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A wavelet-based estimator of the degrees of freedom in denoised fMRI time series for probabilistic testing of functional connectivity and brain graphs a b s t r a c t a r t i c l e i n f o Keywords: fMRI Connectivity Graph theory Wavelet despike Despiking Statistic Probabilistic Inference Degrees of freedom Connectome mapping using techniques such as functional magnetic resonance imaging (fMRI) has become a focus of systems neuroscience. There remain many statistical challenges in analysis of functional connectivity and network architecture from BOLD fMRI multivariate time series. One key statistic for any time series is its (effective) degrees of freedom, df, which will generally be less than the number of time points (or nominal degrees of freedom, N). If we know the df, then probabilistic inference on other fMRI statistics, such as the correlation between two voxel or regional time series, is feasible. However, we currently lack good estimators of df in fMRI time series, especially after the degrees of freedom of the " raw " data have been modified substantially by denoising algorithms for head movement. Here, we used a wavelet-based method both to denoise fMRI data and to estimate the (effective) df of the denoised process. We show that seed voxel correlations corrected for locally variable df could be tested for false positive connectivity with better control over Type I error and greater specificity of anatomical mapping than probabilistic connectivity maps using the nominal degrees of freedom. We also show that wavelet despiked statistics can be used to estimate all pairwise correlations between a set of regional nodes, assign a P value to each edge, and then iteratively add edges to the graph in order of increasing P. These probabilistically thresholded graphs are likely more robust to regional variation in head movement effects than comparable graphs constructed by thresholding correlations. Finally, we show that time-windowed estimates of df can be used for probabilistic connectivity testing or dynamic network analysis so that apparent changes in the functional connectome are appropriately corrected for the effects of transient noise bursts. Wavelet despiking is both an algorithm for fMRI time series denoising and an estimator of the (effective) df of denoised fMRI time series. Accurate estimation of df offers many potential advantages for proba-bilistically thresholding functional connectivity and network statistics tested in the context of spatially variant and non-stationary noise. Code for wavelet despiking, seed correlational testing …


Introduction
Natural speech stimuli give rise to neuronal activity that is phaselocked to the slow modulations of the speech envelope and observable in the neuromagnetic (MEG) and neuroelectric (EEG) response (Abrams et al., 2008;Ahissar et al., 2001;Aiken and Picton, 2008;Deng and Srinivasan, 2010).Phase-locked activity, also referred to as the evoked response (David et al., 2006), appears to be particularly robust within the theta band (3-7 Hz) for these stimuli (Howard and Poeppel, 2010;Luo and Poeppel, 2007;Luo et al., 2010).The neuronal activity underlying the evoked response is not completely understood, but is potentially explained by two competing theories: additive activity and phase-resetting of ongoing oscillations (for review see Sauseng et al., 2007).The additive theory, strictly interpreted, proposes that new phase-locked activity, generated in response to the stimulus, is superimposed on unchanged background activity.
The phase-resetting theory, strictly interpreted, proposes that the (Fell et al., 2004;Q3 Fuentemilla et al., 2006) on the basis of amplitude change analyses.However, critics point out that this approach depends crucially on the assumption that the amplitude of the ongoing oscillation is unaffected by the stimulus (Sauseng et al., 2007).Otherwise, a decrease in the amplitude of the ongoing oscillation could mask an increase in post-stimulus amplitude caused by additive phase-locked activity, while an increase would suggest an additive phase-locked contribution when only phase-resetting is actually involved.Thus, additive phase-locked activity cannot be distinguished from pure phase-resetting solely on the basis of the amplitude change observed in the post-stimulus response.
In this study, we pursue an analytical approach that allows us to distinguish between additive activity and pure phase-resetting even in the presence of changes in background oscillation amplitude.
Specifically, we utilize a fine-grained time-frequency analysis of the response to examine the co-modulation of amplitude change and phase coherence in the post-stimulus theta-band response.The additive activity theory implies that any transient increases in phase coherence observed in the post-stimulus response data will be accompanied by transient increases in amplitude in a manner that can be predicted by a "signal plus background" model.In contrast, the pure phase-resetting theory implies that transient increases in phase coherence will not be reliably associated with transient increases in amplitude.Changes in background oscillation amplitude can be expected to shift the observed amplitude-phase coherence relationship curve relative to that predicted by either theory but have no effect on the relationship characteristics that distinguish the two theories.Thus, amplitude change and phase coherence should be positively correlated in the case of additive activity but uncorrelated in the case of pure phase resetting, regardless of changes in the amplitude of the background oscillation.Using this approach, the change in background amplitude during the post-stimulus response can be estimated from amplitude change values associated with near-zero phase coherence during the period of interest.
This study serves as an investigation of a hypothesis derived from recent research on the discrimination of attended speech stimuli based on phase and power patterns present in the neuromagnetic response arising in the auditory cortex (Howard and Poeppel, 2010;Luo and Poeppel, 2007).Luo and Poeppel (2007) found that spoken, attended sentences can be discriminated on the basis of theta-band phase, but not power, patterns present in single-trial response data.
Howard and Poeppel (2010) confirmed these findings and utilized a signal plus background model to demonstrate that the experimental results for both phase and power are consistent with the presence of additive phase-locked theta activity.The modeling results led to the hypothesis that additive phase-locked theta activity is present in the response to attended speech such that a significant positive correlation will be observed between transient changes in phase coherence and amplitude across a large number of trials.Here, we test this hypothesis by examining the theta-band response in a large trial set for evidence of phase coherence and amplitude co-modulation as predicted by the signal plus background model.A finding of significant co-modulation would imply that the phase-locked theta response to the attended speech stimulus cannot be explained as a pure resetting of the ongoing oscillation but must reflect additive activity, even if no significant pre-to-post stimulus amplitude enhancement is observed.

