Probing hair cell’s mechano-transduction using two-tone suppression measurements

When two sound tones are delivered to the cochlea simultaneously, they interact with each other in a suppressive way, a phenomenon referred to as two-tone suppression (2TS). This nonlinear response is ascribed to the saturation of the outer hair cell’s mechano-transduction. Thus, 2TS can be used as a non-invasive probe to investigate the fundamental properties of cochlear mechano-transduction. We developed a nonlinear cochlear model in the time domain to interpret 2TS data. The multi-scale model incorporates cochlear fluid dynamics, organ of Corti (OoC) mechanics and outer hair cell electrophysiology. The model simulations of 2TS show that the threshold amplitudes and rates of low-side suppression are dependent on mechano-transduction properties. By comparing model responses to existing 2TS measurement data, we estimate intrinsic characteristics of mechano-transduction such as sensitivity and adaptation. For mechano-transduction sensitivity at the basal location (characteristic frequency of 17 kHz) at 0.06 nm−1, the simulation results agree with 2TS measurements of basilar membrane responses. This estimate is an order of magnitude higher than the values observed in experiments on isolated outer hair cells. The model also demonstrates how the outer hair cell’s adaptation alters the temporal pattern of 2TS by modulating mechano-electrical gain and phase.


Methods
Nonlinear model in the time domain. A nonlinear model that incorporates cochlear fluid dynamics, OoC mechanics and OHC electrophysiology was developed from a previous linear model 45 . This model takes into account the interaction between intracochlear fluid and the OoC complex, i.e. the OoC, the basilar membrane and the tectorial membrane. The transduction current saturates with large hair bundle deflections, which is the main nonlinear component. Unlike our previous studies that used a finite difference method in the fluid domain and a finite element (FE) method in the structural domain, a FE method was applied in both the structural and fluid domains for the new model. The global equations were solved in the time domain. Figure 1 shows a schematic of the cochlear model. A long thin tube filled with fluid (12.1 mm long and 0.6 mm high), which represents the uncoiled gerbil cochlea, is divided into two compartments by the OoC. The upper compartment represents the scala vestibuli and scala media, and the lower one represents the scala tympani. The fluid spaces are connected through the helicotrema (Fig. 1A). As our model does not incorporate the middle ear, stimulations are applied at the oval window in the upper compartment near the basal end of the cochlea, and the round window in the lower compartment serves as a pressure release. The rest of the chamber boundaries are stationary. The x-, y-and z-axes correspond to the longitudinal, radial and transverse directions, respectively. The fluid domain is approximated as 2-D in the numerical simulation to save computational costs.
The structures in the OoC complex are modeled by beam elements (Fig. 1C). The radial section (Fig. 1B) is repeated every 10 µm along the cochlear length. The radial sections are coupled through longitudinal elements of the basilar membrane, the tectorial membrane, the reticular lamina and the Y-shaped structure associated with the OHC and the Deiters' cell. The medial edge of the basilar membrane is hinged, while the lateral edge is clamped. The tectorial membrane is clamped at the spiral limbus along its medial edge, and its lateral edge is hinged to the tips of the OHC hair bundles. The OoC complex model interacts with the active OHCs through the cells' MET channel kinetics and electromotility (Fig. 1D). The MET current due to the OHC hair bundle deflection is the input to the electrical circuit of the OHC. The active forces generated from both hair bundle and somatic motility modulate the vibration of the OoC complex. While the fluid and structural mechanics are considered linear, nonlinearities of the OHC physiology were incorporated, such as the saturation of MET current and voltage-dependent membrane conductance 46 . We observed little effect of the latter nonlinearity on the mechanics, because under our simulated conditions the OHCs' membrane potential did not change enough (<5 mV at CF = 17 kHz) to cause nonlinear changes in the capacitance/conductance. Fluid-structure interaction. After assuming the fluid is inviscid and incompressible, the Naiver-Stokes equation reduces to 2 where ∇ 2 is the Laplacian operator and p is the fluid pressure. The boundaries of the fluid domain are as follows.
