The experimental OHC elongation phase is not inconsistent with effective transfer of mechanical power to the BM by the active force

The experimental OHC elongation is not inconsistent with effective transfer of mechanical power to the BM Renata Sisto1, a) and Arturo Moleti2, b) 1 Department of Occupational and Environmental Medicine, Epidemiology and Hygiene, INAIL-National Research Centre for Safety and Prevention at Workplace, Monteporzio Catone(Rome), ITALY. 2 Department of Physics and NAST Center, University of Rome ‘Tor Vergata’ , Rome, ITALY. a)Corresponding author: [email protected] b)[email protected] Abstract. The amazing development of the OCT techniques permits spatially well-defined measurement of the motion of a number of different solid structures within the Organ of Corti. Recent measurements [1] show the motion of the BM and of the RL in correspondence of each OHC row, as well as the motion of the junction between the OHC body and the Deiter’s cells. In the peak region, at low stimulus levels, the motion of the RL is larger than that of the BM. In a piezoelectric actuator model of the OHCs [2], the electric current is supposed to be proportional to the OHC elongation. Considering the low-pass phase rotation, the OHC force exerted on the BM may act as an anti-damping force, proportional to the RL velocity, and transfer power to both the RL and the BM. In this study, a WKB transmission-line cochlear model with two mechanical degrees of freedom at each tonotopic place is used, in which the OHC force is represented by a low-pass filtered elastic term proportional to the OHC elongation. With respect to our previous studies, this WKB model also includes the fluid focusing effect and the viscous damping effect in the peak region. In the model simulations, with a suitable parameter choice, the phase difference between the RL and the BM near the BF is consistent with some experimental findings, and the OHC power is effectively transmitted to the BM motion. MODEL In a recent 2DOF 1-D transmission line cochlear model [3], the local mechanical element consisted of two oscillators, with masses M1>>M2, that can be identified, respectively, with the basilar membrane (BM) and the reticular lamina (RL), coupled by the internal active nonlinear OHC force. The hair bundle shear was simply schematized by the OHC elongation, or the relative displacement of RL and BM, giving the input to the electromotility of the OHC system, represented by an instantaneous nonlinear function of the local relative displacement and velocity of the RL and BM. The model predicted rather well the observed nonlinearity of the RL response over a large basal region, and respected quite naturally the zero-crossing intensity invariance of the impulse response [3]. The main drawbacks of that model were: 1) an insufficiently physiologically-based representation of the OHC micromechanics, in particular having neglected the low-pass characteristic of the electrical response of the OHCs; 2) having neglected 2-D fluid phenomena that enhance the fluid pressure near the BM (pressure focusing), significantly boosting the cochlear gain though pressure amplification, without the need for a peaked admittance; 3) having neglected the viscous damping associated with the development of sharp vertical velocity gradients in the peak region at the fluid-BM boundary. Three papers may help solving the two above issues: About issue 1), a more physiologically-based model was proposed by Lu et al. [2] to schematize the OHC piezoelectric function (direct and reverse), involving a double feedback loop, and including the low-pass behaviour associated with the slow build-up of the OHC transmembrane potential. About issue 2), Shera et al. [4] had used the WKB linear approximation to provide an estimate of the pressure focusing effect on the BM gain. About issue 3), Steele and Taber [5] provided a vector potential formulation of the viscous cochlear fluids that can be used to schematize the viscous force on the moving elements of the OoC. The M1 and M2 impedances can be written as: 𝑍1,2 =𝑠𝑀1,2+ 𝐶1,2 + 𝐾1,2 𝑠 , (1) with s = j𝜔. The impedance associated to the passive interaction term is: 𝑍3=𝐶3+𝐾3 𝑠 . (2) The active force term is written here as an internal force (so the third principle of dynamics is satisfied) proportional to the OHC elongation. The low-pass filtering and the viscous fluid damping are also kept into account. For