The Origin of Binaural Interaction in the Modulation Domain

The purpose of these experiments was to assess whether the detection of diotic 5 Hz " probe " modulation of a 4000 Hz sinusoidal carrier was influenced by binaural interaction of " masker " modulators presented separately to each ear and applied to the same carrier. A 50 Hz masker modulator was applied to one ear and the masker modulator applied to the other ear had a frequency of 55 or 27.5 Hz. The starting phase of the masker modulators was fixed, and the starting phase of the probe modulator was varied. For both pairs of masker modulators, the threshold for detecting the probe modulation varied slightly but significantly with probe starting phase. Further experiments measuring probe detectability as a function of probe modulation depth did not provide clear evidence to support the idea that the internal representations of the masker modulators interacted binaurally to produce a weak distortion component in the internal representation of the modulation at a 5 Hz frequency. Also, the obtained phase effects were not correctly predicted using a model based on short-term loudness fluctuations.


I. INTRODUCTION
Several recent models for the perception of amplitude modulation ͑AM͒ in sounds are based on the idea that the envelopes of the outputs of the ͑peripheral͒ auditory filters are fed to a second array of overlapping bandpass filters tuned to different envelope modulation frequencies ͑Kay, 1982; Martens, 1982;Dau et al., 1997aDau et al., , 1997b;;Ewert and Dau, 2000;Ewert et al., 2002;Verhey et al., 2003͒.This set of filters is usually called a "modulation filter bank" ͑MFB͒.Psychoacoustical evidence consistent with the concept of a MFB has come from experiments involving detection of "probe" modulation in the presence of masker modulation; these experiments appear to show frequency selectivity in the modulation domain ͑Bacon and Grantham, 1989;Houtgast, 1989;Ewert et al., 2002͒.Also, listeners appear to have some ability to "hear out" the sinusoidal components of a complex modulator, provided that the components are widely spaced in frequency ͑Sek and Moore, 2003 The present experiments are concerned with binaural interactions in the modulation domain, and specifically whether such interactions are influenced by modulation distortion products.
Several lines of evidence suggest that human listeners are sensitive to the interaural phase of the envelope of amplitude-modulated sounds for modulation frequencies up to several hundred hertz.First, sounds can be lateralized based on the interaural envelope delay, for envelope rates up to at least 400 Hz ͑Henning, 1974;Nuetzel and Hafter, 1981͒. Second, McFadden and Pasanen ͑1975͒ described an analog of binaural beats ͑Licklider et al., 1950͒ in the modulation domain.They presented a high-frequency two-tone complex to each ear.Each complex produced beats at a rate equal to the frequency difference between the two tones.Subjects were required to distinguish between two stimuli.In one, the beat rate was the same at the two ears ͑e.g., 50 Hz͒.In the other, the beat rate was chosen to be slightly different in the two ears ͑e.g., 50 Hz in one ear and 51 Hz in the other͒.When the difference in beat rate between the two ears was relatively small ͑5 Hz or less͒, subjects performed well above chance on this task.Subjects reported hearing a fluctuation at a rate corresponding to the difference in beat rate between the two ears: 1 Hz in the example given above.The effect occurred at low levels, or in the presence of an intense low-pass noise, so combination tones cannot account for the effect.McFadden and Pasanen ͑1975͒ also found that the carrier frequencies in the two ears did not have to be very close.For example, sinewaves of 2000 and 2050 Hz in one ear and 3000 and 3051 Hz in the other led to a beat sensation with a 1 Hz rate.They concluded that "the auditory system is apparently able to extract envelope periodicities monaurally and compare their temporal relations binaurally, and this ability gives rise not only to time-based lateralization performance at high frequencies, but also to a binaural beat similar in many respects to that heard with low-frequency sinusoids." Related experiments were reported by Bernstein and Trahiotis ͑1996͒.They presented two-tone complexes centered at 3500 Hz with a different beat rate at each ear.In one experiment, listeners were required to distinguish between rightward or leftward directions of intracranial movement produced by the binaural beat.The stimuli and experimental paradigm were designed so that such judgments would be made based on the dynamically varying, envelope-based interaural temporal disparities.Listeners were able to perform reasonably well on this task, at least for a very low interaural beat rate of 0.25 Hz; performance was close to chance for an interaural beat rate of 1 Hz.In a second experiment, listeners a͒ Author to whom correspondence should be addressed.Electronic mail: bcjm@cam.ac.uk were required only to distinguish between the presence or absence of an envelope-based binaural beat, as in the experiment of McFadden and Pasanen ͑1975͒.In this case, the results could be explained by assuming that listeners base their decisions on the presence or absence of dynamically varying interaural intensity disparities.
Other experiments have demonstrated directly that human listeners are sensitive to changes in the interaural phase of the envelopes of high-frequency carriers.For example, Grantham ͑1984͒ showed that subjects could discriminate an amplitude-modulated bandpass filtered noise in which the modulating sinusoid was interaurally in phase from the same AM noise in which the modulator was interaurally phase reversed.However, the modulation depth required for 71% discriminability did tend to increase as the modulation frequency was increased up to 50 Hz.In a related experiment, Thompson and Dau ͑2008͒ found that discrimination of interaural modulator phase was better for a 5000 Hz sinusoidal carrier than for narrowband-noise diotic carriers at the same center frequency.For all carriers, the task could be performed for modulation rates up to 128 Hz ͑the highest tested͒.
The present experiments were intended to assess whether some of the binaural interactions described above, especially envelope-based binaural beats, might be explained in terms of a distortion product in the modulation domain, produced at some point in the auditory system where the modulators at the two ears interact.The experiments were similar in design to experiments that have been conducted previously to examine the possible influence of "distortion" in the modulation domain for monaural stimuli, so we give next a brief overview of such experiments.
Moore et al. ͑1999͒ examined masking in the amplitudemodulation domain when the probe modulation frequency was remote from any spectral frequency in the masker modulation, but there was nevertheless a similarity between the temporal pattern of the masker modulation and the probe modulation.This was achieved by using a two-component modulator.The "beats" between these two components occurred at a rate that was equal to or close to the probe frequency.A similar method had been used earlier by Sheft and Yost ͑1997͒ to examine modulation detection interference.Moore et al. ͑1999͒ found that the threshold for detecting 5 Hz probe modulation was affected by the presence of a pair of masker modulators beating at a 5 Hz rate ͑40 and 45 Hz, 50 and 55 Hz, or 60 and 65 Hz͒.The threshold was dependent on the phase of the probe modulation relative to the beat cycle of the masker modulators.Moore et al. ͑1999͒ proposed an explanation for their results based on the idea that nonlinearities within the auditory system introduce distortion in the internal representation of the envelopes of the stimuli.This notion was initially suggested by Shofner et al. ͑1996͒ based on a study of neural responses in the cochlear nucleus to two-component modulators.In the case of two-component beating modulators, a weak component, corresponding to the simple difference component, would be introduced at the beat rate.Several other researchers have demonstrated modulation masking effects related to the beat rate of complex modulators ͑Verhey et al., 2003;Sek and Moore, 2004;Füllgrabe et al., 2005͒.The results have been explained in terms of distortion in the modulation domain ͑Sek and Moore, 2004;Füllgrabe et al., 2005;Uchanski et al., 2006͒, the combined  The present experiments were similar to the experiments described above, except that the masker modulator presented to each ear was a single sinewave.The 5 Hz probe modulation that the subject was asked to detect was presented diotically, but the masker modulator had a different frequency at the two ears, for example, 50 Hz in one ear and 55 Hz in the other ear.The threshold for detecting the probe modulation was measured as a function of the starting phase of the probe modulation.We reasoned that if a modulation distortion product at 5 Hz was generated following binaural interaction, then a phase effect should be found.This turned out to be the case.Further experiments were performed to assess more specifically whether the phase effect could be explained in terms of a distortion component in the internal representation of the modulation.

