Animal Preparation

Recordings were made from adult Mongolian gerbils (Meriones unguiculatus; three males and 12 females; 21 ears; body weight, 46–66 g). Animals were anesthetized by intraperitoneal injection of ketamine/xylazine solution (effective dosage, 80 and 12 μg/g body weight, respectively). Supplementary anesthesia was administered subcutaneously at ∼¹/₃ of the initial dose in 1-h intervals. A small metal rod was attached to the dorsal surface of the skull and was used to fix the head of the animal. Body temperature was maintained at 37°C using a thermocontrolled heating pad. The pinna and cartilaginous ear canal were removed and a custom-built probe was sealed over the bony ear canal using Vaseline. In eight of the animals a small hole was drilled in the wall of the inferior posterior mastoid chamber of the ipsilateral bulla to prevent pressure buildup in the middle ear cavity. We did not notice apparent differences between recordings with or without such venting, and no distinction between the data from these two conditions is made. Animal procedures were in accordance with guidelines provided by the animal committee of the Erasmus MC.

Stimuli

To evoke otoacoustic emissions, acoustic stimuli that consisted of multiple frequency components (each with a random starting phase) were used. For convenience of description, the components are separated into two constituents: (1) a single frequency component and (2) a tone complex consisting of M tones. In this study, M varied between 5 and 9. The frequency of the single component will be denoted by f; the frequencies of the tone complex by g₁, g₂...g_M. The stimulus frequencies were chosen such that all possible difference and sum frequencies were unique (Victor et al. ¹⁹⁷⁷; van der Heijden and Joris ²⁰⁰³; ²⁰⁰⁶). Following van der Heijden and Joris (²⁰⁰³), we will refer to this stimulus property as “zwuis”. Nonlinear interaction between any combinations of the stimulus components can give rise to third-order DPOAEs, and different groups of third-order DPOAEs will be identified and analyzed in Results.

The primary frequencies were further restricted as described in Table 1 (the terminology used in this table anticipates the naming of the DPOAE groups described in Results). This restriction served to avoid the coincidence of DP components across groups. Combined, the restrictions in the choice of stimulus frequencies (“zwuis” and “periodicity”) ensure that all third-order DPOAEs of the type g_i ± g_j ± f are unique and never equal to a primary component. Consequently, based on its frequency, each DPOAE component can be unambiguously identified with the unique triplet (f,g_i,g_j) of stimulus components whose nonlinear interaction produced it.

The tone complex and the single frequency component were presented over separate D/A channels (Tucker-Davis Technologies RP2.1). To minimize transients, stimuli were gated using a raised cosine window (10–90% in 10 ms). The output of each channel was fed through a stereo power amplifier (TDT SA1) and broadcast from a separate driver (2× TDT CF1 or 2× Visaton FRS-7). These drivers where connected to the recording probe by means of plastic tubes. The correct sound pressure levels were attained by calibrating the drivers in situ while taking the probe transfer characteristics into account. The recording probe contained a 1/2″ pressure-microphone (GRAS 40AG) that was used to record the ear canal sound pressure synchronously with stimulus delivery. The recorded signal was band-pass filtered (0.02–100 kHz, NEXUS 2690), acquired via an A/D channel (TDT RP2.1), and stored on computer disk for offline analysis. Signal generation and data acquisition were both done at a rate of 48.8 kHz.

Data Analysis

All stimulus frequencies were chosen such that an integer number of sample points held an even number of periods for these tones (see Table 1). As a consequence, these periodic segments (typically 97,656 samples, i.e., ∼2 s) also held an even number of periods for all third-order DPOAEs. This allowed for averaging of a single, continuous recording by breaking it down into its periodic segments, and eliminated any spectral leakage when extracting frequency components using Fourier analysis (Papoulis ¹⁹⁶²).

After excluding the first and last periodic segment of each recording (to exclude the ramps and transient phenomena), the remaining signal was averaged over the periodic segments and the magnitude and phase spectra were calculated via Fourier analysis. No form of artifact rejection was used. The stimulus design resulted in the even components of these spectra to be commensurate with the stimulus (and any DPOAEs evoked by it). The odd components of the spectra, which are not commensurate with the stimulus, were used to estimate the noise floor. Additional recordings were made in an artificial ear to check for distortion in the hardware. For the stimuli used in this study, these system distortions were well below the DPOAEs, often not exceeding the noise floor.