Subjects
Thirteen subjects (6 male, mean age 23 yr, range 19-32 yr) took part in the experiment after providing informed consent.All were right-handed (Oldfield, 1971) with 20 selected from each quadrant (Fig. 1).

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The recorded responses to the experimental stimuli were noise- and phase pattern dissimilarity effects using the same methods de-213 scribed in Luo and Poeppel (2007) and Howard and Poeppel (2010).

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In order to maximize consistency with the analyses performed in 229 230 where θ knij and A knij are, respectively, the phase and amplitude for 231 frequency bin i and time bin j in trial n and group k with N =18 232 (total number of trials) for this analysis.The cross-trial coherence values were used to compute a dissimilarity function for each frequency bin i defined as: For this analysis, J = 32 (total number of time bins) and K = 3 (total number of groups).A positive index value means that a phase or power pattern is more dissimilar between the withinsentence groups than between the across-sentence groups.Dissimilarity index values were computed for each of the 157 MEG channels and averaged across all subjects to produce grand average index values for each channel.For each subject, mean dissimilarity index values were computed across the 80 auditory cortex activity channels (see Fig. 1).Grand averages across subjects were then computed for the 80-channel mean dissimilarity index values.

Amplitude and phase modulation analyses
The analysis of co-modulation of phase coherence and amplitude was performed utilizing those subjects whose response data exhibited cross-trial theta-band phase coherence, as reflected in positive thetaband phase dissimilarity index values that significantly exceed the random effects level.Because all negative phase dissimilarity index results are attributable solely to random effects, they were used to establish an upper bound on the random effects range.A conservative criterion for statistical significance was determined by defining the upper bound of random effects as the absolute value of the lower confidence limit on the bootstrap grand average [10,000 iterations, biascorrected and accelerated method (Efron, 1987)] of the minimum (negative) 80-channel phase dissimilarity index value across all frequencies for each subject, with α = 0.001.The resulting random effects level criterion of 0.0074 was then applied to the 80-channel theta phase dissimilarity index value for each subject (i.e.,0.0180, 0.0149, 0.0118, 0.01093, 0.0096, 0.0094, 0.0080, 0.0074, 0.0052, 0.0046, 0.0023, 0.0017, 0.0014) to select those subjects (7 in total) exhibiting significant phase dissimilarity reflective of cross-trial phase coherence.For each selected subject, the analyses were performed on the data from the auditory channel exhibiting the greatest theta phase dissimilarity index value for that subject (auditory channel quadrant locations: 5 left posterior, 2 right posterior).Thus, both subject and channel selection were used to focus the analysis on response data that exhibited the most stimulus-differentiated theta phase patterns, consistent with the presence of a robust phaselocked response in the theta band.The frequency analysis range had an upper limit of 50 Hz such that only frequencies likely to reflect phase-locked responses to speech stimuli were examined.
The first part of the analysis investigated differences in the amplitude of the response data between the pre-and post-stimulus periods.For each subject, the 4500 ms of epoch data retained for each trial of every sentence (360 trials in total) was divided into nine 500 ms segments.Amplitude spectra for the segments of each indi-