A. Stimuli
The carrier was a 4000 Hz sinusoid with a level of 70 dB sound pressure level, presented to both ears.This relatively high carrier frequency was chosen so that the spectral sidebands produced by the modulation would not be resolved by the peripheral auditory filters.The probe modulation frequency was 5 Hz, and the probe modulator was applied to both ears, with the same phase at the two ears.A sinusoidal masker modulator with a frequency of 50 Hz was applied to the carrier in the left ear, and a sinusoidal masker modulator with a frequency of 55 Hz was applied to the right ear.The modulation index for each masker modulator was 0.33.The equation describing the envelope of the masker plus probe in the left ear, E L ͑t͒, is where m m is the depth of the masker modulator ͑0.33͒, f mL is the frequency of the masker modulator in the left ear ͑50 Hz͒, t is time, f p is the frequency of the probe modulator ͑5 Hz͒, m p is the probe modulation depth, and is the starting phase of the probe modulator.The starting phase of the masker modulator was fixed.The equation describing the envelope of the masker plus probe in the right ear, E R ͑t͒, is where f mR is the frequency of the masker modulator in the right ear ͑55 Hz͒.Values of were 0°, 45°, 90°, 135°, 180°, 225°, 270°, and 315°.The left and right columns of Fig. 1 show the envelope ͑without dc component͒ of the masker alone ͑top͒, probe alone ͑middle͒, and masker plus probe ͑bottom͒ when = 90°, for the left and right ears, respec-tively.Figure 2 shows corresponding envelopes for = 225°.Thresholds for detecting the probe modulation were also measured for the probe alone, and for the probe in the presence of a 50 Hz masker modulator presented to the left ear only, with masker modulation depths of 0.33 and 0.5; the latter is comparable to the root-mean-square modulation depth that would be obtained if two modulation maskers were applied to the same ear, each with a modulation depth of 0.33.Note that the probe modulation was applied to both ears, even when the masker modulation was applied to one ear only.
On each trial, the carrier was presented in two bursts separated by a silent interval of 300 ms.Each burst had 20-ms raised-cosine rise and fall ramps, and an overall duration ͑including rise/fall times͒ of 1000 ms.The modulation was applied during the whole of the carrier, and the starting phase of the modulation was defined relative to the start of the carrier.
Stimuli were generated using a Tucker-Davis Technologies array processor ͑TDT-AP2͒ in a host PC and two channels of a 16 bit digital to analog converter ͑TDT-DD1͒ operating at a 50 kHz sampling rate.The stimuli were attenuated ͑TDT-PA4͒ and sent through an output amplifier ͑TDT-HB6͒ to Sennheiser HD580 headphones.Subjects were seated in a double-walled sound-attenuating chamber.

