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] 1 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. 37 2 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 3 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. 77 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 4 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: 5 109 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).