frequencies above the local cut-off frequency, two competing effects occur: 1) the output amplitude is attenuated by the filter response with a 6 dB/oct slope, 2) the up-to-90° phase shift of the OHC active term yields effective anti-damping action, increasing the local admittance. The local cut-off frequency of the OHC low-pass filter can be set according with physiological data [6], which show a strong spatial dependence of membrane conductance and capacitance. The low-pass frequency of the filter monotonically changes along the BM, as the CF, but with a slightly slower spatial gradient. As a consequence, the local characteristic frequency is close to the local cut-off frequency in the apical end of the BM, and well above it in the basal cochlea. The active force on the BM may be written in the frequency domain as: 𝐹𝑂𝐻𝐶 = 𝑍𝑂𝐻𝐶(𝜉󰇗2−𝜉󰇗1) , (3) where a hybrid notation is used, with 𝜉󰇗1,2 meaning the Fourier Transform of the velocity. The impedance of the OHC term, including the low-pass effect on amplitude and phase [7] associated with the characteristic time  , is: 𝑍𝑂𝐻𝐶 =𝐺(𝐶4+𝐾4 𝑠)1 (1+𝑠𝜏) . .(4) where G is a parameter controlling the strength of the active force, roughly describing the nonlinear behavior of the response at different stimulus levels. Similarly to [8] and to [9], the nonlinear dependence of the active force on the instantaneous displacement was roughly schematized by solving N linear models with different strengths of the OHC active term. The equations of motion for the system of coupled oscillators are written as: {𝑍1𝜉󰇗1+ 𝑍3(𝜉󰇗1−𝜉󰇗2)+𝑍𝑂𝐻𝐶(𝜉󰇗1−𝜉󰇗2)=𝑃𝑑 𝑍2𝜉󰇗2− 𝑍3(𝜉󰇗1−𝜉󰇗2)−𝑍𝑂𝐻𝐶(𝜉󰇗1−𝜉󰇗2)=0 . (5) where Pd is the differential pressure on the BM. Starting from the system (5), the BM admittance can be easily calculated: 𝑌𝑏𝑚 = 𝑍2+ 𝑍3+ 𝑍𝑂𝐻𝐶 𝑍1𝑍2+(𝑍𝑂𝐻𝐶+𝑍3)(𝑍1+𝑍2) . (6) The ratio between the RL and the BM velocity is: 𝜉󰇗2 𝜉󰇗1=𝑍3+ 𝑍𝑂𝐻𝐶 (𝑍2 +𝑍3+ 𝑍𝑂𝐻𝐶) . (7) Eq.(7) suggests that the model parameters can be set such that the ratio is larger than unity or, in other words, the RL moves more than BM at low stimulus levels, i.e., in the linearized solution, for the maximal strength of the active term. This objective can be obtained by setting 𝑍𝑂𝐻𝐶~−(𝑍2+ 𝑍3) in the large gain limit, so Eq.(7) tends to a value exceeding unity. In the same limit, Eq.(6) suggests that a large admittance is obtained if the following condition is fulfilled: 𝜉󰇗2 𝜉󰇗1=−𝑍1 𝑍2 , (8) which could be obtained by suitably setting the resonance frequencies of the two oscillators. We did not pursue such fine tuning, which could yield unphysical models, but we are mentioning these properties of the 2 DOF active model to highlight the fact that even with 2 DOF only, the response of the two masses is not easily predictable. In principle, in such 2DOF models, both the BM and the RL contributions to the development of the traveling wave (TW) should be considered. The TW generation is due to both the BM displacement and to the OoC section area deformation that produce longitudinal velocity gradients, due to the fluid incompressibility. When the RL moves with larger oscillations that the BM (the RL/BM velocity ratio could be of order 10, and function of the stimulus level), its contributions to the TW can be larger than that of the BM. The possible effect of a traveling wave component generated in the endocochlear fluid or in the sub-tectorial space, as hypothesized by [10], is beyond the scope of this study. The hydrodynamic coupling permits to keep into account [9, 11]: 1) The fluid focusing effect or, in other words, the boost given to the pressure by the shape factor deriving from the short-wave effect. 2) The stabilizing effect deriving from the viscosity or, in other words, the damping term proportional to the wave vector coming from the viscous friction force acting on the BM in the vertical direction. The fluid focusing effect is kept into account by the factor  representing the ratio between the pressure at the fluid-membrane interface and the pressure averaged over the vertical direction as in Eq.(9) of [9]: 𝑝= 𝑝(𝑥,0,𝜔) 𝛼, with 𝛼= 𝑘𝐻 tanh (𝑘𝐻). (9) The effect of the viscous force was inserted in the impedance following the strategy proposed in [8]. A WKB strategy is proposed for solving a model with a 3D fluid and the linearized 2DOF mechanics written in Eq.(5). An iterative procedure was implemented in order to find self-consistent solutions for the wave vector. At the first iteration: 𝑘2=−2𝑖𝜔𝜌 𝐻(1+𝛿𝜉󰇗2 𝜉󰇗1 )𝑌𝑏𝑚 (10) where the BM admittance is given by Eq.