,

319 as opposed to being an effect of theta harmonics as we first interpreted. To test this

320 possibility, we next revisited the significance of phase-phase plots of actual LFPs.

321 For epochs of up to 100 seconds, we found similar results as in white noise, namely,

322 bin counts were no longer statistically significant after correcting for multiple

323 comparisons (Holm-Bonferroni method); this was true when using either the Time

324 Shift or Random Permutation procedures (Figure 10A). Surprisingly, however, when

325 analyzing much longer time series (10 or 20 minutes of concatenated periods of

326 REM sleep), several bin counts became statistically significant when compared to

327 randomly permutated, but not time-shifted, surrogates (Figure 10A). Moreover, this

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328 result reflected in the Rn:m curves: the Original Rn:m curve fell within the distribution of

329 Time Shift/Single Run surrogate Rn:m values for all analyzed lengths, but outside the

330 distribution of Random Permutation/Single Run surrogates for the longer time series

331 (Figure 10B). We believe such a finding relates to what we observed for synthetic

332 sawtooth waves, in which Random Permutation was more sensitive than Time Shift

333 to detect the significance of the artifactual coupling caused by waveform asymmetry

334 (Figure 4 – figure supplement 2). In this sense, the n:m phase-locking between

335 fundamental and harmonic frequencies would persist for small time shifts (±200 ms),

336 albeit in different phase relations, while it would not resist the much larger time shifts

337 obtained through random permutations. However, irrespective of this explanation, it

338 should be noted that since the n:m phase-locking metrics cannot separate artifactual

339 from true coupling, the possibility of the latter cannot be discarded. But if this is the

340 case, we consider unlikely that the very low coupling level (~0.03) would have any

341 physiological significance.

342 We conclude that the diagonal stripes in phase-phase plots of both white

343 noise and actual LFPs are mainly caused by a temporary n:m alignment of the phase

344 time-series secondary to the filtering-induced sinusoidality, and as such they are also

345 apparent in surrogate data (Figure 8 – figure supplement 1 and Figure 10 – figure

346 supplement 1). However, for actual LFPs there is a second influence, which can only

347 be detected when analyzing very long epoch lengths, and which we believe is due to

348 theta harmonics.

349

350 Discussion

351 Theta and gamma oscillations are hallmarks of hippocampal activity during

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352 active exploration and REM sleep (Buzsáki et al., 2003; Csicsvari et al., 2003;

353 Montgomery et al., 2008). Theta and gamma are well known to interact by means of

354 phase-amplitude coupling, in which the instantaneous gamma amplitude waxes and

355 wanes as a function of theta phase (Bragin et al., 1995; Scheffer-Teixeira et al.,

356 2012; Caixeta et al., 2013). This particular type of cross-frequency coupling has

357 been receiving large attention and related to functional roles (Canolty and Knight,

358 2010; Hyafil et al., 2015). In addition to phase-amplitude coupling, theta and gamma

359 oscillations can potentially interact in many other ways (Jensen and Colgin, 2007;

360 Hyafil et al., 2015). For example, the power of slow gamma oscillations may be

361 inversely related to theta power (Tort et al., 2008), suggesting amplitude-amplitude

362 coupling. Recently, it has been reported that theta and gamma in hippocampal LFPs

363 would also couple by means of n:m phase-locking (Belluscio et al., 2012; Zheng et

364 al., 2013; Xu et al., 2013; Xu et al., 2015; Zheng et al., 2016). Among other

365 implications, this finding was taken as evidence for network models of working

366 memory (Lisman and Idiart, 1995; Jensen and Lisman, 2005; Lisman, 2005) and for

367 a role of basket cells in generating cross-frequency coupling (Belluscio et al., 2012;

368 Buzsáki and Wang, 2012). However, our results show lack of convincing evidence for

369 n:m phase-locking in the two hippocampal datasets analyzed here, and further

370 suggest that previous work may have spuriously detected phase-phase coupling due

371 to an improper use of surrogate methods, a concern also raised for phase-amplitude

372 coupling (Aru et al., 2015).

373 Statistical inference of phase-phase coupling

374 When searching for phase-phase coupling between theta and gamma, we

375 noticed that our Rn:m values differed from those reported in previous studies

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376 (Belluscio et al., 2012; Xu et al., 2013; Xu et al., 2015; Zheng et al., 2016). We

377 suspected that this could be due to differences in the duration of the analyzed

378 epochs. We then investigated the dependence of Rn:m on epoch length, and found a

379 strong positive bias for shorter epochs. In addition, Rn:m values exhibit greater

380 variability across samples as epoch length decreases for both white noise and actual

381 data (e.g., in Figure 5B compare data dispersion in Original R1:5 or R1:8 boxplots for

382 different epoch lengths). Since theta and gamma peak frequencies are not constant

383 in these signals, the longer the epoch, the more the theta and gamma peak

384 frequencies are allowed to fluctuate and the more apparent the lack of coupling. On

385 the other hand, ∆φ distribution becomes less uniform for shorter epochs. The

386 dependence of n:m phase-coupling metrics on epoch length has important

387 implications in designing surrogate epochs for testing the statistical significance of

388 actual Rn:m values. Of note, methodological studies on 1:1 phase-synchrony have

389 properly used single surrogate runs of the same length as the original signal (Le Van

390 Quyen et al., 2001; Hurtado et al., 2004). As demonstrated here, spurious detection

391 of phase-phase coupling may occur if surrogate epochs are longer than the original

392 epoch. This is the case when one lumps together several surrogate epochs before

393 computing Rn:m. When employing proper controls, our results show that Rn:m values

394 of real data do not differ from surrogate values in theta epochs of up to 100 seconds.

395 Moreover, the prominent bump in the Rn:m curve disappears when subsampling data

396 at a lower frequency than gamma for both white noise and hippocampal LFPs (see

397 Figure 2 – figure supplement 1 and Figure 5 – figure supplement 7), which suggests

398 that it is due to the statistical dependence among contiguous data points introduced

399 by the filter (which we referred to as “filtering-induced sinusoidality”).

400 Therefore, even though the n:m phase-locking metric Rn:m is theoretically well-

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401 defined and varies between 0 and 1, an estimated Rn:m value in isolation does not

402 inform if two oscillations exhibit true phase-coupling or not. This can only be inferred

403 after testing the statistical significance of the estimated Rn:m value against a proper

404 surrogate distribution (but notice that false-positive cases may occur due to

405 waveform asymmetry; Figure 4C). While constructing surrogate data renders the

406 metric computationally more expensive, such an issue is not specific for measuring

407 n:m phase-locking but also happens for other metrics commonly used in the analysis

408 of neurophysiological data, such as coherence, spike-field coupling, phase-amplitude

409 coupling, mutual information and directionality measures, among many others (Le

410 Van Quyen et al., 2001; Hurtado et al., 2004; Pereda et al., 2005; Tort et al., 2010).

411 The recent studies assessing theta-gamma phase-phase coupling in

412 hippocampal LFPs have not tested the significance of individual Rn:m values against

413 chance (Belluscio et al., 2012; Zheng et al., 2013; Xu et al., 2013; Xu et al., 2015;

414 Zheng et al., 2016). Two studies (Belluscio et al., 2012; Zheng et al., 2016)

415 statistically inferred the existence of n:m phase-locking by comparing empirical

416 phase-phase plots with those obtained from the average of 1000 time-shifted

417 surrogate runs. Specifically, Belluscio et al. (2012) established a significance

418 threshold for each phase-phase bin based on the mean and standard deviation of

419 individual surrogate counts in that bin, and showed that the bin counts leading to

420 diagonal stripes were statistically significant. Here we were able to replicate these

421 results (Figure 7 – figure supplement 1 and Figure 8). However, we note that a

422 phase-phase bin count is not a metric of n:m phase-locking; it does not inform

423 coupling strength and even coupled oscillators have bins with non-significant counts.

( ) 424 A bin count would be analogous to a phase difference vector (∆ ), which is

425 also not a metric of n:m phase-locking per se, but used to compute one. That is, in

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426 the same way that the Rn:m considers all phase difference vectors, n:m phase-locking

427 can only be inferred when considering all bin counts in a phase-phase plot. In this

428 sense, by analyzing the phase-phase plot as a whole, it was assumed that the

429 appearance diagonal stripes was due to theta-gamma coupling; no such stripes were

430 apparent in phase-phase plots constructed from the average over all surrogate runs

431 (see Figure 6A in Belluscio et al., 2012). However, here we showed that single time-

432 shifted surrogate runs do exhibit diagonal stripes (Figure 8 – figure supplement 1

433 and Figure 10 – figure supplement 1), that is, similar stripes exist at the level of a

434 Single Run surrogate analysis, in the same way that Single Run surrogates also

435 exhibit a bump in the Rn:m curve. Averaging 1000 surrogate phase-phase plots

436 destroys the diagonal stripes since different time shifts lead to different phase lags.

437 Moreover, since the average is the sum divided by a scaling factor (the sample size),

438 computing the average phase-phase plot is equivalent to computing a single phase-

439 phase plot using the pool of all surrogate runs, which is akin to the issue of

440 computing a single Rn:m value from a pooled surrogate distribution (Figure 2). Note

441 that even bin counts in phase-phase plots of white noise are considered significant

442 under the statistical analysis introduced in Belluscio et al. (2012) (Figure 9A).

443 Nevertheless, this was no longer the case when adapting their original framework to

444 include a Holm-Bonferroni correction for multiple comparisons (Figure 9A).

445 Here we showed that the presence of diagonal stripes in phase-phase plots is

446 not sufficient to conclude the existence of phase-phase coupling. The diagonal

447 stripes are simply a visual manifestation of a maintained phase relationship, and as

448 such they essentially reflect what Rn:m measures: that is, the “clearer” the stripes, the

449 higher the Rn:m. Therefore, in addition to true coupling, the same confounding factors

450 that influence Rn:m also influence phase-phase plots, such as filtering-induced

19

451 sinusoidality and frequency harmonics. Our results suggest that the former is a main

452 factor, because white-noise signals have no harmonics but nevertheless display

453 stripes in phase-phase plots (Figure 9A). In accordance, no stripes are observed in

454 phase-phase plots of white noise when subsampling the time series (Figure 10 –

455 figure supplement 2; see also Figure 2 – figure supplement 1). However, in actual

456 LFPs filtering is not the only influence: (1) for the same filtered gamma band (30-50

457 Hz), the number of stripes relates to theta frequency (Figure 8); (2) for very long time

458 series (i.e., 10-20 min of concatenated data), the stripes in phase-phase plots of

459 actual data – but not of white noise – persist after correcting for multiple comparisons

460 when employing Random Permutation/Single Run surrogates (Figure 10A); (3) a

461 striped-like pattern remains in phase-phase plots of actual LFPs after subsampling

462 the time series (Figure 10 – figure supplement 2). Consistently, Rn:m values of actual

463 LFPs are greater than those of white noise in 1200-s epochs (~0.03 vs ~0.005,

464 compare the bottom right panels of Figures 9B and 10B). Interestingly, Original Rn:m

465 values of actual LFPs are not statistically different from the distribution of Time

466 Shift/Single Run surrogates even for the very long epochs (Figure 10B), which

467 suggests that Random Permutation is more powerful than Time Shift and should

468 therefore be preferred. Though a very weak but true coupling effect cannot be

469 discarded, based on our analysis of sawtooth waves (Figure 4 and Figure 4 – figure

470 supplement 2), we believe these results can be explained by theta harmonics, which

471 would remain phase-locked to the fundamental frequency under small time shifts.

