A flexible anatomical set of mechanical models for the organ of Corti

We build a flexible platform to study the mechanical operation of the organ of Corti (OoC) in the transduction of basilar membrane (BM) vibrations to oscillations of an inner hair cell bundle (IHB). The anatomical components that we consider are the outer hair cells (OHCs), the outer hair cell bundles, Deiters cells, Hensen cells, the IHB and various sections of the reticular lamina. In each of the components we apply Newton’s equations of motion. The components are coupled to each other and are further coupled to the endolymph fluid motion in the subtectorial gap. This allows us to obtain the forces acting on the IHB, and thus study its motion as a function of the parameters of the different components. Some of the components include a nonlinear mechanical response. We find that slight bending of the apical ends of the OHCs can have a significant impact on the passage of motion from the BM to the IHB, including critical oscillator behaviour. In particular, our model implies that the components of the OoC could cooperate to enhance frequency selectivity, amplitude compression and signal to noise ratio in the passage from the BM to the IHB. Since the model is modular, it is easy to modify the assumptions and parameters for each component.

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Royal Society Open Science openscience@royalsociety.org on behalf of Pietro Cicuta (Subject Editor) openscience@royalsociety.org Reviewer comments to Author: Reviewer: 1 Comments to the Author(s) This study aims to develop a simplified analytical model to study the OoC behaviour. The text needs to be reviewed by a person fluent in English. In addition, the text needs to be revised structurally. For example, some terms used in the text is not scientific (e.g., hoping to find… instead of making a hypothesis), referencing is not adequate at all, the abstract looks like part of the introduction and does not convey the summary of the work adequately. The introduction section is a blend of abstract, methods and conclusions. The introduction section does not state the current knowledge and most up-to-date findings that should be used as a base for most of the assumptions made for the rest of work. The assumptions made to simplify the model are not justified properly and casts a doubt on the consistency and accuracy of the results. Unfortunately, there is no logical and rationale base to believe in this work at this stage and a major revision is needed. A complete review of the most recent data which have changed our understanding of the OoC mechanics is needed prior to developing a new model based on assumptions and hypotheses that are not valid anymore.

Some minor comments:
The line numbering is not good to make a clear address to a specific section of the text. below, the referencing was made by the page (P) number, line (L) number, left or right (L/R) column (C). P1, L11, LC: 'built' P1, L13, LC: motion P1, L14, LC: 'anatomical' P1, L17, LC: [, can nevertheless] P1, L24, LC: What is 'critical oscillator behaviour'? P1, L29, LC: The model does not 'provide a mechanism', it explains, predicts or an existing mechanism.
First sentence of Introduction: This does not convey any info just by stating the obvious and referring to bunch of papers Reference 4 is broken Page 1, Line 36: please rewrite the sentence. It does not follow logical the cause and effect chain. For example, the middle ear does not use the acoustic input to shake the fluid, it passively transfers the vibrations. P1, L39, LC: 'The wavelength of sound in perilymph is longer than the entire'. The wavelength is frequency dependent, at frequencies above 10 kHz which are still audible by human, the wavelength is shorter than 34mm, an average cochlear length in human. P1, L40, LC: What is the 'partitioned structure of the cochlea'? It can refer to the 3 fluid channels or discrete cellular arrangement longitudinally, for example. The rest of the sentence is vague too. What is 'a short segment of the partition'? P1, L42, LC: 'most of the elastic energy…' is there any evidence for this? P1, L45, LC: P1, L49, LC: … the output. P1, L49, LC: it is not clear why in this modelling approach OoC 'does not include the BM'. It is not based on any reason and 'accordingly' does not fit here. P1, L52, LC: what specific characteristics of basal region is taken into account in this model? geometry, material properties etc.? P1, L54, LC: 'In the basal region … where the OoC has the greatest impact on amplification and frequency selectivity'. Do you have any reference for this statement? Why would OoC has less impact on the amplification at the apex? P1, L57, LC: … (TM) whereas the IHB is not. Fig.1 what is the reference for the OoC arrangement? The OHCs have angle with the DC (tilted toward the base) any probably not visible with this angle of the view. Although it is not to scale, but OHB scale is not proportional at all. Use different color for CP. The figure does not show rest state in which the OHB are supposed to be vertical but it is not stated in the caption. HC's are a number of cells with different orientation. This figure shows a green curvature as a plate. P1, L26, RC: use anatomical orientations instead of 'clockwise'. P1, L26, RC: 'motion of the reticular lamina (RL) has no direct effect on the inclination of the IHB.' Why this is assumed? P1, L29, RC: 'performance' is a vague term here. P1, L32, RC: 'We would like to answer questions such as: Why the IHB is not attached to the TM?...' this kind of studies do not answer such questions. The aim of this study should be explaining and identifying the benefits of such structures. P1, L39, RC: 'Many theoretical treatments fall into an extreme category.' Is very vague and noninformative. It is not clear that authors are comparing their method with previous work. P1, L40, RC: 'At one extreme …' what extreme? Which one? 'OoC is [presented] by…' P1, L45, RC: 'Neither of these approaches enables us to answer'. First of all, the authors defined a wrong question, secondly, why the other methods, specially FEM cannot explain the OoC behaviour? 'with idealised geometry and with as few elements and forces as possible, hoping to capture the features'. 'Hope' is not a scientific term! Any study whether experimental or computational, tries to reduce the previous assumptions and improve a realistic representation to explain a phenomenon. Page 2, 1st paragraph: frequency tuning and amplitude compressions should be defined first. P2, L9, LC: this kind of referencing is neither scientific nor informative. P2, L12, LC: what is the purpose of this sentence and bunch of references? P2, L22, LC: 'Another salient difference is that the RL is not regarded as a completely rigid body' Is the RL rigid in your model or the others? why should it be rigid? P2, L37, LC: 'but we believe that the important fact is that…' what makes you believe this? P2, L40, LC: Most recent measurements contradict this. E.g., 'Ren, T., He, W., & Kemp, D. (2016). Reticular lamina and basilar membrane vibrations in living mouse cochleae. Proceedings of the National Academy of Sciences, 113(35), 9910-9915.' P2, L50, LC: [showed that the x-…] P2, L51, LC: This is not correct. Ref. 20 (Fig.2) shows that x&y motions depend on the measurement location and the frequency. 'which is usually disregarded' references for both assumptions are required. 'Pressure exerted by endolymph on OoC components is expected' why? Your rationale or a reference?
Reviewer: 2 Comments to the Author(s) While the author's ability to construct the model and get it to be stable is commendable, contribution to new methodology is not strong. Would like to see more detailed comparison with existing data, which is what most cochlear modelers would be interested, and this is actually the advantage of a numerical model. Fig1. The schematic drawing does not reflect the anatomical features in the OC. The Hensen cells seem to have a free end which is not true. The motion the BM is due to pressure difference between the scala media and scala tympani (ST), but I do not see ST in Fig 1. "There are normally three rows of outer hair cells, but we believe that the important fact is that there is more than one, and include just two outer hair cells in our explicit models." But the BM displacement shape is complicated and two OHCs may not address this feature well.

