- Email: [email protected]
The relationship between coherence and the phase-locking value
Journal of Theoretical Biology 435 (2017) 106–109
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/jtbi
Letter to Editor The relationship between coherence and the phase-locking value The coupling (or association) of oscillatory activity is thought to indicate a coordination between brain regions (Varela et al., 2001). As a result, many experiments in neuroscience have assessed the relationship between pairs of recorded neural oscillations. Of these, some early (but by no means the earliest) studies include Achermann and Borbély (1998), Zaveri et al. (1999), Bullock et al. (1995), Kristeva et al. (2007). Two statistics commonly used to measure the relationship between neural oscillations are coherence (a term coined by Wiener1 Priestley (1981, p. 661)) and the phase-locking value (Lachaux et al., 1999). These are both frequency dependent quantities that measure the strength of statistical association between two discrete-time continuousvalued recordings. The performance of these two measures of association have been discussed in various contexts—some recent examples include Jarvis and Mitra (2001), Bokil et al. (2007), Celka (2007), Aydore et al. (2013), Srinath and Ray (2014). Recently, the efficacy of coherence as a measure of the synchrony of neural activity between two brain regions has been challenged (Lachaux et al., 1999; Srinath and Ray, 2014). Their argument is that the coherence between two oscillations depends upon the amplitude of the oscillations in addition to the phase difference between the two oscillations. They further show, in simulation, by randomly drawing amplitudes across trials while keeping the difference between the phases of the oscillations consistent, that coherence is decreased while the phase-locking value is not. This result is consistent with the fact that coherence is a measure of the extent to which one time-series can be linearly-predicted from another2 (e.g., Priestley, 1981). Further, this argument is focused upon a form of bias (“amplitude-dependent bias”); however, it is known that both the bias and variance of phase estimates (which are central to the phase-locking value statistic) are both adversely affected when oscillation amplitude is small relative to noise (e.g., Lepage et al., 2013). Furthermore, it is difficult to contrive a physically realistic scenario where neural activity synchronizes the phase of oscillation while randomizing oscillation amplitude. It is thus plausible that the discussion is focused upon edge cases which, while important, serve to complicate the comparison of analyses computed using the two statistics. More generally, coherence bias decreases with an increasing number of trials,3 and also depends upon how rapidly relevant spectral quantities vary with frequency (Brillinger, 2001, p. 308). 1
The original term is coherency. As a function of frequency. 3 The referred to quantity in Brillinger (2001, p.308) is not trials but is rather the time-bandwidth product. This quantity is linked to the number of uncorrelated estimates that are averaged to produce a spectrum estimate. Two times the timebandwidth product is approximately the number of tapers used in the multitaper 2
https://doi.org/10.1016/j.jtbi.2017.08.029 0022-5193/© 2017 Elsevier Ltd. All rights reserved.
Note that when testing for change in coherence between two conditions, under the null hypothesis of no difference, a test statistic that subtracts the coherence estimated from each of the conditions will not be affected by this bias, and given an alternate hypothesis, the effect of this bias on the test statistic will be reduced. As already mentioned, both the phase locking value and coherence have been employed in neuroscience using a wide variety of experimental methodologies and behavioral paradigms. The two measures are often employed to assess whether or not two areas are functionally connected or communicating with one another. For instance, they have been employed in experimental studies to probe whether two brain regions are communicating via rhythmic interactions while processing sensory stimuli, performing a behavioral task, under anesthesia, or sleeping, and to investigate whether such interactions are disrupted in diseased states. For example, these measures were used to examine the presence or absence of a phase relationship in the oscillatory activity of (1) the primary and secondary somatosensory cortices during nerve stimulation (Simões et al., 2003), (2) the lateral occipital complex and the hippocampus during visual object recognition (Sehatpour et al., 2008), and (3) the cingulate and dorsolateral prefrontal cortex during rapid-eye movement sleep (Vijayan et al., 2017). However, in many studies employing these two measures the motivation for employing PLV versus coherence is not given. Experimentalists may be well served by considering the differences in these two measures when choosing between them. For example, the experimentalist may want to examine if a phase relationship emerges between two areas where there are transient rises in the amplitude of an oscillation during sleep; such periods may indicate times during which there is interareal communication serving some functional role of sleep, such as memory consolidation. In this instance, the coherence measure would likely be more appropriate. Furthermore, a deeper understanding of the differences between these two measures will allow experimentalists to gain a deeper understanding of their results when they employ these measures. The purpose of this letter is two-fold. First, we remind the audience that both of the measures of neural synchrony are weighted averages of complex-valued phase factors, a fact often obscured by mathematical details useful to establishing theoretical estimator performance.4 Second, we broaden the discussion re-
method of spectrum estimation when estimating the spectrum from a single trial. When there are J trials, this value is multiplied by J. 4 An estimate is computed from data to obtain a value that is hoped to be representative of a quantity of interest. To assess the accuracy5 of the estimate, the function that was applied to the data to obtain the estimate is applied to a stochastic model of the data to obtain an associated estimator. The estimate is a realization of the estimator, which is a random variable. The concentration of the probability density function of the estimator, when it exists, about the associated theoretical quantity of interest can be used to assess the accuracy of the estimate.
