Our results reveal that the SFOAE and loudness iso-suppression curves have a very similar slope and contrast with the distinctly shallower slope of the DPOAE(L2) iso-suppression curve. Because L1 has to grow 0.5 dB per 1 dB of L2 increase (Fig. 1) to achieve maximum 2F1–F2 level, the iso-suppression curve for DPOAE compares better with the other two when converted so that L1 becomes the varied parameter (utilizing the linear fit to the mean data shown in Fig. 1; L1 = 0.52*L2 + 37.1 dB). The resulting DPOAE(L1) iso-suppression curve presents a noteworthy alignment with the loudness and SFOAE iso-suppression curves (Fig. 3B). The slope of DPOAE(L1) was 0.43 dB/dB when all subjects were included, and 0.40 dB/dB for the subset of subjects with SFOAE data. (The latter slope was not statistically different from the corresponding SFOAE iso-suppression slopes; mean, 0.41 dB/dB, t-test: T11 = − 0.079, p = 0.94). The better alignment might indicate that the F1-tone is of primary interest when studying DPOAE suppression. This is plausible because the 2F1–F2 emission is predominantly generated in the BM region surrounding the peak of the F2 TW [32], and this peak lies within the active region of the F1-TW (given F2/F1 = 1.2). One could conceive that this DPOAE component is produced by the F2-tone interfering with the amplification of the F1-tone, which would get modulated at their beat frequency and so, intermodulation products at 2F1–F2 and F2 are produced.

Fig. 3

A Required biasing tone levels to achieve criterion suppression in the subjective loudness of pip trains as a function of pip-train sensation level (dB SL) for 1-kHz (blue) and 2-kHz (red) tone pips. Individual markers correspond to those in Table 2. (Note that this is a different subject group from that of the OAE experiments). Bold lines show mean data. The slope of the grid is the same as in Fig. 2 (0.22 dB/dB). B The comparison of average iso-suppression curves shows good alignment across all three experiments when DPOAE data are plotted as a function of L1. Loudness iso-suppression data for both carrier frequencies were averaged, and probe levels were converted to dB SPL by adding the average of the mean thresholds for the 1-kHz (30.3 dB pSPL) and 2-kHz (33.3 dB pSPL) pip trains to the sensation levels. Grid lines have a slope of 0.4 dB/dB

Although it is per se not clear how the 9-dB criterion for OAE suppression relates to the galloping criterion for loudness suppression, the similarity between the BT levels achieving criterion suppression for DPOAE(L1) and loudness shows that both criteria require displacement biases that are within 2 dB of each other (which was rather coincidental, since the choice of the 9-dB criterion was somewhat arbitrary). The same also holds for the 9-dB SFOAE suppression, assuming the SFOAE iso-suppression curve would continue to grow linearly at ~ 0.4 dB/dB.

Comparison with Iso-Suppression Curves Obtained with Low-Side Suppressors > 200 Hz

Iso-suppression data of OAE by low-side suppressors (i.e., with a frequency below that of the probe) have been obtained abundantly in the past, and there also exist loudness suppression data. We compare our data here, however, only with iso-suppression measurements which utilized suppressor tones at least one octave below the probe so that the BM response to the suppressor can be assumed to grow linearly at the location of the peak and active region of the probe’s TW (like the BM response to a BT; see the “Iso-suppression curves and the non-linear growth of basilar membrane vibrations” section for more details). The difference between biasing experiments and these low-side suppression experiments is that the suppressor cycle in the latter is too short to either lead to phasic suppression or to resolve it, so that the average suppression over a suppressor cycle is analyzed.

Keefe et al. [33] obtained SFOAE suppression tuning curves (STCs) for a wide range of primary levels and also reported iso-suppression curves (3-dB suppression depth) as a function of primary level (30–60 dB SPL) with off-frequency suppressor tones that were an octave below the primary tones. They were linear, like ours, and had a slope of 0.4 dB/dB with a 1-kHz primary tone and 0.28 dB/dB with a 2-kHz primary tone, which we did not observe in our SFOAE biasing data.

Gorga et al. [34] obtained DPOAE(L2) iso-suppression curves with slightly steeper slopes of 0.26 dB/dB (F2 = 4 kHz) than those observed in our study (0.22 dB/dB), but still clearly shallower than the average slope of the SFOAE iso-suppression data reported by Keefe et al. [33]. Later, Gorga et al. [35] reported even steeper DPOAE(L2) iso-suppression slopes of 0.41 dB/dB for F2 = 500 Hz and 0.31 dB/dB for F2 = 4 kHz. The study by Gorga et al. [36] extended the range of tested probe frequencies and reported slopes of 0.2 dB/dB at 8 kHz. The reason why this dependence on probe frequencies is absent in our data is unclear. The optimization of L1 for every L2 and F2 combination could have been an explanation, but this was also done by Gorga et al. [35, 36].

Related to our loudness iso-suppression data is the classic study by Wegel and Lane (1924) [37], who observed that for tones > 2 kHz to remain audible during masking, their levels have to be increased by 2.4 dB per 1-dB increase in the level of a 400-Hz masker tone. In other words, a masker level increase of 1/2.4 = 0.42 dB is required to make a 1-dB increase in the probe level just inaudible again. This value is comparable to the slope of our average loudness iso-suppression curve (0.47 dB/dB).