The pressure at the oval window was given as an input. The pressure at the round window was set equal to zero. All other boundaries were considered rigid. There are two deformable fluid-structure boundaries-the upper and lower surfaces of the OoC complex. The midpoint of the basilar membrane and the lateral end of the tectorial membrane represent the midpoint of the top and bottom fluid interacting surfaces, respectively, because those two points are approximately in the middle of the radial span. At those boundaries, the pressure gradient normal to the surface is proportional to the accelerations of the surfaces (a TM and a BM ): TM TM BM BM 32 32 Because the fluid domain is 2-D and the structural domain is 3-D, conversion factors are needed. The conversion factor α 32 = 0.6 was determined by approximating the deforming pattern of the interacting surfaces along the radial direction as a half sine wave. The fluid forces f Fluid on the tectorial membrane and the basilar membrane were calculated from the following equations: where p TM and p BM are the pressure differences across the fluid-structure interfaces, and S TM and S BM represent the effective interacting surface areas of the top and bottom surfaces of the OoC complex. Since radial-and longitudinal-running elements divide the TM and the BM into tiles, the effective area for a node is the sum of the areas of its 4 adjacent tiles (2 tiles for the nodes on the edges) divided by 4. The conversion factor α 23 = 0.6 is to convert the fluid pressure to equivalent resultant forces acting at the lateral edge of the tectorial membrane and the centerline of the basilar membrane. The fluid within the OoC complex was not explicitly modeled. Instead, the fluid pressure inside the OoC complex was assumed to be the mean of pressure on both interfaces.
OHC channel kinetics and electro motility. OHCs generate active forces through hair bundle (f MET ) and somatic (f OHC ) motility, which are dependent on the MET channel open probability (p o ) and membrane potential (V m ). The channel kinetics are obtained from previous studies based on the gating spring theory 47 . The MET channel's status is governed by first-order kinetics, www.nature.com/scientificreports www.nature.com/scientificreports/ where k CO and k OC are the rate coefficients defined from the elastic potential energy of a putative gating spring with spring constant k G .
where k B is the Boltzmann constant, T is the absolute temperature, A 0 is a rate constant, γ is the geometric gain, b is the gating swing, k G is the gating spring stiffness, x HB is the hair bundle displacement, x A is the bundle displacement due to adaptation, and x 0 is a constant. The MET sensitivity σ near the resting open probability p o,rest is defined as For instance, according to this equation, a hair cell operating near p o,rest = 0.5 is 2.8 times more sensitive than operating near p o,rest = 0.9.
The speed of MET adaptation is determined by a rate constant k A such as where k E is the combined stiffness of the extent springs 48 . The transduction channel introduces a force (f MET ) proportional to the change of open probability.
f MET,max is the maximum gating force. In the FE model, f MET is applied as a coupled force at the tip and root nodes of a hair bundle so that the bundle deflects in the excitatory-inhibitory direction.
The electrical potential difference between endocochlear potential E P and the membrane potential V m determines the MET current I MET , or where C s and G s are the capacitance and conductance of the hair bundle, respectively. The stereocilia conductance is proportional to the maximum saturating conductance and to p 0 . For the basolateral membrane, it was assumed that the membrane equilibrium potential is maintained at E k . Then, the basolateral current I m is where C m and G m are the capacitance and conductance of the OHC basolateral membrane, respectively. The OHC somatic force f OHC is proportional to the change of membrane potential: OHC OHC m m rest , V m,rest is the membrane potential at rest and α OHC is the electromechanical gain.
Governing equations in matrix form. The governing equation for the fluid pressure is discretized and expressed in matrix-vector form: pp pa A pp corresponds to the Laplace operator. A pa a represents the boundary conditions at the OoC complex-fluid interacting surfaces, and a is the acceleration vector of the OoC complex interacting nodes. b represents the boundary conditions of other boundaries including the oval and round windows.
The OoC complex structures represented by mass (M), damping (C) and stiffness (K) matrices are subjected to a fluid force (f FLD ), and two OHC active forces (f MET and f OHC ). M and K are the assembly of elemental mass and stiffness matrices, and C is a linear combination of M and K (Eq. 15). f FLD is determined by the pressure input at the stapes and a matrix A ap relating pressure to forces at structural degrees of freedom at the interacting surfaces (Eq. 16). f MET and f OHC are the collection of OHC active forces in each longitudinal section (Eq. 9 and 12). The global equation FLD M ET OHC is used to solve for the structural displacement vector (x). Rayleigh damping was used to form the damping matrix, where α c = 0 ms and β c = 20 ms −1 . The fluid pressure determines the nodal force vector acting on the OoC complex surfaces:

FLD a p
Since no time derivatives of the pressure appear in the governing equations, the pressure was substituted using Eq. 13: (2019) 9:4626 | https://doi.org/10.1038/s41598-019-41112-5 www.nature.com/scientificreports www.nature.com/scientificreports/ FLD a p pp p a 1 Thus, the combined governing equation for the fluid and structure domains (Eqs 13 and 14) is: where M EFF and f EFF are the effective system mass and load, respectively:  (Tables S1 and S2).