Letter to Editor / Journal of Theoretical Biology 435 (2017) 106–109
107
garding the accuracy5 of the phase-locking value and coherence to further emphasize the role of estimator variance in the accuracy of estimation. 1. PLV and coherence are weighted averages For trial j ∈ {1, . . . , J } and time-index t ∈ {1, . . . , n}, let the pair ( j) ( j) of recorded measurements be respectively xt and yt . Without loss of generality specify the sample period to be equal to one, resulting in a Nyquist frequency equal to 1/2. The discrete-time Fourier transform Xj (Yj ) of x(j) (y(j) ) evaluated at frequency f, is the sum over time-indices,6
Xj( f ) =
n−1 xt( j ) e−i2π f t , t=0
Yj ( f ) =
n−1 yt( j ) e−i2π f t .
(1)
t=0
Ignoring standard tapering and smoothing modifications (Brillinger, 2001; Percival and Walden, 1993; Priestley, 1981; Thomson, 1982), the classical coherence between the x and y trials can be computed as,
J
Cˇxy ( f ) =
X ∗ ( f )Y j ( f ) j=1 j
J j=1
|X j ( f )|
2
J
j =1
|Y j ( f )|
2
12 .
(2)
Here ∗ denotes complex conjugation and ˇ denotes an estimate, to distinguish from an estimator denoted by the ˆ symbol. Eq. (2) is a ratio of the cross-spectrum estimate to a product of estimates of autospectra:
Cˇxy ( f ) =
J 1
J
j=1
X j∗ ( f )Y j ( f )
|X j ( f )|2 j=1
J
Cˇxy ( f ) =
1 J
Sˇxy ( f ) 1
1
2 2 ( f )Sˇyy (f) Sˇxx
1 J
J
|Y j ( f )|2 j =1
.
(4)
j=1
ˇ (2 )
ρ (j,xf) eiφ j, f ,
Yj ( f ) =
ρ (j,yf) eiφ j, f ,
(y )
(5)
(y )
where ρ (j,xf) (ρ j, f ) is equal to the magnitude of Xj (f) (Yj (f)) and (y )
(x )
φ j, f (φ j, f ) is equal to the complex angle of Xj (f) (Yj (f)). With this
(y ) (x ) (x ) (y ) i φ j, f −φ j, f ρ ρ e j=1 j, f j, f Cxy ( f ) = . (y) 2 12 J ( x ) 2 J ρ j, f ρ j , f j=1 j =1
j = 1, . . . , J, the vector sum in (4) is over collinear unit-magnitude vectors such that the resultant vector has a magnitude equal to J. In this case, the PLV is equal to one. When the phase differences are randomly distributed with trials, the vector average in (4) will tend to be near the origin, resulting in a PLV near zero. Both the magnitude of the coherence, |Cˇ|, and the PLV lie in the interval [0, 1], with values near 1 indicating a prominent association, and values near zero indicating the absence of an association. In this letter we use the term accuracy in a colloquial fashion. It is a stand-in for mathematically precise quantities such as the mean-square error. 6 All of the remaining sums in this paper are over trials. 7 The phase estimates φˇ (j,1f) , φˇ (j,2f) , j = 1, . . . , J may be computed with a variety of methods. Possibilities include the method employing the Hilbert transform to compute the analytic signal, or the method using complex-morlet wavelet coefficients. See Le Van Quyen et al. (20 01), Bruns (20 04) and Lepage et al. (2013) for a description and comparison of various methods of phase estimation in neuroscience. Briefly, best estimator performance in the signal plus noise measurement model is achieved by the estimator that employs a model matching the signal component of the data. See Lepage et al. (2013) for details.
(6)
To relate the phase locking value to coherence, it is necessary to relate the phase estimates defining the PLV to the Fourier domain quantities, Xj (f) (Yj (f)). An optimal estimate of the instantaneous phase of a sinusoidal oscillation with frequency f is the complex angle of the discrete-time Fourier transform, see for example, Lepage et al. (2013):
φˇ (j,1f)
ˇ (1 )
When the phase differences φ j, f − φ j, f are consistent across trials,
5
(x )
Xj( f ) =
J
(3)
J 1 i φˇ (j,2f) −φˇ (j,1f) P LV ( f ) = e . J
The weighted-average relationship is established through the use of the magnitude-phase representation of a complex number (i.e. variable substitution). Consider,
substitution, (2) is equal to,
12 ,
Let trial-index j belong to the set {1, . . . , J }, and let f indicate frequency. Given phase estimates φˇ (j,1f) , φˇ (j,2f) ,7 the phase-locking value (PLV), is,
Fig. 1. Two simulated noisy α rhythms of differing amplitudes are recorded on two electrodes. For all illustrated traces, the additive Gaussian noise is identical. The α rhythms are sinusoids that are 180◦ out-of-phase with each other, and the trial-dependent amplitudes differ by a factor of 2. A phase estimate computed from the top recording (low amplitude) is less reliable than a phase estimate computed from the bottom recording (large amplitude). Coherence down-weights less reliable phase-estimates while the PLV assigns the same weight to the phase estimates for all trials.