Dewey and colleagues [5] measured two-tone suppression curves for the mechanical responses of BM and RL, which had almost identical tuning for near CF probe tones. Suppressor tones more than one octave below CF needed to be increased by ~ 0.5 dB per 1-dB probe level increase to maintain the probe-suppression criterion of 1.5 dB. This value is similar to the slopes of the SFOAE and loudness iso-suppression reported here and indicates that both are likely rooted in the suppression of mechanical amplification of the probe’s TW.

In summary, the BT level required to maintain equal maximum probe suppression within the suppressor cycle as the probe level increases is not substantially different from those reported for suppressor tones above 100 Hz for equal average suppression, as long as their frequencies were at least an octave below the probe frequency.

Iso-Suppression as a Function of Probe Frequency

It must be noted that the 8.9 dB/octave slope in probe-frequency dependence that we found for loudness iso-suppression disagrees with our previous findings of just ~ 3 dB/octave obtained with the same loudness iso-suppression technique [20]. Although our in situ calibration may cause inaccurate absolute levels at the eardrum above ~ 2 kHz, this cannot explain the striking difference between these two studies, which used identical in-ear probes and in situ calibration. We hypothesized in our previous paper that the shallow slope of loudness iso-suppression is seen because very low-frequency BT suppression might be caused by OHC-generated electrical potentials, which decay with a lower spatial gradient toward the cochlear base. Note, however, that the BTs in [20] were lower than 40 Hz, so that the BM moved in-phase along its entire length and thus likely created larger potentials. Since the intracochlear potential saturates at high stimulus levels [38], we cannot be sure that BTs lower than 55 Hz would produce the same loudness iso-suppression slopes as a function of probe level as found in our present study.

The relatively steep primary-frequency dependence of both OAE iso-suppression curves (of ~ 7–8 dB/octave on average) is in line with that previously observed for DPOAE iso-suppression (8.4 dB/octave, for BTs < 40 Hz; [20]) and is thought to reflect the spatial slope of the BM excitation pattern caused by the BT [20]. The fact that with a 55-Hz BT the probe-frequency dependence of loudness iso-suppression was similar lets us conclude that the loudness suppression in this study is caused by mechanical effects on the OHC MET. It is unlikely that biasing of the velocity-sensitive inner hair cell MET by the 55-Hz BT played a role in the observed loudness suppression.

Iso-Suppression Curves and the Non-Linear Growth of Basilar Membrane Vibrations

Researchers have commonly explained the shallow slope of OAE iso-suppression curves as a function of probe level by the compressive growth of the BM response to the probe that contrasts with the linear growth of the response to the suppressor tone at the location of the probe’s characteristic place. So, BM compression estimates were made from these data (e.g., [33, 34]). Also, the mentioned psychoacoustical finding by Wegel and Lane [37] has been interpreted in this way [39]. Of course, the suppressive interaction between tones most likely takes place at the MET of the OHC stereocilia, located at the RL, for which motion data have become recently available with OCT technology (for a review, see Olson et al. [40]). Thus, making a link between suppression data and compressive BM growth is somewhat of a simplification. However, the interpretation of the OCT-based data from within the organ of Coti is still in flux [40] because structures of the organ of Corti, including the RL, apparently move in all three spatial dimensions, but phase-sensitive OCT technology records the motion projected onto the optical axis of the beam. So, recently available semi-transverse RL data are likely poor estimates of the stimulus to the METs, which are activated by shear motion between the RL and the TM [5]. Thus, despite available RL data, some researchers (e.g., [41]) still prefer BM data when conveying basic concepts of cochlear function, as their interpretation of being largely transversal has not been challenged. We will also use the classic simplification of considering BM displacement to relate our suppression data to cochlear mechanics, although it has been shown that RL suppression of characteristic frequency (CF) tones is similarly tuned to the BM [5].

Figure 4 illustrates schematically the TW envelopes for the tones involved in our DPOAE biasing experiment. The envelopes are shown only in the longitudinal section from just basal to the active region of the F2 TW to where the F1 TW has decayed by more than 40 dB (i.e., a range of ~ 2 octaves of CF). They are drawn by assuming (a) linear growth in the tail, (b) compressive growth near their peak determined from our data in Figs. 1 and 2B, and (c) identical F1 and F2 responses at the location of the F2–TW peak. Thus, on the left (basal) side of the illustration, the BM vibrations grow for all tones linearly with stimulus level, so that the sound pressure levels (taken from our results shown in Figs. 1 and 2B) are scaled correctly on the vertical BM displacement axes in decibels. The blue curves illustrate the TW envelopes in response to the F1-tone and those in red in response to the F2-tone. The growth of these TWs with stimulus level becomes increasingly compressed as they progress to their characteristic place, culminating here in a rate of 0.2 dB per dB of stimulus level increase, a value taken from the slope of our DPOAE(L2) iso-suppression curve (Fig. 2B).