(6) and the RL contribution to the wave propagation equation is 𝛿𝜉󰇗2 𝜉󰇗1𝑌𝑏𝑚. The factor  could also be a complex number, due to the compliance of the Reissner Membrane. In the real cochlea a vertical gradient of the fluid pressure and velocity is also present within the Scala Media. The second iteration consists in correcting the wavenumber by the factor alpha (fluid focusing effect), computed for the unperturbed value of the wavenumber: 𝑘2=−2𝑖𝜔𝜌𝛼 𝐻(1+𝛿𝜉󰇗2 𝜉󰇗1 )𝑌𝑏𝑚, (11) then, the expression of alpha is corrected using the corrected value of the wavenumber until convergence is reached. During the iteration procedure, the admittance was corrected by keeping into account the viscosity. In particular the BM impedance was corrected by adding the contribution of a viscous vertical force acting at the interface between the BM and the fluid: 𝑍1→𝑍1+2𝐶𝑑𝑎𝑚𝑝 𝜇𝑏 𝜎𝑏𝑚|𝑘| (12) where  b is the viscosity coefficient. In addition to the fluid, other OoC structures should contribute to viscous damping, behaving as viscoelastic materials. Among these structures, a good candidate as a source of viscous damping are the body of the OoC and the tectorial membrane. This consideration may help understanding why large values of the viscosity are often necessary in this kind of models to get a better agreement with the observed phenomenology. RESULTS The BM and RL velocity gain functions at a fixed place x are shown in Fig.1 as a function of frequency, along with the amplitude ratio and the phase difference between the RL and BM displacements, for six different values of the activity of the OHC force (G variable in the 0.25-3.5 range, increasing from black to red, to green, blue, cyan, magenta). The lowest stimulus levels would correspond to the maximal gain, as in [8, 9]. Here, the small dimensionless parameter  was set to 0, for simplicity, but the effect of  can be large when it includes a phase factor. It may be noted that such a 2DOF model including focusing and viscous damping, with a suitable choice of the parameters, predicts larger amplitude of the RL motion at the lowest stimulus levels, and larger gain dynamics, along with a nonlinear RL behavior over a wide basal region. Interestingly, the RL-BM amplitude ratio also increases above unity approaching the peak, at the lowest stimulus level. All these features are in reasonable agreement with OCT data of the basal cochlea. The phase difference between RL and BM in the peak region is negative, slightly exceeding quadrature (-110-120°). We note that, with this particular parameter choice, the power transfer to the BM is positive and large too (see Fig.2). DISCUSSION The experimental observation of RL oscillations larger than those of the BM and of phase quadrature between the two motions in the peak region may seem at odds with the idea that OHC power is actually fed to the BM. This is not necessarily true in the general case, the result being dependent on the model parameters and on the functional dependence of the active force on the BM and RL displacements. The motion of each mechanical element is indirectly determined by the internal and external forces through differential equations and boundary conditions, and, even in systems with two DOF only, even the qualitative features of the solution may not be trivially predicted. In particular, the larger amplitude of the motion of the lighter element (RL in this scheme) and the phase quadrature between the two masses motion do not mean that the RL is the only element absorbing power from the OHC active force. This occurrence suggests that, in a model in which different elements of different mass are connected by internal forces, the target element of the main power transfer are not straightforwardly dependent only on the kinematics observed by OCT experiments. In this study, we presented model simulations of a 2DOF system in which, although the RL shows larger oscillations than the BM in the peak region, there is large power absorption by the BM. Interestingly, the ratio between the RL and the BM motion increases approaching the BF, and with increasing amplifier gain. Altoè et al. [12] conclusion that the OHC power is not fed to the BM was based on different arguments. One of them is the observation of phase-lag of the OHC-Deiters’ cell vibration. Another strong argument is the floppiness of the Deiters’ cells, recently experimentally