472 Sharp signal deflections have been previously recognized to generate artifactual

473 phase-amplitude coupling (Kramer et al., 2008; Scheffer-Teixeira et al., 2013; Tort et

474 al., 2013; Aru et al., 2015; Lozano-Soldevilla et al., 2016). Interestingly, Hyafil (2015)

475 recently suggested that the non-sinusoidality of alpha waves could underlie the 1:2

20

476 phase-locking between alpha and beta observed in human EEG (Nikulin and

477 Brismar, 2006; see also Palva et al., 2005). To the best of our knowledge, there is

478 currently no metric capable of automatically distinguishing true cross-frequency

479 coupling from waveform-induced artifacts in collective signals such as LFP, EEG and

480 MEG signals. Ideally, learning how the signal is generated from the activity of

481 different neuronal populations would answer whether true cross-frequency coupling

482 exists or not (Hyafil et al., 2015), but unfortunately this is methodologically

483 challenging.

484

485 Lack of evidence vs evidence of non-existence

486 One could argue that we did not analyze a proper dataset, or else that

487 prominent phase-phase coupling would only occur during certain behavioral states

488 not investigated here. We disagree with these arguments for the following reasons:

489 (1) we could reproduce our results using a second dataset from an independent

490 laboratory (Figure 5 – figure supplement 3), and (2) we examined the same

491 behavioral states in which n:m phase-locking was reported to occur (active waking

492 and REM sleep). One could also argue that there exists multiple gammas, and that

493 different gamma types are most prominent in different hippocampal layers (Colgin et

494 al., 2009; Scheffer-Teixeira et al., 2012; Tort et al., 2013; Schomburg et al., 2014;

495 Lasztóczi and Klausberger, 2014); therefore, prominent theta-gamma phase-phase

496 coupling could exist in other hippocampal layers not investigated here. We also

497 disagree with this possibility because: (1) we examined the same hippocampal layer

498 in which theta-gamma phase-phase coupling was reported to occur (Belluscio et al.,

499 2012); moreover, (2) we found similar results in all hippocampal layers (we recorded

21

500 LFPs using 16-channel silicon probes, see Material and Methods) (Figure 5 – figure

501 supplement 4) and (3) in parietal and entorhinal cortex recordings (not shown).

502 Furthermore, similar results hold when (4) filtering LFPs within any gamma sub-band

503 (Figure 5 and Figure 5 – figure supplements 1 to 6), (5) analyzing CSD signals

504 (Figure 5 – figure supplement 4), or (6) analyzing independent components that

505 maximize activity within particular gamma sub-bands (Schomburg et al., 2014)

506 (Figure 5 – figure supplement 5). Finally, one could argue that gamma oscillations

507 are not continuous but transient, and that assessing phase-phase coupling between

508 theta and transient gamma bursts would require a different type of analysis than

509 employed here. Regarding this argument, we once again stress that we used the

510 exact same methodology as originally used to detect theta-gamma phase-phase

511 coupling (Belluscio et al., 2012). Nevertheless, we also ran analysis only taking into

512 account periods in which gamma amplitude was >2SD above the mean (“gamma

513 bursts”) and found no statistically significant phase-phase coupling (Figure 5 – figure

514 supplement 6).

515 Following Belluscio et al. (2012), other studies also reported theta-gamma

516 phase-phase coupling in the rodent hippocampus (Zheng et al., 2013; Xu et al.,

517 2013; Xu et al., 2015; Zheng et al., 2016) and amygdala (Stujenske et al., 2014). In

518 addition, human studies had previously reported theta-gamma phase-phase coupling

519 in scalp EEG (Sauseng et al., 2008; Sauseng et al., 2009; Holtz et al., 2010). Most of

520 these studies, however, have not tested the statistical significance of coupling levels

521 against chance (Sauseng et al., 2008; Sauseng et al., 2009; Holtz et al., 2010;

522 Zheng et al., 2013; Xu et al., 2013; Xu et al., 2015; Stujenske et al., 2014), while

523 Zheng et al. (2016) based their statistical inferences on the inspection of diagonal

524 stripes in phase-phase plots as originally introduced in Belluscio et al. (2012). We

22

525 further note that epoch length was often not informed in the animal studies. Based

526 on our results, we believe that differences in analyzed epoch length are likely to

527 explain the high variability of Rn:m values across different studies, from ~0.4 (Zheng

528 et al., 2016) down to 0.02 (Xu et al., 2013).

529 Since it is philosophically impossible to prove absence of an effect, the burden

530 of proof should be placed on demonstrating that a true effect exists. In this sense,

531 and to the best of our knowledge, none of previous research investigating theta-

532 gamma phase-phase coupling has properly tested Rn:m against chance. Many

533 studies have focused on comparing changes in n:m phase-locking levels, but we

534 believe these can be influenced by other variables such as changes in power, which

535 affect the signal-to-noise ratio and consequently also the estimation of the phase

536 time series. Interestingly, in their pioneer work, Tass and colleagues used filtered

537 white noise to construct surrogate distributions and did not find significant n:m

538 phase-locking among brain oscillations (Tass et al., 1998; Tass et al., 2003). On the

539 other hand, it is theoretically possible that n:m phase-locking exists but can only be

540 detected by other types of metrics yet to be devised. In any case, our work shows

541 that there is currently no convincing evidence for genuine theta-gamma phase-phase

542 coupling using the same phase-locking metric (Rn:m) as employed in previous studies

543 (Belluscio et al., 2012; Zheng et al., 2013; Xu et al., 2013; Stujenske et al., 2014; Xu

544 et al., 2015; Zheng et al., 2016), at least when examining LFP epochs of up to 100

545 seconds of prominent theta activity. For longer epoch lengths, though, we did find

546 that Rn:m values of hippocampal LFPs may actually differ from those of randomly

547 permuted, but not time-shifted, surrogates (Figure 10B). While we tend to ascribe

548 such result to the effect of theta harmonics, we note that the possibility of true

549 coupling cannot be discarded. In any case, we are particularly skeptical that the very

23

550 low levels of coupling strength observed in long LFP epochs would be physiologically

551 meaningful.

552 Implications for models of neural coding by theta-gamma coupling

553 In 1995, Lisman and Idiart proposed an influential model in which theta and

554 gamma oscillations would interact to produce a neural code. The theta-gamma

555 coding model has since been improved (Jensen and Lisman, 2005; Lisman, 2005;

556 Lisman and Buzsáki, 2008), but its essence remains the same (Lisman and Jensen,

557 2013): nested gamma cycles would constitute memory slots, which are parsed at

558 each theta cycle. Accordingly, Lisman and Idiart (1995) hypothesized that working

559 memory capacity (7±2) is determined by the number of gamma cycles per theta

560 cycle.

561 Both phase-amplitude and phase-phase coupling between theta and gamma

562 have been considered experimental evidence for such coding scheme (Lisman and

563 Buzsáki, 2008; Sauseng et al., 2009; Axmacher et al., 2010; Belluscio et al., 2012;

564 Lisman and Jensen, 2013; Hyafil et al., 2015; Rajji et al., 2016). In the case of

565 phase-amplitude coupling, the modulation of gamma amplitude within theta cycles

566 would instruct a reader network when the string of items represented in different

567 gamma cycles starts and terminates. On the other hand, the precise ordering of

568 gamma cycles within theta cycles that is consistent across theta cycles would imply

569 phase-phase coupling; indeed, n:m phase-locking is a main feature of computational

570 models of sequence coding by theta-gamma coupling (Lisman and Idiart, 1995;

571 Jensen and Lisman, 1996; Jensen et al., 1996). In contrast to these models,

572 however, our results show that the theta phases in which gamma cycles begin/end

573 are not fixed across theta cycles, which is to say that gamma cycles are not precisely

24

574 timed but rather drift; in other words, gamma is not a clock (Burns et al., 2011).

575 If theta-gamma neural coding exists, our results suggest that the precise

576 location of gamma memory slots within a theta cycle is not required for such a code,

577 and that the ordering of the represented items would be more important than the

578 exact spike timing of the cell assemblies that represent the items (Lisman and

579 Jensen, 2013).

580 Conclusion

581 In summary, while absence of evidence is not evidence of absence, our

582 results challenge the hypothesis that theta-gamma phase-phase coupling exists in

583 the hippocampus. At best, we only found significant Rn:m values when examining

584 long LFP epochs (>100 s), but these had very low magnitude (and we particularly

585 attribute their statistical significance to the effects of harmonics). We believe that

586 evidence in favor of n:m phase-locking in other brain regions and signals could

587 potentially also be explained by simpler effects (e.g., filtering-induced sinusoidality,

588 asymmetrical waveform, and improper statistical tests). While no current technique

589 can differentiate spurious from true phase-phase coupling, previous findings should

590 be revisited and, whenever suitable, checked against the confounding factors and

591 the more conservative surrogate procedures outlined here.

592

593 Material and Methods

594 Animals and surgery. All procedures were approved by our local institutional ethics

595 committee (Comissão de Ética no Uso de Animais - CEUA/UFRN, protocol number

596 060/2011) and were in accordance with the National Institutes of Health guidelines.

25

597 We used 7 male Wistar rats (2–3 months; 300–400 g) from our breeding colony, kept

598 under 12h/12h dark-light cycle. We recorded from the dorsal hippocampus through

599 either multi-site linear probes (n=6 animals; 4 probes had 16 4320-μm2 contacts

600 spaced by 100 μm; 1 probe had 16 703-μm2 contacts spaced by 100 μm; 1 probe

601 had 16 177-μm2 contacts spaced by 50 μm; all probes from NeuroNexus) or single

602 wires (n=1 animal; 50-μm diameter) inserted at AP -3.6 mm and ML 2.5 mm. Results

603 shown in the main figures were obtained for LFP recordings from the CA1 pyramidal

604 cell layer, identified by depth coordinate and characteristic electrophysiological

605 benchmarks such as highest ripple power (see Figure 5 – figure supplement 4 for an

606 example). Similar results were obtained for recordings from other hippocampal

607 layers (Figure 5 – figure supplement 4).

608 We also analyzed data from 3 additional rats downloaded from the Collaborative

609 Research in Computational Neuroscience data sharing website (www.crcns.org)

610 (Figure 5 – figure supplement 3). These recordings are a generous contribution by

611 György Buzsáki’s laboratory (HC3 dataset, Mizuseki et al., 2013 and 2014).

612

613 Data collection. Recording sessions were performed in an open field (1 m x 1 m) and

614 lasted 4-5 hours. Raw signals were amplified (200x), filtered between 1 Hz and 7.5

615 kHz (3rd order Butterworth filter), and digitized at 25 kHz (RHA2116, IntanTech). The

616 LFP was obtained by further filtering between 1 – 500 Hz and downsampling to 1000

617 Hz.

618

619 Data analysis. Active waking and REM sleep periods were identified from spectral

620 content (high theta/delta power ratio) and video recordings (movements during active

621 waking; clear sleep posture and preceding slow-wave sleep for REM). Results were

26

622 identical for active waking and REM epochs; throughout this work we only show the

623 latter. The analyzed REM sleep dataset is available at

624 http://dx.doi.org/10.5061/dryad.12t21. MATLAB codes for reproducing our analyses

625 are available at https://github.com/tortlab/phase_phase.

626

627 We used built-in and custom written MATLAB routines. Band-pass filtering was

628 obtained using a least squares finite impulse response (FIR) filter by means of the

629 “eegfilt” function from the EEGLAB Toolbox (Delorme and Makeig, 2004). The filter

630 order was 3 times the sampling rate divided by the low cutoff frequency. The eegfilt

631 function calls the MATLAB “filtfilt” function, which applies the filter forward and then

632 again backwards to ensure no distortion of phase values. Similar results were

633 obtained when employing other types of filters (Figure 5 – figure supplement 8).

634 The phase time series was estimated through the Hilbert transform. To estimate the

635 instantaneous theta phase of actual data, we filtered the LFP between 4 – 20 Hz, a

636 bandwidth large enough to capture theta wave asymmetry (Belluscio et al., 2012).

637 Estimating theta phase by the interpolation method described in Belluscio et al.

638 (2012) led to similar results (Figure 5 – figure supplement 2).