The phase of each DPOAE component was expressed in cycles re. the phase of the stimulus components producing it as determined from the microphone signal. Thus, the phase [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\Phi_{\rm{dp}}}(f,{g_i},{g_j}) $$\end{document}] of a DP component having frequency f_dp = g_i ± g_j ± f becomes

where [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\varphi_{...}} $$\end{document}] denotes the phase values extracted from the Fourier spectrum of the recording.

The relation between DPOAE phase and stimulus frequency was analyzed by fitting the phase data using two linear models. Any DPOAE components not exceeding the noise floor were excluded from the fitting procedure. Weight factors were assigned to the remaining DPOAE components to prevent the phases of weak components (close to the noise floor) from dominating the data fits. The weight factor W was

where σ² is the variance in amplitude of a tonal component caused by the addition of noise. This variance was evaluated using numerical simulations of the effect of adding random components (with phase drawn from a uniform distribution, and amplitude drawn from a Rayleigh distribution) on the phase of single spectral components. The simulations yielded a polynomial description of the variance σ²where NSR denotes the reciprocal of the signal-to-noise amplitude ratio.

The first model was a straight line of DPOAE phase Φ_dp against DPOAE frequency f_dp

The phase offset Φ₀ and group delay τ were estimated by minimizing the weighted sum of squared differences between data and model,

The second model, which was used in the case of two-dimensional representations of the DPOAE phase data (see Fig. 5), is given by

Notice that Eq. 6 does not explicitly include a term for the single stimulus frequency f, because f was not varied within data sets that were fitted. The phase offset Φ₀ and group delays τ₊ and τ₋ were estimated by minimizing the weighted sum of squared residuals (Eq. 5). Group delays will be reported by their mean ±95% confidence interval. All stimulus generation, data acquisition, and offline analysis were done via custom software in MATLAB.

The Far Group

In the frequency spectrum, these DPOAEs (left-pointing triangles in Fig. 1) occur at the “far side” of the tone complex (at the opposite side of the single tone as the tone complex). Using a tone complex having M components results in a far group that consists of M DPOAEs, at frequencies f_far(i) = 2f−g_i, i = 1…M. Each DPOAE thus results from the interaction of only two stimulus components, the single-tone primary and one frequency component from the tone complex. This is analogous to DPOAEs evoked by two-tone stimuli, but instead of sweeping one primary across frequencies to evoke subsequent DPOAEs, all the different stimulus frequencies are here presented simultaneously as a tone complex. More specifically, the far group resembles 2f₁ − f₂ DPOAEs when f₂ is replaced by the tone complex (f < g; Fig. 1B), or 2f₂ − f₁ DPOAEs when f₁ is replaced by the tone complex (f > g; Fig. 1A).

We tested the similarity between the far group and DPOAEs evoked by two-tone stimuli by also recording the latter emissions using either a fixed-f₂ or a fixed-f₁ paradigm (i.e., one stimulus tone was fixed in frequency, while the other was varied across recordings). DPOAEs from the two-tone recordings were then compared to the far group obtained from a single recording in which all of the stimulus components were presented at once. As an example, Figure 2A shows the superimposed amplitude spectra of 10 recordings. In nine of these recordings, two-tone stimuli (light gray) were used, in which the upper stimulus component f₂ was fixed, while the frequency of the lower component f₁ was varied across recordings (fixed-f₂ paradigm). In the 10th recording, all stimulus frequency components were presented simultaneously (dark gray). The intensity of each component in the tone complex was reduced relative to the f₁ level in the two-tone paradigm in such a way that the total power of the tone complex equaled the power of each single f₁ component in the fixed-f₂ paradigm. The amplitude and phase data for the DPOAEs from these recordings are shown in more detail in Figure 2B, C, respectively (triangles for far group, circles for DPOAEs from consecutive two-tone stimuli). Notice that, although the amplitude of each DPOAE component in the far group is smaller than that resulting from the two-tone stimulus, their total power (horizontal dashed line in Fig. 2B) is nearly identical to the power of each of the latter DPOAEs. The phase data exhibit very similar group delays, but show a small vertical offset (∼0.02 cycles) between the two types of recordings. Such an offset was, however, not observed systematically across recordings. Comparisons between the far group and two-tone DPOAEs from similar recordings in four animals are made in Figure 3 (triangles) as scatter plots for the group delay (A) and DPOAE power (B). Depending on the relative primary frequencies, the far group resembled either the 2f₁−f₂ (when f < g) or the 2f₂−f₁ (when f > g) DPOAEs. There appears to be no need to distinguish between these two situations.