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The second part of the analysis examined the relationship be- 335 336 where A fktn is the amplitude for frequency bin f at time bin t in trial n, 337 N is total trials (120) and T is the number of time bins reflecting only 338 the prestimulus portion of the response (26).Mean differences in 339 amplitude relative to the mean prestimulus amplitude (percent) for 340 sentence k were then computed for each frequency f and time bin t as: Apre fk : 341 342 343 Cross-trial phase coherence as a function of frequency and time 344 was computed as: 345 346 where θ fktn is the phase for frequency bin f at time bin t for trial n 347 of sentence k.Within-subject means for relative amplitude change 348 and phase coherence were found by averaging across sentences.

Statistical analysis
Bootstrap resampling (10,000 repetitions) based on the biascorrected and accelerated method (Efron, 1987) was used to establish confidence limits on all grand average results as determined by specified α values (α values associated with particular results are noted in the Results section).Positivity (negativity) in a grand average result was considered significant if both the upper and lower confidence limits were positive (negative) for the specified α level.

Power and phase dissimilarity
The grand average phase dissimilarity index for the auditory channel mean (80 channels) is significantly greater than zero for frequencies below 15 Hz (bootstrap resampling, α = 0.02, p b 0.01) and is particularly prominent for delta-theta band frequencies (1-7 Hz), peaking at 4 Hz (Fig. 2a).The power dissimilarity index is not significantly different from zero for any frequency.The topography of the grand average phase dissimilarity for the theta band (Fig. 2b) demonstrates that theta phase dissimilarity is most evident in the auditory channel set and is consistent with bilateral sources located in auditory cortex (see Fig. (1-500 ms) but not during the ongoing (501-3500 ms) response (Fig. 3c).A significant decrease in amplitude is observed for frequencies in the 13-19 Hz range for the onset response, peaking at 16 Hz.The ongoing response displays a similar amplitude decrease, with significant decreases extending from 9 to 25 Hz and also appearing at some frequencies between 30 and 40 Hz.Amplitude changes for the total response are quite similar to those for the ongoing period, which constitutes most of the total.Results for sub-periods of the ongoing response (Fig. 3d) suggest that amplitude change for frequencies below ~30 Hz consistently moves in a negative direction over the course of the response.
The grand average time-frequency distribution for phase coherence (Fig. 4a) indicates that post-stimulus phase coherence is mainly confined to frequencies below 15 Hz peaking in the theta band at ~4 Hz.The grand average time-frequency distribution for the change in amplitude relative to mean prestimulus amplitude (Fig. 4b) results are consistent with the results from the more coarse-grained time analysis discussed previously (Fig. 3).The time-frequency distributions for phase coherence and amplitude suggest that at low frequencies, particularly in the theta band, phase coherence and amplitude are co-modulated, with points of co-occurring increases appearing not only at onset but also throughout the post-stimulus period.
Co-modulation of phase coherence and amplitude in the theta band was investigated by examining amplitude change as a function of phase coherence using the subject time-frequency distribution data obtained for each sentence (Fig. 5).The data covering the full 4500 ms epoch (Fig. 5a) exhibits a significant positive correlation The relationship between phase coherence and amplitude changes remains similar to that predicted by the model for all time periods but with a reduction in amplitude relative to phase coherence after the onset period that appears to increase slightly with time (Fig. 6a).
Mean amplitude change and phase coherence for data points with phase coherence values close to zero (b0.01) show mean amplitude change ≈0.0 during the prestimulus period, as expected (Fig. 6b).
During the onset period, amplitude change is significantly greater than zero (p b 0.025) with a mean of 1.2%.For later periods, amplitude change is significantly less than zero (p b 0.025) with means of −1.1%, −2.6%, and − 2.4% for periods 2, 3, and 4, respectively.