φˇ (j,2f)
t
t
= ∠{X j ( f )} + 2π f t, =
φ (j,xf) + 2π f t,
= ∠{Y j ( f )} + 2π f t, =
φ (j,yf) + 2π f t,
(7)
Substituting (7) into (4) results in,
1 i(φ (j,yf) −φ (j,xf) ) P LV ( f ) = e . J J
(8)
j=1
Eqs. (6) and (8) equate when the amplitudes of oscillation associated with trial j are equal to 1. Fig. 1 helps the reader gain an intuition about the implications of the relationship between the coherence magnitude and the PLV. To further illustrate the behavior of the magnitude coherence and the PLV, a second simulation is performed. The second simulation is identical to the first except that the number of trials has been increased from two to ten, the random noise has been redrawn to form each recording, the α rhythm phase difference is 135◦ between the two electrodes, and for three of the trials the α rhythm has an amplitude that is 50
108
Letter to Editor / Journal of Theoretical Biology 435 (2017) 106–109
2. The importance of amplitude weighting The difference between the PLV and the coherence magnitude is the weighting of the unit-magnitude 2D vectors
exp i φˇ (j,yf) − φˇ (j,xf)
(9)
in the complex plane by the trial-dependent amplitudes of os(y ) cillation, ρ (j,xf) , and ρ j, f . These weights reduce the influence of phase-differences associated with low-amplitude oscillations upon the coherence magnitude, see Fig. 1. For the PLV these weights are taken to be equal to one, independent of the amplitude of oscillation. From this run of simulated data, the coherence and the PLV are computed for f equal to 10 Hz. In the linear statistical observation model of the noisy mea( j) surement dt at time-index t for trial j of an oscillation with amplitude a, frequency f, and phase offset φ 0 ,
dt( j ) = a cos(2π f t + φ0 ) + ηt , = a cos((φ j, f )t ) + ηt ,
Fig. 2. Noisy α rhythms are recorded on two electrodes over ten trials. On three of the trials the amplitudes of the α rhythm are 50 times larger, simulating the trialto-trial variability present in many neural recordings. The α rhythms are sinusoids with a 10 Hz frequency.
(10)
it is known that the accuracy of the phase estimate increases with increasing oscillation amplitude a and duration of observation, and decreases with the variance of the additive noise η (Eq. (18), Lepage et al. (2013)). Thus, if model (10) produces the data for any one trial, and further if the amplitude a depends in (10) upon trial j, the influence of less-accurate phase estimates associated with the trials with low-amplitudes upon the coherence magnitude is less than that for the PLV. In the model specified in (10) noise contributes to the mea( j) surement dt additively. If, for a different generative measurement (y ) model, the accuracy of the phase difference φˇ − φˇ (x ) were to dej, f
j, f
crease with an increased amplitude of oscillation aj , then the PLV is expected to increase in accuracy relative to the coherence magnitude. In this latter case, one expects the modified coherence,
C (mod ) ( f ) =
J i φˇ (y ) −φˇ (x ) w(j,xf) w(j,yf) e j, f j, f
(11)
j=1
(y )
to be more accurate for some w(j,xf) , w j, f > 0, j = 1, . . . , J.
Fig. 3. The probability densities associated with |Cˇxy | and PLV computed in simulation two. Here, 100, 10-trial, 2-electrode recordings are simulated, and for each simulated set of 10 trials, both the magnitude coherence and the PLV are computed. Empirical probability density functions computed from the 100 magnitude coherence and PLV values are depicted. The magnitude coherence, due to its ability to down-weight low signal-to-noise ratio trials, more clearly indicates the relationship present between electrode 1 and electrode 2.
times larger. This simulates the trial-to-trial variability in recorded response seen in, for example, the superficial layers of the mouse visual cortex to artificial visual stimuli (e.g., Nase et al., 2003), where trial-to-trial variability is common. The recordings for the first run are shown in Fig. 2. From this run of simulated data, the coherence and the PLV are computed for f equal to 10 Hz. One hundred runs are performed, and the computed magnitude coherence and PLV are used to estimate their associated probability density functions. These are shown in Fig. 3. In this example, the magnitude coherence indicates a linear association between the signals at electrode 1 and electrode 2 at 10 Hz, while the PLV does not.