B. Procedure
Thresholds were measured using an adaptive twointerval forced-choice ͑2IFC͒ procedure, with a two-down one-up stepping rule that estimates the 70.7% correct point on the psychometric function.The masker modulation was present in both intervals of a trial, and the probe modulation was presented in either the first or the second interval, selected at random.The task of the subject was to indicate, by pressing one of two buttons, the interval containing the probe modulation.Feedback was provided by lights following each response.At the start of a run, the probe modulation depth, m, was chosen to be well above the threshold value.Following two correct responses, m was decreased, while following one incorrect response it was increased.The step size was 3.5 dB ͑in terms of 20 log m͒ until four reversals occurred, after which it was decreased to 2 dB and eight more reversals were obtained.The threshold for a given run was taken as the mean value of 20 log m at the last eight reversals.Each threshold reported here is based on the mean of four runs.

C. Subjects
Four subjects were tested, all of whom were paid for their services.All subjects had absolute thresholds less than 20 dB HL at all audiometric frequencies from 250 to 8000 Hz and had no history of hearing disorders.All had extensive previous experience in psychoacoustic tasks, including tasks similar to the one used here.They received 4 h of training on the task used here before data collection started.

D. Results
Figure 3 shows the individual results and the mean results across subjects ͑bottom left͒.The filled square indicates the threshold for detecting the probe modulation in the absence of masker modulation.The hexagon and the filled inverted triangle show the probe detection thresholds with 50 Hz masker modulation applied to the left ear only, with masker modulation depths of 0.33 and 0.5, respectively.The masker with the lower depth had almost no effect, as might be expected from the wide separation of the probe and masker frequencies in the modulation domain.The masker with the greater depth did produce a small amount of modulation masking, even though the probe in the right ear was presented without any modulation masker.This means that subjects could not listen only to the right ear stimulus, ignoring the masker modulation in the left ear.
The open squares in Fig. 3 show probe detection thresholds as a function of the starting phase of the probe modulator when the dichotic modulation masker was present, with the 50 Hz modulation presented to the left ear and the 55 Hz modulation presented to the right ear.The dichotic modulation masker produced only a small amount of masking.How- ever, there was a distinct variation in the probe threshold with starting phase, and the pattern of variation was consistent across subjects.The probe threshold was highest when the starting phase was 45°-135°and was lowest when the starting phase was 180°-270°.A within-subjects analysis of variance ͑ANOVA͒ was conducted with factor probe starting phase.The effect of phase was significant: F͑7,21͒ = 6.27, p Ͻ 0.001.These results suggest that the detection of the probe was influenced by interaction in the binaural system of the masker modulator presented to each ear.