Results
To validate our model, we compared both pure tone and impulse responses with existing measurements. It should be noted that our model presents the gerbil cochlea, while referenced experimental data are a cohort of different rodent species (gerbil, chinchilla, and guinea pig). To summarize, nonlinear characteristics observed in impulse ( Fig. 2) and pure-tone (Fig. 3) responses were simulated. In Fig. 4, we demonstrate how the saturation of OHC MET can explain the cochlear nonlinearity. Temporal and spatial patterns of 2TS were analyzed to obtain two measured quantifies of 2TS: suppression thresholds and rates (Figs 5 and 6). We show that these two 2TS quantities and the MET sensitivity are monotonically related (Fig. 6). This monotonic relationship made it possible to exploit experimental data to estimate the MET sensitivity (Fig. 7). The timing of the maximum suppression was explained with MET adaptation (Figs 8 and 9).
Nonlinear characteristics of cochlear mechanics in response to impulse and pure tone. Impulse responses at a basal location (approximately 2 mm from the basal end of the gerbil cochlea) were simulated and compared to experimental data (Fig. 2). Because middle ear transmission was not included in the model, simulation input levels were expressed in dB with respect to 20 µPa at the stapes, hereafter written as "dB re. StP". For a rough comparison with experimental data, a 25 dB middle ear gain 49 was assumed. That is, 0 dB SPL roughly corresponds to 25 dB re. StP. The basilar membrane responses to a 20 µs click and a 17 kHz tone at 65 dB re. StP show characteristics observed in experiments (Figs 2 and 3). For an impulse response, the main response lobe lasts for approximately 1 ms and the response amplitude initially increases and then gradually decreases. The basilar membrane oscillated for more than a dozen cycles before decaying below 5% of the peak displacement. Such a large number of oscillatory cycles, comparable to experimental measurements 50,51 , is a signature of highly tuned systems. Secondary lobes were considered another sign of a sensitive cochlea since they were absent in cochleae that were surgically damaged or overstimulated 50,52 . The amplitudes of the secondary lobes became smaller as the input level increased as was shown in other modeling studies 53,54 . When passive, the first peak was the greatest and the oscillations died away (below 5% of the first peak) within six cycles. A change of instantaneous frequency (IF) with time, termed the frequency glide 51 , has been observed in cochlear impulse responses 55 . At basal cochlear locations, the IF at the onset of the response is lower than the characteristic frequency (CF) and gradually increases with time ( Fig. 2B). When normalized with the CF, the simulation result compares well with the experimental data from the 18 kHz CF location of the guinea pig cochlea 51 and the10 kHz best frequency location of the chinchilla cochlea 50 .
Level-dependence is another well-known characteristic of cochlear physics. For example, responses to small sounds are amplified and tuned more than the responses to loud sounds. Our model reproduces observed characteristics. In a time-domain response ( Fig. 2A), this level-dependent amplification/tuning appears as increased number of oscillations when the SPL is small. When the cochlea is passive, the basilar membrane oscillates only a few cycles after the impulse and the first peak is the greatest. On the contrary, when the active cochlea is subjected to small sounds, it vibrates a few dozen cycles. The amplitude of the initial peak in an impulse response increases linearly regardless of the input level (Fig. 2C). For later peaks, the slope of the response-SPL curve decreases at high input levels (>95 dB re. StP). The simulated response of the 7 th peak shows a similar level of nonlinearity (in the slope of the amplitude vs. input level) as compared to experimental data (Fig. 2C). The lowest growth rate is 0.11 and 0.15 dB/dB from the simulation and experimental data 50 , respectively. The peak amplitude of the simulated response was 10 to 20 dB smaller than the experimental data. Such differences are ascribed to different species and best frequency locations (17 kHz in gerbil versus 10 kHz in chinchilla).