The general observation is that the far group is practically identical to DPOAEs evoked by consecutive two-tone stimuli. Thus, for this group of DPOAEs there is little difference between sweeping one primary across a set of frequencies (fixed-f₁ or fixed-f₂ paradigm) and presenting the whole set together as a tone complex.

The Near Group

Spectrally, this group of emissions occurs at the side of the tone complex opposite of the single stimulus tone. In analogy with the far group we will refer to it as the “near group.” The frequencies of this group are given by [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {f_{\rm{near}}}\left( {i,j} \right) = {g_i} + {g_j} - f $$\end{document}] . Figure 4 gives an example for which f > g (see Fig. 1B for an example with f < g).

This DPOAE group contains more frequency components than there are stimulus components. In the example of Figure 4, a tone complex that consisted of nine components evoked a total of 36 unique DPOAE components in the near group. Also, the near group appears to be composed of two subsets that systematically differ in their amplitudes by ∼6 dB (compare close circle and triangles in Fig. 4A).

Both the number of DPOAE components and the apparent existence of two subsets can be explained by realizing that the generation of each DPOAE component involves three stimulus components: the single stimulus component f, and two components (g_i, g_j) from the tone complex. The single frequency component f is constant within the near group, so all possible combinations are represented by the square matrix (g_i, g_j). Graphically, these combinations (g_i, g_j) are given by the lattice points in Figure 5A. The subset of M combinations for which i = j produce DPOAE components (circles) involving only one frequency component from the tone complex. This subgroup is the “mirror image” of the far group of DPOAEs. The remaining ¹/₂  M(M−1) combinations for which i ≠ j (triangles in Fig. 5A) produce the additional frequency components of the near group of DPOAEs. Their generation involves two different components from the tone complex. Notice that not all possible combinations result in a unique DPOAE component. Since g_i + g_j = g_j + g_i, only half of all i ≠ j combinations result in a unique frequency. In combinatorial terms, both g_i + g_j and g_j + g_i contribute to the same DPOAE component. This combinatorial effect corresponds to the coefficient 2 of the xy term in the binomial expansion of (x+y)², and causes the DPOAEs for which i ≠ j to have their amplitudes doubled (+6 dB) re. the amplitude of DPOAEs for which i = j. Apart from this combinatorial factor 2 in amplitude, there is no fundamental difference between the two groups, as is shown by the phase data (Fig. 4B) in which all near components are well described by a single function of DPOAE frequency.

To investigate the analogy between the near group and DPOAEs evoked by two-tone stimuli, recordings identical to those described for Figure 2 were made: in a series of two-tone recordings, DPOAEs were obtained using either a fixed-f₂ or a fixed-f₁ paradigm. These two-tone data were compared with a single recording in which all stimulus components were presented at once. Results are given in Figure 3 (circles) together with the data for the far group. As before, the grouping of all data around the diagonals suggests that the DPOAEs evoked by the two stimulus protocols are essentially the same.

The similarity between the DPOAEs obtained using a tone complex or a series of two-tone recordings again shows that there are no essential differences between DPOAEs recorded sequentially and simultaneously. They are generated by the same nonlinear mechanism, and there are no indications that they propagate along different cochlear paths.

Given the close similarity between the near and far groups and consecutive two-tone stimuli, the multitone data may be considered as 2f₁ − f₂ or 2f₂ − f₁ DPOAEs as if recorded using a fixed-f₁ or a fixed-f₂ paradigm. Specifically, the near group resembles either 2f₁ − f₂’s with f₂ fixed (f > g) or 2f₂ − f₁’s with f₁ fixed (f < g), while the far group corresponds to either 2f₂ − f₁’s with f₂ fixed (f > g) or 2f₁ − f₂’s with f₁ fixed (f < g). Using this interpretation of the multitone data, DPOAE group delays were calculated for these four different {DPOAE, stimulus} configurations (Fig. 6A, B). For the “2f₁ − f₂” DPOAEs, the fixed-f₁ paradigm yielded systematically larger group delays than the fixed-f₂ paradigm. A comparison of the former group delays with 2f₁−f₂ group delays from two-tone data of the gerbil (adapted from Faulstich and Kössl ²⁰⁰⁰), shows that the absolute group delays as well as the trend for these delays to decrease with increasing frequency are very similar, confirming the similarity between the two-tone and multitone DPOAEs. In contrast, group delays for the “2f₂ − f₁” DPOAEs were similar for the fixed-f₁ and fixed-f₂ paradigm, and are comparable to the {“2f₁ − f₂”/fixed-f₂} group delays.