A. Rationale
A possible explanation for the results presented above is that the masker modulators produced a weak 5 Hz distortion component in the modulation domain at a level in the auditory system where binaural interaction occurs.The probe de-tection threshold for a probe phase of about 90°might be relatively high because the probe is almost in opposite phase to the distortion component and is partially canceled.Conversely, for a probe phase of about 225°, the probe and distortion components might be in phase, and their addition would enhance detection of the probe.If this explanation is correct, then, for a probe with very small modulation depth, the detectability of the probe might actually become negative for a probe phase of 90°; in other words, subjects would hear the probe as being in the "wrong" interval of a forced-choice trial.This could happen because the probe and distortion component would almost cancel each other in the signal interval, but the distortion component would remain in the nonsignal interval.This idea was tested in experiment 2 by measuring the detectability of the probe as a function of probe modulation depth, using two starting phases of the probe which led to relatively high and low thresholds for detection of the probe modulation in experiment 1.One might expect that the phases leading to the highest and lowest thresholds would differ by 180°, so we could have chosen phases of 90°and 270°, or 45°and 225°.However, since we did not know which pair might give the most clear cut results, we decided to use the phases which led to the highest and lowest empirically measured thresholds, namely, 90°and 225°.A similar method has been used to check for the presence of a distortion component in the modulation domain produced by monaural interaction of masker modulator components ͑Sek and Moore, 2004;Füllgrabe et al., 2005͒.

B. Method
Three of the subjects from experiment 1 took part ͑the fourth was no longer available͒.Because the putative distortion product in the modulation domain was likely to be very weak, we increased the modulation depth of each masker to 0.5 ͑compared to the value of 0.33 in experiment 1͒, so as to increase the likely magnitude of the distortion product.The timing of the stimuli was the same as for experiment 1.A 2IFC procedure was again used.However, instead of using an adaptive procedure, we measured the percent correct in blocks of 55 trials using a fixed probe modulation depth.Responses for the first five trials in each block were regarded as "warm up" and were discarded.At least four blocks of trials were run for each probe modulation depth.Pilot runs showed that the probe modulation depth needed to be very small to obtain negative detectability for the probe starting phase of 90°.This guided the choice of fixed probe modulation depths, which were Ϫ37, Ϫ40, Ϫ43, Ϫ46, and Ϫ48 dB ͑in terms of 20 log m p ͒.No feedback was given because it was anticipated that the probe might sometimes be heard in the wrong interval.

C. Results
The percent correct scores were converted to values of the detectability index, dЈ ͑Hacker and Ratcliff, 1979͒.The individual and mean values of dЈ are shown in Fig. 4. All of the dЈ values were close to zero, as would be expected given the very small probe modulation depths.Nevertheless, there was a consistent phase effect, dЈ values being higher for the probe phase of 225°than for the phase of 90°.Also, the values of dЈ for the probe phase of 90°tended to fall below zero, especially for modulation depths around Ϫ43 dB.A within-subjects ANOVA with factors probe phase and probe modulation depth gave a significant effect of phase, F͑1,2͒ = 100.3,p = 0.01, and of modulation depth, F͑4,8͒ = 5.21, p = 0.023.The interaction was not significant.However, given that we used only three subjects with 200 observations per subject per condition, the 95% confidence interval for the proportion correct values is about 0.04, which means that for a dЈ value to be significantly below zero it would have to be less than Ϫ0.14.None of the measured mean dЈ values fell below Ϫ0.14.Thus, subjects did not score significantly below chance for these very small modulation depths.The results do not provide clear support for the idea that there was a weak distortion component in the modulation domain, which partially or completely canceled the probe modulation for the probe phase of 90°and led to the subjects identifying the probe in the wrong interval.

A. Rationale
The stimuli in experiments 1 and 2 might be thought of as analogous to stimuli which lead to envelope distortion, but with the distortion produced following binaural interaction.The modulation distortion component corresponds to f 2 -f 1 , where f 1 and f 2 are the frequencies of the primary modulator components and f 2 Ͼ f 1 .In experiments 3 and 4, we pursued the distortion analogy using as dichotic modulation maskers components with frequencies of 27.5 and 50 Hz.In other words, the experiments were similar to experiments 1 and 2, except that f mR , the frequency of the masker modulator component in the right ear was 27.5 Hz, rather than 55 Hz.Modulators with frequencies of 50 and 27.5 Hz might interact in the binaural system to produce a modulation distortion component at 5 Hz, corresponding to 2f 2 -f 1 .