The level-dependence is also observed in pure tone responses. In Fig. 3, the model responses to a pure tone (17 kHz) at different stimulation levels are shown. Traveling wave patterns at low SPL (45 dB re. StP), high SPL (115 dB re. StP) and in a passive condition (no OHC active forces) are shown in Fig. 3A. As the input level www.nature.com/scientificreports www.nature.com/scientificreports/ increases, the hair cell's MET saturates over a greater span (see Supporting Materials Fig. S2). Thus, the amplification due to OHC active feedback decreases and the spatial envelope asymptotes to the passive condition (Fig. 3B). The peak location shifts towards the base as the stimulation level increases, consistent with experimental observations (circles on the curves in Fig. 3B). Simulated basilar membrane displacement at the best frequency location is comparable to a measured result (Fig. 3C). At 115 dB re. StP, the experimental data still shows compressive nonlinearity, which indicates OHC feedback is not yet saturated. The response amplitude gain (Fig. 3C) is comparable to measurements at the basal turn of the gerbil cochlea 56 . Although not presented in the plot, the model response became linear when the stimulation level was >145 dB re. StP. Responses at a more apical location are shown in the Supporting Materials (Fig. S1).
These nonlinear responses originate from nonlinear MET. Displacement amplitudes of impulse and pure tone responses and the amplitude of MET response as a function of the input level were plotted in Fig. 4. The MET response was represented by the amplitude of MET channel open probability (|∆p O | in Fig. 4). For both pure tone and impulse responses, the displacement amplitudes are linear at low SPLs, but becomes compressed at higher levels due to saturation of the MET current (Fig. 4A,C). That is, the slope is 1 dB/dB when the stimulus level is <65 dB re. StP, and is <1 dB/dB for higher stimulus levels. The intersection of the two tangential lines taken at the lowest and highest stimulus levels defines the critical level, which indicates the transition from linear to nonlinear growth in the response amplitude (Fig. 4B,D). For an impulse response, because it takes time for the OHC feedback to develop, the displacement amplitudes of later peaks show stronger nonlinear effects than the ones right after the onset. The nonlinear response indicates the overall effect of MET saturation of the OHCs around the peak responding location. While the MET current of some OHCs is saturated at the peak location, there is room for those OHCs off the peak region to increase the MET current. Also, the energy coming from fluid-structure interaction becomes dominate as the MET is saturated. Therefore, the nonlinearity in amplitude curves are less dramatic than the nonlinearity of a single OHC's MET. www.nature.com/scientificreports www.nature.com/scientificreports/ Two-tone suppression with low-side suppressors. An example case of 2TS was simulated, and its temporal and spatial patterns are presented in Fig. 5. In 2TS, if the suppressor frequency is lower than that of the probe tone, it is called a low-side suppressor. The probe tone response exhibits both tonic (overall averaged suppression) and phasic (in a certain phase relationship with the suppressor tone) suppression with low-side suppressors 38 . The maximum suppression of the probe tone occurs near the largest displacement of the suppressor tone towards the scala tympani (Fig. 5C). The asymmetry of maximum suppression suggests that the hair cell MET saturates more readily when the hair bundle is deflected toward its inhibitory direction. The simulated asymmetry occurs when the resting p o is less than 0.5 (e.g., the model used p o,rest = 0.4). When p o > 0.5, the maximum suppression occurs while the suppressor tone deflects the basilar membrane toward the scala vestibuli. There is a secondary suppression while the basilar membrane is displaced towards the scala vestibuli side. Snapshots of the basilar membrane traveling waves (Fig. 5A,B) were taken when the hair bundle deflection was most (t = t 1 ) and least (t = t 2 ) suppressed. The amplitude of the probe tone response is reduced by 90% and 20% compared to the unsuppressed state in the two cases. The shape of probe tone traveling waves changes depending on suppressor tone level: The shape (envelope) of probe tone traveling waves at the most suppressed state is similar to the shape of pure tone traveling waves at high SPL, while the least suppressed situation is close to pure tone responses at low SPL (Figs 3A and 5B). This change of response shape due to suppressor tone level occurs because the least and most suppressed timings correspond to the least and most sensitive states of MET operation, respectively.