The weights ρ j, f in (6) are implicated in the claim that Cˆxy is biased due to amplitude correlation (see for example Srinath and Ray (2014)). It is important to recognize that the mean-square error of an estimator is equal to a sum of estimator variance with the square of the estimator bias. Thus, coherence pays for decreased variance in the model (10) with increased bias. As acknowledged late in Srinath and Ray (2014), it remains to be determined if this trade-off is advantageous. The coherence estimate discussed, Cˆxy , differs from the multitaper estimate that is typically computed (Bokil et al., 2010; 2007; Jarvis and Mitra, 2001; Thomson, 1982). Specifically, no tapering is used (this is equal to the use of a single, constant-valued, taper). Because of this, Cˆxy is the realization of an estimator that is more variable, can exhibit more out-of-band bias, and exhibits less in-band bias relative to the standard multitaper estimator. Finally, though obvious, the accuracy of the phase estimates, φˇ (j,xf) and φˇ (j,yf) involved in the computation of the PLV depends upon the choice of the computed phase estimate. A comparison of the sampling properties associated with three commonly used phase estimates is provided in Lepage et al. (2013). 3. Conclusions In neuroscience, two popular methods of assessing oscillatory synchrony between two recordings are the phase-locking value
Letter to Editor / Journal of Theoretical Biology 435 (2017) 106–109
and the coherence. Here we note that both of these statistics are a weighted average with each statistic associated with different weightings, providing a conceptual framework for comparing results using the two measures. We also bring attention to and expand the discussion of how the accuracy of these two statistics is affected by estimator variance. This effect on accuracy will be most pronounced for low signal-to-noise ratio situations exhibiting pronounced trial-to-trial variability. We hope this discussion will lead to further dialogue and future work in this area. Conflict of interest The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Acknowledgments KQL and SV acknowledge NSF award (DMS-1042134). References Achermann, P., Borbély, A., 1998. Coherence analysis of the human sleep electroencephalogram. Neuroscience 85 (4), 1195–1208. Aydore, S., Pantazis, D., Leahy, R.M., 2013. A note on the phase locking value and its properties. Neuroimage 74, 231–244. Bokil, H., Andrews, P., Kulkarni, J.E., Mehta, S., Mitra, P.P., 2010. Chronux: a platform for analyzing neural signals. J. Neurosci. Methods 192 (1), 146–151. Bokil, H., Purpura, K., Schoffelen, J.-M., Thomson, D., Mitra, P., 2007. Comparing spectra and coherences for groups of unequal size. J. Neurosci. Methods 159 (2), 337–345. Brillinger, D.R., 2001. Time Series: Data Analysis and Theory. Society for Industrial and Applied Mathematics. Bruns, A., 2004. Fourier-, Hilbert- and wavelet-based signal analysis: are they really different approaches? J. Neurosci. Methods 137 (2), 321–332. Bullock, T.H., Mcclune, M.C., Achimowicz, J.Z., Iragui-Madoz, V.J., Duckrow, R.B., Spencer, S.S., 1995. Temporal fluctuations in coherence of brain waves. Proc. Natl. Acad. Sci. USA 92 (25), 11568–11572. Celka, P., 2007. Statistical analysis of the phase-locking value. IEEE Signal Process. Lett. 14 (9), 577–580. Jarvis, M., Mitra, P., 2001. Sampling properties of the spectrum and coherency of sequences of action potentials. Neural Comput. 13 (4), 717–749. Kristeva, R., Patino, L., Omlor, W., 2007. Beta-range cortical motor spectral power and corticomuscular coherence as a mechanism for effective corticospinal interaction during steady-state motor output. Neuroimage 36 (3), 785–792. doi:10. 1016/j.neuroimage.2007.03.025.
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Lachaux, J.-P., Rodriguez, E., Martinerie, J., Varela, F.J., et al., 1999. Measuring phase synchrony in brain signals. Human Brain Mapp. 8 (4), 194–208. Le Van Quyen, M., Foucher, J., Lachaux, J.-P., Rodriguez, E., Lutz, A., Martinerie, J., Varela, F.J., 2001. Comparison of Hilbert transform and wavelet methods for the analysis of neuronal synchrony. J. Neurosci. Methods 111 (2), 83–98. Lepage, K.Q., Kramer, M.A., Eden, U.T., 2013. Some sampling properties of common phase estimators. Neural Comput. 25 (4), 901–921. Nase, G., Singer, W., Monyer, H., Engel, A.K., 2003. Features of neuronal synchrony in mouse visual cortex. J. Neurophysiol. 90 (2), 1115–1123. Percival, D., Walden, A., 1993. Spectral Analysis For Physical Applications. Cambridge University Press. Priestley, M.B., 1981. Spectral Analysis and Time Series. Elsevier Academic Press. Sehatpour, P., Molholm, S., Schwartz, T.H., Mahoney, J.R., Mehta, A.D., Javitt, D.C., Stanton, P.K., Foxe, J.J., 2008. A human intracranial study of long-range oscillatory coherence across a frontal-occipital-hippocampal brain network during visual object processing. Proc. Natl. Acad. Sci. 105 (11), 4399–4404. Simões, C., Jensen, O., Parkkonen, L., Hari, R., 2003. Phase locking between human primary and secondary somatosensory cortices. Proc. Natl. Acad. Sci. 100 (5), 2691–2694. Srinath, R., Ray, S., 2014. Effect of amplitude correlations on coherence in the local field potential. J. Neurophysiol. 112 (4), 741–751. Thomson, D.J., 1982. Spectrum estimation and harmonic analysis. Proc. IEEE 70, 1055–1096. Varela, F., Lachaux, J.-P., Rodriguez, E., Martinerie, J., 2001. The brainweb: phase synchronization and large-scale integration. Nat. Rev. Neurosci. 2 (4), 229–239. Vijayan, S., Lepage, K.Q., Kopell, N.J., Cash, S.S., 2017. Frontal beta-theta network during rem sleep. Elife 6, e18894. Zaveri, H.P., Williams, W.J., Sackellares, J.C., Beydoun, A., Duckrow, R.B., Spencer, S.S., 1999. Measuring the coherence of intracranial electroencephalograms. Clin. Neurophysiol. 110 (10), 1717–1725.
