Reducing tectorial membrane viscoelasticity enhances spontaneous otoacoustic emissions and compromises the detection of low level sound

The mammalian cochlea is able to detect faint sounds due to the presence of an active nonlinear feedback mechanism that boosts cochlear vibrations of low amplitude. Because of this feedback, self-sustained oscillations called spontaneous otoacoustic emissions (SOAEs) can often be measured in the ear canal. Recent experiments in genetically modified mice have demonstrated that mutations of the genes expressed in the tectorial membrane (TM), an extracellular matrix located in the cochlea, can significantly enhance the generation of SOAEs. Multiple untested mechanisms have been proposed to explain these unexpected results. In this work, a physiologically motivated computational model of a mammalian species commonly studied in auditory research, the gerbil, is used to demonstrate that altering the viscoelastic properties of the TM tends to affect the linear stability of the cochlea, SOAE generation and the cochlear response to low amplitude stimuli. These results suggest that changes in TM properties might be the underlying cause for SOAE enhancement in some mutant mice. Furthermore, these theoretical findings imply that the TM contributes to keeping the mammalian cochlea near an oscillatory instability, which promotes high sensitivity and the detection of low level stimuli.

1 Supporting materials and methods 1

.1 Cochlear model Parameters
The addition of longitudinal coupling to the TM bending mode required a slight modification to the model parameters compared to our previous work [1]. The numerical values of the present cochlear model parameters are given in Tables S1 and S2. Some of these parameters appear in Eqs. 4-13 of the manuscript. The other equations of the model are given in previous papers [22,14,17,1].

Electrical Parameters and Model Formulation
The saturating mechanoelectrical transduction current, I max hb , appears in Eq. 4 of the manuscript. I max hb , is related to the parameters given in Table S2 by the following equation: where G max hb is the saturating hair bundle mechanoelectrical conductance and ∆V 0 hb is the resting value of the difference between the scala media potential and intracellular OHC potential.
As discussed in [15], cochlear amplification depends on the slope of the mechanoelectrical conductance vs HB deflection function; as in our previous work [1], the parameters of the mechanoelectrical transduction channel were chosen such that the sensitivity of the BM response is similar to experimental measurements. Because experimental measurements have shown that the mutations have very limited effect on the cochlear microphonic in T ecta Y 1870C/+ and T ectb KO mice [11,26] and on cubic and quadratic distortion product otoacoustic emissions in the Ceacam16 KO mice [2], the operating point of the transduction channel was assumed to be the same for all models.

Middle ear model and parameters
As in our previous work [1], the cochlear model is coupled to a one DOF model of the middle ear with mass, damping and stiffness coefficients. While sound transmission through the middle ear can be represented by more realistic multi-DOF models [19,12], the advantage of a simpler model is that it makes it possible to easily vary the reflection of reverse waves at the stapes to study the influence of the middle ear properties on SOAE generation (see for example Fig. S7). In the frequency domain, the reverse middle ear impedance (i.e, the impedance looking out from the stapes in the reverse direction [21]) is given by: where M s , C s , and K s are the mass, damping coefficient, and stiffness coefficient of the stapes, respectively; ω is the radian frequency and A s is the area of the stapes footplate.
In the theory of coherent reflection [27], the reflection of the reverse traveling wave by the stapes plays a critical role in SOAE generation: SOAEs are expected to be generated only when the amplitude of the stapes reflection coefficient is sufficiently close to 1. This hypothesis was tested in our model. The stapes reflection coefficient, R st , can be calculated using the equation [28,1]: where Z c is the input impedance of the cochlear model, and * denotes the complex conjugate. Two parameter sets for the middle ear model, given in Table S3, were considered: a baseline parameter set (used in all of the numerical results except for some results in Fig. S7), in which |R st | is approximately equal to 0.7 from 10 to 25 kHz; a low |R st | parameter set (used only for Fig. S7) in which |R st | is below 0.2 from 10 to 25 kHz (see Fig. S2 in [1]). At a longitudinal position, x, and frequency, ω, the gain of the BM velocity relative to the stapes velocity is defined as where v bm and v s are the velocities of the BM and stapes, respectively. The pure tone response of the smooth model is compared to experimental data from Refs. [20] and [23] in Fig. S1. As mentioned earlier, model parameters were adjusted so that the gain of the model matched that reported from experimental measurements. At the 34.6 kHz peak position (CF=34.6 kHz, Fig. S1a), the model results are similar to the experimental data in terms of the sharpness of the peaks and relative levels of the gain (both show a ≈ 21 dB difference between low and high SPL). At CF=13.3 kHz (Fig. S1b), the both model and experiment show a ≈27 dB difference between low and high SPL and the peaks are similarly broad. However, there is a noticeable difference of 11 dB between the model and experiment that is possibly due to a mismatch in the structural properties or assumptions of the model. As is the case with many cochlear models (e.g. [30,1,18]), the phase rolls-off or decreases somewhat more rapidly than what is observed experimentally (Figs. S1c and S1d) despite effort during calibration to mitigate this. The mismatch between the model and the experimental data is, however, noticeably smaller than that of the previous version of the model (Fig. 3 in [1]). The pure tone response of the smooth model at all longitudinal positions with the place-frequency map, ratio of the BM gain at CF, and quality factor, Q 10dB are shown in Fig. S2. The model results in Fig. S2 were obtained using a linear formulation of the model; the active and passive models represent the model responses at low and high stimulus levels, respectively. The passive model is obtained by setting G max hb = 0 in Eq. S1. As shown in the figures, the model predictions agree well with the measurements from multiple longitudinal locations.  2 Supporting results