Summary
The results support the hypothesis that theta-band phase coher- response to single tones (Mäkinen et al., 2005).The amplitude increase appears to be mainly attributable to phase-locked activity, with non-phase-locked activity contributing approximately 1.2% to the total 7.7% amplitude increase observed for the 4 and 6 Hz bins (bin average).
During the onset response, we also observed a significant decrease in lower beta-band amplitude that became more pronounced for the ongoing response, extending into the upper alpha and beta bands.
This may be related to the motor response that was required following each stimulus presentation.Event-related amplitude decreases (event-related desynchronization) at these frequencies have been observed previously in association with voluntary movement starting about 2 s prior to the movement (Leocani et al., 1997;Pfurtscheller and Lopes da Silva, 1999;Stancák and Pfurtscheller, 1996), which could account for the amplitude decreases observed during the ongoing response.Furthermore, this beta suppression is succeeded by amplitude increases (event-related synchronization) that peak approximately 1 s after movement completion (Pfurtscheller and Lopes da Silva, 1999).Such increases would be reflected in the prestimulus amplitudes used as the baseline in our analysis, which could explain the amplitude decreases relative to baseline observed during the onset response.However, amplitude suppression in this same frequency range has been observed in electrocorticographical (ECoG) responses to spoken syllables recorded in the mid-superior temporal gyrus (STG) during a passive listening task (Edwards et al., 2009), suggesting that the suppression may be attributable, at least partially if not totally, to the auditory stimulus.In addition, the overall shape of  13 Hz that were consistently significant across stimulus levels and 572 subjects for the 1-7 Hz range.Furthermore, they observed significant 573 amplitude increases at ~1-7 Hz that were consistently significant across stimulus levels and subjects for the 3-5 Hz range.Jansen et al. (2003) utilized pairs of identical tone bursts with an intra-pair interval of ~500 ms and an inter-pair interval ≥ 8 s, analyzing 25 responses from each of 20 subjects.They found significant phase coherence but no significant amplitude increase at ~2-8 Hz for both tones.In a similar study, Fuentemilla et al. (2006) utilized stimulus trains consisting of three identical tone bursts with an intra-train interval of 584 ms and an inter-train interval of 30 s, analyzing 54 to 88 (mean 71) responses in each of 16 subjects.They found a significant increase in phase coherence for all tones at ~3-16 Hz (analysis did not extend to frequencies below 3 Hz), accompanied by a significant increase in amplitude for the first tone only.The phase coherence and amplitude increases were most prominent at ~3-7 Hz.Despite differences in stimuli and methods, these studies were consistent in finding significant phase coherence in the response to the first (or single) tone in the stimulus for theta band frequencies (3-7 Hz).Furthermore, two of the studies found significant amplitude increases in the theta band response to the first tone, consistent with amplitude increases observed in the MEG response to a single tone (Mäkinen et al., 2005).The failure to find a significant increase in theta amplitude in the response to the first tone by Jansen et al. (2003) is likely attributable to the small number of responses per subject analyzed in that study, given that increases in phase coherence resulting from the addition of a small amount (relative to background) of phaselocked activity are detectible over just a few responses while increases in amplitude are not (Howard and Poeppel, 2010).Similarly, the extended range over which significant increases in phase coherence and amplitude were observed in the response to the first tone in Fuentemilla et al. (2006) investigation may be attributable to the larger number of responses per subject analyzed in that study, which would provide more sensitivity to the weaker effects found at higher frequencies.
Because our study used continuous speech rather than discrete, repeated tones as stimuli, there is not an exact correspondence between our results and those just described.However, the response to a first/single tone can be considered an onset response to an auditory stimulus and, therefore, comparable to the onset response for speech stimuli.In addition, the envelopes of both the tone trains and the speech signals are dominated by slow modulations such that the responses to the second and third tones may be comparable to the ongoing response for speech stimuli.We observed that, like the response to the first/single tones, the onset response to speech exhibited significant amplitude increase for theta frequencies.Further, like the response to second and third tones, the ongoing response exhibited no significant amplitude increase.
