American Institute of Mathematical Sciences

Advanced
2014, 11(6): 1357-1373. doi: 10.3934/mbe.2014.11.1357

A model for the nonlinear mechanism responsible for cochlear amplification

Kimberly Fessel 1, and  Mark H. Holmes 1,

1. 

Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, United States, United States

Received  September 2013 Revised  July 2014 Published  September 2014

A nonlinear model for the mechanism responsible for the amplification of the sound wave in the ear is derived using the geometric and material properties of the system. The result is a nonlinear beam equation, with the nonlinearity appearing in a coefficient of the equation. Once derived, the beam problem is analyzed for various loading conditions. Based on this analysis it is seen that the mechanism is capable of producing a spatially localized gain, as required by any amplification mechanism, but it is also capable of increasing the spatial contrast in the signal.
Keywords: basilar membrane, Cochlear amplification, nonlinear beam..
Mathematics Subject Classification: Primary: 92C10, 74H10; Secondary: 35C2.
Citation: Kimberly Fessel, Mark H. Holmes. A model for the nonlinear mechanism responsible for cochlear amplification. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1357-1373. doi: 10.3934/mbe.2014.11.1357
References:
[1]

J. Ashmore, Cochlear outer hair cell motility, Physiol. Rev., 88 (2008), 173-210. doi: 10.1152/physrev.00044.2006.

[2]

J. Ashmore, P. Avan, W. E. Brownell, P. Dallos, K. Dierkes, R. Fettiplace, K. Grosh, C. M. Hackney, A. J. Hudspeth, F. Jülicher, B. Lindner, P. Martin, J. Meaud, C. Petit, J. R. Santos Sacchi and B. Canlon, The remarkable cochlear amplifier, Hearing Res., 266 (2010), 1-17. doi: 10.1016/j.heares.2010.05.001.

[3]

J. F. Ashmore, A fast motile response in guinea-pig outer hair cells: The cellular basis of the cochlear amplifier, J. Physiol., 388 (1987), 323-347.

[4]

I. A. Belyantseva, H. J. Adler, R. Curi, G. I. Frolenkov and B. Kachar, Expression and localization of prestin and the sugar transporter glut-5 during development of electromotility in cochlear outer hair cells, J. Neurosci., 20 (2000), RC116.

[5]

R. S. Chadwick, Studies in cochlear mechanics, in Mathematical Modeling of the Hearing Process (eds. M. H. Holmes and L. A. Rubenfeld), Lecture Notes in Biomathematics, Springer-Verlag, New York, 1981, 369-386. doi: 10.1007/978-3-642-46445-4_2.

[6]

R. S. Chadwick, Compression, gain, and nonlinear distortion in an active cochlear model with subpartitions, Proc. Nat. Acad. Sci., 95 (1998), 14594-14599. doi: 10.1073/pnas.95.25.14594.

[7]

P. Dallos and B. Fakler, Prestin, a new type of motor protein, Nature Reviews Molecular Cell Biology, 3 (2002), 104-111. doi: 10.1038/nrm730.

[8]

D. Y. Gao, Nonlinear elastic beam theory with application in contact problems and variational approaches, Mech. Res. Commun., 23 (1996), 11-17. doi: 10.1016/0093-6413(95)00071-2.

[9]

R. Glueckert, K. Pfaller, A. Kinnefors, A. Schrott-Fischer and H. Rask-Andersen, High resolution scanning electron microscopy of the human organ of Corti: A study using freshly fixed surgical specimens, Hearing Res., 199 (2005), 40-56. doi: 10.1016/S0378-5955(04)00184-4.

[10]

M. H. Holmes, Frequency discrimination in the mammalian cochlea: Theory vs. experiment, J. Acoust. Soc. Amer., 81 (1987), 103-114.

[11]

M. H. Holmes and J. D. Cole, Cochlear mechanics: Analysis for a pure tone, J. Acoust. Soc. Amer., 76 (1984), 767-778. doi: 10.1121/1.391300.

[12]

A. J. Hudspeth and D. P. Corey, Sensitivity, polarity, and conductance change in the response of vertebrate hair cells to controlled mechanical stimuli, Proc. Nat. Acad. Sci., 74 (1977), 2407-2411. doi: 10.1073/pnas.74.6.2407.