demonstrated. In [13] another pathway, alternative to the coupling between the OHC bodies and the Deiters’ cells, is proposed to explain the power transfer to the BM by the active force. This pathway could involve the pillar cells instead than the Deiters’ ones. Although it was not the main argument against the hypothesis that the OHC active force could make work directly on the BM, it may be worth mentioning that [12] showed data of the RL and BM moving in quadrature. These data refer to a more apical region of the cochlea, where the physics could be different from that of the basal region, but [14] also show BM and RL motion in phase below the BF, and close to quadrature at the BF. However, the experimental situation is not clear, and more complex than what can be included in our model. Indeed, Cooper and coworkers [15] had showed that the hot spot (close to the RL) and the BM move approximately in phase at the response peak, and approximately in quadrature two octaves below. In [1], different behaviour was shown for the three OHC rows. In [16], similar BM and RL phase is observed at the BF for the transverse motion component. One must consider that the OoC motion is not only transverse [15,16], which means it is not fully predictable or even describable within the oversimplified geometry of our model. FIGURE 1. Top: BM and RL velocity gain functions at a fixed place x, with a BF of 6kHz. Bottom: amplitude and phase relation between the BM and RL response functions. The magenta line corresponds to the largest strength of the active mechanism. FIGURE 2. Power absorbed by the BM (thin lines) and RL (thick lines). Assuming that the phase relation between RL and BM is not well-established yet, let us calculate the work made by the active force on the two masses, 1 and 2, respectively, for the BM and RL, in the case of different phase relations between the motion of the two masses, for the functional dependence of the OHC force proposed in the present study: ℒ1,2 = ±∫𝑑𝑡 𝐹𝑂𝐻𝐶 𝑇 0𝜉󰇗1,2 (13) and, substituting the active force equation and expressing the power in the frequency domain: ℒ𝐵𝑀 = ∫𝑑𝑡 𝐹𝑂𝐻𝐶 𝑇 0𝜉󰇗1 ∝ℛ𝑒 {(𝐶4+𝐾4 𝑠)( 1 (1+𝑠𝜏))|𝑠|2(𝜉2𝜉1 ∗−|𝜉1|2)} (14) ℒ𝑅𝐿 = −∫𝑑𝑡 𝐹𝑂𝐻𝐶 𝑇 0𝜉󰇗2∝ℛ𝑒{(𝐶4+𝐾4 𝑠)( 1 (1+𝑠𝜏))|𝑠|2(𝜉1𝜉2 ∗−|𝜉2|2)} (15) In the limit  >>1, in which the low pass phase rotation is full (90°), some general consideration can be made: If the RL and BM were in phase, the term in C4 would not contribute. In that case, if the RL moved more than the BM, the work on the RL would be positive and that on the BM would be negative. If the RL and the BM were in quadrature, as found by some experiments, with RL - BM = -90°, the work done on the BM would have a positive contribution from the K4 term, and a negative contribution from the C4 term. The work done on the RL would have positive contributions from both terms. As the limit  >>1 is not fully achieved in the real case, an intermediate behaviour occurs in the general case, with positive and negative work contributions whose relative size depends on the model parameters. We point out that this model is designed to get large RL motion and positive work on the BM, so we are NOT demonstrating that positive OHC work is done on the BM in the real cochlea. We are just showing the existence of a specific model with reasonable BM and RL gain functions, in which, although the RL moves more than the BM in the peak region, positive work is done on the BM, and the two motions are approximately in quadrature in the peak region. CONCLUSION A 3D model containing the full hydrodynamical coupling between the cochlear fluid and 2DOF mechanical elements, is presented and solved in the frequency domain in the WKB approximation. The OHC activity is modeled as an internal force between the BM and the RL. The low pass filtering associated with the slow build-up of the transmembrane potential of the OHC, acting as the source for the active actuator, provides an up-to-90° phase rotation, partially transforming an elastic force into an antidamping one. The absorption of power from the actuator mainly occurs through the BM with a suitable parameter choice, suggesting that it may be difficult to identify the target element of the OHC power absorption when the functional form of the force between the different interacting elements of the system is not exactly known and only elongation and phase of the interacting elements of the system can be measured. REFERENCES 1. Cho, N.H., Puria, S. Cochlear motion across the reticular lamina implies that it is not a stiff plate. Sci Rep 2022, 12, 18715. 