For the near group of DPOAEs, the zwuis character of the stimulus becomes indispensable (see Materials and Methods). It ensures that all possible combinations g_i + g_j are unique, so that each DPOAE frequency g_i + g_j−f can be unambiguously attributed to a unique pair (g_i, g_j) of stimulus components from the tone complex. This is illustrated in Figure 5B, in which the (g_i,g_j)-matrix of Figure 5A is transformed (rotated and translated) into the equivalent matrix ([g_i + g_j−f],[g_i−g_j]), where g_i + g_j − f equals the DPOAE frequency. The zwuis character of the stimulus ensures that none of the symbols are vertically aligned: projecting them on the abscissa (i.e., DPOAE frequency) will never result in two coinciding points. This is necessary to disentangle the different contributions of the stimulus components to the individual DPOAEs.

The wealth of components in the near group offers an extra “degree of freedom” in the analysis of phase data. The presence of i ≠ j components enables an analysis of DPOAE phase that extends beyond the dependence on DPOAE frequency analyzed thus far. The additional degree of freedom is illustrated by representing the DPOAE components of the near group in the two-dimensional format of Figure 5B. In this representation, the dependence of DPOAE phase on DPOAE frequency corresponds to variation along horizontal lines (i.e., parallel to the abscissa). In contrast, variation along vertical lines indicates a dependence on stimulus frequencies in a way that does not correspond to changes in emission frequency.

Figure 7A shows the phase data from Figure 4B in the two-dimensional representation in the form of a contour plot. It is important to keep in mind that the phase data shown here were obtained from a single recording. In discussing them, it is hard to avoid phrases like “variations in stimulus frequencies”, which might suggest the use of frequency sweeps. In reality, all stimulus components were presented together, and the “variations” refer to comparisons across DPOAE components produced by different triplets drawn from the M + 1 stimulus components.

The contours in Figure 7A are iso-phase lines; the maximum variation of DPOAE phase occurs along the direction perpendicular to them. The vertical orientation of the contours thus indicates that DPOAE phase changes with variation along the abscissa. Thus, DPOAE phase only depends on DPOAE frequency. In contrast, variations in stimulus frequencies (g_i, g_j) that keep the DPOAE frequency fixed (variations along the ordinate) have no systematic effect on DPOAE phase. This observation is quantified by fitting the two-dimensional phase data with the model given in Eq. 6 (see Materials and Methods), which yielded τ₊ = 665 ± 16 μs, and τ₋ = 16 ± 26 μs (Fig. 7B). Notice that τ₊ is the coefficient describing the phase variations with DPOAE frequency, i.e., the slope of the frequency-phase plot of Figure 4B. As explained above, τ₊ matches the group delay obtained with a two-tone stimulation paradigm. In contrast, τ₋ is the coefficient for those variations in the frequencies of the tone complex g that keep the (g_i + g_j), and thus DPOAE frequency, fixed.

Figure 7C is a scatter plot of similarly calculated group delays for 998 recordings (830 when f > g; 168 when f < g) from 15 animals. These data show that DPOAE phase for the near group is well described by their variation with DPOAE frequency. This holds true for both stimulus configurations f > g and f < g. Variations in stimulus frequency that do not correspond with changes in DPOAE frequency (i.e., varying g_i − g_j while fixing g_i + g_j) have no explanatory contribution to the observed DPOAE phase. This observation is consistent with the good fit of the straight line to the phase-versus-frequency data (Fig. 4B), which leaves little residual variance to be accounted for.