Kyle Q. Lepage∗ Allen Institute for Brain Science, Seattle, WA, USA Sujith Vijayan Virginia Tech, Blacksburg, VA, USA ∗ Corresponding author. E-mail address: [email protected] (K.Q. Lepage)
Received 24 September 2016 Revised 18 July 2017 Accepted 31 August 2017
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/jtbi
Letter to Editor The relationship between coherence and the phase-locking value The coupling (or association) of oscillatory activity is thought to indicate a coordination between brain regions (Varela et al., 2001). As a result, many experiments in neuroscience have assessed the relationship between pairs of recorded neural oscillations. Of these, some early (but by no means the earliest) studies include Achermann and Borbély (1998), Zaveri et al. (1999), Bullock et al. (1995), Kristeva et al. (2007). Two statistics commonly used to measure the relationship between neural oscillations are coherence (a term coined by Wiener1 Priestley (1981, p. 661)) and the phase-locking value (Lachaux et al., 1999). These are both frequency dependent quantities that measure the strength of statistical association between two discrete-time continuousvalued recordings. The performance of these two measures of association have been discussed in various contexts—some recent examples include Jarvis and Mitra (2001), Bokil et al. (2007), Celka (2007), Aydore et al. (2013), Srinath and Ray (2014). Recently, the efficacy of coherence as a measure of the synchrony of neural activity between two brain regions has been challenged (Lachaux et al., 1999; Srinath and Ray, 2014). Their argument is that the coherence between two oscillations depends upon the amplitude of the oscillations in addition to the phase difference between the two oscillations. They further show, in simulation, by randomly drawing amplitudes across trials while keeping the difference between the phases of the oscillations consistent, that coherence is decreased while the phase-locking value is not. This result is consistent with the fact that coherence is a measure of the extent to which one time-series can be linearly-predicted from another2 (e.g., Priestley, 1981). Further, this argument is focused upon a form of bias (“amplitude-dependent bias”); however, it is known that both the bias and variance of phase estimates (which are central to the phase-locking value statistic) are both adversely affected when oscillation amplitude is small relative to noise (e.g., Lepage et al., 2013). Furthermore, it is difficult to contrive a physically realistic scenario where neural activity synchronizes the phase of oscillation while randomizing oscillation amplitude. It is thus plausible that the discussion is focused upon edge cases which, while important, serve to complicate the comparison of analyses computed using the two statistics. More generally, coherence bias decreases with an increasing number of trials,3 and also depends upon how rapidly relevant spectral quantities vary with frequency (Brillinger, 2001, p. 308). 1
The original term is coherency. As a function of frequency. 3 The referred to quantity in Brillinger (2001, p.308) is not trials but is rather the time-bandwidth product. This quantity is linked to the number of uncorrelated estimates that are averaged to produce a spectrum estimate. Two times the timebandwidth product is approximately the number of tapers used in the multitaper 2
https://doi.org/10.1016/j.jtbi.2017.08.029 0022-5193/© 2017 Elsevier Ltd. All rights reserved.