2. Lu TK, Zhak S, Dallos P, Sarpeshkar R. Fast cochlear amplification with slow outer hair cells. Hear Res. 2006 214, 45. 3. Sisto R, Shera CA, Altoè A, Moleti A. Constraints imposed by zero-crossing invariance on cochlear models with two mechanical degrees of freedom. J Acoust Soc Am. 2019, 146, 1685. 4. Shera CA, Tubis A, Talmadge CL. Coherent reflection in a two-dimensional cochlea: Short-wave versus longwave scattering in the generation of reflection-source otoacoustic emissions. J Acoust Soc Am. 2005 118, 287. 5. Steele CR, Taber LA. Comparison of WKB calculations and experimental results for three-dimensional cochlear models. J Acoust Soc Am. 1979 65, 1007. 6. Nam JH, Fettiplace R. Optimal electrical properties of outer hair cells ensure cochlear amplification. PLoS One. 2012, 7, e50572. 7. Sisto R and Moleti A. Low-passed outer hair cell response and apical-basal transition in a nonlinear transmissionline cochlear model. J Acoust Soc Am. 2021, 149, 1296. 8. Wang Y, Steele CR, Puria S. Cochlear Outer-Hair-Cell Power Generation and Viscous Fluid Loss. Sci Rep. 2016 6, 19475. 9. Sisto R, Belardinelli D, Moleti A. Fluid focusing and viscosity allow high gain and stability of the cochlear response. J Acoust Soc Am. 2021, 150, 4283. 10. Guinan J, Jr. How does mechanics in the cochlear base translate to mechanics in the low-frequency cochlear apex? Proceedings of the International Workshop on Mechanics of Hearing 2024, Ann Arbor, Michigan USA, 08-14 June 2024 11. Sisto R, Belardinelli D, Altoè A, Shera CA, Moleti A. Crucial contributions of 3-D viscous Hydrodynamics to cochlear amplification. J. Acoust. Soc. Am. 2023, 153, 77. 12. Altoè A, Dewey JB, Chariaziak K, Oghalai JS and Shera CA. Overturning the mechanism of cochlear amplification via area deformation of the organ of Corti. J.Acoust. Soc.Am 2022, 152, 2227. 13. Samaras G, Meaud J. Compliant Deiters’ Cells: A Challenge for Outer Hair Cell-based Cochlear Amplification? Proceedings of the International Workshop on Mechanics of Hearing 2024, Ann Arbor, Michigan USA, 08-14 June 2024 14. Cooper NP, Vavakou A and van der Heijden M. Vibration hotspots reveal longitudinal funneling of sound – evoked motion in the mammalian cochlea. Nat. Commun. 2018, 9, 3054. 15. He, W., Burwood, G., Porsov, E.V. et al. The reticular lamina and basilar membrane vibrations in the transverse direction in the basal turn of the living gerbil cochlea. Sci Rep 2022, 12, 19810. 16. Lee HY, Raphael PD, Xia A, Kim J, Grillet N, Applegate BE, Ellerbee Bowden AK, Oghalai JS. TwoDimensional Cochlear Micromechanics Measured In Vivo Demonstrate Radial Tuning within the Mouse Organ of Corti. J Neurosci. 2016, 36, 8160-73. COMMENTS AND DISCUSSION [Alessandro Altoè]: Hi Renata and Arturo, nice work! A couple of comments: —You state that the model implements full hydrodynamic coupling; can you elaborate on how the RL/TM vibration is coupled to the fluid? I had the impression that, because you take delta=0, in practice only BM motion is coupled to the pressure wave in your simulations. —I really appreciate your conclusion regarding the uncertainty in the data and its interpretation. I would like to point out that our argument regarding the “power-not-fed” on BM, was not primarily based on the RL moving in quadrature re BM, but was sparkled by the observation of phase-lag of the (hopefully transverse enough) OHC-Deiters’ cell vibration, and by the floppiness of the Deiters’ cells. While I agree with you that we should keep an open mind, given objective difficulties of interpreting the data, I believe that preservation of electricallyevoked emission in animal models with detached TM is a critical observation in support of "overturned amplification" (we gave a lengthy explanation in the paper you cite). It would be great to know to what extent a 2DOF model fluid-coupled on both BM and RL can explain those results [Renata Sisto]: thank you so much for your interest in our work. It will be nice having the opportunity of discussing at the meeting. I thought more thoroughly to what you asked me in your comment about the RL contribution to the traveling wave. I and Arturo definitely agree with you that the contribution to the TW generation comes from the OoC section area deformation. This deformation occurs both in the up