Collectively, these findings suggest that additive power in the theta band contributes to the onset, but not to the ongoing, portion of the response to auditory stimuli containing low frequency modulations of the envelope.Thus, they support the view that enhanced phase coherence during the ongoing period is attributable to pure phase-resetting.However, the co-modulation results contradict this conclusion, indicating that even for the ongoing response, amplitude and phase coherence co-vary in a manner consistent with the presence of additive activity in the theta band.Specifically, both mean amplitude and phase coherence decline during the ongoing response, while crucially, transient phase coherence increases continue to be reliably accompanied by the transient amplitude increases predicted by the additive activity model.
Although the theta-band phase coherence observed in the MEG response to speech cannot be explained as pure phase-resetting of the ongoing theta oscillation present in the prestimulus activity, this does not imply that phase-resetting plays little or no role in the neuronal response.On the contrary, additive activity may reflect not only additional or stronger neuronal activity but also the phase alignment of multiple theta oscillations arising in somewhat different locales that contribute to the composite ongoing theta oscillation.Such   (Sauseng et al., 2007;Telenczuk et al., 2010).Differentiation of the 644 possible sources of additive activity requires information on neuronal 645 activity at a granularity that is not, in principle, accessible in MEG/EEG 646 data, which reflect the composite activity of large neuronal popula-647 tions (Telenczuk et al., 2010).However, more fine-grained views of neuronal activity in response to auditory stimuli obtained via intracortical electrodes situated in the primary auditory cortex of awake macaques also found robust theta-band power enhancement in local field potential response (Chandrasekaran et al., 2010) and provided evidence that inhibitory responses to tones involve mainly phase resetting in the delta, theta, and gamma bands, while excitatory responses involve both additive and phase reset activity (O'Connell et al., 2011).With respect to stimulus type, our hypothesis is consistent with our earlier finding of robust theta band phase discrimination for time-reversed speech stimuli (Howard and Poeppel, 2010).This result demonstrated that stimuli that are similar to speech in their spectral-temporal characteristics, but that do not involve speech comprehension, produce the same the same sort of stimulus-driven theta band phase patterns as speech stimuli.Our hypothesis also finds support in a study that examined CSD (current source density) and LFP (local field potential) responses in the rat auditory cortex (A1) to non-naturalistic stimuli, namely, rock music and 1/f distributed random dynamic tone complexes (Szymanski et al., 2011).
This study found that for such stimuli, stimulus-dependent phase information is maximal in low frequency (b16 Hz) LFP responses, just as was observed for naturalistic stimuli that include vocalizations (Chandrasekaran et al., 2010;Kayser et al., 2009).Furthermore, Szymanski et al. discovered that the phase-locked LFP responses originate in transient CSD "events", time-locked to the stimulus, that occur at a frequency of ~2-4 Hz, which is near the peak of phase discrimination for speech.Such convergence of phase-related response characteristics across stimulus types suggests that the response to nonspeech stimuli will exhibit the same phase coherence and mean amplitude co-modulation as was observed for speech; however, further investigation is required to determine the validity of this prediction.
With respect to attention, our hypothesis is consistent with findings on the effects of attention on the EEG response to speech (Kerlin et al., 2010).Utilizing methods conceptually similar to, but analytically distinct from, those of Luo and Poeppel (2007), this study confirmed that attended spoken sentences can be robustly distinguished based on phase-locked responses at lower frequencies, with discrimination performance peaking in the theta band (4-8 Hz).
Kerlin et al. further discovered that selective attention to a particular stimulus of a simultaneously-presented stimulus pair produces gains in discrimination performance compared with that observed for the single-stimulus, attended condition.The gain was statistically significant for the theta band response.The results demonstrate that selective attention increases discrimination performance that depends on phase coherence but does not fundamentally alter the frequencyrelated discrimination profile.It seems reasonable to conjecture that task-related attention to a single stimulus, which is known to enhance the auditory evoked response over that observed for the passive listening condition (Keating and Ruhm, 1971), produces gain effects of the same nature.Thus, we would expect unattended stimuli to produce weaker phase coherence than attended stimuli, but not phase coherence increases in the absence of mean amplitude increases, in conflict with our current findings.Verification of this contention requires a replication of our phase coherence-mean amplitude analysis with response data obtained using unattended stimuli.