Note that when testing for change in coherence between two conditions, under the null hypothesis of no difference, a test statistic that subtracts the coherence estimated from each of the conditions will not be affected by this bias, and given an alternate hypothesis, the effect of this bias on the test statistic will be reduced. As already mentioned, both the phase locking value and coherence have been employed in neuroscience using a wide variety of experimental methodologies and behavioral paradigms. The two measures are often employed to assess whether or not two areas are functionally connected or communicating with one another. For instance, they have been employed in experimental studies to probe whether two brain regions are communicating via rhythmic interactions while processing sensory stimuli, performing a behavioral task, under anesthesia, or sleeping, and to investigate whether such interactions are disrupted in diseased states. For example, these measures were used to examine the presence or absence of a phase relationship in the oscillatory activity of (1) the primary and secondary somatosensory cortices during nerve stimulation (Simões et al., 2003), (2) the lateral occipital complex and the hippocampus during visual object recognition (Sehatpour et al., 2008), and (3) the cingulate and dorsolateral prefrontal cortex during rapid-eye movement sleep (Vijayan et al., 2017). However, in many studies employing these two measures the motivation for employing PLV versus coherence is not given. Experimentalists may be well served by considering the differences in these two measures when choosing between them. For example, the experimentalist may want to examine if a phase relationship emerges between two areas where there are transient rises in the amplitude of an oscillation during sleep; such periods may indicate times during which there is interareal communication serving some functional role of sleep, such as memory consolidation. In this instance, the coherence measure would likely be more appropriate. Furthermore, a deeper understanding of the differences between these two measures will allow experimentalists to gain a deeper understanding of their results when they employ these measures. The purpose of this letter is two-fold. First, we remind the audience that both of the measures of neural synchrony are weighted averages of complex-valued phase factors, a fact often obscured by mathematical details useful to establishing theoretical estimator performance.4 Second, we broaden the discussion re-
method of spectrum estimation when estimating the spectrum from a single trial. When there are J trials, this value is multiplied by J. 4 An estimate is computed from data to obtain a value that is hoped to be representative of a quantity of interest. To assess the accuracy5 of the estimate, the function that was applied to the data to obtain the estimate is applied to a stochastic model of the data to obtain an associated estimator. The estimate is a realization of the estimator, which is a random variable. The concentration of the probability density function of the estimator, when it exists, about the associated theoretical quantity of interest can be used to assess the accuracy of the estimate.
Letter to Editor / Journal of Theoretical Biology 435 (2017) 106–109
107
garding the accuracy5 of the phase-locking value and coherence to further emphasize the role of estimator variance in the accuracy of estimation. 1. PLV and coherence are weighted averages For trial j ∈ {1, . . . , J } and time-index t ∈ {1, . . . , n}, let the pair ( j) ( j) of recorded measurements be respectively xt and yt . Without loss of generality specify the sample period to be equal to one, resulting in a Nyquist frequency equal to 1/2. The discrete-time Fourier transform Xj (Yj ) of x(j) (y(j) ) evaluated at frequency f, is the sum over time-indices,6
Xj( f ) =
n−1 xt( j ) e−i2π f t , t=0
Yj ( f ) =
n−1 yt( j ) e−i2π f t .
(1)
t=0
Ignoring standard tapering and smoothing modifications (Brillinger, 2001; Percival and Walden, 1993; Priestley, 1981; Thomson, 1982), the classical coherence between the x and y trials can be computed as,
J
Cˇxy ( f ) =
X ∗ ( f )Y j ( f ) j=1 j
J j=1
|X j ( f )|
2
J
j =1
|Y j ( f )|
2
12 .
(2)
Here ∗ denotes complex conjugation and ˇ denotes an estimate, to distinguish from an estimator denoted by the ˆ symbol. Eq. (2) is a ratio of the cross-spectrum estimate to a product of estimates of autospectra:
Cˇxy ( f ) =
J 1
J
j=1
X j∗ ( f )Y j ( f )
|X j ( f )|2 j=1
J
Cˇxy ( f ) =
1 J
Sˇxy ( f ) 1
1
2 2 ( f )Sˇyy (f) Sˇxx
1 J
J
|Y j ( f )|2 j =1
.
(4)
j=1
ˇ (2 )
ρ (j,xf) eiφ j, f ,
Yj ( f ) =
ρ (j,yf) eiφ j, f ,
(y )
(5)
(y )
where ρ (j,xf) (ρ j, f ) is equal to the magnitude of Xj (f) (Yj (f)) and (y )
(x )
φ j, f (φ j, f ) is equal to the complex angle of Xj (f) (Yj (f)). With this
(y ) (x ) (x ) (y ) i φ j, f −φ j, f ρ ρ e j=1 j, f j, f Cxy ( f ) = . (y) 2 12 J ( x ) 2 J ρ j, f ρ j , f j=1 j =1
j = 1, . . . , J, the vector sum in (4) is over collinear unit-magnitude vectors such that the resultant vector has a magnitude equal to J. In this case, the PLV is equal to one. When the phase differences are randomly distributed with trials, the vector average in (4) will tend to be near the origin, resulting in a PLV near zero. Both the magnitude of the coherence, |Cˇ|, and the PLV lie in the interval [0, 1], with values near 1 indicating a prominent association, and values near zero indicating the absence of an association. In this letter we use the term accuracy in a colloquial fashion. It is a stand-in for mathematically precise quantities such as the mean-square error. 6 All of the remaining sums in this paper are over trials. 7 The phase estimates φˇ (j,1f) , φˇ (j,2f) , j = 1, . . . , J may be computed with a variety of methods. Possibilities include the method employing the Hilbert transform to compute the analytic signal, or the method using complex-morlet wavelet coefficients. See Le Van Quyen et al. (20 01), Bruns (20 04) and Lepage et al. (2013) for a description and comparison of various methods of phase estimation in neuroscience. Briefly, best estimator performance in the signal plus noise measurement model is achieved by the estimator that employs a model matching the signal component of the data. See Lepage et al. (2013) for details.