Stability of organ of Corti model
To determine whether the spontaneous oscillations in the model were generated by local instabilities or a global phenomenon, the linear stability of an isolated longitudinal crosssection of the organ of Corti was analyzed (Fig. S3). This organ of Corti model neglects longitudinal coupling and fluid loading. Both the fully active model and a passive model (obtained by setting the mechanoelectrical transduction current to 0) were analyzed. As seen in Fig. S3b, both the passive and active organ of Corti models are linearly stable. The parameters of the cochlear model vary spatially (Tables S1 and S2

Influence of mesh on linear stability
Throughout the results shown in the manuscript, a finite element mesh with elements of longitudinal length ∆x = 25 µm long was used (total of 448 elements along the BM). Furthermore, the roughness function, r(x), is assumed to be a piecewise linear function with breakpoints that are separated by a distance ∆x r . For the results shown in the manuscript, ∆x r was chosen to be also equal to 25 µm. The influence of ∆x and ∆x r on the results was examined. Two different cases were considered: (1) ∆x r was fixed at 25 µm and the element size, ∆x, was varied, shown in Fig.  S4 and (2) the roughness lengthscale, ∆x r , was varied for a fixed element size (∆x = 25 µm) shown in Figs. S5.
The results for the first case (Fig. S4), indicate a slight decrease in stability for models with ∆x = 12.5 µm long elements. When the element size is increased to ∆x = 50 µm, there is a noticeable decrease in the number of linearly unstable models (Fig. S4d). The same roughness variations were used to allow for a meaningful comparison ( Fig. S4a and S4b); using different roughness variations would result in slightly different numbers of linearly unstable modes. For many of the models considered, using the finer mesh results in at most one additional linearly unstable mode than using the mesh used in the manuscript. Shown in Fig. S4c is a less common case, in which two nearly unstable modes for ∆x = 25 µm (indicated by the arrows) become unstable when the element size is reduced to ∆x = 12.5 µm. Such a small decrease in stability (increase in σ) for the finer mesh also affects n inst when N = 20 models are considered, as shown in Fig. S4d. Despite the slight decrease in stability for the finer mesh, the results in Fig. S4d are consistent with results shown in Figs. 2-5 and confirm that the effects of altering viscoelastic coupling on cochlear stability do not change if the size of the element is doubled or divided by two.
Varying the length scale of the roughness, ∆x r , results in different roughness functions, r(x) (Fig. S5). Using a roughness length scale of 10-25 µm is reasonable given the value of the OHCs diameter (8-10 µm [3]) and the variability in OHC organization [13]. While identifying the related individual poles for the three meshes was straightforward in Fig. S4c, there is no such correlation between poles for the results shown in Fig. S5c since different r(x) are used. This difference is due to the roughness having very different spatial variations ( Fig. S5a and S5b). More importantly, the average numbers of linearly unstable modes is not affected much by the changes in ∆x r (Fig. S5d). The results of Fig. S5d show that reducing TM viscoelastic coupling results in a reduction in cochlear stability and an increase in the average number of linearly unstable modes at all ∆x r values that were considered. (c) Linear stability diagram for the "Reduced Both" model for RS=2. (d) Average number of linearly unstable models, n inst , for N=20 RS for the four compling cases, "Baseline" (B), "Reduced Elastic" (RE), "Reduced Viscous" (RV), and "Reduced Both" (RB). The error bar corresponds to ± one standard deviation.