Following the Geisler-Nuttall idea 38 (their Fig. 11), the MET transfer function is shown together with 2TS in Fig. 5C. The decreased MET sensitivity due to large suppressor displacement reduced the active feedback of OHCs, thus the MET current of the probe tone became 'phasic'-the probe tone response amplitude changes depending on the phase of the suppressor tone. Note that the maximum suppression does not coincide with the peak of the suppressor tone (*In Fig. 5C).
The proximity between two tones affects the extent of suppression. According to the literature, the extent of suppression depends on the proximity between the two tones (represented by the frequency ratio), and the level of the suppressor tone 36,37 . To quantify the extent of suppression, 2TS with suppressors at various www.nature.com/scientificreports www.nature.com/scientificreports/ frequencies and levels were simulated. The probe tone response is represented by the amplitude of the basilar membrane, corresponding to the probe tone frequency component. The probe tone response decreases as the suppressor level increases (Fig. 6A). For the suppressor tone to affect the probe tone response, the suppressor tone must be loud enough to affect the MET sensitivity in the region of the probe tone response. There exists a threshold level below which the suppressor tone does not affect the probe tone response. Following Rhode and Cooper 36 , the threshold is defined as the suppressor level (represented by the basilar membrane displacement amplitude) at which the probe response is suppressed by 1 dB. The frequency dependence of the threshold is shown in Fig. 6C. The threshold is 2 nm when the frequency ratio (f S /f P ) is <0.15. As the suppressor frequency (f S ) increases, the threshold remains at a certain level and then quickly rolls off. This characteristic is consistent with experimental observations which show that the threshold amplitude is nearly constant when the suppressor frequency is one octave below the CF and that the threshold decreases as the suppressor frequency approaches the CF 39,57 .
The rate of suppression (ROS), the slope of the curve in Fig. 6A, is another quantity used to characterize 2TS (Fig. 6B). At low SPLs (<65 dB re. StP), the probe amplitude remains unchanged and the ROS is zero. The suppression becomes most effective when the suppressor increases to a certain level. ROS peaks at a certain suppressor level (Fig. 6B). The maximum rate of suppression (max ROS) decreases as the frequency ratio (f S /f P ) increases (Fig. 6D). Similar ROS patterns were reported in auditory nerve responses 33 . Here the suppressor level is not limited (e.g., max ROS occurs at >125 dB re. StP with 1 kHz suppressor). In experiments, however, the highest suppressor level hardly exceeded 100 dB SPL. To compare with experimental data, max ROS was calculated with an upper limit of 120 dB re. StP on the suppressor level in the following sections.
Estimating the MET sensitivity from 2TS threshold and maximum rate of suppression. 2TS data provide a means to investigate hair cell MET because the 2TS occurs as a result of saturating (nonlinear) MET. We simulated the effect of the MET sensitivity on the probe tone response (17 kHz, 65 dB re StP). To adjust the MET sensitivity, the value of b in Eqs 5 and 6 was increased or decreased.
The threshold and max ROS were obtained for different levels of the MET sensitivity (Fig. 7A,B). The MET sensitivity, defined by Eq. 7, has the physical unit of nm −1 . In the present model, the MET sensitivity varies from 0.08 nm −1 (basal end) to 0.02 nm −1 (apical end), and the change of sensitivity applies to all locations (multiplied by the same factor). As the MET sensitivity increased, the suppressor displacement required to saturate the MET www.nature.com/scientificreports www.nature.com/scientificreports/ current became smaller. That is, as MET became more sensitive, the 2TS threshold decreased monotonically. At two asymptotic cases, where the MET sensitivity is zero and infinite, the threshold amplitude should approach infinite and zero, respectively. Also, the slope of the current-displacement curve changed more dramatically over a narrower displacement range with a higher MET sensitivity, resulting in an increase of the ROS. The simulation results were curve-fitted in terms of the MET sensitivity (σ) at the best frequency location of the probe tone, or x = 2 mm, using simple powerlaw functions: Threshold = 0.062 σ −1.31 and max ROS = 718 σ 2.06 . When σ was greater than 0.075 nm −1 , the model became unstable. For a given 2TS threshold and max ROS, the inverse functions were used to estimate the MET sensitivity.