In summary, the near group results from the combination of either two or three stimulus components. The “combinatorial effects” of these components explain both the number of DPOAE components and the emergence of two subgroups whose magnitudes are 6 dB apart. There is little difference between sweeping one primary across a set of frequencies and presenting the whole set together as a tone complex. In this respect the near group is similar to the far group. The “off-diagonal” (i ≠ j) components permit an analysis of the DPOAE phase that goes beyond their dependence on DPOAE frequency. This analysis shows that, within the range of frequencies tested, the phase of DPOAE components only varies with those variations in stimulus frequencies that correspond to changes in the DPOAE frequency.

The “Sideband Group”

The third and final group of DPOAEs does not have a two-tone equivalent; it only arises when the stimulus consists of more than two frequency components. It consists of a set of sidebands around the single stimulus component f (e.g., diamonds in Fig. 1A, B) at frequencies f + g_i − g_j, with i ≠ j.

As before, the origin of the DPOAE components (in terms of the stimulus components from the tone complex) is illustrated by considering the matrix (g_i,g_j) (Fig. 5C). Each i ≠ j combination results in a DPOAE component, and the zwuis character of the stimulus ensures that these components are all unique (see Fig. 5D). Because (g_i − g_j) ≠ (g_j − g_i), the combinatorial effect described for the near group does not apply here: no subgroups of DPOAEs arise. DPOAE components for which i = j coincide with the stimulus component f, and cannot be resolved in the spectra of the recorded signals. Thus an M-component tone complex results in M²−M = M(M−1) different DPOAE components within the sideband group.

Since the DPOAE frequencies of the sideband group depend on the frequency difference g_i − g_j, a collective frequency shift of the entire tone complex g does not affect the frequencies of the DPOAEs. Recently, we used this property to study reverse intracochlear propagation of otoacoustic emissions in the gerbil ear (Meenderink and van der Heijden ²⁰¹⁰). In the phase analysis of the sideband group, we distinguish the two cases f > g (single tone above tone complex) and f < g (single tone below tone complex).

DPOAE Phase; f > g

As an example, Figure 8A shows phase data for the sideband group of DPOAEs as a function of DPOAE frequency. These data correspond to the amplitude spectrum in Figure 4A, and were obtained using a stimulus for which f > g. A straight line (Eq. 4) fitted to these data yields a group delay of 536 ± 13 μs. As was the case for the near group, the phase data can be rearranged in a two-dimensional format (Fig. 8B) with DPOAE frequency along the abscissa and non-DPOAE frequency variations represented along the ordinate. Similar to the near group, the contour lines are vertical, indicating that the variation in DPOAE phase only occurs with those changes in stimulus frequency that cause the DPOAE frequency to change. Fitting Eq. 6 to these data (Fig. 8C) yielded τ₊ = 22 ± 14 μs and τ₋ = 536 ± 13 μs. Note that compared to the near group, the roles of τ₊ and τ₋ are reversed: this time it is the τ₋ coefficient that represents the group delay along the DPOAE frequency axis. A scatter plot (Fig. 8D) of group delays for phase data obtained from all 15 animals (N = 1,065) generalizes this observation. DPOAE phase varies systematically with DPOAE frequency (captured in τ₋), whereas non-DPOAE-related frequency variations (quantified by τ₊) do not improve the explanatory power of the linear model Eq. 6 for stimulus conditions with f > g.

DPOAE Phase; f < g

From the DPOAE amplitude spectra, no differences are apparent between the sideband DPOAEs obtained with f > g (Fig. 1A, diamonds) versus those evoked with f < g (Fig. 1B, close diamonds). The phase data, however, do reveal a contrast between the two stimulus configurations. As an example, Figure 9A, B show amplitude and phase spectra obtained with f < g. As before, the phase data change with DPOAE frequency, and this trend is largely captured by a straight line fit to these data. On closer inspection, however, the phase also show a systematic patterning along near-vertical lines; a trend that cannot be explained by their dependence on DPOAE frequency alone. Rearranging the phase data in the two-dimensional format as described earlier (see Fig. 5D) clarifies the nature of the dependence (Fig. 9C). This time, the contours are not vertical, but tilted clockwise. The tilting indicates that, although a major portion of the phase variation is explained by the DPOAE frequency, phase is also affected by the stimulus frequencies in a different way. This is quantified by fitting Eq. 6 to these data, which yielded τ₊ = −113 ± 20 μs and τ₋ = 642 ± 19 μs. The resulting plot (Fig. 9D) clearly shows non-vertical contours that reflect the nonzero group delay τ₊. Analysis of a total of 298 recordings obtained in 13 gerbils generalizes this observation. Unlike the corresponding analysis for the f > g case (Fig. 8D), the scatter plot of the group delays τ₊ and τ₋ (Fig. 9E) is no longer distributed around the line τ₊ = 0. Instead, it shows a systematic trend towards negative group delays for the non-DPOAE-related frequency components (τ₊ < 0). In this respect, these DPOAEs (i.e., sideband group obtained with f < g) are different from all other groups, including the sideband DPOAEs obtained with f > g.