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Author's Response to Decision Letter for (RSOS-200675.R0)
See Appendix A.

Recommendation?
Accept with minor revision (please list in comments)

Comments to the Author(s)
General comments and suggestions: The presentation of the work has been improved significantly. Authors make a clever use of dimensionless analysis, although at the end I would suggest scaling the frequency and spatial dimensions to a specific specimen (animal or human) to provide a realistic perception of the numbers (e.g., critical frequency). The unsynchronized work generated by the OHCs are assumed to be the reason for lower vibrations, but I could not distinguish it from mechanical energy dissipation caused by the mechanical and passive damping of the system. The amplification mechanism for the slice model is thoroughly studied, but extrapolating it to the full-length cochlea needs a continuous segmentation or at least wave-dependent repetition of the slice model. For example, the amplification of the OHCs shifts the resonance to higher frequencies e.g., in fig.2M of (Lee, H. Y., Raphael, P. D., Xia, A., Kim, J., Grillet, N., Applegate, B. E., ... & Oghalai, J. S. (2016). Two-dimensional cochlear micromechanics measured in vivo demonstrate radial tuning within the mouse organ of Corti. Journal of Neuroscience, 36(31), 8160-8173.) the resonance peak has shifted from 5kHz at 80 dB to 10 kHz at 10 dB. It was not clear to me that model can predict the shift to make conclusion about the amplification of the traveling wave. Specific comments: Maybe it is the RSOS style, but the line numbering is still troubling. P1,L38, RC: 'typical audible frequencies' is a spectrum from 20 to 20,000Hz, the wavelength varies by 3 order of magnitudes. P1,L40, RC: First you refer to <fluid> chambers as 'partitions', then in the same sentence partition is referring to a small cross-sectional region of the BM where the "energy" is "deposited". The switch between reference of the partition continues in the next sentences. The definition of 'partition' could be different as mentioned in different papers (fluid chambers partitions, cross-sectional partitions of the cochlea etc.), but should be consistent in the text. L55: 'separate collagen fibers, with length, width and stiff-ness that gradually vary' could you put a reference for this? I believe the collagen fibers vary in concentration but not in width and length thorough the cochlear length (Cabezudo, L. M. (1978). The ultrastructure of the basilar membrane in the cat. Acta oto-laryngologica, 86(1-6), 160-175.) LC, L11: Please rewrite this sentence: 'exposed only to pressure differences within the SM, is exposed to the large pressure difference between'. L22: 'frequency tuning (output sharply peaked at some frequency for a given input) and amplitude compression (input changes by several orders of magnitude give rise to significantly smaller changes of the output).' What are inputs and outputs? I assume the input is the mechanical excitation that is delivered to the cochlea at the oval window and the output is electrical signals of the auditory neurons. Please revise the definition of the compression. L28: what does 'theoretical treatments' mean? L29: the representation of mechanical systems with electrical circuit models is well-accepted method in mechanics of hearing. The newton law is a simple relationship between force and acceleration which analogously could be represented by Maxwell's equations in electrical systems. I am a mechanical engineer, but I don't think this is an issue to question the method. L33: this is not true, there are FE models in which the cochlear amplification and nonlinearity have been implemented (e.g., Motallebzadeh, H., Soons, J. A., & Puria, S. (2018). Cochlear amplification and tuning depend on the cellular arrangement within the organ of Corti. Proceedings of the National Academy of Sciences, 115 (22), 5762-5767.). Any modelling study is basically a trade of simplicity and realistic representation of a system. Even with detailed FE models, one can differentiate contribution of different components on the overall response. When you make a realistic system the interactions between the components get more complicated, but the model is still following the physical governing laws, you make a simple model, you neglect more details, and deviate from the realistic behaviour and you may not see the features that simplification cause, but what you see is easier to understand. This paragraph is not a strong argument to question other methods to advertise your method. L43: 'Substantial evidence(s) has(have) led … L57: In recent models (full length cochlear models) the pressure is a function of place and time as well (e.g., Sasmal, A., & Grosh, K. (2019). Unified cochlear model for low-and high-frequency mammalian hearing. Proceedings of the National Academy of Sciences, 116 (28) : Please define 'sgn' function P4, L52: please elaborate 'work performed by the bundle motility vanishes'. Work is performed by force do you mean that hair bundles, in addition to the OHC body, generate force? Line 27: Please elaborate on 'non vanishing work'. Damped or dissipated energy refers to energy loss due to the damping or friction but here I believe you are referring to the work due to the OHC motility. P5, L11: remind the reader that the length 1 is in the adopted normalized coordinate system. L46: please elaborate on 'we do not consider noise that arises in the OoC itself'. What kind of noise the OoC can produce and what exactly noise mean here? Do you mean perturbations due to the nonuniform distribution of the mechanical properties? 'A. Main Results' is not a specific subtitle to differentiate from the rest of the results. Maybe 'summary of key findins'? L 37, LC: please elaborate on 'self-oscillations (non zero output for zero input)'. Do you mean the system is unstable or underdamped in this specific satiation? P5, L42, RC: Could you translate the units and dimensions for a specific specimen so the reader can have an idea of the critical frequency? 'corresponds to a contraction of a few percent' of a dimensionless parameter is not enough to evaluate its value. L52, RC: 'then the OoC would provide additional tuning; if it is not, the OoC would provide an alternative mechanism for tuning' what is the difference of 'additional tuning' and 'alternative mechanism for tuning'? P8, L53 LC: 'is in the range between 0° and 180° (or equivalent)' please clarify. What is equivalent to 0-180? It could be either pi plus/minus or 2pi plus/minus, depending on the periodicity function. P8, L41 RC: 'Within a more realistic model, the energy exchange described here should be regarded as a contribution.' Contribution to? P8, L43 RC: The attenuation according to the calculations here, is coming from unsynchronized OHC force, but is there any direct calculation of the attenuations due to the damping of the system? Is it possible even when OHC provides constructive force (and work) that work is dissipated due to the mechanical damping? P10, 5, LC: although HC's are outside the subtectorial space, but do to their large deformations (because of lower stiffness), could they also pump fluid into the subtectorial space?