2TS threshold from Cooper et al. 's measurements 37 ranged between 1 and 5 nm with a logarithmic mean of 2.2 nm (n = 9). Geisler and Rhode 38,39 both reported a threshold of 2 nm. When compared to our simulations, these threshold values corresponded to a MET sensitivity of 0.065-0.070 nm −1 (Fig. 7A). Max ROS were derived from the probe amplitude-suppressor level curve if not provided directly. Cooper 37 reported max ROS at around 1 dB/dB. Geisler and Nuttall 38 presented similar results. More recent measurements showed max ROS at 2 dB/dB 57 and at least 2.5 dB/dB 39 at the 8 and 6 kHz CF locations. When compared to our simulations, these max ROS values were equivalent to a MET sensitivity between 0.045 and 0.060 nm −1 (Fig. 7B). The MET sensitivity obtained from the threshold amplitude was slightly higher than the upper limit obtained from the ROS (only 1-2 nm threshold data included).
Since we defined the MET sensitivity using channel open probability, varying σ would change both the current sensitivity (defined as the slope of MET current vs. hair bundle displacement curve) and the saturation displacement (the hair bundle displacement required for 90% saturation). Here we show that 2TS characteristics are mainly determined by the saturation displacement. To isolate the effect of saturation, 2TS responses were simulated with constant current sensitivity (Fig. 7C,D). This means that a lower MET sensitivity was compensated by a larger MET current. Saturation displacement was estimated to be 13-15 nm with threshold and 16-24 nm with max ROS (comparable to 15-22 nm derived from the MET sensitivity estimated in Fig. 7A,B).
Temporal pattern of 2TS explained by MET adaptation. The temporal pattern of low-side 2TS is modulated by the time course in active processes. The activation of the MET channel results in a phase-lag of the MET current (low-pass filter), while adaptation results in phase-lead (high-pass filter) 58 . Together the channel activation and adaptation form a band-pass filter, whose bandwidth is determined by the cut-off frequencies of the activation and the adaptation (Fig. 8A,B). As a consequence of the adaptation, the MET current (I MET ) can lead the hair bundle (x HB ) deflection by 90 degrees at low frequencies (Fig. 8B). When there is no adaptation, I MET and x HB are in phase at low frequencies. www.nature.com/scientificreports www.nature.com/scientificreports/ If the suppressor frequency is lower than the cut-off frequency of the adaptation, the effect of the high-pass filtering will be prominent in the temporal pattern, especially for the phasic suppression (Fig. 8C,D). In the present model, the adaptation cut-off frequency f Ad is 4 kHz at the CF location of the probe tone. For a suppressor at 400 Hz, the adaptation lowered MET current gain to about 10% of its maximal value, reducing suppression to the extent that secondary suppression was absent. Because the MET current leads the bundle displacement at low frequencies due to adaptation, the maximum suppression can occur prior to the maximum suppressor displacement. That is, due to adaptation, the phase of maximum suppression (φ MS ) can be greater than 0 (Fig. 8D).
According to measurements, maximum suppression occurred either before or after the maximum suppressor displacement 37,38 . The phase of maximum suppression is indicated as φ MS in Fig. 8. The phase is dependent on the relationship between the probe (f P ), suppressor (f S ), and adaptation cut-off frequency (f Ad ).
The effect of the frequency ratio on the suppression phase was analyzed when the adaptation cut-off frequency remained at 4 kHz (Fig. 9). Positive φ MS was obtained when the frequency ratio <0.05. In contrast, φ MS was negative when there was no adaptation, independent of f S /f P (Fig. 9A, solid and broken curves). φ MS increases as the frequency ratio decreases in both cases, consistent with experimental observations 37,38 . When the two frequencies are close enough (f S /f P > 0.1), tonic suppression dominates the response and phasic modulation is shadowed: with fewer cycles of probe tone vibration within one cycle of the suppressor tone, it is difficult to determine the timing of maximum suppression accurately. Temporal responses of 2TS with 1.3, 1.5, and 1.7 kHz suppressors are demonstrated in Supporting Materials (Fig. S3).