B. Method
The subjects were the same as for experiment 1.The stimuli and method were also the same as for experiment 1, except that f mR , the frequency of the modulator component applied to the carrier in the right ear, was 27.5 Hz, rather than 55 Hz.Figures 5 and 6 illustrate the modulator waveforms ͑without dc component͒ for probe starting phases of 90°and 225°, respectively.

C. Results
Figure 7 shows the individual and mean results.The filled square indicates the mean threshold for detecting the probe modulation in the absence of masker modulation.The hexagon and the filled inverted triangle show the probe detection thresholds with 27.5 Hz masker modulation applied to the left ear only, with masker modulation depths of 0.33 and 0.5, respectively.Overall, the masker with the lower depth had a slightly greater effect than found for the 50 Hz modulator in experiment 1, but, based on a t-test, this effect was not statistically significant ͑p = 0.86͒.As expected, the masker with higher depth produced somewhat more modulation masking, but a t-test showed that this effect was also not statistically significant ͑p = 0.23͒.
The open squares in Fig. 7 show probe detection thresholds as a function of the starting phase of the probe when the dichotic modulation masker was present: 50 Hz to the left ear and 27.5 Hz to the right ear.As in experiment 1, the dichotic modulation masker produced only a small amount of masking, but there was a distinct variation in the probe threshold with starting phase, and the pattern of variation was consistent across subjects.The probe threshold was highest when the starting phase was in the range 0°-90°and was lowest when the starting phase was in the range 180°-270°.A within-subjects ANOVA was conducted with factor probe starting phase.The effect of phase was significant: F͑7,21͒ = 11.32,p Ͻ 0.001.These results suggest that the detection of the probe was influenced by interaction in the binaural system of the masker modulator presented to each ear.

A. Rationale
The rationale for this experiment was similar to that for experiment 2. If the masker modulators produced a weak 5 Hz distortion component in the modulation domain, the detectability of the probe might become negative at very low probe modulation depths for the probe phase that led to the highest threshold in experiment 3, which was 90°; subjects would hear the probe as being in the wrong interval of a forced-choice trial.

B. Method
The subjects and procedure were the same as for experiment 2. The stimuli were also the same as for experiment 2, except that the masker modulator frequency in the right ear was 27.5 Hz.

C. Results
The individual and mean values of dЈ are shown in Fig. 8.All of the dЈ values are close to zero, as would be expected given the very small probe modulation depths.There appears to be a very small effect of probe phase for subjects S2 and S4, but not for S3.A within-subjects ANOVA showed no significant effect of probe modulation depth or probe starting phase, and no significant interaction.The dЈ values did not fall consistently below zero for the probe phase of 90°.Thus, these results do not support the idea that interaction of the masker modulators in the binaural system led to a distortion product in the modulation domain with frequency corresponding to 2f 2 -f 1 .