direction (towards the SV) and in the down direction (towards the ST). In other words, both the fluid compression in the SV and in the ST produces a longitudinal motion, due to the fluid incompressibility. In our paper we set delta to zero per simplicity. In any case , we estimated that the RL vibration amplitude should contribute for a 25% and the BM for the 75%. This esteem is based on geometrical considerations (and we are not perfectly sure to have completely understood what the exact geometry is). When the RL moves with larger oscillations that the BM (we read this ratio RL/Bm could be of order 10) , its contributions to the TW can be larger than that of the BM and the fluid compression in the upper scalae can be more relevant. I hope this makes sense for you [Brian Frost]: Hi Renata - Really nice model. I have two suggestions for the revised manuscript regarding the truly 3-D motion in the OCC. 1. You have correctly mentioned that the motion in the OCC is not purely transverse, and it is perfectly fine that the model doesn't exhibit this 3-D motion. I am interested in whether the model could reasonably be extended to 2-D in future work, for example by adding longitudinal feedforward. I don't suggest that you do this work necessarily, rather I would like the possibility to be mentioned in the article so that the path to a more physical but still low-DOF model could be seen. 2. You are correct that there is disagreement in the field regarding the relative phase of BM and RL transverse motion. However, I believe there are recent measurements of this in gerbil by Cho and Puria (which you cite in the abstract), as well as Strimbu et al. that can be considered as "trustworthy". The Cooper et al. data, while wonderful, is taken at a very longitudinal angle. I think that the aforementioned transverse data should be referenced in at least the final section as it is the most direct comparison to your work. Thanks so much for the lovely manuscript and talk! [Renata Sisto]: Dear Brian thank you so much for your review! it's very important for me having your feedback I am interested in whether the model could reasonably be extended to 2-D in future work, for example by adding longitudinal feedforward. 1. It is very easy to add a longitudinal feedforward in my 1DOF , 3D hydrodynamical coupling cochlear model. I have done this implementation and if you were interested, I could provide you the code. But I am not sure this is what you are interested in. You are interested mostly in the longitudinal, lineal motion and I did not wrote the equations for these variables. In other words, my oscillators are described only by the transversal displacement and velocity. I should have to add other dynamical variables related to the longitudinal (and radial) degrees of freedom. I could try with my oversimplified FEM model in which, at the moment, the BM is described as an orthotropic plate with negligible longitudinal coupling. Maybe, relaxing this condition, I will be able to look at something related to what you are asking me. 2. thank you so much for providing me a guide in this intricate labyrinth of the data. I'll definitely keep in mind what you suggested in the extended version of my work. I'll compare the results of my model to data shown in Strimbu et al. [Yanli Wang]: Hi Renata, Thanks for this work in adding fluid focusing, damping, and a more physiologically-based OHC model to the existing low degree of freedom model, to provide insight and being able to explain the relative motion between RL and BM with new data. I appreciate the iterative methods for solving wavenumber and fluid focusing factor alpha. Some possible improvements below: 1) I would like to see more explanation on how you controlled the "strength" of the active mechanism. Where does this occur in your equations? Which parameter did you adjust? 2) Can you explain the symbol tau in Eq. 4 and P_d in Eq. 5? 3) Can you provide a table with the actual parameters you used for Z1 Z2 etc... and how they are determined? 4) you mentioned that high viscosity was needed to get a better agreement with observed phenomenology. I wonder how much viscosity you need to put in. This is related to 3). 