639 The CSD signals analyzed in Figure 5 – figure supplement 4 were obtained as

640 −A+2B−C, where A, B and C denote LFP signals recorded from adjacent probe sites.

641 In Figure 5 – figure supplement 5, the independent components were obtained as

642 described in Schomburg et al. (2014); phase-amplitude comodulograms were

643 computed as described in Tort et al. (2010).

644

645 n:m phase-locking. We measured the consistency of the phase difference between

27

646 accelerated time series (∆φ()=∗φ −∗φ()). To that end, we

647 created unitary vectors whose angle is the instantaneous phase difference

∆φ ( ) 648 ( ), where j indexes the time sample, and then computed the length of the

∆φ ( ) 649 mean vector: = ∑ , where N is the total number of time :

650 samples (epoch length in seconds x sampling frequency in Hz). Rn:m equals 1 when

651 ∆φ is constant for all time samples tj, and 0 when ∆φ is uniformly distributed.

652 This metric is also commonly referred to as “mean resultant length” or “mean radial

653 distance” (Belluscio et al., 2012; Stujenske et al., 2014; Zheng et al., 2016).

654 Qualitatively similar results were obtained when employing the framework introduced

655 in Sauseng et al. (2009), which computes the mean radial distance using gamma

656 phases in separated theta phase bins, or the pairwise phase consistency metric

657 described in Vinck et al. (2010) (Figure 5 – figure supplement 1). Phase-phase plots

658 were obtained by first binning theta and gamma phases into 120 bins and next

659 constructing 2D histograms of phase counts, which were smoothed using a

660 Gaussian kernel of σ = 10 bins.

661

662 Surrogates. In all cases, theta phase was kept intact while gamma phase was

663 mocked in three different ways: (1) Time Shift: the gamma phase time series is

664 randomly shifted between 1 and 200 ms; (2) Random Permutation: a contiguous

665 gamma phase time series of the same length as the original is randomly extracted

666 from the same session. (3) Phase Scrambling: the timestamps of the gamma phase

667 time series are randomly shuffled (thus not preserving phase continuity). For each

668 case, Rn:m values were computed using either ∆φ distribution for single surrogate

669 runs (Single Run Distribution) or the pooled distribution of ∆φ over 100 surrogate

28

670 runs (Pooled Distribution).

671 For each animal, behavioral state (active waking or REM sleep) and epoch length,

672 we computed 300 Original Rn:m values using different time windows along with 300

673 mock Rn:m values per surrogate method. Therefore, in all figures each boxplot was

674 constructed using the same number of samples (= 300 x number of animals). For

675 instance, in Figure 5B we used n = 7 animals x 300 samples per animal = 2100

676 samples (but see Statistics below). In Figure 2, boxplot distributions for the white-

677 noise data were constructed using n = 2100.

678

679 Simulations. Kuramoto oscillators displaying n:m phase-locking were modeled as

680 described in Osipov et al. (2007):

= + sin −

681 = + sin( −),

682 where is the coupling strength and and are the natural frequencies of theta

683 and gamma, respectively, which followed a Gaussian probability (σ = 5 Hz) at each

684 time step. We used = 10, n = 1, m = 5, and dt = 0.001 s. The mean theta and

685 gamma frequencies of each simulation are stated in the main text. For uncoupled

686 oscillators, we set = 0.

687 For implementing the O-I cell network (Figure 3B), we simulated the model

688 previously described in Kopell et al. (2010). We used the same parameters as in

689 Figure 3A of Kopell et al. (2010), with white noise (σ = 0.001) added to the I cell drive

690 to create variations in spike frequency. NEURON (https://www.neuron.yale.edu/)

691 codes for the model are available at ModelDB (https://senselab.med.yale.edu/).

692 The sawtooth wave in Figure 4C was simulated using dt = 0.001 s. Its instantaneous

29

693 frequency followed a Gaussian distribution with mean = 8 Hz and σ = 5 Hz; white

694 noise (σ = 0.1) was added to the signal.

695 In Figures 3 and 4C, boxplot distributions for simulated data were constructed using

696 n = 300.

697

698 Statistics. For white noise data (Figure 2F), given the large sample size (n=2100)

699 and independence among samples, we used one-way ANOVA with Bonferroni post-

700 hoc test. For statistical analysis of real data (Figure 5B), we avoided nested design

701 and inflation of power and used the mean Rn:m value per animal. In this case, due to

702 the reduced sample size (n = 7) and lack of evidence of normal distribution (Shapiro-

703 Wilk normality test), we used the Friedman’s test and Nemenyi post-hoc test. In

704 Figures 3 and 4C, we tested if Rn:m values of simulated data were greater than the

705 distribution of surrogate values using one-tailed t-tests.

706

707 Acknowledgment

708 The authors are indebted to the reviewers for many constructive comments and

709 helpful suggestions. The authors are grateful to Jurij Brankačk and Andreas Draguhn

710 for donation of NeuroNexus probes. Supported by Conselho Nacional de

711 Desenvolvimento Científico e Tecnológico (CNPq) and Coordenação de

712 Aperfeiçoamento de Pessoal de Nível Superior (CAPES). The authors declare no

713 competing financial interests.

714

715 Author contributions

30

716 A.B.L.T. designed research; R.S.T. acquired and analyzed data; A.B.L.T. and R.S.T.

717 wrote the manuscript.

718

719 720 721 References 722 723 724 Aru, J., Aru, J., Priesemann, V., Wibral, M., Lana, L., Pipa, G., Singer, W., Vicente, R. 725 (2015). Untangling cross-frequency coupling in neuroscience. Curr. Opin. 726 Neurobiol. 31, 51–61. doi: 10.1016/j.conb.2014.08.002

727 Axmacher, N., Henseler, M.M., Jensen, O., Weinreich, I., Elger, C.E., Fell, J. (2010). Cross- 728 frequency coupling supports multi-item working memory in the human hippocampus. 729 Proc. Natl. Acad. Sci. U. S. A. 107, 3228-3233. doi: 10.1073/pnas.0911531107

730 Belluscio, M.A., Mizuseki, K., Schmidt, R., Kempter, R., Buzsáki, G. (2012). Cross-frequency 731 phase-phase coupling between θ and γ oscillations in the hippocampus. J. Neurosci. 732 32, 423–435. doi: 10.1523/JNEUROSCI.4122-11.2012

733 Bragin, A., Jandó, G., Nádasdy, Z., Hetke, J., Wise, K., Buzsáki, G. (1995). Gamma (40-100 734 Hz) oscillation in the hippocampus of the behaving rat. J. Neurosci. 15, 47–60.

735 Burns, S.P., Xing, D., Shapley, R.M. (2011). Is gamma-band activity in the local field 736 potential of V1 cortex a "clock" or filtered noise? J. Neurosci. 31, 9658–9664. doi: 737 10.1523/JNEUROSCI.0660-11.2011

738 Buzsáki, G., Buhl, D.L., Harris, K.D., Csicsvari, J., Czéh, B., Morozov, A. (2003). 739 Hippocampal network patterns of activity in the mouse. Neuroscience 116, 201–211. 740 doi: 10.1016/S0306-4522(02)00669-3

741 Buzsáki, G., Wang, X.-J. (2012). Mechanisms of gamma oscillations. Annu. Rev. Neurosci. 742 35, 203–225. doi: 10.1146/annurev-neuro-062111-150444

743 Caixeta, F.V., Cornélio, A.M., Scheffer-Teixeira, R., Ribeiro, S., Tort, A.B. (2013) Ketamine 744 alters oscillatory coupling in the hippocampus. Sci. Rep. 3, 745 2348. doi: 10.1038/srep02348

746 Canavier, C.C., Gurel Kazanci, F., Prinz, A.A. (2009). Phase resetting curves allow for 747 simple and accurate prediction of robust N:1 phase locking for strongly coupled 748 neural oscillators. Biophys. J. 97, 59–73. doi: 10.1016/j.bpj.2009.04.016

749 Canolty, R.T., Knight, R.T. (2010). The functional role of cross-frequency coupling. Trends 750 Cogn. Sci. 14, 506–515. doi: 10.1016/j.tics.2010.09.001

751 Colgin, L.L., Denninger, T., Fyhn, M., Hafting, T., Bonnevie, T., Jensen, O., Moser, M.B., 752 Moser, E.I. (2009). Frequency of gamma oscillations routes flow of information in the 753 hippocampus. Nature 462, 353–457. doi: 10.1038/nature08573

31

754 Csicsvari, J., Jamieson, B., Wise, K.D., Buzsáki, G. (2003). Mechanisms of gamma 755 oscillations in the hippocampus of the behaving rat. Neuron 37, 311–322. doi: 756 10.1016/S0896-6273(02)01169-8

757 DeCoteau, W.E., Thorn, C., Gibson, D.J., Courtemanche, R., Mitra, P., Kubota, Y., Graybiel, 758 A.M. (2007). Learning-related coordination of striatal and hippocampal theta rhythms 759 during acquisition of a procedural maze task. Proc. Natl. Acad. Sci. U. S. A. 104, 760 5644–5649. doi: 10.1073/pnas.0700818104

761 Delorme, A., Makeig, S. (2004). EEGLAB: an open source toolbox for analysis of single-trial 762 EEG dynamics including independent component analysis. J. Neurosci. Methods 763 134, 9–21. doi: 10.1016/j.jneumeth.2003.10.009

764 Fell, J., Axmacher, N. (2011). The role of phase synchronization in memory processes. Nat. 765 Rev. Neurosci. 12, 105–118. doi: 10.1038/nrn2979

766 Fries, P. (2005). A mechanism for cognitive dynamics: neuronal communication through 767 neuronal coherence. Trends Cogn. Sci. 9, 474–480. doi: 10.1016/j.tics.2005.08.011

768 García-Alvarez, D., Stefanovska, A., McClintock, P.V.E. (2008). High-order synchronization, 769 transitions, and competition among Arnold tongues in a rotator under harmonic 770 forcing. Phys. Rev. E Stat. Nonlin. Soft. Matter. Phys. 77, 056203. doi: 771 10.1103/PhysRevE.77.056203

772 Guevara, M.R., Glass, L. (1982). Phase locking, period doubling bifurcations and chaos in a 773 mathematical model of a periodically driven oscillator: a theory for the entrainment of 774 biological oscillators and the generation of cardiac dysrhythmias. J. Math. Biol. 14, 1– 775 23.

776 Holz, E.M., Glennon, M., Prendergast, K., Sauseng, P. (2010). Theta–gamma phase 777 synchronization during memory matching in visual working memory. Neuroimage 52, 778 326–335. doi: 10.1016/j.neuroimage.2010.04.003

779 Hurtado, J.M., Rubchinsky, L.L., Sigvardt, K.A. (2004). Statistical method for detection of 780 phase-locking episodes in neural oscillations. J. Neurophysiol. 91, 1883–1898. doi: 781 10.1152/jn.00853.2003

782 Hyafil, A., Giraud, A.-L., Fontolan, L., Gutkin, B. (2015). Neural cross-frequency coupling: 783 connecting architectures, mechanisms, and functions. Trends Neurosci. 38, 725– 784 740. doi: 10.1016/j.tins.2015.09.001

785 Hyafil, A. (2015). Misidentifications of specific forms of cross-frequency coupling: three 786 warnings. Front. Neurosci. 9, 370. doi: 10.3389/fnins.2015.00370.

787 Jensen, O., Colgin, L.L. (2007). Cross-frequency coupling between neuronal oscillations. 788 Trends Cogn. Sci. 11, 267–269. doi: 10.1016/j.tics.2007.05.003

789 Jensen, O., Idiart, M.A., Lisman, J.E. (1996). Physiologically realistic formation of 790 autoassociative memory in networks with theta/gamma oscillations: role of fast 791 NMDA channels. Learn. Mem. 3, 243–256.