In the other set of simulations (Fig. 9B), the effect of adaptation frequency (f Ad ) on the suppression phase (φ MS ) was analyzed. To our knowledge, there is no in vivo measurement of the adaptation time constant at the base of the cochlea. While probe and suppressor frequencies were fixed, f Ad was changed from 30 Hz to 12 kHz. φ MS increased as f Ad increased. The effect of the high-pass filtering by MET adaptation is twofold: (1) MET current leads the hair bundle deflection by 90 degrees, which adds 90 degrees to φ MS (see the shift between the solid and broken lines in Fig. 9A); (2) the MET gain for the suppressor decreases, resulting in less suppression (Fig. 8C,D).  www.nature.com/scientificreports www.nature.com/scientificreports/ Geisler and Nuttall 38 showed that φ MS is also dependent on the level of suppression (stronger suppression causes more delay). Our results indicate that positive φ MS is more likely to be observed when f Ad ≫ f S (Fig. 9B). Measured φ MS is best explained when f Ad is between 1 and 10 kHz. This adaptation speed is consistent with whole-cell patch clamp measurement results of the rodent outer hair cells 25,59,60 . To conclude, the phase lead of suppression with respect to suppressor displacement can be explained when the MET adaptation affects the 2TS.

Discussion
Nonlinearity in cochlear model studies. There have been theoretical studies regarding cochlear nonlinearity. The source of nonlinearity is becoming more specific as more is known about the active feedback from the OHCs. In the 1970s, before the active feedback from the OHCs was discovered, pioneering nonlinear models 9,61,62 used nonlinear damping imposed on the basilar membrane to explain the mechanical nonlinearity first observed by Rhode 4 . Geisler 63 incorporated amplitude factors to limit displacement-proportional OHC motile forces, so that the model could account for 2TS. Steele, Puria and their colleagues 42,64 modeled the nonlinear force of the OHCs in the form of a feed-forward gain factor to reproduce 2TS and the distortion product. Verhulst and Shera 65 used a level-dependent basilar membrane impedance that saturates at high SPL. Liu and Neely 66 considered the sensitivity of MET as a nonlinear function of the displacement and velocity of the reticular lamina. Meaud and Grosh 67 used the first-order Boltzmann function to represent the OHC mechano-transduction. This nonlinear mechano-transduction was combined with an OHC somatic electromotility model, analogous to a piezoelectric circuit.
Our model is similar to Meaud-Grosh's in that the OHC MET is explicitly described. Compared to other studies, our cochlear model incorporates more detail regarding the OoC micromechanics (Supporting Materials, Tables S1 and S2). The model reproduced signature nonlinear responses in cochlear mechanics, such as a broadened response envelope with increasing stimulus level and the frequency glide (Figs 2 and 3). Taking advantage of the details in MET and micro-mechanics, in this study, we used our model to explore the MET properties of the OHC by analyzing existing data of 2TS.
Limitations of this study. There are some limitations of our study. First, assorted data from different locations, frequencies and species were used for comparison. For example, in the literature, different probe/suppressor frequencies and levels were used to measure 2TS responses from different rodent species. While our study is focused on a specific location (x = 2 mm) of the gerbil cochlea, the probe tones in experiments ranged from 6 to 28 kHz. The MET sensitivity and level of nonlinearity may vary depending on the location. Second, the model details present an opportunity and a challenge at the same time. Due to the large parameter space, there can be different combinations of model parameters that could generate similar results. Although most model parameters are constrained by existing physiological data (such as geometry, mechanical properties of the basilar and tectorial membranes, etc.), in many cases, some literature data have large variations. www.nature.com/scientificreports www.nature.com/scientificreports/ The MET sensitivity estimated from 2TS. Existing 2TS data are consistent with a MET sensitivity of 0.06 nm −1 at the base of the cochlea (CF near 17 kHz, Fig. 7). According to electrophysiological measurements of isolated hair cells 26,59,[68][69][70] , the MET sensitivity ranges from 0.0033 to 0.045 nm −1 for hair cells in rodent cochleae. Most of the single cell measurements gave roughly one order of magnitude smaller MET sensitivity than our estimation. The discrepancy can be ascribed to two aspects.