Fig. 4

Schematic showing three triplets of TW envelopes that have equal displacement at the location of the TW peak in response to the F2 tone (red) along a section of the BM. The TWs of the F1 primary tone are shown in blue. The black lines show the BM displacement bias magnitude in response to the BT. The triplets are illustrated for L2 levels of 30 dB SPL (dotted line), 45 dB SPL (dashed line), and 60 dB SPL (solid line). Values in green show the primary level differences that on average produced the maximum 2F1–F2 emission (Fig. 1). Values in blue give the corresponding L1 levels (in dB SPL). Values in black are the BT levels (in dB SPL) that on average produced a 9-dB suppression of the 2F1–F2 emission (Fig. 2B). Locations labeled “CP” are the characteristic places where the BM is most sensitive to either the F1 or F2 primary tone. Note that the upper end points of the green brackets touch the F1–TWs accordingly more apically

Based on the widely accepted assumption that the 2F1–F2 emission reaches its maximum level when the primary tones produce equal displacement at its generation site, so that the amplitude modulation due to beating of the primary responses is 100% (reviewed [42]), the F1–TW shapes (blue) were drawn to fulfill this condition at the F2–TW peak, the approximate center of the DPOAE generation region. This is, of course, a simplification, because the DPOAE is not generated exactly at the F2–TW peak but in the surrounding it, and, in reality, the condition of equal TW amplitudes is likely a weighted compromise within this region. In accordance with the growth of the optimum L1 (Fig. 1), their amplitudes increase at half the rate of the F2–TW amplitude in the linear region (left-hand side). Motivated by the slope of our DPOAE(L1) iso-suppression curve, the growth of F1–TW amplitude at the location of the F2–TW peak shows a compression rate of 0.4 dB/dB. Measurements in gerbil cochleae agree with such spatial build-up of compression, showing that compression has not yet reached its maximum at the location where a tone of 1.2 × higher frequency would peak but is roughly just half of that at the TW peak (Fig. 6 in [41]; although with lower overall compression). Note that the TW peaks shift basally with an increasing tone level (e.g., [5, 43], Fig. 2), and we assume here that the region of DPOAE generation.

The schematic illustrates that the difference in growth rates of L1 and L2 required to maximize the 2F1–F2 DPOAE might be, in the first approximation, a compensation for the differing compression rates of the F1- and F2–TWs in the DPOAE-generation region surrounding the F2–TW peak. The absolute compression rates of the F1- and F2–TW growths in the F2–TW peak regions might be revealed by a low-frequency suppressor tone that produces a linearly growing TW amplitude within the DPOAE-generation region. The linearly growing magnitudes of the three suppressor displacements that caused 9-dB DPOAE suppression for the three primary level pairs (from Fig. 2B mean data) are shown as black lines. Although their amplitudes at the F2–TW peak are shown here equal to those of the primary-tone TWs, we actually assume only that the 9-dB DPOAE suppression is maintained when the suppressor amplitude is kept proportional to the amplitudes of the primary TWs at the F2–TW peak. The vertical spacing of the suppressor TWs shows that the suppressor level needs to rise by just 6 dB for a 30-dB increase in L2 for this condition to hold. This corresponds to a 0.2 dB/dB compression of the F2–TW growth at its peak, while an L1 increase of 15 dB corresponds to a ~ 0.4 dB/dB compression of F1–TW growth at the same location. Comparable compression rates have been derived from BM measurements in sensitive, pristine cochlear preparations in response to a tone at the CF of the measurement location and to a tone with a frequency a factor of 1.2 below the CF [44].

The similar slopes of SFOAE and loudness iso-suppression to the DPOAE(L1) iso-suppression slope indicate that both SFOAE suppression and loudness suppression are also taking place in the active region of the probe-tone TW, where the suppressor tone likely hampers its amplification. Recent experiments by Goodman and colleagues [45] confirm that the SFOAE is largely generated in the active region of the primary’s TW. Evidence for a SFOAE generation site slightly basal to the primary’s characteristic place also comes from suppression tuning curves (STC) obtained by Keefe and colleagues [33], showing that the lowest suppression thresholds were obtained with suppressor tones slightly higher than the probe’s frequency. BM and RL responses to a CF tone are also most efficiently reduced by a suppressor tone that has its characteristic place in the active region of the CF tone [5]. Not surprisingly, the same is found in the auditory nerve [46]. In summary, only the DPOAE(L2) iso-suppression slope (~ 0.2 dB/dB) can be thought to reflect the compression rate at the TW peak because the F2–TW peak is located in the active region of the F1 tone.

The association between the compression rate in the growth of the BM response in the active region and the slopes of both the loudness iso-suppression curves and the OAE iso-suppression curves assumes that equal suppression occurs when suppressor vibration grows proportionally to probe vibration in the active region of the probe’s TW. The physiological reason why this should be the case is not obvious to us. Furthermore, the compression rate of BM growth typically increases with stimulus level, whereas iso-suppression slopes (also those published by others) are fairly invariant with probe level. Thus, confirmation by simultaneous measurements of OAE suppression and direct BM displacement in an animal cochlea is still required to confirm that the slopes of these iso-suppression curves indeed reveal rates of cochlear compression.