5) you mentioned that fluid focusing effect is diminished at RL. I suspect that with a real physical OoC modeled, the fluid focusing will be as strong right above TM. Could you comment on this? 6) Similar to Brian's comment, I wonder if you could comment on the possible effect of longitudinal motion in the OoC, the Y shape structure on the modeling results. Thanks! [Renata Sisto]: Dear Yanli it's always a great pleasure discussing with you. It reminds to me of Charles Steele that is always in my hearth. As regards your comments 1) the parameter I adjust is the strength of the active coupling force between the RL and BM The corresponding impedance Z4 is multiplied by a factor act with min =0.1 max = 0.8 , step =0.1. This linear parameter mimics the active behavior when the stimulus decreases 2) Thank you for asking! the active force term is, in this model, low pass filtered. This low pass filtering is very important in my view as it is in agreement with the low pass characteristics of the transmembrane potential. In other words, the amplification is performed cycle by cycle but the transmembrane potential acts as an RC circuit. Tau represents the characteristic time of this circuit and 1/tau is the cutoff frequency. When omega*tau (in equation (4) you find the Laplace variable s=1i*omega ) <<1, omega greater than the cutoff, the filter response is 1. On the opposite in the limit omega*tau >>1 , at frequency larger than the cutoff, the response of the filter is 1/(s*tau). As you can see, when you are at frequencies larger than the cutoff, the phase is rotated up to pi/2. This means that the stiffness term in z4 becomes an antidamping and the damping c4 becomes a stiffness term. In our view the low pass filter is able to change the phase, converting the elastic force into the antidamping term that amplifies the masses motion Pd in eq (5) is just the differential pressure between the scalae (Scala Vestibuli and Scala Tympani) 3) Many models parameters can be found in Sisto R, Belardinelli D, Altoè A, Shera CA, Moleti A. Crucial contributions of 3-D viscous Hydrodynamics to cochlear amplification. J. Acoust. Soc. Am. 153 (1), January 2023, pag. 77 – 86. In any case, we developed the 2DOF model during the time. You can find the parameters with 2DOF in Sisto et al. J. Acoust. Soc. Am. 146 (3), September 2019. After the 2DOF model, we introduced another equation for the low pass filter J Acoust Soc Am 2021 Feb;149(2):1296 doi: 10.1121/10.0003569. You can find listed the parameters of the 2DOF with and without the low pass filter 4) how much viscosity you have to put in the model depends on how much the model is active. If you just reduce the damping, but the model remains stable (positive damping term) you don't need a large viscosity contribution. If the model becomes very active, (the damping becomes an antidamping or, in other words, your poles cross the imaginary axis) you have to increase the viscosity to stabilize the model, especially if you don't use a true nonlinear model in which the active force cubically saturate. In the model you are looking at, the viscosity is that of the water. 5) In the model I considered the values of the wavenumber are quite large, consequently the fluid focusing effective height is quite less than 100 micron. In addition, in the model I presented, the contribution of the upper part of the OoC to the taveling wave is not so far included. I think that, as only the BM is strongly coupled to the fluid, and the k values are very high, the fluid focusing mainly occurs close to the BM. This is a consequence of how the model is built. I can adjust it so that it could be more close to the data and to more realistic models As I responded to Brian, the 2DOF model I presented does not contain the longitudinal (nor radial) degrees of freedom for the oscillators. As consequence, unfortunately, I cannot say nothing about the longitudinal motions of the OoC. I cannot comments on Y shape as regards the longitudinal motion. I can only say if the feed-forward feed-backward mechanism is able to produce an amplification of the traveling wave and this is surely true. The presence of a low pass filter is a major issue in the micromechanics community. Someone doubts that, due to the low pass filter, a cycly-by-cycle amplification could occur. The fluid focusing combined with the fact that the cutoff frequencies change along the BM (Nam & Fettiplace data) permit such an amplification. In our model, in addition, it is shown that the low pass filter has beneficial effects as it provides the right phase rotation in order to produce the necessary antidamping term. I hope this argument will convince you