Conclusion
Based on our analysis of phase coherence-amplitude co-modulation, the theta-band post-stimulus activity associated with attended spoken sentences, as observed at an MEG channel over auditory cortex, seems best explained as the addition of a phase-locked response to non-phased-locked background activity.The exact relationship of the phase-locked response to the auditory signal is not known but, conceivably, can be characterized as a series of theta-dominant responses produced in correspondence to the low frequency (≤ 8 Hz) modulations of the acoustic envelope (Aiken and Picton, 2008;Howard and Poeppel, 2010).If this is the case, then the phaselocked theta-band response could represent a segmenting of the near-continuous acoustic signal into individual elements, approximately 150-300 ms in length, which provide the basis for speech, melody, rhythm, and prosodic analysis.However, the amplitude

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reduced using the time-shift principle component analysis algorithm 205 (TSPCA) (de Cheveigné and Simon, 2007).Only the first 3600 ms sec-206 tion of each post-trigger response was retained so that the responses 207 to the two versions of a sentence could be combined in subsequent 208 analyses.A pre-trigger response of 900 ms was also retained for 209 each trial, resulting in a total epoch length of 4500 ms.210 Power and phase dissimilarity analysis 211 Post-trigger responses for each sentence were analyzed for power 212 215 these earlier studies, which were based on 21 responses presented 216 in a single block, the first 18 responses (multiple of three required 217 for analysis as explained below) of the first block (20 responses per 218 block) were used in the analysis.Specifically, spectrograms of the 219 3600 ms response, based on a 500 ms time window and 100 ms 220 steps, were computed for all 157 channels.For each channel, three 221 within-sentence groups were formed, each comprising the spectro-222 gram response data for a particular sentence (18 trials).Three across-223 sentence groups were also constructed, each comprising the spectro-224 gram response data for six randomly selected trials for each of the 225 three sentences (18 trials in total).Cross-trial phase and power coher-226 ence were computed as: vidual trial were obtained via a 500 point fast-Fourier transform and then averaged over all trials to obtain a mean amplitude spectrum for each segment.The mean amplitude spectrum for the prestimulus response (1-1000 ms) was computed by averaging the mean spectra obtained for the first two segments.Similarly, the mean amplitude spectrum for the post-stimulus response (1001-4500 ms) was computed by averaging the mean spectra for segments 3 through 9.The post-stimulus response was also broken down into the onset response (1001-1500 ms) with an amplitude spectrum corresponding amplitude spectrum of segment 3, and the ongoing 293 response (1501-4500 ms) with an amplitude spectrum computed by 294 averaging the mean spectra for segments 4 through 9.In addition, the 295 ongoing response was broken down into successive one second time 296 periods by averaging the mean spectra for segments 4 and 5, 6 and 7, 297 and 8 and 9, respectively.Percent changes in the mean amplitude of 298 the post-stimulus response relative to the prestimulus response were 299 determined for all post-stimulus segments described above.Grand 300 averages for each of the segments were also obtained by averaging 301 across the subject results.In addition, the mean prestimulus amplitude 302 spectrum for each subject was normalized by dividing by the sum of 303 amplitude values in the frequency range 2 ≤ f ≤50, and then averaged 304 across subjects to obtain a grand average.305 For each subject, the evoked response for each sentence was com-306 puted by averaging the 4500 ms epoch data across the 120 trials.307 Amplitude spectra for the prestimulus, onset and ongoing periods of 308 the evoked response were obtained by applying a fast Fourier trans-309 form (500 points) to successive 500 ms segments and then averaging 310 the results over sentences and the segment groups associated with 311 the response periods as described above.Evoked response amplitude 312 as a percent of the mean prestimulus amplitude was then computed 313 and the results averaged across subjects to determine the grand aver-314 age.Although the evoked response for the prestimulus period should 315 be zero at all frequencies when computed over an infinite number of 316 trials, it may be greater than zero when computed over 120 trials.To 317 find the expected value of the evoked response for the prestimulus 318 period when computed over 120 trials for comparison with the actual 319 results, prestimulus activity was simulated as 1000 ms of white noise, 320 generated in sets of 120 trials representing data for seven subjects 321 presented with three stimuli.The simulated prestimulus data was 322 then subjected to the same evoked response analysis as the actual 323 prestimulus data to produce the grand average evoked response am-324 plitude as a percent of mean prestimulus amplitude.This process was 325 repeated 1000 times and the results averaged across both repetitions 326 and frequency bins from 2 to 50 Hz to determine the expected value.