Review form: Reviewer 2
Is the manuscript scientifically sound in its present form? No

Do you have any ethical concerns with this paper? No
Have you any concerns about statistical analyses in this paper? No

Recommendation?
Major revision is needed (please make suggestions in comments) Comments to the Author(s) Some of the assumptions made to the model are not justified properly and lack necessary supporting materials, which makes it difficult to follow the logic of the work. In the abstract and throughout the text, verb tense was used incorrectly. Some other comments can be found below: To explain comments clearly, the referencing was made by the page (P) number, line (L) number, and left or right (L/R) column. P1, L38~40, L: "For typical audible frequencies, the wave length of sound in the perilymph is of the same order of magnitude as the length of the entire cochlea." This refers to the human cochlea, but there is no clear explanation that the model proposed in this paper was for the human or other species. P1, L55, L: the BM thickness also varies along its length. P1, L52, L: "The partition is composed of the BM, the organ of Corti (OoC) and the tectorial membrane (TM)". This description includes only three main components, but there are other important components within the cochlear partition. P1, L57, L: "The BM has a large Young modulus", the term Young modulus is usually used in the cochlear model but not for describing the physiological feature of the BM. P1, L11, R: The statement "pressure differences within the SM" is not proper, there is a pressure difference between the SM and ST, what does pressure difference within the SM refer to? There is another conflict with this point (P2, L25~L27, R) where the pressure in the SM will be taken equal to the pressure in the tissues under the RL and the HC.
P2, L26, R: "the RL pivots as a rigid beam around the pillar cells head". This was mentioned several times in the paper and was used as a reference for setting the origin. But this somewhat conflicts with the point that the authors introduced cuticular plates into the model. P2, L40~42, L: "Accordingly, except for rotational and for fluid motion, motion will be restricted to the y-direction." The authors mentioned that the amplitudes of the x and of the y components of the relative motion are of the same order of magnitude, which implies that motions in both directions are important. (P10, L2~3, L) "According to our models, the fluid flow at the IHB region is driven by the vertical motion of the CPs," This is because the current setting only considers motion in the y-direction.