792 Jensen, O., Lisman, J.E. (2005). Hippocampal sequence-encoding driven by a cortical multi- 793 item working memory buffer. Trends Neurosci. 28, 67–72. doi: 794 10.1016/j.tins.2004.12.001

32

795 Jensen, O., Lisman, J.E. (1996). Theta/gamma networks with slow NMDA channels learn 796 sequences and encode episodic memory: role of NMDA channels in recall. Learn. 797 Mem. 3, 264–278.

798 Kopell, N., Borgers, C., Prevouchine, D., Malerba, P., Tort, A.B. (2010). Gamma and theta 799 rhythms in biophysical models of hippocampal circuits. In: Cutsuridis, V., Graham, B., 800 Cobb, S., Vida, I. (Org.). Hippocampal Microcircuits: A Computational Modeler s 801 Resource Book. New York: Springer-Verlag, 5, 423–457. doi: 10.1007/978-1-4419- 802 0996-1_15

803 Kramer, M.A., Tort, A.B., Kopell, N.J. (2008). Sharp edge artifacts and spurious coupling in 804 EEG frequency comodulation measures. J. Neurosci. Methods 170, 352–357. 805 doi:10.1016/j.jneumeth.2008.01.020

806 Lachaux, J.-P., Rodriguez, E., Martinerie, J., Varela, F.J. (1999). Measuring phase 807 synchrony in brain signals. Hum. Brain Mapp. 8, 194–208. doi: 10.1002/(SICI)1097- 808 0193(1999)8:4<194::AID-HBM4>3.0.CO;2-C

809 Lasztóczi, B., Klausberger, T. (2014). Layer-specific GABAergic control of distinct gamma 810 oscillations in the CA1 hippocampus. Neuron 81, 1126–1139. doi: 811 10.1016/j.neuron.2014.01.021

812 Le Van Quyen, M., Foucher, J., Lachaux, J., Rodriguez, E., Lutz, A., Martinerie, J., Varela, 813 F.J. (2001). Comparison of Hilbert transform and wavelet methods for the analysis of 814 neuronal synchrony. J. Neurosci. Methods 111, 83-98. doi: 10.1016/S0165- 815 0270(01)00372-7

816 Lisman, J. (2005). The theta/gamma discrete phase code occuring during the hippocampal 817 phase precession may be a more general brain coding scheme. Hippocampus 15, 818 913–922. doi: 10.1002/hipo.20121

819 Lisman, J., Buzsáki, G. (2008). A neural coding scheme formed by the combined function of 820 gamma and theta oscillations. Schizophr. Bull. 34, 974-980. doi: 821 10.1093/schbul/sbn060

822 Lisman, J.E., Idiart, M.A. (1995). Storage of 7+/-2 short-term memories in oscillatory 823 subcycles. Science 267, 1512–1515. doi: 10.1126/science.7878473

824 Lisman, J.E., Jensen, O. (2013). The θ-γ neural code. Neuron 77, 1002-1016. doi: 825 10.1016/j.neuron.2013.03.007

826 Lozano-Soldevilla, D., Ter Huurne, N., Oostenveld, R. (2016) Neuronal oscillations with non- 827 sinusoidal morphology produce spurious phase-to-amplitude coupling and 828 directionality. Front Comput Neurosci. 10, 87. doi: 10.3389/fncom.2016.00087

829 Mizuseki, K., Diba, K., Pastalkova, E., Teeters, J., Sirota, A., Buzsáki, G. (2014). 830 Neurosharing: large-scale data sets (spike, LFP) recorded from the hippocampal- 831 entorhinal system in behaving rats. F1000Research 3, 98. 832 doi: 10.12688/f1000research.3895.1

833 Mizuseki, K., Sirota, A., Pastalkova, E., Diba, K., Buzsáki, G. (2013). Multiple single unit 834 recordings from different rat hippocampal and entorhinal regions while the animals 835 were performing multiple behavioral tasks. CRCNS.org. doi: 10.6080/K09G5JRZ

33

836 Montgomery, S.M., Buzsáki, G. (2007). Gamma oscillations dynamically couple hippocampal 837 CA3 and CA1 regions during memory task performance. Proc. Natl. Acad. Sci. U. S. 838 A. 104, 14495–14500. doi: 10.1073/pnas.0701826104

839 Montgomery, S.M., Sirota, A., Buzsáki, G. (2008). Theta and gamma coordination of 840 hippocampal networks during waking and rapid eye movement sleep. J. Neurosci. 841 28, 6731–6741. doi: 10.1523/JNEUROSCI.1227-08.2008

842 Nácher, V., Ledberg, A., Deco, G., Romo, R. (2013). Coherent delta-band oscillations 843 between cortical areas correlate with decision making. Proc. Natl. Acad. Sci. U. S. A. 844 110, 15085–15090. doi: 10.1073/pnas.1314681110

845 Nikulin, V.V., Brismar, T. (2006). Phase synchronization between alpha and beta oscillations 846 in the human electroencephalogram. Neuroscience 137, 647–657. doi: 847 10.1016/j.neuroscience.2005.10.031

848 Osipov, G.V., Kurths, J., Zhou, C. (2007). Synchronization in oscillatory networks. Springer 849 Science & Business Media.

850 Palva, J.M., Palva, S., Kaila, K. (2005). Phase synchrony among neuronal oscillations in the 851 human cortex. J. Neurosci. 25, 3962–3972. doi: 10.1523/JNEUROSCI.4250-04.2005

852 Pereda, E., Quiroga, R.Q., Bhattacharya, J. (2005). Nonlinear multivariate analysis of 853 neurophysiological signals. Prog. Neurobiol. 77, 1–37. doi: 854 0.1016/j.pneurobio.2005.10.003

855 Rajji, T.K., Zomorrodi, R., Barr, M.S., Blumberger, D.M., Mulsant, B.H., Daskalakis, Z.J. 856 (2016). Ordering information in working memory and modulation of gamma by theta 857 oscillations in humans. Cereb. Cortex Ahead of print. doi: 10.1093/cercor/bhv326

858 Sauseng, P., Klimesch, W., Gruber, W.R., Birbaumer, N. (2008). Cross-frequency phase 859 synchronization: a brain mechanism of memory matching and attention. Neuroimage 860 40, 308–317. doi: 10.1016/j.neuroimage.2007.11.032

861 Sauseng, P., Klimesch, W., Heise, K.F., Gruber, W.R., Holz, E., Karim, A.A., Glennon, M., 862 Gerloff, C., Birbaumer, N., Hummel, F.C. (2009). Brain oscillatory substrates of visual 863 short-term memory capacity. Curr. Biol. 19, 1846–1852. 864 doi: 10.1016/j.cub.2009.08.062

865 Schack, B., Weiss, S. (2005). Quantification of phase synchronization phenomena and their 866 importance for verbal memory processes. Biol. Cybern. 92, 275–287. doi: 867 10.1007/s00422-005-0555-1

868 Scheffer-Teixeira, R., Belchior, H., Caixeta, F.V., Souza, B.C., Ribeiro, S., Tort, A.B.L. 869 (2012). Theta phase modulates multiple layer-specific oscillations in the CA1 region. 870 Cereb. Cortex 22, 2404–2414. doi: 10.1093/cercor/bhr319

871 Scheffer-Teixeira, R., Belchior, H., Leão, R.N., Ribeiro, S., Tort, A.B. (2013). On high- 872 frequency field oscillations (>100 Hz) and the spectral leakage of spiking activity. J 873 Neurosci. 33, 1535–1539. doi: 10.1523/JNEUROSCI.4217-12.2013

874 Sheremet, A., Burke, S.N., Maurer, A.P. (2016). Movement enhances the nonlinearity of 875 hippocampal theta. J. Neurosci. 36, 4218–4230. doi: 10.1523/JNEUROSCI.3564- 876 15.2016

34

877 Schomburg, E.W., Fernández-Ruiz, A., Mizuseki, K., Berényi, A., Anastassiou, C.A., Koch, 878 C., Buzsáki, G. (2014). Theta phase segregation of input-specific gamma patterns in 879 entorhinal-hippocampal networks. Neuron 84, 470–485. 880 doi: 10.1016/j.neuron.2014.08.051

881 Stujenske, J.M., Likhtik, E., Topiwala, M.A., Gordon, J.A. (2014). Fear and safety engage 882 competing patterns of theta-gamma coupling in the basolateral amygdala. Neuron 883 83, 919–933. doi: 10.1016/j.neuron.2014.07.026

884 Tass, P., Rosenblum, M.G., Weule, J., Kurths, J., Pikovsky, A., Volkmann, J., Schnitzler, A., 885 Freund, H.-J. (1998). Detection of n:m phase locking from noisy data: application to 886 magnetoencephalography. Phys. Rev. Lett. 81, 3291–3294. doi: 887 10.1103/PhysRevLett.81.3291

888 Tass, P.A., Fieseler, T., Dammers, J., Dolan, K., Morosan, P., Majtanik, M., Boers, F., 889 Muren, A., Zilles, K., Fink, G.R. (2003). Synchronization tomography: a method for 890 three-dimensional localization of phase synchronized neuronal populations in the 891 human brain using magnetoencephalography. Phys. Rev. Lett. 90, 088101. doi: 892 10.1103/PhysRevLett.90.088101

893 Tort, A.B.L., Rotstein, H.G., Dugladze, T., Gloveli, T., Kopell, N.J. (2007). On the formation 894 of gamma-coherent cell assemblies by oriens lacunosum-moleculare interneurons in 895 the hippocampus. Proc. Natl. Acad. Sci. 104, 13490–13495. doi: 896 10.1073/pnas.0705708104

897 Tort, A.B.L., Kramer, M.A., Thorn, C., Gibson, D.J., Kubota, Y., Graybiel, A.M., Kopell, N.J. 898 (2008). Dynamic cross-frequency couplings of local field potential oscillations in rat 899 striatum and hippocampus during performance of a T-maze task. Proc. Natl. Acad. 900 Sci. U. S. A. 105, 20517–20522. doi: 10.1073/pnas.0810524105

901 Tort, A.B.L., Komorowski, R., Eichenbaum, H., Kopell, N. (2010). Measuring phase- 902 amplitude coupling between neuronal oscillations of different frequencies. J. 903 Neurophysiol. 104, 1195–1210. doi: 10.1152/jn.00106.2010

904 Tort, A.B.L., Scheffer-Teixeira, R., Souza, B.C., Draguhn, A., Brankačk, J. (2013). Theta- 905 associated high-frequency oscillations (110-160Hz) in the hippocampus and 906 neocortex. Prog. Neurobiol. 100, 1–14. doi: 10.1016/j.pneurobio.2012.09.002

907 Varela, F., Lachaux, J.-P., Rodriguez, E., Martinerie, J. (2001). The brainweb: phase 908 synchronization and large-scale integration. Nat. Rev. Neurosci. 2, 229–239. doi: 909 10.1038/35067550

910 Vinck, M., van Wingerden, M., Womelsdorf, T., Fries, P., Pennartz, C.M.A. (2010). The 911 pairwise phase consistency: a bias-free measure of rhythmic neuronal 912 synchronization. Neuroimage 51, 112–122. doi: 10.1016/j.neuroimage.2010.01.073

913 Xu, X., Zheng, C., Zhang, T. (2013). Reduction in LFP cross-frequency coupling between 914 theta and gamma rhythms associated with impaired STP and LTP in a rat model of 915 brain ischemia. Front. Comput. Neurosci. 7, 27. doi: 10.3389/fncom.2013.00027

916 Xu, X., Liu, C., Li, Z., Zhang, T. (2015). Effects of hydrogen sulfide on modulation of theta- 917 gamma coupling in hippocampus in vascular dementia rats. Brain Topogr. 28, 879– 918 894. doi: 10.1007/s10548-015-0430-x

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919 Zheng, C., Zhang, T. (2013). Alteration of phase-phase coupling between theta and gamma 920 rhythms in a depression-model of rats. Cogn. Neurodyn. 7, 167–172. 921 doi:10.1007/s11571-012-9225-x

922 Zheng, C., Bieri, K.W., Hsiao, Y.-T., Colgin, L.L. (2016). Spatial sequence coding differs 923 during slow and fast gamma rhythms in the hippocampus. Neuron 89, 398–408. doi: 924 10.1016/j.neuron.2015.12.005

36

925 Figure Legends

926 Figure 1. Measuring cross-frequency phase-phase coupling.

927 (A) Traces show 500 ms of the instantaneous phase time series of two Kuramoto

928 oscillators (see Material and Methods). When uncoupled (top panels), the mean

929 natural frequencies of the “theta” and “gamma” oscillator are 8 Hz (blue) and 43 Hz

930 (red), respectively. When coupled (bottom panels), the oscillators have mean

931 frequencies of 8.5 Hz and 42.5 Hz.