(6)
To relate the phase locking value to coherence, it is necessary to relate the phase estimates defining the PLV to the Fourier domain quantities, Xj (f) (Yj (f)). An optimal estimate of the instantaneous phase of a sinusoidal oscillation with frequency f is the complex angle of the discrete-time Fourier transform, see for example, Lepage et al. (2013):
φˇ (j,1f)
ˇ (1 )
When the phase differences φ j, f − φ j, f are consistent across trials,
5
(x )
Xj( f ) =
J
(3)
J 1 i φˇ (j,2f) −φˇ (j,1f) P LV ( f ) = e . J
The weighted-average relationship is established through the use of the magnitude-phase representation of a complex number (i.e. variable substitution). Consider,
substitution, (2) is equal to,
12 ,
Let trial-index j belong to the set {1, . . . , J }, and let f indicate frequency. Given phase estimates φˇ (j,1f) , φˇ (j,2f) ,7 the phase-locking value (PLV), is,
Fig. 1. Two simulated noisy α rhythms of differing amplitudes are recorded on two electrodes. For all illustrated traces, the additive Gaussian noise is identical. The α rhythms are sinusoids that are 180◦ out-of-phase with each other, and the trial-dependent amplitudes differ by a factor of 2. A phase estimate computed from the top recording (low amplitude) is less reliable than a phase estimate computed from the bottom recording (large amplitude). Coherence down-weights less reliable phase-estimates while the PLV assigns the same weight to the phase estimates for all trials.
φˇ (j,2f)
t
t
= ∠{X j ( f )} + 2π f t, =
φ (j,xf) + 2π f t,
= ∠{Y j ( f )} + 2π f t, =
φ (j,yf) + 2π f t,
(7)
Substituting (7) into (4) results in,
1 i(φ (j,yf) −φ (j,xf) ) P LV ( f ) = e . J J
(8)
j=1
Eqs. (6) and (8) equate when the amplitudes of oscillation associated with trial j are equal to 1. Fig. 1 helps the reader gain an intuition about the implications of the relationship between the coherence magnitude and the PLV. To further illustrate the behavior of the magnitude coherence and the PLV, a second simulation is performed. The second simulation is identical to the first except that the number of trials has been increased from two to ten, the random noise has been redrawn to form each recording, the α rhythm phase difference is 135◦ between the two electrodes, and for three of the trials the α rhythm has an amplitude that is 50
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Letter to Editor / Journal of Theoretical Biology 435 (2017) 106–109
2. The importance of amplitude weighting The difference between the PLV and the coherence magnitude is the weighting of the unit-magnitude 2D vectors
exp i φˇ (j,yf) − φˇ (j,xf)
(9)
in the complex plane by the trial-dependent amplitudes of os(y ) cillation, ρ (j,xf) , and ρ j, f . These weights reduce the influence of phase-differences associated with low-amplitude oscillations upon the coherence magnitude, see Fig. 1. For the PLV these weights are taken to be equal to one, independent of the amplitude of oscillation. From this run of simulated data, the coherence and the PLV are computed for f equal to 10 Hz. In the linear statistical observation model of the noisy mea( j) surement dt at time-index t for trial j of an oscillation with amplitude a, frequency f, and phase offset φ 0 ,
dt( j ) = a cos(2π f t + φ0 ) + ηt , = a cos((φ j, f )t ) + ηt ,
Fig. 2. Noisy α rhythms are recorded on two electrodes over ten trials. On three of the trials the amplitudes of the α rhythm are 50 times larger, simulating the trialto-trial variability present in many neural recordings. The α rhythms are sinusoids with a 10 Hz frequency.
(10)
it is known that the accuracy of the phase estimate increases with increasing oscillation amplitude a and duration of observation, and decreases with the variance of the additive noise η (Eq. (18), Lepage et al. (2013)). Thus, if model (10) produces the data for any one trial, and further if the amplitude a depends in (10) upon trial j, the influence of less-accurate phase estimates associated with the trials with low-amplitudes upon the coherence magnitude is less than that for the PLV. In the model specified in (10) noise contributes to the mea( j) surement dt additively. If, for a different generative measurement (y ) model, the accuracy of the phase difference φˇ − φˇ (x ) were to dej, f
j, f
crease with an increased amplitude of oscillation aj , then the PLV is expected to increase in accuracy relative to the coherence magnitude. In this latter case, one expects the modified coherence,
C (mod ) ( f ) =
J i φˇ (y ) −φˇ (x ) w(j,xf) w(j,yf) e j, f j, f
(11)
j=1
(y )
to be more accurate for some w(j,xf) , w j, f > 0, j = 1, . . . , J.