Decision letter (RSOS-210016.R0)
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Reviewer comments to Author: Reviewer: 2 Comments to the Author(s) Some of the assumptions made to the model are not justified properly and lack necessary supporting materials, which makes it difficult to follow the logic of the work. In the abstract and throughout the text, verb tense was used incorrectly. Some other comments can be found below: To explain comments clearly, the referencing was made by the page (P) number, line (L) number, and left or right (L/R) column. P1, L38~40, L: "For typical audible frequencies, the wave length of sound in the perilymph is of the same order of magnitude as the length of the entire cochlea." This refers to the human cochlea, but there is no clear explanation that the model proposed in this paper was for the human or other species. P1, L55, L: the BM thickness also varies along its length. P1, L52, L: "The partition is composed of the BM, the organ of Corti (OoC) and the tectorial membrane (TM)". This description includes only three main components, but there are other important components within the cochlear partition. P1, L57, L: "The BM has a large Young modulus", the term Young modulus is usually used in the cochlear model but not for describing the physiological feature of the BM. P1, L11, R: The statement "pressure differences within the SM" is not proper, there is a pressure difference between the SM and ST, what does pressure difference within the SM refer to? There is another conflict with this point (P2, L25~L27, R) where the pressure in the SM will be taken equal to the pressure in the tissues under the RL and the HC.
P2, L3, L: Is there any supporting evidence? P2, L19~23, L: It would be interesting to see the difference of using three OHCs.
P2, L26, R: "the RL pivots as a rigid beam around the pillar cells head". This was mentioned several times in the paper and was used as a reference for setting the origin. But this somewhat conflicts with the point that the authors introduced cuticular plates into the model. P2, L40~42, L: "Accordingly, except for rotational and for fluid motion, motion will be restricted to the y-direction." The authors mentioned that the amplitudes of the x and of the y components of the relative motion are of the same order of magnitude, which implies that motions in both directions are important. (P10, L2~3, L) "According to our models, the fluid flow at the IHB region is driven by the vertical motion of the CPs," This is because the current setting only considers motion in the y-direction.
Reviewer: 1 Comments to the Author(s) General comments and suggestions: The presentation of the work has been improved significantly. Authors make a clever use of dimensionless analysis, although at the end I would suggest scaling the frequency and spatial dimensions to a specific specimen (animal or human) to provide a realistic perception of the numbers (e.g., critical frequency). The unsynchronized work generated by the OHCs are assumed to be the reason for lower vibrations, but I could not distinguish it from mechanical energy dissipation caused by the mechanical and passive damping of the system. The amplification mechanism for the slice model is thoroughly studied, but extrapolating it to the full-length cochlea needs a continuous segmentation or at least wave-dependent repetition of the slice model. For example, the amplification of the OHCs shifts the resonance to higher frequencies e.g., in fig.2M of (Lee, H. Y., Raphael, P. D., Xia, A., Kim, J., Grillet, N., Applegate, B. E., ... & Oghalai, J. S. (2016). Two-dimensional cochlear micromechanics measured in vivo demonstrate radial tuning within the mouse organ of Corti. Journal of Neuroscience, 36(31), 8160-8173.) the resonance peak has shifted from 5kHz at 80 dB to 10 kHz at 10 dB. It was not clear to me that model can predict the shift to make conclusion about the amplification of the traveling wave. Specific comments: Maybe it is the RSOS style, but the line numbering is still troubling. P1,L38, RC: 'typical audible frequencies' is a spectrum from 20 to 20,000Hz, the wavelength varies by 3 order of magnitudes. P1,L40, RC: First you refer to chambers as 'partitions', then in the same sentence partition is referring to a small cross-sectional region of the BM where the "energy" is "deposited". The switch between reference of the partition continues in the next sentences. The definition of 'partition' could be different as mentioned in different papers (fluid chambers partitions, crosssectional partitions of the cochlea etc.), but should be consistent in the text. L55: 'separate collagen fibers, with length, width and stiff-ness that gradually vary' could you put a reference for this? I