First, in vitro experimental limitations. Experimental preparation may alter physiological conditions of hair bundle excitation due to removal of the tectorial membrane from the OoC complex. Fettiplace and Kim 20 demonstrated that with different stimulating methods, the current-displacement curve slope can be changed by an order of magnitude. Nam et al. 21 showed both glass probe and fluid jet stimulation may cause splay between stereociliary columns and result in underestimation of the sensitivity. Often in experiments, the calcium concentration (>1 mM) is higher than that in the endolymph 71 . Beurg et al. 59 showed that higher calcium concentration broadens the operating range of a MET channel. In experiments, the speed of the stimulation system is limited 25 , which may have a time constant larger than that of the channel activation. Thus, the recorded MET current may be reduced by adaptation with either step or low-frequency sinusoidal stimulation.
Second, the MET sensitivity measurement data and 2TS data used for the estimations were not all from the same species or location. Most of the sensitivity measurements were from mouse or rat cochleae and hair cells used for recordings were located at the apical turn (CF ~4 kHz), while our estimation is at the high-frequency (17 kHz) basal location and 2TS data are from guinea-pig and chinchilla cochleae (CFs ranging from 6-26 kHz). The estimation depends on available data from 2TS, thus more 2TS measurements will help us to probe MET properties accurately. Variation of the MET sensitivity at different locations and among different species could be a future study.

Time courses affecting maximum suppression phase.
For the low-frequency suppressor, maximum suppression of the probe tone would lag the maximum displacement of the suppressor given the time courses needed for the OHC active feedback to affect basilar membrane vibration. Channel activation, the RC constant of the OHC membrane, and viscoelastic delay were the sources for the delay in the simulation. The relative contribution of each component depends on the rate constant A 0 in Eqs 5 and 6, conductance and capacitance of the OHC in Eq. 11, and mechanical properties of the basilar membrane (Supporting Materials, Tables S1 and S2). Some of the parameters may not be completely accurate, yet the fact that these processes result in phase delay remains.
Although adaptation has been commonly observed in excised cochlea or hair cells 25,26,58,72,73 , there is no direct evidence of adaptation in mammalian hair cells under physiological conditions. However, in experimental data that showed positive phase, such delay is most likely compensated through adaptation of MET for the suppressor tone, since MET adaptation may cause up to a 90-degree phase lead at low frequencies (Fig. 8B). Geisler 38 also reported level-dependent changes of the φ MS . Less suppression due to smaller transduction currents of the suppressor component also helps to shift the phase towards the positive direction.
MET adaptation estimated from 2TS temporal pattern. We used two extreme cases for simulation, one without adaptation and one with complete adaptation (MET current decays to zero given enough time). In practice, both extent and time constant of adaptation affect the phase shift (Supporting Materials Fig. S5). With lower adaptation extents and larger time constants, the phase falls within the two bounds in Fig. 9A. The linear fit (both phase and frequency are on a linear scale) to the data from Cooper's study 37 gives a zero-frequency intercept of ~20 degrees, less than 90 degrees with complete adaptation. This suggests partial adaptation at the base in a natural condition.
Some studies 74,75 suggested adaptation in hair cells helps to maintain mechanotransduction sensitivity and to widen its dynamic range. Ricci et al. 58 claimed that it works as a "prefilter" to increase the signal to noise ratio. Others 25,28,76 argued that adaptation may contribute to frequency tuning and amplification. We did a parametric study to see how different adaptation extents and time constants affect tuning and amplification at the basal location. No significant change was seen to support its role for amplification/tuning enhancement. However, we cannot exclude the possibility that adaptation may work as a secondary filter at the apical region, where the CF is lower and mechanical tuning is broader.

Bias of rest open probability and polarity of suppressor displacement at maximum suppression.
It is still controversial whether maximum suppression occurs most closely with maximum displacement of the suppressor towards the scala tympani or the scala vestibuli. Some studies showed maximum suppression on the scala tympani side [36][37][38] , while others reported maximum suppression scala vestibuli side 30 and the rest exhibited both 39 . The asymmetry between maximum and secondary suppression has been attributed to the bias of the resting open probability 38,44

Conclusion. A computer model of the cochlear mechano-transduction was created incorporating detailed
OoC mechanics and nonlinear kinetics of hair cell MET. Using the model, we showed that 2TS data can be interpreted and analyzed to investigate the hair cell MET. The estimation of a key parameter -the MET sensitivity at the basal cochlea (0.06 nm −1 ) through comparison between simulation results and experimental data of 2TS helps to gain new insights about the sensory cell. The temporal pattern of 2TS represented by φ MS could be explained with the hair cell's MET adaptation.