VI. DISCUSSION
A possible confounding factor in our experiments is that the threshold for detecting 5 Hz probe modulation might depend on the starting phase of the modulation, independent of the characteristics of the maskers.Such sensitivity is possible when the modulator has a very low frequency, given that subjects appear to be sensitive to the starting phase of a single sinusoidal modulator when the modulation rate is below about 12 Hz ͑Dau, 1996; Sheft and Yost, 2007͒.However, it has been shown that psychometric functions for the detection of 5 Hz sinusoidal AM are not affected by the starting phase of the AM for a wide range of carrier frequen- cies ͑Sek and Skrodzka, 1999͒.Therefore, we believe that the pattern of phase effects found in experiments 1-3 cannot be explained by a dependence of the ͑absolute͒ threshold for probe detection on the starting phase of the probe.
Our results appear to reflect an interaction in the binaural system of the stimuli presented to each ear.Consistent with this interpretation, for the stimuli of experiment 1 subjects reported hearing a weak fluctuation at a relatively low rate when listening to the masker modulators alone ͑the 50 Hz modulator in one ear and the 55 Hz modulator in the other ear͒.It was not clear to the subjects whether the fluctuation was in loudness or in spatial position.The reported fluctuation is comparable to that reported by McFadden and Pasanen ͑1975͒, as described in the Introduction.
The results of our experiment 1 are consistent with the idea that listeners are sensitive to dynamic variations in interaural time or intensity.The envelope periodicities of 50 Hz in the left ear and 55 Hz in the right would have given rise to fluctuations in interaural intensity at a 5 Hz rate.Also, the interaural time difference associated with the envelope would fluctuate at a 5 Hz rate.Both of these fluctuations might influence the detection of 5 Hz probe modulation.However, it is not obvious why the fluctuations would lead to an effect of relative modulator phase.
One possible explanation of our results is related to short-term fluctuations in loudness.The loudness of the masker may have fluctuated at a 5 Hz rate, owing to the fluctuating interaural level difference.The addition of the probe modulation might change the perceived amount of loudness fluctuation, and this might be the cue used to detect the probe modulation.To assess this possibility we used a model of loudness for time-varying sounds.The model was similar to that described by Glasberg and Moore ͑2002͒ but modified to incorporate the concept of binaural inhibition proposed by Moore and Glasberg ͑2007͒.The model described by Glasberg and Moore ͑2002͒ starts by calculating the "instantaneous loudness" from the short-term spectrum of the stimulus.This is an intervening variable, assumed not to be accessible to conscious perception.The instantaneous loudness is subjected to an initial stage of smoothing or averaging over time using a mechanism similar to an automatic gain control system, with an attack time and a release time.This gives the short-term loudness.The model also includes a second stage of averaging, using longer attack and release times, to give an estimate of the overall loudness impression of a fluctuating sound.However, here we considered only the output of the first stage of averaging, i.e., we assumed that only the short-term loudness estimate was relevant.In the version of the model used here, the instantaneous loudness was calculated separately for each ear, and then the instantaneous loudness was combined across ears using the method described by Moore and Glasberg ͑2007͒, so as to include the effect of binaural inhibition.Summation across ears prior to averaging meant that rapid fluctuations in amplitude were preserved at the point of binaural interaction.Then the binaural instantaneous loudness was smoothed as described above to give the short-term loudness as a function of time.
Figure 9 shows the output of the model for the case when the masker modulation had a frequency of 50 Hz in one ear and 55 Hz in the other.The solid line shows the short-term loudness level in phons for the masker alone, and the dashed line shows the short-term loudness when probe modulation with a starting phase of 0°was added; the probe modulation depth was equal to the mean measured threshold value for that probe starting phase.The curve for the masker alone shows a weak rapid fluctuation corresponding roughly to the mean of the two masker modulation frequencies and a slower fluctuation corresponding to the 5 Hz beat rate of the masker modulators.However, the amount of fluctuation is small in both cases.The small fluctuation at a 5 Hz rate is consistent with the subjective reports of the subjects.The addition of the probe modulation caused an increase in the amount of fluctuation at the 5 Hz rate which might have been used as a cue for detection of the probe.
To assess whether the pattern of phase effects could be explained in terms of fluctuations in the short-term loudness, we used as a decision variable the difference in short-term loudness for the masker alone and the masker plus probe.For the example in Fig. 9, this corresponds to the difference between the solid and dashed curves on a point-by-point basis.We denote this difference STL M-P ͑t͒.We assumed that performance was related to the peak-to-valley difference of STL M-P ͑t͒.We initially used as input to the model the mean value of the probe modulation depth at threshold for each probe starting phase.Averaged across starting phases, the mean value of the peak-to-valley difference of STL M-P ͑t͒ was 0.61 phons.Then for each phase condition, we iteratively adjusted the probe modulation depth so that the obtained value of the peak-to-valley difference of STL M-P ͑t͒ was 0.61 phons.The probe modulation depth obtained in this way was taken as the predicted probe modulation depth at threshold.
The outcome is shown in Table I.The obtained thresholds differed by 2.3 dB across conditions, whereas the predicted thresholds differed by only 0.7 dB across conditions.Furthermore, the phase effects were not predicted correctly.The obtained threshold was highest for the probe starting phase of 45°and lowest for the phase of 225°, while the predicted threshold was highest for the phase of 0°and lowest for the phase of 135°.Predictions were generated using several other decision variables, but none led to correct pre- dictions of the observed phase effects.Overall, it appears that the results cannot be adequately explained in terms of the short-term fluctuations in loudness predicted by the model.It should be noted that the model of Glasberg and Moore ͑2002͒ was designed to account for the loudness perception of time-varying monaural or diotic sounds, while the model of Moore and Glasberg ͑2007͒ was designed to account for the loudness perception of static diotic and dichotic sounds.It is possible that the model used here does not correctly account for the loudness of time-varying dichotic sounds and that a different model might give results that fit the data better.However, we tried several other versions of the model, including one in which the instantaneous loudness was simply summed across ears, and none of them predicted the phase effects correctly.Furthermore, it is difficult to think of any way in which the loudness model could be modified so as to predict the phase effects correctly.
Given that the results could not be predicted using the loudness models that we tried, it is worth considering again the idea that there might have been a weak "distortion component" in the modulation domain with a frequency corresponding to 5 Hz, even though our results did not provide clear evidence to support the existence of such a distortion component.Sek and Moore ͑2004͒ conducted an experiment similar to experiment 2, except that the 50 and 55 Hz masker modulators were presented to the same ear.They found negative dЈ values for some conditions.The minimum value of dЈ was about Ϫ0.43 and it occurred for a probe modulation depth of about Ϫ30 dB.Füllgrabe et al. ͑2005͒ used a white noise carrier and a second-order masker modulator ͑a "carrier" modulator with frequency of 64, 180, or 200 Hz, whose depth was sinusoidally varied at a 5 Hz rate͒.For the 64 Hz carrier modulation frequency, the minimum value of dЈ ͑ϷϪ1͒ occurred for a probe modulation depth of about Ϫ23 dB.In contrast, in experiment 2 of the present paper, the minimum value of dЈ was Ϫ0.106, and it occurred for a probe modulation depth of about Ϫ43 dB.These comparisons suggest that, if there is a distortion component produced by binaural interaction, its effective level is much lower than that produced by monaural presentation of the masker modulators.
It is not clear how to explain the results of experiment 3.In that experiment, the envelope modulation rate was 50 Hz in one ear and 27.5 Hz in the other.A significant effect of the phase of the 5 Hz probe on detectability of the probe was found, but the results of experiment 4 did not provide support for the idea that there was a distortion component in the modulation domain corresponding to 2f 2 -f 1 .Assuming that the time pattern of the modulators is preserved at the point of binaural interaction, it is relevant to consider the modulation waveform produced by summing the 50 and 27.5 Hz components.The result of this summation is illustrated in the top panel of Fig. 10.This waveform is reproduced in the middle panel of Fig. 10, which also shows the Hilbert envelope of the summed modulation ͑thick line͒ ͑The Hilbert envelope is used here merely as a convenient way of estimating the en-velope͒.The Hilbert envelope is periodic, repeating 22.5 times per second, which corresponds to the difference in frequency between the two modulator components.Not surprisingly, the Hilbert envelope does not show any periodicity at a 5 Hz rate.However, the major peaks in the summed modulation are spaced at approximately 0.2 s, corresponding to the 5 Hz period of the probe modulation.This is illustrated in the bottom panel of Fig. 10, which shows the summed modulator waveform together with a 5 Hz sinewave which has been scaled so as to coincide approximately with the main peaks in the waveform.Thus, the temporal pattern of the summed modulator waveform includes temporal features occurring at a 5 Hz rate, even though the Hilbert envelope does not reveal these features.
Füllgrabe and Lorenzi ͑2005͒ investigated the perception of a noise carrier, amplitude modulated at frequency f m , when the AM depth of this "carrier" modulation was itself sinusoidally modulated by a "second-order" modulator with a 5 Hz rate.They included conditions where f m was an integer multiple of 5 Hz ͑so that the envelope was strictly periodic with a 5 Hz repetition period͒, and where f m was shifted in frequency, so as to create a more complex modulator ͑but where the Hilbert envelope of the modulation waveform was still periodic with a 5 Hz repetition period͒.The perceived envelope "beat" rate was estimated using a matching procedure.The results indicated that the perceived beat rate was influenced by the frequency shift, at least for f m Ͻ 20 Hz.Füllgrabe and Lorenzi ͑2005͒ suggested that the perceived envelope beat rate is determined by the time intervals between major peaks in the first-order envelope and not by the repetition rate of the Hilbert envelope of the modulator or by a distortion component in the modulation domain.Something similar could be the case for our stimuli of experiment 3, except that the internal representations of the modulators presented separately to the two ears would have to be summed at a point in the auditory system where their temporal structure was preserved.The detection of the 5 Hz probe modulation could be influenced by its phase relative to the perceived beat in the complex modulator.For some phases, the probe might make the perceived beat more or less salient, while for other phases it might make the beat qualitatively different, for example, by doubling the apparent beat rate.When questioned about whether they could hear a relatively slow ͑5 Hz͒ beat when listening to the 50 Hz modulator in one ear and the 27.5 Hz modulator in the other ear, the reports of our subjects were mixed; two reported hearing such a beat and two did not.