believe the collagen fibers vary in concentration but not in width and length thorough the cochlear length (Cabezudo, L. M. (1978). The ultrastructure of the basilar membrane in the cat. Acta oto-laryngologica, 86(1-6), 160-175.) LC, L11: Please rewrite this sentence: 'exposed only to pressure differences within the SM, is exposed to the large pressure difference between'. L22: 'frequency tuning (output sharply peaked at some frequency for a given input) and amplitude compression (input changes by several orders of magnitude give rise to significantly smaller changes of the output).' What are inputs and outputs? I assume the input is the mechanical excitation that is delivered to the cochlea at the oval window and the output is electrical signals of the auditory neurons. Please revise the definition of the compression. L28: what does 'theoretical treatments' mean? L29: the representation of mechanical systems with electrical circuit models is well-accepted method in mechanics of hearing. The newton law is a simple relationship between force and acceleration which analogously could be represented by Maxwell's equations in electrical systems. I am a mechanical engineer, but I don't think this is an issue to question the method. L33: this is not true, there are FE models in which the cochlear amplification and nonlinearity have been implemented (e.g., Motallebzadeh, H., Soons, J. A., & Puria, S. (2018). Cochlear amplification and tuning depend on the cellular arrangement within the organ of Corti. Proceedings of the National Academy of Sciences, 115(22), 5762-5767.). Any modelling study is basically a trade of simplicity and realistic representation of a system. Even with detailed FE models, one can differentiate contribution of different components on the overall response. When you make a realistic system the interactions between the components get more complicated, but the model is still following the physical governing laws, you make a simple model, you neglect more details, and deviate from the realistic behaviour and you may not see the features that simplification cause, but what you see is easier to understand. This paragraph is not a strong argument to question other methods to advertise your method. L43: 'Substantial evidence(s) has(have) led … L57: In recent models (full length cochlear models) the pressure is a function of place and time as well (e.g., Sasmal, A., & Grosh, K. : Please define 'sgn' function P4, L52: please elaborate 'work performed by the bundle motility vanishes'. Work is performed by force do you mean that hair bundles, in addition to the OHC body, generate force? Line 27: Please elaborate on 'non vanishing work'. Damped or dissipated energy refers to energy loss due to the damping or friction but here I believe you are referring to the work due to the OHC motility. P5, L11: remind the reader that the length 1 is in the adopted normalized coordinate system. L46: please elaborate on 'we do not consider noise that arises in the OoC itself'. What kind of noise the OoC can produce and what exactly noise mean here? Do you mean perturbations due to the nonuniform distribution of the mechanical properties? 'A. Main Results' is not a specific subtitle to differentiate from the rest of the results. Maybe 'summary of key findins'? L 37, LC: please elaborate on 'self-oscillations (non zero output for zero input)'. Do you mean the system is unstable or underdamped in this specific satiation? P5, L42, RC: Could you translate the units and dimensions for a specific specimen so the reader can have an idea of the critical frequency? 'corresponds to a contraction of a few percent' of a dimensionless parameter is not enough to evaluate its value. L52, RC: 'then the OoC would provide additional tuning; if it is not, the OoC would provide an alternative mechanism for tuning' what is the difference of 'additional tuning' and 'alternative mechanism for tuning'? P8, L53 LC: 'is in the range between 0° and 180° (or equivalent)' please clarify. What is equivalent to 0-180? It could be either pi plus/minus or 2pi plus/minus, depending on the periodicity function. P8, L41 RC: 'Within a more realistic model, the energy exchange described here should be regarded as a contribution.' Contribution to? P8, L43 RC: The attenuation according to the calculations here, is coming from unsynchronized OHC force, but is there any direct calculation of the attenuations due to the damping of the system? Is it possible even when OHC provides constructive force (and work) that work is dissipated due to the mechanical damping? P10, 5, LC: although HC's are outside the subtectorial space, but do to their large deformations (because of lower stiffness), could they also pump fluid into the subtectorial space?