932 (B) Top blue traces show the instantaneous phase of the coupled theta oscillator for

933 the same period as in A but accelerated m times, where m= 3 (left), 5 (middle) and 7

934 (right). Middle red traces reproduce the instantaneous phase of the coupled gamma

935 oscillator (i.e., n=1). Bottom black traces show the instantaneous phase difference

936 between gamma and accelerated theta phases (∆). Notice roughly constant

937 ∆ only when theta is accelerated m=5 times, which indicates 1:5 phase-locking.

938 See Figure 1 – figure supplement 1 for the uncoupled case.

939 (C) ∆ distributions for the coupled case (epoch length = 100 s). Notice uniform

940 distributions for n:m=1:3 and 1:7, and a highly concentrated distribution for n:m=1:5.

941 The black arrow represents the mean resultant vector for each case (see Material

942 and Methods). The length of this vector (Rn:m) measures the level of n:m phase-

943 locking. See Figure 1 – figure supplement 1 for the uncoupled case.

944 (D) Phase-locking levels for a range of n:m ratios for the uncoupled (left) and

945 coupled (right) oscillators (epoch length = 100 s).

946

947 Figure 2. Detection of spurious n:m phase-locking in white-noise signals due

948 to inappropriate surrogate-based statistical testing.

37

949 (A) Example white-noise signal (black) along with its theta- (blue) and gamma- (red)

950 filtered components. The corresponding instantaneous phases are also shown.

951 (B) n:m phase-locking levels for 1- (left) and 10-second (right) epochs, computed for

952 noise filtered at theta (θ; 4 – 12 Hz) and at three gamma bands: slow gamma (γS; 30

953 – 50 Hz), middle gamma (γM; 50 – 90 Hz) and fast gamma (γF; 90 – 150 Hz). Notice

954 Rn:m peaks in each case.

955 (C) Boxplot distributions of θ−γS R1:5 values for different epoch lengths (n=2100

956 simulations per epoch length). The inset shows representative ∆ distributions

957 for 0.3- and 100-s epochs.

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

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

960 (Time Shift; 10 runs of 1-s epochs), along with the corresponding Rn:m values. The

961 bottom panel shows the pooled ∆ distribution; the Rn:m of the pooled distribution

962 is lower than the Rn:m of single runs (compare with values for 1- and 10-s epochs in

963 panel C).

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

965 either the Original or 5 surrogate methods (insets are a zoomed view of Rn:m peaks).

966 Bottom, R1:5 values for white noise filtered at θ and γS. Original Rn:m values are not

967 different from Rn:m values obtained from single surrogate runs of Random

968 Permutation and Time Shift procedures. Less conservative surrogate techniques

969 provide lower Rn:m values and lead to the spurious detection of θ−γS phase-phase

970 coupling in white noise. *p<0.01, n=2100 per distribution, one-way ANOVA with

971 Bonferroni post-hoc test.

38

972

973 Figure 3. True n:m phase-locking leads to significant Rn:m values.

974 (A) The left panels show mean Rn:m curves and distributions of R1:5 values for original

975 and surrogate (Random Permutation/Single Run) data obtained from the simulation

976 of two coupled Kuramoto oscillators (n=300; epoch length = 30 s; *p<0.001, t-test).

977 The right panels show the same, but for uncoupled oscillators. In these simulations,

978 each oscillator has instantaneous peak frequency determined by a Gaussian

979 distribution; the mean natural frequencies of the theta and gamma oscillators were

980 set to 8 Hz and 40 Hz, respectively (coupling does not alter the mean frequencies

981 since they already exhibit a 1:5 ratio; compare with Figure 1).

982 (B) Top panels show results from a simulation of a model network composed of two

983 mutually connected interneurons, O and I cells, which emit spikes at theta and

984 gamma frequency, respectively (Tort et al., 2007; Kopell et al., 2010). Original n:m

985 phase-locking levels are significantly higher than chance (n = 300; epoch length = 30

986 s; *p<0.001, t-test). The bottom panels show the same, but for unconnected

987 interneurons. In this case, n:m phase-locking levels are not greater than chance.

988

989 Figure 4. Waveform asymmetry may lead to artifactual n:m phase-locking.

990 (A) The top traces show a theta sawtooth wave along with its decomposition into a

991 sum of sinusoids at the fundamental (7 Hz) and harmonic (14 Hz, 21 Hz, 28 Hz, 35

992 Hz, etc) frequencies. The bottom panel shows the power spectrum of the sawtooth

993 wave. Notice power peaks at the fundamental and harmonic frequencies.

994 (B) Phase-phase plots (2D histograms of phase counts) for the sawooth wave in A

995 filtered at theta (7 Hz; x-axis phases) and harmonic frequencies (14, 21, 28 and 35

996 Hz; y-axis phases).

39

997 (C) The left traces show 500 ms of a sawtooth wave along with its theta- and

998 gamma-filtered components and corresponding phase time series. The sawtooth

999 wave was set to have variable peak frequency, with mean = 8 Hz; no gamma

1000 oscillation was added to the signal. Notice that the sharp deflections of the sawtooth

1001 wave give rise to artifactual gamma oscillations in the filtered signal (Kramer et al.,

1002 2008), which have consistent phase relationship to the theta cycle. The right panels

1003 show that artifactual n:m phase-coupling levels induced by the sharp deflections are

1004 significantly higher than the chance distribution (n = 300; epoch length = 30 s;

1005 *p<0.001, t-test).

1006

1007 Figure 5. Spurious detection of theta-gamma phase-phase coupling in the

1008 hippocampus.

1009 (A) n:m phase-locking levels for actual hippocampal LFPs. Compare with Figure 2B.

1010 (B) Original and surrogate distributions of Rn:m values for slow (R1:5; left) and middle

1011 gamma (R1:8; right) for different epoch lengths. The original data is significantly

1012 higher than the pooled surrogate distribution, but indistinguishable from the

1013 distribution of surrogate values computed using single runs. Similar results hold for

1014 fast gamma. *p<0.01, n=7 animals, Friedman’s test with Nemenyi post-hoc test.

1015

1016 Figure 6. Phase-phase plots of hippocampal LFPs display diagonal stripes.

1017 Phase-phase plot for theta and slow gamma (average over animals; n=7 rats).

1018 Notice diagonal stripes suggesting phase-phase coupling.

1019

40

1020 Figure 7. Phase–phase coupling between theta and gamma oscillations may be

1021 confounded by theta harmonics.

1022 (A) Top, representative LFP epoch exhibiting prominent theta activity (~7 Hz) during

1023 REM sleep. Bottom, power spectral density. The inset shows power in dB scale.

1024 (B) Phase–phase plots for theta and LFP band-pass filtered at harmonic frequencies

1025 (14, 21, 28 and 35 Hz), computed using 20 minutes of concatenated REM sleep.

1026 Also shown are phase-phase plots for the conventional gamma band (30 – 90 Hz)

1027 and for the average over individual surrogate runs. Notice that the former mirrors the

1028 phase-phase plot of the 4th theta harmonics (35 Hz).

1029

1030 Figure 8. The number of stripes in phase-phase plots is determined by the

1031 frequency of the first theta harmonic within the filtered gamma range.

1032 (A) Representative example in which theta has peak frequency of 7.1 Hz. The

1033 phase-phase plot between theta and slow gamma (30-50 Hz) exhibits 5 stripes,

1034 since the 4th theta harmonic (35.5 Hz) is the first to fall within 30 and 50 Hz. The

1035 rightmost panels show the average phase-phase plot computed over all time-shifted

1036 surrogate runs (n=1000) and the significance of the original plot when compared to

1037 the mean and standard deviation over individual surrogate counts, respectively.

1038 (B) Example in which theta has peak frequency of 8.4 Hz and the phase-phase plot

1039 exhibits 4 stripes, which correspond to the 3rd theta harmonic (33.6 Hz).

1040

1041 Figure 9. Phase-phase plots of white-noise signals display diagonal stripes.

1042 (A) Representative phase-phase plots computed for white-noise signals. Notice the

1043 presence of diagonal stripes for both 100-s (left) and 1200-s (right) epochs. The

41

1044 colormaps underneath show the p-values of the original bin counts when compared

1045 to the mean and standard deviation over bin counts of single time-shifted surrogate

1046 runs. Also shown are significance maps after correcting for multiple comparisons

1047 (Holm-Bonferroni) using either time-shifted (top) or randomly permutated (bottom)

1048 surrogate runs. No bin count was considered statistically significant after the

1049 correction.

1050 (B) The top panels show original Rn:m curves (green) plotted along with Single Run

1051 distributions of Rn:m curves of time-shifted (orange) and randomly permutated (red)

1052 surrogates for different epoch lengths (shades denote the 2.5th – 97.5th percentile

1053 interval; n = 2100 per distribution). The bottom panels show the same in a zoomed

1054 scale.

1055

1056 Figure 10. Weak but statistically significant n:m phase-locking can be detected

1057 when analyzing long LFP epochs (>100 s).

1058 (A) Panels show the same as in Figure 9A but for a representative hippocampal LFP.

1059 Notice that several bin counts leading to the striped pattern in the phase-phase plot

1060 of the 1200-s epoch remain statistically significant after correction for multiple

1061 comparisons (Holm-Bonferroni) when compared to the distribution of individual

1062 counts from randomly permutated, but not time-shifted, surrogates (bottom right

1063 plot).

1064 (B) As in Figure 9B, but for actual LFPs (n = 300 samples per animal; the number of

1065 analyzed animals is stated in each panel). For the very long epochs, notice that the

1066 original Rn:m curve falls within the distribution of time-shifted surrogates but outside

1067 the distribution of randomly permutated ones.

42

1068

1069 Figure Supplement Legends

1070

1071 Figure 1 – figure supplement 1. Uncoupled oscillators display uniform ∆

1072 distribution.

1073 (A,B) Panels show the same as in Figure 1B,C, but for the uncoupled oscillators.

1074 Notice roughly uniform ∆ distributions.

1075

1076 Figure 2 – figure supplement 1. Filtering induces quasi-linear phase shifts in

1077 white-noise signals.

1078 (A) Distribution of the phase difference between two consecutive samples for white

1079 noise band-pass filtered at theta (4-12 Hz, top) and slow gamma (30-50 Hz, bottom).

1080 Epoch length = 100 s; sampling rate = 1000 Hz (dt = 0.001 s). Notice that the top

1081 histogram peaks at ~0.05, which corresponds to 2*3.14*8*0.001 (i.e., 2*π*fc*dt,

1082 where fc is the center frequency), and the bottom histogram peaks at ~0.25 =

1083 2*3.14*40*0.001.

1084 (B) Rn:m curves computed for theta- and slow gamma-filtered white-noise signals.