VII. SUMMARY AND CONCLUSIONS
To assess the possible existence of distortion components in the modulation domain, produced following binaural interaction, the detection of diotic 5 Hz probe modulation of a 4000 Hz sinusoidal carrier was measured in the presence of two higher-frequency masker modulators, one presented to each ear.The following are the main results.
͑1͒ When the masker modulator frequencies were 50 and 55 Hz, the probe modulation depth at threshold varied slightly but significantly with the starting phase of the probe.The probe threshold was highest when the starting phase was 45°-135°and was lowest when the starting phase was 180°-270°.The pattern of the phase effects could not be predicted based on short-term fluctuations in loudness at the output of a loudness model.͑2͒ When the masker modulator frequencies were 50 and 55 Hz, the detectability of the probe modulation, dЈ, for very low probe modulation depths was lower when the probe starting phase was 90°than when it was 225°.For the 90°starting phase, dЈ was slightly but not significantly below zero for probe modulation depths close to Ϫ43 dB.The results indicate that if the internal representations of the 50 and 55 Hz masker modulators interacted binaurally to produce a weak distortion component in the internal representation of the modulation at a 5 Hz frequency, then that distortion component was very weak.͑3͒ When the masker modulator frequencies were 50 and 27.5 Hz, the probe modulation depth at threshold varied slightly but significantly with the starting phase of the probe.The probe threshold was highest when the starting phase was 0°-90°and was lowest when the starting phase was 180°-270°.͑4͒ When the masker modulator frequencies were 50 and 27.5 Hz, the detectability of the probe modulation, dЈ, for very low probe modulation depths was not significantly different for probe starting phases of 90°and 225°.The value of dЈ did not fall significantly below zero.These results do not provide evidence for a 5 Hz distortion component in the internal representation of the modulation.