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Recommendation?
Accept with minor revision (please list in comments)

Comments to the Author(s)
Minor comments: P1,4, RC: "The BM acts as an elastic barrier and is exposed to the large pressure difference between the SM and the ST", the pressure gradient between SM and ST varies along the length as a function of frequency. Apical to the characteristic frequency there is almost no pressure difference between these two chambers. P2 26-46, LC: In my opinion the main feature of this work in implementing analytical approach to solve the mechanics of the OoC. FE studies use numerical discretion methods and lumped element models use electrical analogy. P2 2-4 RC: RL and CPs are making a continues frame, which is flexible at CPs and rigid at RL segments. Authors emphasize on the RL characteristics 'A salient' feature of their method and a significant deviation from 'featureless body' from the whole studies in the literature. There are studies of both rigid and flexible RL and I suggest to emphasize on more important characteristics on the study. P3 14-15 RC: "fluid velocity as being similar to that of the BM velocity". Due to the no-slip boundary condition of the NS equations on the BM and fluid interface, their velocities are exactly the same (not similar) at the BM surface. P5 51, RC: Is this instability part of the OoC physiology which results in spontaneous otoacoustic emissions? P6 19 LC: The hearing frequency is more common that angular frequency (omega) in hearing acoustics, although they are linearly related, but I would suggest converting them to actual frequency (as done in line 12 the same page). P9 F. Nonlinearity: I could not follow the argument of nonlinearity due to the non-harmonic theta. What kind of nonlinearity is referred here? How does non-sinusoidal theta and the Fourier transfer of it establish nonlinearity?

Decision letter (RSOS-210016.R1)
We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.

Dear Dr Berger
On behalf of the Editors, we are pleased to inform you that your Manuscript RSOS-210016.R1 "A flexible anatomical set of mechanical models for the organ of Corti" has been accepted for publication in Royal Society Open Science subject to minor revision in accordance with the referees' reports. Please find the referees' comments along with any feedback from the Editors below my signature.
We invite you to respond to the comments and revise your manuscript. Below the referees' and Editors' comments (where applicable) we provide additional requirements. Final acceptance of your manuscript is dependent on these requirements being met. We provide guidance below to help you prepare your revision.
Please submit your revised manuscript and required files (see below) no later than 7 days from today's (ie 09-Aug-2021) date. Note: the ScholarOne system will 'lock' if submission of the revision is attempted 7 or more days after the deadline. If you do not think you will be able to meet this deadline please contact the editorial office immediately.
Please note article processing charges apply to papers accepted for publication in Royal Society Open Science (https://royalsocietypublishing.org/rsos/charges). Charges will also apply to papers transferred to the journal from other Royal Society Publishing journals, as well as papers submitted as part of our collaboration with the Royal Society of Chemistry (https://royalsocietypublishing.org/rsos/chemistry). Fee waivers are available but must be requested when you submit your revision (https://royalsocietypublishing.org/rsos/waivers). Comments to the Author(s) Minor comments: P1,4, RC: "The BM acts as an elastic barrier and is exposed to the large pressure difference between the SM and the ST", the pressure gradient between SM and ST varies along the length as a function of frequency. Apical to the characteristic frequency there is almost no pressure difference between these two chambers. P2 26-46, LC: In my opinion the main feature of this work in implementing analytical approach to solve the mechanics of the OoC. FE studies use numerical discretion methods and lumped element models use electrical analogy. P2 2-4 RC: RL and CPs are making a continues frame, which is flexible at CPs and rigid at RL segments. Authors emphasize on the RL characteristics 'A salient' feature of their method and a significant deviation from 'featureless body' from the whole studies in the literature. There are studies of both rigid and flexible RL and I suggest to emphasize on more important characteristics on the study. P3 14-15 RC: "fluid velocity as being similar to that of the BM velocity". Due to the no-slip boundary condition of the NS equations on the BM and fluid interface, their velocities are exactly the same (not similar) at the BM surface. P5 51, RC: Is this instability part of the OoC physiology which results in spontaneous otoacoustic emissions? P6 19 LC: The hearing frequency is more common that angular frequency (omega) in hearing acoustics, although they are linearly related, but I would suggest converting them to actual frequency (as done in line 12 the same page). P9 F. Nonlinearity: I could not follow the argument of nonlinearity due to the non-harmonic theta. What kind of nonlinearity is referred here? How does non-sinusoidal theta and the Fourier transfer of it establish nonlinearity? ===PREPARING YOUR MANUSCRIPT=== Your revised paper should include the changes requested by the referees and Editors of your manuscript. You should provide two versions of this manuscript and both versions must be provided in an editable format: one version identifying all the changes that have been made (for instance, in coloured highlight, in bold text, or tracked changes); a 'clean' version of the new manuscript that incorporates the changes made, but does not highlight them. This version will be used for typesetting.
Please ensure that any equations included in the paper are editable text and not embedded images.
Please ensure that you include an acknowledgements' section before your reference list/bibliography. This should acknowledge anyone who assisted with your work, but does not qualify as an author per the guidelines at https://royalsociety.org/journals/ethicspolicies/openness/.
While not essential, it will speed up the preparation of your manuscript proof if you format your references/bibliography in Vancouver style (please see https://royalsociety.org/journals/authors/author-guidelines/#formatting). You should include DOIs for as many of the references as possible.
If you have been asked to revise the written English in your submission as a condition of publication, you must do so, and you are expected to provide evidence that you have received language editing support. The journal would prefer that you use a professional language editing service and provide a certificate of editing, but a signed letter from a colleague who is a native speaker of English is acceptable. Note the journal has arranged a number of discounts for authors using professional language editing services (https://royalsociety.org/journals/authors/benefits/language-editing/).