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tween phase and amplitude over the course of the response using a 329 spectrogram analysis like that utilized in the power and phase dissim-330 ilarity analysis, but with a finer-grained time scale.For each subject, a 331 spectrogram of the 4500 ms epoch obtained for each trial was com-332 puted based on a 500 ms Hamming window with 20 ms steps.The 333 mean prestimulus amplitude across trials for sentence k frequency 334 bin f was computed as: 349  Grand average time-frequency distributions of amplitude change and phase coherence were then determined by averaging across the subject means.The relationship between phase coherence and amplitude change relative to mean prestimulus amplitude in the theta band was examined using the pool of 8442 phase coherence-amplitude change data points obtained for all subjects (7), sentences (3), and time bins(201)      for the 4 Hz and 6 Hz frequency bins as described above.First, the theoretical relationship between phase coherence and amplitude change was derived for a "signal plus background" model in which theta-band responses were simulated as the sum of a 500 ms phaselocked signal and a 1000 ms background oscillation of random phase (both represented as 4 Hz sine waves) for sets of 120 trials.Simulated background data, corresponding directly to the empirical prestimulus data for the 4 Hz frequency bin (7 subjects × 3 stimuli × 26 time points × 120 trials), were generated by adjusting the amplitude of the background oscillation relative to its mean value to match the empirical data.Signal amplitudes (seven linearly increasing values plus zero) were chosen to produce phase coherence in the range observed experimentally, i.e., 0 to 0.6.A total of 546 sets (7subjects × 3 stimuli × 26 time points) of 120 trials were generated for each signal amplitude value.Then phase coherence and amplitude change were computed across each trial set using the same methodology previously applied to the empirical data.Phase coherence and amplitude change values were averaged across each 546 data point cluster produced for a given signal level to produce the theoretical model data points.Next, comparable estimates of the empirical amplitude change-phase coherence relationship were derived from the pooled data by binning the empirical data points falling within specific phase coherence and amplitude change boundaries and computing the mean phase-coherence and amplitude change for each bin containing at least 10 points.Bin boundaries were derived as lines through the theoretical model data points with slopes equal to the inverse slope of the linear regression fit to each simulated data cluster resulting from a non-zero additive signal (a linear fit was also applied to this set of slopes to smooth slope progression as a function of additive signal level).Estimates of the amplitude change-phase coherence relationship were computed for the entire epoch period and also for successive time periods 0 (prestimulus), 1 (onset), and 2 through 4 associated with time bin subsets 1-26, 27-76, 77-126,     127-176, and 177-201.
1 for comparison).These results are consistent with earlier findings(Howard and Poeppel, 2010;Luo and Poeppel, et al., 2010) showing that responses to attended spoken sen-412 tences in MEG channels reflecting activity in the auditory cortex can be 413 discriminated based on differences in low frequency (b15 Hz) phase 414 but not power patterns present in single-trial data.As observed 415 previously, phase discrimination is strongest in the theta band on 416 average, peaking at 4 Hz, implying that theta band activity is robustly 417 phase-locked to stimulus features that differ in their timing across phase modulation analyses that follow were 421 conducted on the response data exhibiting the strongest theta-band 422 phase-locked activity (top channel for the top seven subjects), identi-423 fied on the basis of the theta phase dissimilarity results as described 424 in Material and methods.The grand average of the normalized am-425 plitude spectra for the prestimulus activity (Fig. 3a) shows the 426 decline with increasing frequency characteristic of cerebral oscillato-427 ry activity as well as a prominent peak at ~10 Hz associated with an 428 eyes-closed state of alert readiness (Dockree et al., 2007; Hari and 429 Salmelin, 1997).As expected, the grand average spectrum for the 430 evoked response as a percent of mean prestimulus amplitude shows a 431 peak in the theta band consistent with robust phase-locked activity in 432 the 3-7 Hz frequency range (Fig. 3b).Evoked activity significantly 433 greater than that expected by chance (bootstrap resampling, α =0.02, 434 p b 0.01) can be observed consistently at frequencies below 15 Hz for 435both the onset and ongoing periods.Significant evoked activity is largely absent in the beta band(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30) but is observed at low gamma frequencies for both the onset (b40 Hz) and ongoing (b50 Hz) responses.The grand average results for the change in amplitude in the poststimulus response relative to mean prestimulus amplitudes show a significant increase (bootstrap resampling, α = 0.02, p b 0.01) for frequencies in the 3-9 Hz range with a 4 Hz peak during onset