Theoretical Analysis of the Stimulus Dependence of DPOAE Phase: Residual Phase Effects

The multitone stimuli provided an extra degree of freedom in the analysis of DPOAE phase beyond the straightforward, one-dimensional graphs of DPOAE phase versus DPOAE frequency. In the two-dimensional phase plots (Figs. 7–9), the phase gradients in the horizontal direction correspond to the straightforward dependence on DPOAE frequency, which is the dominant effect in all of the phase data of the near and sideband groups. The two-dimensional phase data, however, also allowed the assessment of phase gradients in the vertical direction, which may be interpreted as data obtained with a “fixed-f_dp” recording paradigm. These gradients allowed us to assess whether DPOAE phase changes systematically when the stimulus frequencies (g_i, g_j) are “varied” in a way that keeps the DPOAE frequency fixed. (They are not really varied, because the various frequencies g₁…g_M are all presented simultaneously. For ease of explanation, however, we will discuss the data as if collected while varying (g_i,g_j) according to a fixed-f_dp sweep.)

Because the dependence on DPOAE frequency (i.e., the phase gradient along the horizontal direction) is the main effect in our data (often it is also the only dependence observed), we will term the second type of DPOAE phase dependency the residual phase dependency. It is an interesting type of dependence, because travel times common to all components (stimulus and DPOAEs alike) do not contribute to it; they are absorbed into the main effect. As explained below, this makes the residual phase dependence a sensitive probe of the details of DPOAE generation.

Consider a generic model of DPOAE generation, in which the phase of a stimulus component f at a certain location x the cochlea is described by a function Φ(f,x), and the generation site of a third-order DP component f_dp is X_G(f₁,f₂,f₃; f_dp). No specific assumptions are made on the functions Φ and X_G. In the Appendix it is shown that, to first-order approximation, the phase of the DP at its place of generation is given by

where κ and τ, and Φ₀ are constants. Consider now a particular joint variation of the stimulus frequencies that leaves f_dp fixed, i.e. Δf_dp = 0. For such fixed-f_dp variations, the only term contributing to variations in Φ_dp is the second term of Eq. 7, resulting in

Therefore, what we have called the residual phase dependence, directly corresponds to shifts ΔX_G of the cochlear location X_G at which the DP is generated. If the generation site is unchanged (ΔX_G = 0), fixed-f_dp “sweeps” will leave DPOAE phase unchanged. Conversely, when a residual phase effect is observed, it signals a shift of the generation site of the DPOAE.

With our particular stimulus design, two-dimensional phase representations were possible for two DPOAE types (near and sideband groups) in two stimulus configurations (f > g and f < g). The analysis of DPOAE phase in the resulting four situations is summarized in Table 2. The only case showing a systematic residual phase effect was the sideband DPOAE when f > g (lower right entry of the table). In order to relate this observation to the theoretical analysis of the residual phase effect, it is necessary to consider how, for each of the four situations listed in Table 2, the site of DPOAE generation is affected by the fixed-f_dp “sweeps”. To this end, we make the following assumption: When the nonlinear interaction of three stimulus frequencies (f₁,f₂,f₃), produces a DP at frequency f_dp, the site of DP generation corresponds to the region with characteristic frequency f_max = max(f₁,f₂,f₃,f_dp). In other words, DP generation is at the most basal best site among the four frequency components. We will refer to this as the assumption of the most basal generation site.

The assumption of the most basal generation site is a generalization of the evidence that, for two-tone stimuli, the 2f₁ − f₂ DP is generated near the f₂ site, whereas the 2f₂ − f₁ DP is generated near its own best site (Brown and Kemp ¹⁹⁸⁴; Martin et al. ¹⁹⁸⁷). The assumption is further motivated by the extremely steep high-frequency flank of mechanical tuning (Robles and Ruggero ²⁰⁰¹), implying that tones (whether presented acoustically or generated inside the cochlea) barely propagate in cochlear regions more apical than their own best site.