1085 The black curve was obtained using continuous 1-s long time series sampled at

1086 1000 Hz. The red curve was obtained by also analyzing 1000 data points, but which

1087 were subsampled at 20 Hz (subsampling was performed after filtering). Notice Rn:m

1088 peak at n:m=1:5 only for the former case. See also Figure 5 – figure supplement 7

1089 for similar results in hippocampal LFPs.

1090

43

1091 Figure 2 – figure supplement 2. Filter bandwidth influences n:m phase-locking

1092 levels in white-noise signals.

1093 (A) Mean Rn:m curves computed for 1-s long white-noise signals filtered into different

1094 bands (same color labels as in B; n=2100). Notice that the narrower the filter

1095 bandwidth, the higher the Rn:m peak.

1096 (B) Mean Rn:m peak values for different filter bandwidths and epoch lengths (n =

1097 2100 simulations per filter setting and epoch length).

1098

1099 Figure 2 – figure supplement 3. Uniform p-value distributions upon multiple

1100 testing of Original Rn:m values against Single Run Rn:m surrogates.

1101 The histograms show the distribution of p-values (bin width = 0.02) for 10000 t-tests

1102 of Original Rn:m vs Single Run surrogate values (n = 30 samples per group; epoch

1103 length = 1 second). The red dashed line marks p = 0.05. The p-value distributions do

1104 not statistically differ from the uniform distribution (Kolmogorov-Smirnov test).

1105

1106 Figure 4 – figure supplement 1. Waveform asymmetry may lead to spurious

1107 phase-amplitude coupling.

1108 (A) A theta sawtooth wave along with its theta- (7 Hz) and gamma-filtered (35 Hz)

1109 components. Notice that no gamma oscillations exist in the original sawtooth wave,

1110 but they spuriously appear when filtering sharp deflections (Kramer et al., 2008). The

1111 amplitude of the spurious gamma waxes and wanes within theta cycles.

1112 (B) Mean gamma amplitude as a function of theta phase.

1113

44

1114 Figure 4 – figure supplement 2. Statistical significance of artifactual n:m

1115 phase-locking levels induced by waveform asymmetry depends on epoch

1116 length and peak frequency variability.

1117 Shown are the median R1:5 computed between theta and slow gamma for sawtooth

1118 waves simulated as in Figure 4C, but of different epoch lengths and peak frequency

1119 variability. Dashed area corresponds to the interquartile range (n = 300). Surrogate

1120 data were obtained either by Radom Permutation (top row) or Time Shift (bottom

1121 row) procedures. Notice that the longer the epoch or the peak frequency variability,

1122 the larger the difference between original and surrogate data, and that this difference

1123 is greater for randomly permutated than time-shifted surrogates.

1124

1125 Figure 5 – figure supplement 1. Lack of evidence for cross-frequency phase-

1126 phase coupling between theta and gamma oscillations using alternative

1127 phase-locking metrics.

1128 (A) The left plots show the mean radial distance (R) computed for gamma phases in

1129 different theta phase bins, as described in Sauseng et al. (2009). The lines denote

1130 the mean ± SD over all channels across animals (n = 16 channels per rat x 7 rats);

1131 300 1-s long epochs were analyzed for each channel. Note that original and

1132 surrogate R values overlap. The variations of R values within a theta cycle are

1133 explained by the different number of theta phase bins (right bar plot), which leads to

1134 different number of analyzed samples; the higher the number of analyzed samples,

1135 the lower the R (see also Figure 2C).

1136 (B) The first column shows the mean pairwise phase consistency (PPC) between

1137 gamma and accelerated theta phases as a function of the number of ∆ samples

45

1138 (dashed lines denote SD over individual PPC estimates; n = 112 channels x 1000

1139 PPC estimates per channel). Since PPC requires independent observations (Vinck

1140 et al., 2010), ∆ was randomly sampled to avoid the statistical dependence

1141 among neighboring data points imposed by the filter (Figure 2 – figure supplement 1;

1142 see also Figure 5 – figure supplement 7). The second column shows mean PPC as

1143 function of n:m ratio (individual PPC estimates were computed using 1000 ∆

1144 samples); the boxplot distributions show PPC values at selected n:m ratios, as

1145 labeled. PPC values are very low for all analyzed frequency pairs and not statistically

1146 different from zero.

1147

1148 Figure 5 – figure supplement 2. Spurious detection of theta-gamma phase-

1149 phase coupling when theta phase is estimated by interpolation.

1150 (A) n:m phase-locking levels for actual hippocampal LFPs (same dataset as in

1151 Figure 5). Theta phase was estimated by the interpolation method described in

1152 Belluscio et al. (2012).

1153 (B) Original and surrogate distributions of Rn:m values. The original data is

1154 significantly higher than surrogate values obtained from pooled ∆, but

1155 indistinguishable from single run surrogates. *p<0.01, n=7 animals, Friedman’s test

1156 with Nemenyi post-hoc test.

1157

1158 Figure 5 – figure supplement 3. Spurious detection of theta-gamma phase-

1159 phase coupling (second dataset).

1160 (A) n:m phase-locking levels for actual hippocampal LFPs.

46

1161 (B) Original and surrogate distributions of Rn:m values. Results obtained for 3 rats

1162 recorded in an independent laboratory (see Material and Methods).

1163

1164 Figure 5 – figure supplement 4. Lack of evidence for theta-gamma phase-

1165 phase coupling in all hippocampal layers.

1166 (Left) Example estimation of the anatomical location of a 16-channel silicon probe by

1167 the characteristic depth profile of sharp-wave ripples (inter-electrode distance = 100

1168 μm). (Middle) Original and surrogate (Random Permutation/Single Run) distributions

1169 of Rn:m values computed between theta phase and the phase of three gamma sub-

1170 bands (1-s long epochs). Different rows show results for different layers. (Right)

1171 Distribution of original and surrogate Rn:m values computed for current-source

1172 density (CSD) signals (1-s long epochs) in three hippocampal layers: s. pyramidale

1173 (top), s. radiatum (middle), and s. lacunosum-moleculare (bottom). Notice no

1174 difference between original and surrogate values. Similar results were found in all

1175 animals.

1176

1177 Figure 5 – figure supplement 5. Lack of theta-gamma phase-phase coupling in

1178 independent components of gamma activity.

1179 (Left) Average phase-amplitude comodulograms for three independent components

1180 (IC) that maximize coupling between theta phase and the amplitude of slow gamma

1181 (top row), middle gamma (middle row) and fast gamma (bottom row) oscillations

1182 (n=4 animals). Each IC is a weighted sum of LFPs recorded in different hippocampal

1183 layers (see Schomburg et al., 2014). (Middle) n:m phase-locking levels for theta

1184 phase and the phase of ICs filtered at the gamma band maximally coupled to theta in

1185 the phase-amplitude comodulogram. (Right) Original and surrogate distributions of

47

1186 Rn:m values. Rn:m values were computed for 1-s long epochs (n=4 animals); surrogate

1187 gamma phases were obtained by Random Permutation/Single Run.

1188

1189 Figure 5 – figure supplement 6. Lack of theta-gamma phase-phase coupling

1190 during transient gamma bursts.

1191 (A) Examples of slow-gamma bursts. Top panels show raw LFPs, along with theta-

1192 (thick blue line) and slow gamma-filtered (thin red line) signals. The amplitude

1193 envelope of slow gamma is also shown (thick red line). The bottom rows show

1194 gamma and accelerated theta phases (m=5), along with their instantaneous phase

1195 difference (∆:). For each gamma sub-band, a “gamma burst” was defined to

1196 occur when the gamma amplitude envelope was 2SD above the mean. In these

1197 examples, periods identified as slow-gamma bursts are marked with yellow in the

1198 amplitude envelope and phase difference time series. Notice variable ∆: across

1199 different burst events.

1200 (B) The left panel shows n:m phase-locking levels for theta phase and the phase of

1201 different gamma sub-bands (1-s epochs); for each gamma sub-band, Rn:m values

1202 were computed using only theta and gamma phases during periods of gamma

1203 bursts. The right panels show original and surrogate (Random Permutation/Single

1204 Run) distributions of Rn:m values (n=4 animals).

1205

1206 Figure 5 – figure supplement 7. The bump in the Rn:m curve of hippocampal

1207 LFPs highly depends on analyzing contiguous phase time series data.

1208 Average Rn:m curves computed for theta- and gamma-filtered hippocampal LFPs.

1209 The green curves were obtained using 1-s (top) or 10-s (bottom) continuous epochs

48

1210 of the phase time series, sampled at 1000 Hz (same analysis as in Figure 5A). The

1211 blue curves were obtained by analyzing 1000 data points subsampled from the

1212 phase time series at 20 Hz (i.e., 50 ms sampling period, longer than a gamma cycle).

1213 The red curves were obtained by analyzing 1000 (top) or 10000 (bottom) data points

1214 randomly sampled from the phase time series. These plots show that the prominent

1215 bump in the Rn:m curve of actual LFPs only occurs for continuously sampled data

1216 (1000 Hz sampling rate), and therefore probably reflects the “sinusoidality” imposed

1217 by the filter (see also Figure 2 – figure supplement 1). But notice that a small Rn:m

1218 bump remains for θ−γS (see Figure 10 – figure supplement 2). Due to limitation of

1219 total epoch length, we could not perform the 20 Hz subsampling analysis for 10000

1220 points, but notice that the blue and red curves coincide for 1000 points.

1221

1222 Figure 5 – figure supplement 8. Different filter types give rise to similar results.

1223 (A) Original (green) and surrogate (red) n:m phase-locking levels for actual

1224 hippocampal LFPs (same dataset as in Figure 5) filtered at theta and slow gamma

1225 (1-s epochs). Different rows show results for different types of filters. FIR

1226 corresponds to the same finite impulse response filter employed in all other figures.

1227 For the infinite impulse response filters (Butterworth and Bessel), the digit on the

1228 right denotes the filter order. Wavelet filtering was achieved by convolution with a

1229 complex Morlet wavelet with a center frequency of 7 Hz.

1230 (B) Original and surrogate distributions of R1:5 values. For each filter type, the

1231 original data is significantly higher than surrogate values obtained from pooled

1232 ∆, but indistinguishable from single run surrogates. *p<0.01, n=7 animals,

1233 Friedman’s test with Nemenyi post-hoc test.

49

1234

1235 Figure 7 – figure supplement 1. Histogram counts leading to diagonal stripes

1236 in phase-phase plots are statistically significant when compared to the

1237 distribution of surrogate counts.

1238 Panels show the significance of the phase-phase plots in Figure 7 when compared to

1239 the mean and standard deviation of pooled surrogate counts.

1240

1241 Figure 8 – figure supplement 1. Individual time-shifted surrogate runs exhibit

1242 diagonal stripes in phase-phase plots.

1243 (A) The middle panels show phase-phase plots for theta and slow gamma computed

1244 for different time shifts of the example epoch analyzed in Figure 8A. Notice diagonal

1245 stripes in individual surrogate runs. The left- and the rightmost panels show the

1246 original and average surrogate phase-phase plots, respectively (same panels as in

1247 Figure 8A).

1248 (B) Same as above, but for the example epoch analyzed in Figure 8B.

1249

1250 Figure 10 – figure supplement 1. Random Permutation leads to less visible

1251 diagonal stripes than Time Shift in phase-phase plots of long LFP epochs.

1252 Examples of phase-phase plots computed for single Time Shift (top) and Random

1253 Permutation (bottom) surrogate runs of 100 (left) and 1200 seconds (right) for the

1254 same hippocampal LFP as in Figure 10A. Notice that, for the longer LFP epoch, the

1255 randomly permutated surrogate exhibits less discernible stripes than the time-shifted

1256 one.

1257

50

1258 Figure 10 – figure supplement 2. Diagonal stripes in phase-phase plots depend

1259 on analyzing contiguous phase time series data.