Fig. 3. The probability densities associated with |Cˇxy | and PLV computed in simulation two. Here, 100, 10-trial, 2-electrode recordings are simulated, and for each simulated set of 10 trials, both the magnitude coherence and the PLV are computed. Empirical probability density functions computed from the 100 magnitude coherence and PLV values are depicted. The magnitude coherence, due to its ability to down-weight low signal-to-noise ratio trials, more clearly indicates the relationship present between electrode 1 and electrode 2.
times larger. This simulates the trial-to-trial variability in recorded response seen in, for example, the superficial layers of the mouse visual cortex to artificial visual stimuli (e.g., Nase et al., 2003), where trial-to-trial variability is common. The recordings for the first run are shown in Fig. 2. From this run of simulated data, the coherence and the PLV are computed for f equal to 10 Hz. One hundred runs are performed, and the computed magnitude coherence and PLV are used to estimate their associated probability density functions. These are shown in Fig. 3. In this example, the magnitude coherence indicates a linear association between the signals at electrode 1 and electrode 2 at 10 Hz, while the PLV does not.
The weights ρ j, f in (6) are implicated in the claim that Cˆxy is biased due to amplitude correlation (see for example Srinath and Ray (2014)). It is important to recognize that the mean-square error of an estimator is equal to a sum of estimator variance with the square of the estimator bias. Thus, coherence pays for decreased variance in the model (10) with increased bias. As acknowledged late in Srinath and Ray (2014), it remains to be determined if this trade-off is advantageous. The coherence estimate discussed, Cˆxy , differs from the multitaper estimate that is typically computed (Bokil et al., 2010; 2007; Jarvis and Mitra, 2001; Thomson, 1982). Specifically, no tapering is used (this is equal to the use of a single, constant-valued, taper). Because of this, Cˆxy is the realization of an estimator that is more variable, can exhibit more out-of-band bias, and exhibits less in-band bias relative to the standard multitaper estimator. Finally, though obvious, the accuracy of the phase estimates, φˇ (j,xf) and φˇ (j,yf) involved in the computation of the PLV depends upon the choice of the computed phase estimate. A comparison of the sampling properties associated with three commonly used phase estimates is provided in Lepage et al. (2013). 3. Conclusions In neuroscience, two popular methods of assessing oscillatory synchrony between two recordings are the phase-locking value
Letter to Editor / Journal of Theoretical Biology 435 (2017) 106–109
and the coherence. Here we note that both of these statistics are a weighted average with each statistic associated with different weightings, providing a conceptual framework for comparing results using the two measures. We also bring attention to and expand the discussion of how the accuracy of these two statistics is affected by estimator variance. This effect on accuracy will be most pronounced for low signal-to-noise ratio situations exhibiting pronounced trial-to-trial variability. We hope this discussion will lead to further dialogue and future work in this area. Conflict of interest The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Acknowledgments KQL and SV acknowledge NSF award (DMS-1042134). References Achermann, P., Borbély, A., 1998. Coherence analysis of the human sleep electroencephalogram. Neuroscience 85 (4), 1195–1208. Aydore, S., Pantazis, D., Leahy, R.M., 2013. A note on the phase locking value and its properties. Neuroimage 74, 231–244. Bokil, H., Andrews, P., Kulkarni, J.E., Mehta, S., Mitra, P.P., 2010. Chronux: a platform for analyzing neural signals. J. Neurosci. Methods 192 (1), 146–151. Bokil, H., Purpura, K., Schoffelen, J.-M., Thomson, D., Mitra, P., 2007. Comparing spectra and coherences for groups of unequal size. J. Neurosci. Methods 159 (2), 337–345. Brillinger, D.R., 2001. Time Series: Data Analysis and Theory. Society for Industrial and Applied Mathematics. Bruns, A., 2004. Fourier-, Hilbert- and wavelet-based signal analysis: are they really different approaches? J. Neurosci. Methods 137 (2), 321–332. Bullock, T.H., Mcclune, M.C., Achimowicz, J.Z., Iragui-Madoz, V.J., Duckrow, R.B., Spencer, S.S., 1995. Temporal fluctuations in coherence of brain waves. Proc. Natl. Acad. Sci. USA 92 (25), 11568–11572. Celka, P., 2007. Statistical analysis of the phase-locking value. IEEE Signal Process. Lett. 14 (9), 577–580. Jarvis, M., Mitra, P., 2001. Sampling properties of the spectrum and coherency of sequences of action potentials. Neural Comput. 13 (4), 717–749. Kristeva, R., Patino, L., Omlor, W., 2007. Beta-range cortical motor spectral power and corticomuscular coherence as a mechanism for effective corticospinal interaction during steady-state motor output. Neuroimage 36 (3), 785–792. doi:10. 1016/j.neuroimage.2007.03.025.
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Kyle Q. Lepage∗ Allen Institute for Brain Science, Seattle, WA, USA Sujith Vijayan Virginia Tech, Blacksburg, VA, USA ∗ Corresponding author. E-mail address: [email protected] (K.Q. Lepage)
Received 24 September 2016 Revised 18 July 2017 Accepted 31 August 2017