The schematic diagrams in Figure 10 depict the overlapping cochlear excitation patterns for each of the four different situations of Table 2. The generation site of DPOAEs (as predicted from the assumption of most basal generation) is labeled X_G in each case. The fixed-f_dp “sweeps” that underlie the analysis of the residual phase effect are indicated by the arrows placed over the peaks. These arrows indicate the joint shift in primaries g_i and g_j that keep f_dp fixed. Notice that the requirement of a fixed f_dp calls for different types of joint shifts. For the near group (upper two panels), where [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {f_{\rm{dp}}} = {g_i} + {g_j} - f $$\end{document}] , primaries g_i and g_j must move in opposite directions; for the sideband group (lower two panels), where [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {f_{\rm{dp}}} = f + {g_i} - {g_j} $$\end{document}] , the primaries g_i and g_j must move in the same direction. A straightforward application of the above assumption of the most basal generation site now immediately leads to the result that the only situation in which a fixed-f_dp “sweep” causes a shift of the intracochlear location of DP generation X_G, is the sideband group with f < g (lower right panel of Fig. 10). This is consistent with our observation that the sideband group with f < g is the only one showing a residual phase effect (Table 2).

Appendix

Denote the phase of the cochlear vibration evoked by a single tone by

where f is the frequency of the tone and x is the cochlear location. No specific assumptions are made regarding this function, except sufficient smoothness. Consider a triplet of stimulus tones at frequencies f₁, f₂, and f₃, which need not be all different. The frequency f_dp of a third-order DP produced by the nonlinear interaction of the three tones is where w_k = ±1 are weight factors. For the three DPOAE groups considered in this study (e.g., far, near, and sideband groups), the weight factors always add up to unity:

With the assumption that a DP component is generated at a single cochlear location X_G, its phase Φ_dp at this location becomes

In general, X_G will be a function of all three primary frequencies and the DP frequency, f_dp:

For sufficiently narrowband signals, the phase of a frequency component f close to f₀ at a place x close to x₀ is well described by the first-order approximation

where the constants [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \kappa = \partial \Phi /\partial x({x_0},{f_0}) $$\end{document}] and [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tau = - \partial \Phi /\partial f({x_0},{f_0}) $$\end{document}] are the average wave number (inverse wavelength) and group delay, respectively. Similarly, if the primary frequencies and DP frequency are sufficiently close together, one may approximate where X_G(f₀) is shorthand for X_G(f₀,f₀,f₀;f₀).

The combination of A4, A5, and A6 yields

where various constant terms were absorbed into Φ₀.

Equation A7 shows that variations in primary frequency f_k affect DP phase in two possible ways. Variations in f_k that cause f_dp to change result in a “direct” effect on Φ_dp via the last term. This simply reflects the average group delay τ of the collective incoming traveling waves toward the generation site. It is the main effect of changing the primary frequencies, which incorporates the travel times toward the generation site. The main effect does not require the generation site X_G to shift with the variation in stimulus frequencies. The other two contributions to changes in Φ_dp, the second and third terms of A7, occur through shifts of the generation site X_G. Both terms contain the factor κ, the wave number of the traveling waves at the generation site. The third term describes the changes of X_G with f_dp; this contribution merges into the main effect of the fourth term. The second term is the only term that can cause phase changes mediated by fixed-f_dp variations of the stimulus frequencies. This establishes the link between residual phase effects and shifts of the location of DP generation.

All stimulus components were chosen to have an even number of periods (rows 1–2, with integer k) over an integer number of sample points (typically 97,656). This results in the third-order DPOAEs of the different groups (rows 3–5; see text) to be periodic over the same number of samples, and never coincide with any stimulus component. The choice for the number of periods also rules out any coincidences across members of the three groups and the fourth group (bottom row) of the third-order DPOAEs arising from interaction of triplets (g_i, g_j, g_k) within the tone complex

Each cell displays mean (±standard deviation) of the fixed-f_dp group delay (see text). The numbers between parentheses give the number of recordings. Only for the sideband group, recorded with f < g, does the residual phase contribute to the observed DPOAE phase variation