(
R = 0.533, p b 0.001) between amplitude change and phase coherence.Furthermore, the empirical function derived from the binned data is quite similar to the modeled function derived from simulated data constructed from the addition of phase-locked responses to random-phase background activity.The data subset restricted to the prestimulus period (Fig.5b) exhibits minimal phase coherence and no significant correlation between amplitude change and phase coherence (R = 0.048, p b 0.113) as expected in the absence of a stimulus.Data subsets covering successive post-stimulus periods (Figs.5b-f) reveal that the amplitude-phase coherence correlation decreases over time but remains highly significant even for the last post-stimulus period (period 1: R =0.664, p b 0.001, period 2: R =0.523, p b 0.001, period 3: R = 0.471, p b 0.001, period 4: R = 0.315, p b 0.001).The decreasing correlation appears to be associated with decreasing phase coherence and amplitude change over time (see also Fig.6a).
Fig. 2. a) Grand averages for phase and power dissimilarity index spectra based on 80 auditory channels.Circle markers indicate results that are significantly greater (less) than zero (p b 0.01).b) Topography of grand-average phase dissimilarity in the theta band.

U
Fig. 3. a) Grand average of normalized mean prestimulus amplitude (1-1000 ms) for selected subjects and channels.Dashed lines represent 95% confidence levels.b) Grand average of evoked response amplitude as a percent of mean prestimulus amplitude.Solid line without markers shows expected value of prestimulus evoked response levels based on Monte Carlo simulation.c) Grand average of percent change in amplitude relative to mean prestimulus amplitude.Solid markers indicate values that are significantly different from zero for p b 0.01.Total post-stimulus response includes both onset (1001-1500 ms) and ongoing response (1501-3500 ms).d) Same as c but with post-stimulus results broken down into successive periods.
amplitude as a function of frequency curve obtained in 550 our experiment closely matches the post-stimulus LFP (local field 551 potential) power enhancement curves obtained in primary auditory 552 cortex of macaques listening to natural auditory stimuli with robust 553 enhancement at theta frequencies and little or no enhancement at 554 lower beta frequencies (Chandrasekaran et al., 2010).This pattern 555 suggests that the MEG data may reflect the summation of power 556 enhancement attributable to phase-locked activity in auditory cortex 557 (concentrated mainly in the theta band) and broadband power 558 suppression in background activity induced by the auditory stimulus.559 If so, then the apparent lower-beta power suppression would actually 560 reflect the absence of stimulus-evoked power enhancement in lower-561 beta band relative to adjacent frequency bands rather than suppres-562 sion concentrated in this band.563 Additive activity versus phase-resetting 564 Several EEG studies have sought evidence of additive phase-565 locked activity in response to auditory stimuli by looking for coin-566 cidental, statistically-significant increases in phase coherence and 567 amplitude relative to prestimulus levels.Jervis et al. (1983) analyzed 568 the responses to a single tone presented 64 times each at two in-569 tensity levels to three subjects (inter-tone interval unknown but 570 ≥924 ms).They found significant phase coherence increases at ~1-571

Fig. 4 .
Fig. 4. Grand average time-frequency distributions across all sentence stimuli and subjects.a) Phase coherence across 120 trials for each stimulus.b) Percent change in amplitude relative to mean prestimulus level averaged over 120 trials for each stimulus.
would increase the amplitude of the theta oscillation 641 in the post-stimulus response (on average) by reducing cancellation 642 effects associated with phase differences among the reset oscillations 643

Fig. 5 .
Fig. 5. Change in amplitude versus phase coherence for individual subject/stimulus time-frequency distribution data points for the theta band (gray circles).Dashed line connects bin means based on bin boundaries represented as polygons surrounding the solid center line.The solid center line represents the theoretical amplitude change-phase coherence relationship based on the pure additive model.a) Data for entire 4500 ms epoch.b-f) Data for successive time periods across epoch.
Fig. 6. a) Composite plot of the amplitude change-phase coherence relationship across successive time periods based on bin means.b) For successive time periods, mean amplitude change for data points for which phase coherence b0.01.