1260 Phase-phase plots of a white noise (left) and an actual LFP (right) computed using

1261 100 seconds of total data, but subsampled at 83.3 Hz (we used a subsampling

1262 period of 12 ms because the total data length of the actual LFP was 1200 seconds).

1263 Notice no diagonal stripes in the phase-phase plot of white noise, while some

1264 striped-like pattern persists for the actual LFP data.

1265

1266

51

AB n:m = 1:3 n:m = 1:5 n:m = 1:7 3∗φθ 5∗φθ 7∗φθ γ +180o +180o +180o θ Uncoupled 0o 0o 0o

-180o -180o -180o

+180o 1∗φγ 1∗φγ 1∗φγ +180o +180o +180o 0o φθ 0o 0o 0o -180o 8 Hz -180o -180o -180o +180o 0o φγ ∆φ = (n*φ - m*φ ) o n:m γ θ -180 43 Hz +180o +180o +180o 1:3 0o 1:5 0o 1:7 0o

o o o ∆φ -180 ∆φ -180 ∆φ -180

γ Coupled θ CD∆φ 1:3 ∆φ 1:5 ∆φ 1:7 Uncoupled Coupled

) 1 1 90 o 0.1 90 o 0.3 90 o 0.1 o o o o o o

120 60 120 60 120 60 n:m

o 150o 0.05 30 o o 30 o 150o 30 o +180 θ γ θ γ 0o φ θ 180o 0o 180o 0o 180o 0o -180o 8.5 Hz o +180 210o 330o 210 o 210 o o Theta Frequency: 8 Hz Theta Frequency: 8.5 Hz 0o Gamma Frequency: 43 Hz Gamma Frequency: 42.5 Hz φγ 240o 300o o 240o o 270o 270o 270o -180o 42.5 Hz phase-locking n:m (R 0 0 R1:3 = 0.001 R1:5 = 0.868 R1:7 = 0.001 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:1 1:5 1:10 1:15 1:20 1:25 1:30 n:m n:m -phase - -phase coupling θ γS A B 1-s epochs 10-s epochs C .65 .18 .60 θ-filtered 0.3 second 100 seconds γ .55 90º 90º θ- S 120º 60º 120º 60º ) γ S 30-50 Hz .50 150º 30º 150º 30º

n:m γ θ- M γM 50-90 Hz .45 180º 0º 180º 0º θ-γ .12 F γF 90-150 Hz .40 210º 330º 210º 330º φθ 240º 300º 240º 300º .35 270º 270º 1:5

R .30 γ-filtered .25 .06 .20 .15

n:m phase-locking n:m (R .10 white noise φγ .05 0 0 1:1 1:5 1:10 1:15 1:20 1:25 1:1 1:5 1:10 1:15 1:20 1:25 0.3 1 52 10 20806040 100 n:m n:m epoch length (s)

DE F θ-phase -γ -phase coupling Original Surrogate Runs S R (per surrogate run): 90º n:m 1-s epochs 10-s epochs 120º 60º .18 φ 150º 30º θ .16 Original

run 1 180º 0º ) .14 Time Shift (Pool. Dist.) x2 n:m R1 = 0.233 Phase Scramble (Pool. Dist.) φγ 210º 330º n:m .173 .12 Phase Scramble (Single Run Dist.) 240º 300º .172 270º .10 Rand. Perm. (Single Run Dist.) 90º .171 Time Shift 120º 60º Time Shift (Single Run Dist.) .08 1:5 150º 30º .055 .06 .054 φθ 180º 0º run 2 2 .04 R = 0.101 phase-locking n:m (R .053 210º 330º n:m 1:5

240º 300º .02 φγ 270º . 0 . 1:1 1:5 1:10 1:15 1:20 1:25 1:1 1:5 1:10 1:15 1:20 1:25 n:m n:m 90º Random Permutation 120º 60º

150º 30º .45 n.s. φθ * run 10 180º 0º .40 Original R10 = 0.174 Time Shift (Pool. Dist.) 210º 330º n:m .35 Phase Scramble (Pool. Dist.) φγ 240º 300º 270º j .30 Average over single R n:m = 0.127 Phase Scramble (Single Run Dist.) .25 Rand. Perm. (Single Run Dist.)

90º 1:5 Phase Scramble 120º 60º R .20 Time Shift (Single Run Dist.) 150º 30º n.s. φ .15 * θ 180º 0º .10 210º 330º pooled .05 φγ 240º 300º distribution 270º R of pool. distrib. = 0.047 n:m 0 A 1 1 1 1 * ) ) Original n:m n:m Rand. Perm. (Single Run Dist.) 1:5 1:5 θ γ Coupled R θ γ Uncoupled R

Theta Frequency: 8 Hz Theta Frequency: 8 Hz n.s. Gamma Frequency: 40 Hz Gamma Frequency: 40 Hz n:m phase-locking (R n:m phase-locking (R

0 0 0 0 1:1 1:5 1:10 1:15 1:20 1:25 1:1 1:5 1:10 1:15 1:20 1:25 Original Random Original Random n:m n:m

B 1 1 * I )

n:m Original I Rand. Perm. (Single Run Dist.)

O 1:6 R

100 ms O LFP n:m phase-locking (R 0 0 1:1 1:5 1:10 1:15 1:20 1:25 Original Random n:m

1 .40

I ) n:m I .35 n.s. X X 1:5

O R

100 ms .30 O n:m phase-locking (R LFP 0 .25 1:1 1:5 1:10 1:15 1:20 1:25 Original Random n:m A B i ii 7 Hz sawtooth wave 14 Hz-filtered 21 Hz-filtered +180º +180º

+90º +90º

= 0º 0º

7 Hz

12-16HzPhase −90º 19-23HzPhase −90º + −180º −180º 14 Hz −180º −90º 0º +90º +180º −180º −90º 0º +90º +180º + Theta Phase Theta Phase 21 Hz iii iv + 28 Hz-filtered 35 Hz-filtered 28 Hz +180º +180º + 35 Hz + +90º +90º max ... 100 ms

0º 0º counts

min 26-30HzPhase −90º 33-37HzPhase −90º

0 10 20 30 40 50 −180º −180º −180º −90º 0º +90º +180º −180º −90º 0º +90º +180º Frequency (Hz) Theta Phase Theta Phase

C 1 .70 * ) Original +180o n:m Rand. Perm. (Single Run Dist.) 0o φθ -180o .351:5 +180o R θ 0o φγ -180o

γ 100 ms phase-locking n:m (R 0 0 1:1 1:5 1:10 1:15 1:20 Original Random n:m A 1-s epochs 10-s epochs 100-s epochs .18

) .16

n:m γ θ- S .14 γ γS 30-50 Hz θ- M .12 γ 50-90 Hz θ-γ M .10 F γF 90-150 Hz .08 .06 .04 .02 n:m phase-locking n:m (R 0 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 n:m n:m n:m

-phase - -phase coupling -phase - -phase coupling B θ γS θ γM

n.s. .40 .40 * Original Original .35 .35 Time Shift (Pool. Dist.) n.s. Time Shift (Pool. Dist.) .30 Rand. Perm. (Single Run Dist.) .30 * Rand. Perm. (Single Run Dist.) .25 .25 1:5 .20 n.s. 1:8 .20 R R .15 * .15 n.s. n.s. .10 .10 * n.s. * .05 .05 * 0 0 1-s epochs 10-s epochs 100-s epochs 1-s epochs 10-s epochs 100-s epochs +180º

+90º max

0º counts

−90º min SlowGammaPhase −180º −180º −90º 0º +90º +180º Theta Phase 14 Hz-filtered 21 Hz-filtered 28 Hz-filtered A B i +180o ii +180o iii +180o 1 2 3 4 5 6 7

160 140 120 +90o +90o +90o 110 140 120 120 100 100 90 1000 ms 0o 100 0o 0o 3 80 x10 counts 80 counts counts 80 70 10 7 Hz 60 60 60 -90o -90o -90o 9 12-16 Hz Phase 40 19-23 Hz Phase 40 26-30 Hz Phase 50

40 8 -180o -180o -180o -180o -90o 0o +90o +180o -180o -90o 0o +90o +180o -180o -90o 0o +90o +180o 7 35 Theta Phase Theta Phase Theta Phase 35 Hz-filtered γ-filtered (30−90 Hz) Surrogate Mean (Time Shift) 6 30 iv +180o v +180o vi +180o

5 Power (db) 25 100 100 100 +90o +90o +90o 4 95 95 95 Power (a.u.) 20 90 90 90 85 85 85 3 0o 0o 0o 5 10 15 20 25 30 80 80 80 Frequency (Hz) counts counts 75 75 75

2 mean counts 70 70 70 -90o -90o -90o 33-37 Hz Phase 30-90 Hz Phase 1 65 30-90 Hz Phase 65 65

0 -180o -180o -180o 0 5 10 15 20 25 30 -180o -90o 0o +90o +180o -180o -90o 0o +90o +180o -180o -90o 0o +90o +180o Frequency (Hz) Theta Phase Theta Phase Theta Phase Original Surrogate Mean (Time Shift) A +180 45 7.1 Hz 40 1 2 3 4 5 35 +90 max 30 p < 0.025 25 counts 20 0 0.025 < p < 0.975 15 p > 0.975

Power(a.u.) min 10 −90 5 (30-50 Hz)Phase (º) S

0 γ 0 2 4 6 8 101214 161820 1 Frequency (Hz) −180 −180 −90 0 +90 +180 −180 −90 0 +90 +180 −180 −90 0 +90 +180 Theta Phase (º) Theta Phase (º) Theta Phase (º) B Original Surrogate Mean (Time Shift) +180 35 8.4 Hz 1 2 3 4 30 +90 25 max p < 0.025 20 counts 0 0.025 < p < 0.975 15 p > 0.975 10 min Power(a.u.) 5 −90 (30-50 Hz)Phase (º) S

0 γ 1 0 2 4 6 8 101214 161820 −180 Frequency (Hz) −180 −90 0 +90 +180 −180 −90 0 +90 +180 −180 −90 0 +90 +180 Theta Phase (º) Theta Phase (º) Theta Phase (º)

A n:m = 1:3 n:m = 1:5 n:m = 1:7 3∗φθ 5∗φθ 7∗φθ +180o +180o +180o

0o 0o 0o

-180o -180o -180o 1∗φγ 1∗φγ 1∗φγ +180o +180o +180o

0o 0o 0o

-180o -180o -180o

∆φ n:m = (n*φγ - m*φθ) +180o +180o +180o 1:3 0o 1:5 0o 1:7 0o

o o o ∆φ -180 ∆φ -180 ∆φ -180 B ∆φ 1:3 ∆φ 1:5 ∆φ 1:7 90 o 0.1 90 o 0.1 90 o 0.1 120o 60 o o 60 o 120o 60 o

150o 0.05 30 o 0.05 30 o 150o 30 o

180o 0o 180 0o 180o 0o

210 330o 210 330o 210o o

240o 300o 240o 300o 240o 300o 270o 270o 270o

R1:3 = 0.003 R1:5 = 0.003 R1:7 = 0.012 A 4-12 Hz filtered B 4000 0.16 0.05 rad

3000 0.14 1000 Hz sampling rate 2000 20 Hz sampling rate

Counts 0.12

1000 0.10 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 φ − φ (rad) n+1 n n m 0.08 R 30-50 Hz filtered 2000 0.06 0.25 rad 1500

0.04 1000 Counts 0.02 500

0 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 1:51:10 1:15 1:20 1:25 1:30 1:35 1:40 (rad) φn+1 − φn n:m A B 1-s epochs .20 .20 Filter Settings .18 .18 9-11 Hz; 55-65 Hz .16 .16 7-13 Hz; 45-75 Hz .14 .14 5-15 Hz; 35-85 Hz

