Reduced equations for the hydroelastic waves in the cochlea: The spring model
Authors:
Lydia Peres Hari, Jacob Rubinstein and Peter Sternberg
Journal:
Quart. Appl. Math. 74 (2016), 647-670
MSC (2010):
Primary 92C10
DOI:
https://doi.org/10.1090/qam/1443
Published electronically:
July 18, 2016
MathSciNet review:
3539027
Full-text PDF Free Access
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Abstract: Hydroelastic waves in the cochlea are studied through modeling a passive basilar membrane as an elastic spring. A rigorous reduction of the three-dimensional equations for the fluid pressure and deflection of the basilar membrane to a one- dimensional ordinary differential equation for the pressure jump across the membrane is derived. The one-dimensional reduced model is then critically examined and limits on its validity are discussed. An approximate solution of the reduced equations is in agreement with the experimental Greenwood formula for a proper selection of elastic parameters. The model is used to compute the effect of cochlear implants on the Place Principle governing the spectral decomposition of sound by the cochlea. Numerics are also carried out to see the effect of a cochlear implant on the mechanical response of the cochlea.
References
- J. B. Allen, Cochlear micromechanics - a mechanism for transferring mechanical to neural tuning within the cochlea, J. Acoust. Soc. Amer., 62, 930-940, 1977.
- J. Ashmore et al., The remarkable cochlear amplifier, Hearing Research, 266, 1-17, 2010.
- Gregory Berkolaiko, Robert Carlson, Stephen A. Fulling, and Peter Kuchment (eds.), Quantum graphs and their applications, Contemporary Mathematics, vol. 415, American Mathematical Society, Providence, RI, 2006. MR 2279143, DOI 10.1090/conm/415
- T. Fukaawa, A model of cochlear micromechanics, Hearing Research, 113, 182-190, 1997.
- E. Givelberg and J. Bunn, A comprehensive three-dimensional model of the cochlea, J. Comp. Phys. 191, 377-391, 2003.
- T. Gold, Hearing II: The physical basis for the action of the cochlea, Proc. R. Soc. Lond. 135, 492-298, 1948.
- D. Geisler and C. Sang, A cochlear model using feed-forward outer hair cell forces, Hearing Research, 86, 132-146, 1995.
- D. D. Greenwood, Critical bandwidth and the frequency coordinates of the basilar membrane, J. Acoust. Soc. Amer., 33, 1344-1356, 1961.
- D. D. Greenwood, A cochlear frequency-position function for several species-29 years later, J. Acoust. Soc. Amer., 87, 2592-2605, 1990.
- William R. Holmes, Michael Jolly, and Jacob Rubinstein, Hydro-elastic waves in a cochlear model: numerical simulations and an analytically reduced model, Confluentes Math. 3 (2011), no. 3, 523–541. MR 2847242, DOI 10.1142/S1793744211000382
- T. S. A. Jaffer, H. Kunov, and W. Wong, A model cochlear partition involving longitudinal elasticity, J. Acoust. Soc. Amer., 112, 576-589, 2002.
- Joseph B. Keller and John C. Neu, Asymptotic analysis of a viscous cochlear model, J. Acoust. Soc. Amer. 77 (1985), no. 6, 2107–2110. MR 791675, DOI 10.1121/1.391735
- James Keener and James Sneyd, Mathematical physiology, Interdisciplinary Applied Mathematics, vol. 8, Springer-Verlag, New York, 1998. MR 1673204
- Yongsam Kim and Jack Xin, A two-dimensional nonlinear nonlocal feed-forward cochlear model and time domain computation of multitone interactions, Multiscale Model. Simul. 4 (2005), no. 2, 664–690. MR 2162871, DOI 10.1137/040612464
- K. M. Lim and C. R. Steele, A three-dimensional nonlinear active cochlear model analyzed by the WKB method, Hearing Research, 170, 190-205, 2002.
- D. Manoussaki, E. K. Dimitriadis, and R. S. Chadwick, Cochlea’s graded curvature effect on low frequency waves, Phys. Rev. Lett., 96, 088701, 2006.
- Jindřich Nečas, Direct methods in the theory of elliptic equations, Springer Monographs in Mathematics, Springer, Heidelberg, 2012. Translated from the 1967 French original by Gerard Tronel and Alois Kufner; Editorial coordination and preface by Šárka Nečasová and a contribution by Christian G. Simader. MR 3014461, DOI 10.1007/978-3-642-10455-8
- W. M. Siebert, Ranke revisited-a simple short wave cochlear model, J. Acoust. Soc. Amer., 56, 594-601, 1974.
- C. R. Steele and L. A. Taber, Comparison of WKB and finite difference calculations for a two dimensional cochlear model, J. Acoust. Soc. Amer., 65, 1001-1006, 1979.
- R. Szalai, K. Tsaneva-Atanasova, M. E. Homer, A. R. Champneys, H. J. Kennedy, and N. P. Cooper, Nonlinear models of development, amplification and compression in the mammalian cochlea, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 369 (2011), no. 1954, 4183–4204. MR 2843876, DOI 10.1098/rsta.2011.0192
- M. A. Viergever, A two dimensional model for the cochlea: The heuristic approach and numerical results, J. Engn. Math., 11, 11-28, 1971.
- M. A. Viergever, Mechanics of the inner ear, Thesis, Technical University Delft, 1980.
- J. Xin, Dispersive instability and its minimization in time-domain computation of steady-state responses of cochlear models, J. Acoust. Soc. Amer., 115, 2173-2177, 2004.
- Y. Yoon, S. Puria, and C. R. Steele, A cochlear model using the time-averaged Lagrangian and the push-pull mechanism in the organ of Corti, J. Mech. Mater. Struct., 4, 977-986, 2009.