===PREPARING YOUR REVISION IN SCHOLARONE===
To revise your manuscript, log into https://mc.manuscriptcentral.com/rsos and enter your Author Centre -this may be accessed by clicking on "Author" in the dark toolbar at the top of the page (just below the journal name). You will find your manuscript listed under "Manuscripts with Decisions". Under "Actions", click on "Create a Revision".
Attach your point-by-point response to referees and Editors at Step 1 'View and respond to decision letter'. This document should be uploaded in an editable file type (.doc or .docx are preferred). This is essential.
Please ensure that you include a summary of your paper at Step 2 'Type, Title, & Abstract'. This should be no more than 100 words to explain to a non-scientific audience the key findings of your research. This will be included in a weekly highlights email circulated by the Royal Society press office to national UK, international, and scientific news outlets to promote your work.

At
Step 3 'File upload' you should include the following files: --Your revised manuscript in editable file format (.doc, .docx, or .tex preferred). You should upload two versions: 1) One version identifying all the changes that have been made (for instance, in coloured highlight, in bold text, or tracked changes); 2) A 'clean' version of the new manuscript that incorporates the changes made, but does not highlight them. --If you are requesting a discretionary waiver for the article processing charge, the waiver form must be included at this step.
--If you are providing image files for potential cover images, please upload these at this step, and inform the editorial office you have done so. You must hold the copyright to any image provided.
--A copy of your point-by-point response to referees and Editors. This will expedite the preparation of your proof.

At
Step 6 'Details & comments', you should review and respond to the queries on the electronic submission form. In particular, we would ask that you do the following: --Ensure that your data access statement meets the requirements at https://royalsociety.org/journals/authors/author-guidelines/#data. You should ensure that you cite the dataset in your reference list. If you have deposited data etc in the Dryad repository, please only include the 'For publication' link at this stage. You should remove the 'For review' link.
--If you are requesting an article processing charge waiver, you must select the relevant waiver option (if requesting a discretionary waiver, the form should have been uploaded at Step 3 'File upload' above).
--If you have uploaded ESM files, please ensure you follow the guidance at https://royalsociety.org/journals/authors/author-guidelines/#supplementary-material to include a suitable title and informative caption. An example of appropriate titling and captioning may be found at https://figshare.com/articles/Table_S2_from_Is_there_a_trade-off_between_peak_performance_and_performance_breadth_across_temperatures_for_aerobic_sc ope_in_teleost_fishes_/3843624.

At
Step 7 'Review & submit', you must view the PDF proof of the manuscript before you will be able to submit the revision. Note: if any parts of the electronic submission form have not been completed, these will be noted by red message boxes.