.12 n:m .12 0-20 Hz; 25-95 Hz

n:m .10 .10 R .08 .08 Peak R .06 .06

.04 .04

.02 .02

0 0 1:1 1:5 1:10 1:15 1:20 1 20 40 60 80 100 n:m epoch length (s) - - - θ γS θ γM θ γF 250 250 250

200 200 200

150 150 150

100 100 100 Counts

50 50 50

0 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 P-value P-value P-value A B

7 Hz

35 Hz 10x Amplitude(a.u.) 35-Hz

100 ms 0 90 180 270 360 450 540 630 720 7-Hz Phase (º) 1-s epochs 10-s epochs 100-s epochs 1 1 1 .9 Original .9 Original .9 Original .8 Rand. Perm. .8 Rand. Perm. .8 Rand. Perm. .7 .7 .7 .6 .6 .6 1:5 1:5 .5 1:5 .5 .5 R R R .4 .4 .4 .3 .3 .3 .2 .2 .2 .1 .1 .1 0 0 0 0 0.5 1 1.5 2 2.5 3 Hz 0 0.5 1 1.5 2 2.5 3 Hz 0 0.5 1 1.5 2 2.5 3 Hz Peak Frequency Noise (σ) Peak Frequency Noise (σ) Peak Frequency Noise (σ)

1-s epochs 10-s epochs 100-s epochs 1 1 1 .9 Original .9 Original .9 Original .8 Time Shift .8 Time Shift .8 Time Shift .7 .7 .7 .6 .6 .6

1:5 .5 1:5 .5 1:5 .5 R R R .4 .4 .4 .3 .3 .3 .2 .2 .2 .1 .1 .1 0 0 0 0 0.5 1 1.5 2 2.5 3 Hz 0 0.5 1 1.5 2 2.5 3 Hz 0 0.5 1 1.5 2 2.5 3 Hz Peak Frequency Noise (σ) Peak Frequency Noise (σ) Peak Frequency Noise (σ) A γ γ γ S M F 50 0.38 0.34 0.28 Original 45 0.36 0.32 0.26 Random Permutation 40 35 0.34 0.30 0.24 30 R R 0.32 0.28 R 0.22 25

counts 20 0.30 0.26 0.20 15 0.28 0.24 0.18 10 0.26 0.22 0.16 5 0 0 90 180 270 360 450 540 630 720 0 90 180 270 360 450 540 630 720 0 90 180 270 360 450 540 630 720 0 90 180 270 360 450 540 630 720 Theta Phase (º) Theta Phase (º) Theta Phase (º) Theta Phase (º) B 1000 samples 0.8 0.1 γ γ θ- S θ- S 1:5

0 0

PPC 1000 samples PPC 0.01

0.008 −0.8 −0.1 0 50 100 150 200 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 0.006 0.8 0.1 γ γ 0.004 θ- M θ- M

1:8 0.002

0 0

PPC 0 PPC

−0.002

−0.8 −0.1 −0.004 0 50 100 150 200 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 PairwisePhase Consistency 0.8 0.1 −0.006 -γ -γ θ F θ F −0.008 1:13 −0.01 0 0 PPC

PPC γ γ γ θ- S θ- M θ- F 1:5 1:8 1:13

−0.8 −0.1 0 50 100 150 200 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 Number of Samples n:m A 1-s epochs 10-s epochs .18

) .16

n:m γ θ- S .14 γ 30-50 Hz θ-γ S .12 M γ 50-90 Hz γ M θ- F .10 γF 90-150 Hz .08 .06 .04 .02 n:m phase-locking n:m (R 0 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 n:m n:m

B -phase - -phase coupling -phase - -phase coupling θ γS θ γM n.s. .40 * Original .40 .35 Time Shift (Pool. Dist.) .35 n.s. .30 Rand. Perm. (Single Run Dist.) .30 * .25 .25 1:6 .20 1:10 .20

R n.s. R n.s. .15 * .15 * .10 .10 .05 .05 0 0 1-s epochs 10-s epochs 1-s epochs 10-s epochs A 1-s epochs 10-s epochs .18

) .16 n:m θ-γ .14 S γ 30-50 Hz θ-γ S .12 M γ 50-90 Hz θ-γ M .10 F γF 90-150 Hz .08 .06 .04 .02 n:m phase-locking n:m (R 0 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 n:m n:m

B -phase - -phase coupling -phase - -phase coupling θ γS θ γM

Original .40 .40 Time Shift (Pool. Dist.) .35 .35 Rand. Perm. (Single Run Dist.) .30 .30 .25 .25 1:7 1:4 .20 .20 R R .15 .15 .10 .10 .05 .05 0 0 1-s epochs 10-s epochs 1-s epochs 10-s epochs LFP CSD γ γ γ γ γ γ par. cx. θ- S θ- M θ- F θ- S θ- M θ- F .4 .4 .3 s. pyr.

n:m .2 .3 R .1

n:m .2

0 R Original Random Original Random Original Random .1 .4

.3 0 Original Random Original Random Original Random s. or. n:m .2 R .1 s. pyr. 0 Original Random Original Random Original Random .4 s. rad. .4 .3 .3 n:m n:m .2 .2 R s. rad. R .1 .1 0 Original Random Original Random Original Random 0 s. l.-m. .4 Original Random Original Random Original Random .3

n:m .2 R .1 .4 s. l.-m. 0 Original Random Original Random Original Random .3

.4 n:m .2 .3 R

n:m .2 .1 R

m .1 µ 0

100 0 50 ms Original Random Original Random Original Random Original Random Original Random Original Random 200 .18 Original .45 ) 180 3 .16 Rand. Perm. .40 n:m 160 .14 .35 ) -3 .12 .30 140 (x10 120 .10 .25 1:6

100 .08 R .20 80 .06 .15 ModulationIndex 60 .04 .10 0 AmplitudeFrequency (Hz) .02 .05 40 phase-locking n:m (R 0 0 5 10 15 20 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 Phase Frequency (Hz) n:m Original Rand. Perm.

200 .18 Original .45 ) 180 3 .16 Rand. Perm. .40 n:m .14 .35 160 ) -3

140 (x10 .12 .30 120 .10 .25 1:9

100 .08 R .20 .06 .15

80 ModulationIndex 60 .04 .10 0 AmplitudeFrequency (Hz) .02 .05 40 phase-locking n:m (R 0 0 5 10 15 20 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 Phase Frequency (Hz) n:m Original Rand. Perm.

200 .18 Original .45 ) 180 5 .16 Rand. Perm. .40 n:m .14 .35 160 ) -3

140 (x10 .12 .30 120 .10 .25 1:18

100 .08 R .20 .06 .15

80 ModulationIndex 60 .04 .10 0 AmplitudeFrequency (Hz) .02 .05 40 phase-locking n:m (R 0 0 5 10 15 20 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 Phase Frequency (Hz) n:m Original Rand. Perm. A

LFP

γ S θ

+180o 1∗φγ S 0o 5∗φθ -180o

+180o

1:5 0o ∆φ -180o

100 ms

B γ γ γ θ- S θ- M θ- F .18 γ .45 .45 .45 θ- S .16 ) γ .40 .40 .40 θ- M n:m .14 γ .35 .35 .35 θ- F .12 .30 .30 .30 .10 .25 .25 .25 1:5 1:7 1:14 R .08 R .20 .20 R .20 .06 .15 .15 .15 .04 .10 .10 .10 n:m phase-locking n:m (R .02 .05 .05 .05 0 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 0 0 0 n:m Original Random Original Random Original Random γ γ γ θ- S θ- M θ- F .16 .16 .16 Original - 1000 Hz sampling

) .14 ) .14 ) .14 Original - 50 Hz sampling n:m n:m n:m .12 .12 .12 Original - Random sampling .10 .10 .10

.08 .08 .08

.06 .06 .06

1-s epochs 1-s .04 .04 .04

n:m phase-locking n:m (R .02 phase-locking n:m (R .02 phase-locking n:m (R .02

0 0 0 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 n:m n:m n:m γ γ γ θ- S θ- M θ- F .16 .16 .16 ) ) ) .14 .14 .14 n:m n:m n:m .12 .12 .12

.10 .10 .10

.08 .08 .08

.06 .06 .06

.04 .04 .04 10-sepochs n:m phase-locking n:m (R phase-locking n:m (R n:m phase-locking n:m (R .02 .02 .02

0 0 0 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 1:1 1:5 1:10 1:15 1:20 1:25 1:30 1:35 1:40 n:m n:m n:m -phase - -phase coupling (1-s epochs) A θ γS

.20 FIR Butterworth - 2 Butterworth - 5 Bessel - 2 Bessel - 5 Wavelet .18 )

n:m .16 Original .14 Rand. Perm. (Single Run Dist.) .12 .10 .08 .06 .04 n:m phase-locking n:m (R .02 0 1:1 1:5 1:10 1:15 1:20 1:1 1:5 1:10 1:15 1:20 1:1 1:5 1:10 1:15 1:20 1:1 1:5 1:10 1:15 1:20 1:1 1:5 1:10 1:15 1:20 1:1 1:5 1:10 1:15 1:20 n:m n:m n:m n:m n:m n:m

B n.s. n.s. .45 n.s. n.s. n.s. n.s. * * .40 * * * * Original .35 Time Shift (Pool. Dist.) .30 Rand. Perm. (Single Run Dist.) .25 1:5

R .20 .15 .10 .05 0 FIR Butterworth - 2 Butterworth - 5 Bessel - 2 Bessel - 5 Wavelet 14 Hz - filtered 21 Hz - filtered 28 Hz - filtered +180 +180 +180

+90 +90 +90

0 0 0

−90 −90 −90 12 - 1612HzPhase- (°) 19 - 2319HzPhase- (°) 3026HzPhase- (°) −180 −180 −180 −180 −90 0 +90 +180 −180 −90 0 +90 +180 −180 −90 0 +90 +180 Theta Phase (°) Theta Phase (°) Theta Phase (°)

35 Hz - filtered γ-filtered (30-90 Hz) +180 +180

+90 +90 p < 0.025 0 0 0.025 < p < 0.975 p > 0.975 −90 −90 33 - 3733HzPhase- (°) 9030HzPhase- (°) −180 −180 −180 −90 0 +90 +180 −180 −90 0 +90 +180 Theta Phase (°) Theta Phase (°) A Original Shift = 2 ms Shift = 23 ms Shift = 50 ms Shift = 79 ms Shift = 178 ms Surrogate Mean (Time Shift) +180º

max +90º counts

0º

(30-50 Phase Hz) −90º

S min γ

−180º −180º −90º 0º +90º +180º Theta Phase

B Original Shift = 1 ms Shift = 31 ms Shift = 68 ms Shift = 95 ms Shift = 146 ms Surrogate Mean (Time Shift) +180º

max +90º counts

0º

(30-50 Phase Hz) −90º

S min γ

−180º −180º −90º 0º +90º +180º Theta Phase Time Shift (100 s) Time Shift (1200 s) +180 +180

+90 +90

0 0

−90 −90 30 - 5030HzPhase- (°) 5030HzPhase- (°)

−180 8.0 −180 87 −180 −90 0 +90 +180 −180 −90 0 +90 +180 85 Theta Phase (°) 7.5 Theta Phase (°) 83 7.0 81

Rand. Perm. (100 s) counts Rand. Perm. (1200 s) counts 6.5 79 +180 +180 6.0 77

+90 +90

0 0

−90 −90 30 - 5030HzPhase- (°) 5030HzPhase- (°)

−180 −180 −180 −90 0 +90 +180 −180 −90 0 +90 +180 Theta Phase (°) Theta Phase (°) White noise LFP

Subsampled (100 s) Subsampled (100 s) +180 +180

+90 +90 max

0 0 counts

−90 −90 min 30 - 5030HzPhase- (°) 5030HzPhase- (°)

−180 −180 −180 −90 0 +90 +180 −180 −90 0 +90 +180 Theta Phase (°) Theta Phase (°)