FIG. 2 .
FIG.2.As Fig.1, but for a starting probe phase of 225°.FIG.1.Illustration of modulator waveforms ͑without dc component͒ presented to the left and right ears for a starting probe phase ͑as defined in Eqs.͑1͒ and ͑2͒͒ of 90°.The top panels show waveforms for the maskers alone ͓M L ͑t͒ and M R ͑t͒, for the left and right ears, respectively͔, the middle panels show waveforms for the probe alone ͓P L ͑t͒ and P R ͑t͔͒, and the bottom panels show the masker+ probe waveforms ͓E L ͑t͒ and E R ͑t͔͒.

FIG. 3 .
FIG. 3. Individual and mean results for experiment 1.The detection threshold for the probe modulation is plotted as a function of the starting phase of the probe modulator.For the individual results, error bars indicate Ϯ1 standard deviation ͑SD͒ across repeated runs.For the mean results, error bars indicate Ϯ1 SD across subjects.Filled symbols on the left of each panel show probe detection thresholds measured with no masker ͑squares͒, a 50Hz masker with m = 0.5 ͑inverted triangles͒, and a 50 Hz masker with m = 0.33 ͑hexagons͒.In the last two cases, the masker modulator was applied to the left ear only.

FIG. 4 .
FIG. 4. Individual and mean results for experiment 2. The detectability of the probe modulation, dЈ, is plotted as a function of the probe modulation depth for two starting phases of the probe, 90°and 225°.

FIG. 7 .
FIG.7.As Fig.3, but showing results for experiment 3, with the masker modulation frequency for the right ear set to 27.5 Hz.

FIG. 9 .
FIG. 9. Output of the loudness model described in the text, showing shortterm loudness as a function of time.The solid line is for the masker modulator alone ͑50 Hz in one ear and 55 Hz in the other͒.
effects of cochlear filtering and off-frequency listening ͑Füllgrabe et al., 2005͒, the perception of slow fluctuations at the output of a modulation filter tuned to the "primary" modulator components ͑Uchanski et al., 2006͒, or a mechanism that explicitly extracts the envelope of a complex modulator ͑Verhey et al., 2003͒.

TABLE I .
Comparison of mean obtained probe modulation depths at threshold from experiment 1 ͑using masker modulators with frequencies of 50 and 55 Hz͒, with predictions derived using the loudness model described in the text.The data are the same as shown in Fig.3.FIG.10.The top panel shows the modulation waveform dc com-ponent͒ resulting from summing the 50 Hz and 27.5 Hz modulators, each with m = 0.33.The middle panel reproduces that waveform and also shows the Hilbert envelope of the waveform as a thick line.The bottom panel shows the same waveform, together with a 5 Hz sinewave that has been scaled to coincide with the major peaks of the waveform.