- G. Zweig, Finding the impedance of the organ of corti, J. Acoust. Soc. Amer., 89, 1229-1254, 1991.
References
- J. B. Allen, Cochlear micromechanics - a mechanism for transferring mechanical to neural tuning within the cochlea, J. Acoust. Soc. Amer., 62, 930-940, 1977.
- J. Ashmore et al., The remarkable cochlear amplifier, Hearing Research, 266, 1-17, 2010.
- Quantum graphs and their applications, edited by Gregory Berkolaiko, Robert Carlson, Stephen A. Fulling and Peter Kuchment, Contemporary Mathematics, vol. 415, American Mathematical Society, Providence, RI, 2006. MR 2279143, DOI 10.1090/conm/415
- T. Fukaawa, A model of cochlear micromechanics, Hearing Research, 113, 182-190, 1997.
- E. Givelberg and J. Bunn, A comprehensive three-dimensional model of the cochlea, J. Comp. Phys. 191, 377-391, 2003.
- T. Gold, Hearing II: The physical basis for the action of the cochlea, Proc. R. Soc. Lond. 135, 492-298, 1948.
- D. Geisler and C. Sang, A cochlear model using feed-forward outer hair cell forces, Hearing Research, 86, 132-146, 1995.
- D. D. Greenwood, Critical bandwidth and the frequency coordinates of the basilar membrane, J. Acoust. Soc. Amer., 33, 1344-1356, 1961.
- D. D. Greenwood, A cochlear frequency-position function for several species-29 years later, J. Acoust. Soc. Amer., 87, 2592-2605, 1990.
- William R. Holmes, Michael Jolly, and Jacob Rubinstein, Hydro-elastic waves in a cochlear model: numerical simulations and an analytically reduced model, Confluentes Math. 3 (2011), no. 3, 523–541. MR 2847242, DOI 10.1142/S1793744211000382
- T. S. A. Jaffer, H. Kunov, and W. Wong, A model cochlear partition involving longitudinal elasticity, J. Acoust. Soc. Amer., 112, 576-589, 2002.
- Joseph B. Keller and John C. Neu, Asymptotic analysis of a viscous cochlear model, J. Acoust. Soc. Amer. 77 (1985), no. 6, 2107–2110. MR 791675, DOI 10.1121/1.391735
- James Keener and James Sneyd, Mathematical physiology, Interdisciplinary Applied Mathematics, vol. 8, Springer-Verlag, New York, 1998. MR 1673204
- Yongsam Kim and Jack Xin, A two-dimensional nonlinear nonlocal feed-forward cochlear model and time domain computation of multitone interactions, Multiscale Model. Simul. 4 (2005), no. 2, 664–690. MR 2162871, DOI 10.1137/040612464
- K. M. Lim and C. R. Steele, A three-dimensional nonlinear active cochlear model analyzed by the WKB method, Hearing Research, 170, 190-205, 2002.
- D. Manoussaki, E. K. Dimitriadis, and R. S. Chadwick, Cochlea’s graded curvature effect on low frequency waves, Phys. Rev. Lett., 96, 088701, 2006.
- Jindřich Nečas, Direct methods in the theory of elliptic equations, translated from the 1967 French original by Gerard Tronel and Alois Kufner; editorial coordination and preface by Šárka Nečasová and a contribution by Christian G. Simader, Springer Monographs in Mathematics, Springer, Heidelberg, 2012. MR 3014461, DOI 10.1007/978-3-642-10455-8
- W. M. Siebert, Ranke revisited-a simple short wave cochlear model, J. Acoust. Soc. Amer., 56, 594-601, 1974.
- C. R. Steele and L. A. Taber, Comparison of WKB and finite difference calculations for a two dimensional cochlear model, J. Acoust. Soc. Amer., 65, 1001-1006, 1979.
- R. Szalai, K. Tsaneva-Atanasova, M. E. Homer, A. R. Champneys, H. J. Kennedy, and N. P. Cooper, Nonlinear models of development, amplification and compression in the mammalian cochlea, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 369 (2011), no. 1954, 4183–4204. MR 2843876, DOI 10.1098/rsta.2011.0192
- M. A. Viergever, A two dimensional model for the cochlea: The heuristic approach and numerical results, J. Engn. Math., 11, 11-28, 1971.
- M. A. Viergever, Mechanics of the inner ear, Thesis, Technical University Delft, 1980.
- J. Xin, Dispersive instability and its minimization in time-domain computation of steady-state responses of cochlear models, J. Acoust. Soc. Amer., 115, 2173-2177, 2004.
- Y. Yoon, S. Puria, and C. R. Steele, A cochlear model using the time-averaged Lagrangian and the push-pull mechanism in the organ of Corti, J. Mech. Mater. Struct., 4, 977-986, 2009.
- G. Zweig, Finding the impedance of the organ of corti, J. Acoust. Soc. Amer., 89, 1229-1254, 1991.
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Additional Information
Lydia Peres Hari
Affiliation:
Department of Mathematics, Israel Institute of Technology, Haifa 32000, Israel
MR Author ID:
335697
Email:
lydia@fermat.technion.ac.il
Jacob Rubinstein
Affiliation:
Department of Mathematics, Israel Institute of Technology, Haifa 32000, Israel
MR Author ID:
204416
Email:
koby@tx.technion.ac.il
Peter Sternberg
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405
MR Author ID:
167150
Email:
sternber@indiana.edu
Received by editor(s):
December 18, 2015
Published electronically:
July 18, 2016
Article copyright:
© Copyright 2016
Brown University