Influence of large roughness variations on linear stability
In Fig. 3c, the influence of varying the amplitude of the random perturbations, ∆R, on the average number of linearly unstable modes was examined for a relatively narrow range of ∆R values (∆R ≤ 1%). For larger random perturbations (1% ≤ ∆R ≤ 40%), shown in Fig.  S6, the average number of unstable modes continues to increases as ∆R is increased, albeit at a slightly lower rate once ∆R exceeeds 5%. In all cases, for a given random perturbation amplitude, reducing TM viscoelastic coupling results in a reduction in cochlear stability and an increase in the average number of linearly unstable modes. The results for the linear stability of the models, shown in Fig. S7, demonstrate that unstable modes are significantly reduced when the magnitude of the stapes reflection coefficient, |R st |, is low. Since SOAEs are only generated when linear unstable modes are present, this implies that SOAE generation requires the stapes reflection coefficient to have a sufficiently high magnitude. This is a consistent with the theory that are SOAEs are a global phenomenon rather than locally generated. Baseline TM Reduced Both Figure S8: Modeling the effect of holes in the TM on linear stability. The random seed (RS) of the electromechanical coupling coefficient is fixed to RS=13 for these results. N = 20 random seeds were used for the roughness in the TM properties. The error bar corresponds to ± one standard deviation.
Because the presence of holes has been reported in the TM of Ceacam16 KO and T ecta Y 1870C/+ mice [2,11], random inhomogeneities in the TM properties were considered. These inhomogeneities could potentially enhance the generation of reflection-type otoacoustic emissions, including SOAEs. These inhomogeneities were introduced by using the following equations of the longitudinal coupling stiffness and viscosity of the TM: where ∆R tm is the amplitude of the random perturbation (which was set to 5%, which is somewhat similar to the percentage of the TM area occupied by holes in the TM of Ceacam16 KO mice [2]), r tm (x) is a number generated by a random number generator based on a normal distribution of average value 0 and standard deviation 1. Since roughness due to holes in the TM or to inhomogeneities in the OHC properties arises from different mechanisms, different random seed numbers were used for r tm (x) in Eq. S5 and r(x) in Eq. 12. The results shown in Fig. S8 show that adding roughness to the TM properties has a limited effect on the linear stability of the models: the N = 20 different models with roughness in the TM properties with the baseline value of the TM parameters have n inst. = 2.5 unstable modes, while the model without roughness in the TM parameters has no unstable modes; for the model with reduced viscosity, the average number of instabilities is slightly higher in the models with roughness in the TM parameters (average=20.4) than in the model without TM roughness (which has 19 unstable modes). The change corresponds to 50% of the baseline value for C LC tm and C tms , 50% of the baseline value for K LC tm and K tms ; 50% and 90% of the baseline value for C f .

Influence of other TM parameters on cochlear stability and SOAEs
Because TM mutations might affect other properties of the TM, the influence of other TM model parameters on linear stability was also analyzed (Fig. S9). For example, given the effect of the mutations on the intrinsic properties of the TM, TM mutations might also cause a reduction in the stiffness or viscosity of the attachment of the TM to the spiral limbus. Reducing the stiffness of the attachment to the spiral limbus, K tms , reduces the average number of linearly unstable modes; this effect is opposite to the increase in linearly unstable modes observed when the TM longitudinal coupling elastic stiffness is reduced. Reducing the viscosity of the attachment of the TM to the spiral limbus slightly increases the average number of linearly unstable modes; however, reducing the TM viscosity has a much more significant effect. It is possible that due to the loss of the Hensen's stripe or broadening of the subtectorial space, TM mutations tend to reduce power dissipation in the subtectorial space, which would affect the effective damping coefficient due to fluid viscosity in the subtectorial space, C f , and would slightly increase the number of instabilities.