Decision letter (RSOS-210016.R2)
We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.
Dear Dr Berger, I am pleased to inform you that your manuscript entitled "A flexible anatomical set of mechanical models for the organ of Corti" is now accepted for publication in Royal Society Open Science.
Please ensure that you send to the editorial office an editable version of your accepted manuscript, and individual files for each figure and table included in your manuscript. You can send these in a zip folder if more convenient. Failure to provide these files may delay the processing of your proof. You may disregard this request if you have already provided these files to the editorial office.
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Please see the Royal Society Publishing guidance on how you may share your accepted author manuscript at https://royalsociety.org/journals/ethics-policies/media-embargo/. After publication, some additional ways to effectively promote your article can also be found here https://royalsociety.org/blog/2020/07/promoting-your-latest-paper-and-tracking-yourresults/.
On behalf of the Editors of Royal Society Open Science, thank you for your support of the journal and we look forward to your continued contributions to Royal Society Open Science. If you have not already done so, please remember to make any data sets or code libraries 'live' prior to publication, and update any links as needed when you receive a proof to check -for instance, from a private 'for review' URL to a publicly accessible 'for publication' URL. It is good practice to also add data sets, code and other digital materials to your reference list.
You can expect to receive a proof of your article in the near future. Please contact the editorial office (openscience@royalsociety.org) and the production office (openscience_proofs@royalsociety.org) to let us know if you are likely to be away from e-mail Dear Editor, Thank you for your mail from 7-Aug-2020 Re: our manuscript, RSOS-200675, entitled "A flexible anatomical set of mechanical models for the organ of Corti".
We carefully read the comments, corrections and suggestions of the reviewers and amended the paper accordingly. We list below in bold face font our reply to each of the reviewer comments and requests. In addition to addressing these specific points, we made additional changes and improvements throughout the paper following the general philosophy expressed by the reviewers.
We thank the reviewers for their careful reading of the paper and for their productive criticism that has helped us improve the current version significantly. We hope that the revised paper can now be accepted for publication.
Concerning the style of our Introduction, the author guidelines of the Royal Society state that the authors should feel free to subdivide the main text as best suits the article. We followed the standard format in our disciplines, i.e, the first section of the article is the Introduction, which customarily contains a brief review of the background (complemented by a cluster of references that provide the additional required background details), states the goals of the study at hand, cites studies with similar goals, and describes the differences between these previous studies and the one at hand.
Due to the unconventional character of our goal, we took care to not just state what it is, but also what it is not. We did not look for a better description of the OoC in some particular animal. Instead, we built an instrument that enables any researcher to try any particular hypothesis on the mechanical behaviour of some anatomical component of the OoC, and assess the influence that it has on the mechanical activity of OoC as a whole. This being the case, there is room in our article for revisiting hypotheses that have not been experimentally supported. We are confident that the instrument that we provide can be useful to the readers of RSOS.
Answers to the reviewers Note that reference numbers indicated below correspond to those in the present version of the manuscript.

Replies to Reviewer 1
Reviewer: This study aims to develop a simplified analytical model to study the OoC behaviour. The text needs to be reviewed by a person fluent in English. In addition, the text needs to be revised structurally. For example, some terms used in the text is not scientific (e.g., hoping to find… instead of making a hypothesis), referencing is not adequate at all, the abstract looks like part of the introduction and does not convey the summary of the work adequately. The introduction section is a blend of abstract, methods and conclusions. The introduction section does not state the current knowledge and most upto-date findings that should be used as a base for most of the assumptions made for the rest of work.
The assumptions made to simplify the model are not justified properly and casts a doubt on the consistency and accuracy of the results. Unfortunately, there is no logical and rationale base to believe in this work at this stage and a major revision is needed. A complete review of the most recent data which have changed our understanding of the OoC mechanics is needed prior to developing a new model based on assumptions and hypotheses that are not valid anymore.
Authors: We amended the text in many places according to the reviewer comments, suggestions, and general criticism. Changes made in direct response to specific reviewer's comments are detailed below. In the modified text of the paper we highlighted these specific changes as well as those made following the spirit of the reviewers' feedback.
The abstract has been rewritten to convey a concise summary of the work we present in the paper. The paper was professionally reviewed by a language editor and was corrected accordingly. We replaced phrases that Reviewer 1 found as "non scientific" by more precise phrases throughout. To the best of our knowledge, we have comprehensively cited the literature which we consider relevant to the present study, either directly or through some review. The assumptions made in our work are based either on the literature we quote or on the spirit of this study, which favors taking the simplest possible model (what is usually dubbed "the minimal model").
We also carefully followed all the reviewer's suggestions regarding the justification of the models and methodology which we now set forward in the Introduction section.

Some minor comments:
The line numbering is not good to make a clear address to a specific section of the text. below, the referencing was made by the page (P) number, line (L) number, left or right (L/R) column (C). P1, L11, LC: 'built' The text was corrected.

P1, L13, LC: motion
Corrected in the new abstract.

P1, L17, LC: [, can nevertheless]
The abstract was rewritten, and this sentence was removed.

P1, L24, LC: What is 'critical oscillator behaviour'?
This is explained in the main body of the paper (Section 4A and Appendix 3). We are not accustomed to including citations in the abstract.
P1, L29, LC: The model does not 'provide a mechanism', it explains, predicts or an existing mechanism.
The abstract was rewritten, and this term was removed.
First sentence of Introduction: This does not convey any info just by stating the obvious and referring to bunch of papers We think such an opening sentence might be useful to a reader not familiar with the basic literature. In fact, this is a common practice. For instance, the first sentence in a recent paper in RSOS (