[13]

Z. Liao, S. Feng, A. S. Popel, W. E. Brownell and A. A. Spector, Outer hair cell active force generation in the cochlear environment, J. Acoust. Soc. Amer., 122 (2007), 2215-2225. doi: 10.1121/1.2776154.

[14]

M. C. Liberman, J. Gao, D. Z. He, X. Wu, S. Jia and J. Zuo, Prestin is required for electromotility of the outer hair cell and for the cochlear amplifier, Nature, 419 (2002), 300-304. doi: 10.1038/nature01059.

[15]

J. Lighthill, Energy flow in the cochlea, J. Fluid Mechanics, 106 (1981), 149-213. doi: 10.1017/S0022112081001560.

[16]

K. M. Lim and C. R. Steele, A three-dimensional nonlinear active cochlear model analyzed by the WKB-numeric method, Hearing Res., 170 (2002), 190-205. doi: 10.1016/S0378-5955(02)00491-4.

[17]

J. Meaud and K. Grosh, Response to a pure tone in a nonlinear mechanical-electrical-acoustical model of the cochlea, Biophysical Journal, 102 (1996), 1237-1246. doi: 10.1016/j.bpj.2012.02.026.

[18]

K. E. Nilsen and I. J. Russell, The spatial and temporal representation of a tone on the guinea pig basilar membrane, Proc. Natl. Acad. Sci., 97 (2006), 11751-11758. doi: 10.1073/pnas.97.22.11751.

[19]

J. O. Pickles, An Introduction to the Physiology of Hearing, Emerald Group, Bingley, UK, 2008.

[20]

S. Ramamoorthy, N. V. Deo and K. Grosh, A mechano-electro-acoustical model for the cochlea: Response to acoustic stimuli, JASA, 121 (2007), 2758-2773. doi: 10.1121/1.2713725.

[21]

I. J. Russell, A. R. Cody and G. P. Richarson, The responses of inner and outer hair cells in the basal turn of the guinea-pig cochlea and in the mouse cochlea grown in vitro, Hearing Res., 22 (1986), 199-216. doi: 10.1016/0378-5955(86)90096-1.

[22]

I. J. Russell and K. E. Nilsen, The location of the cochlear amplifier: Spatial representation of a single tone on the guinea pig basilar membrane, Proc. Nat. Acad. Sci., 94 (1997), 2660-2664. doi: 10.1073/pnas.94.6.2660.

[23]

C. R. Steele and L. A. Taber, Comparison of WKB calculations and experimental results for three-dimensional cochlear models, J. Acoust. Soc. Amer., 65 (1979), 1007-1018. doi: 10.1121/1.382570.

[24]

I. U. Teudt and C.-P. Richter, The hemicochlea preparation of the guinea pig and other mammalian cochleae, J. Neurosci. Methods, 162 (2007), 187-197. doi: 10.1016/j.jneumeth.2007.01.012.

[25]

J. A. Tolomeo and M. C. Holley, Mechanics of microtubule bundles in pillar cells from the inner ear, Biophys. J., 73 (1997), 2241-2247. doi: 10.1016/S0006-3495(97)78255-9.

[26]

Y. Yoon, S. Puria and C. R. Steele, Frequency and spatial response of basilar membrane vibration in a three-dimensional gerbil cochlear model, J. Mech. Mater. Struct., 2 (2007), 1449-1458. doi: 10.2140/jomms.2007.2.1449.

show all references

References:
[1]

J. Ashmore, Cochlear outer hair cell motility, Physiol. Rev., 88 (2008), 173-210. doi: 10.1152/physrev.00044.2006.

[2]

J. Ashmore, P. Avan, W. E. Brownell, P. Dallos, K. Dierkes, R. Fettiplace, K. Grosh, C. M. Hackney, A. J. Hudspeth, F. Jülicher, B. Lindner, P. Martin, J. Meaud, C. Petit, J. R. Santos Sacchi and B. Canlon, The remarkable cochlear amplifier, Hearing Res., 266 (2010), 1-17. doi: 10.1016/j.heares.2010.05.001.

[3]

J. F. Ashmore, A fast motile response in guinea-pig outer hair cells: The cellular basis of the cochlear amplifier, J. Physiol., 388 (1987), 323-347.

[4]

I. A. Belyantseva, H. J. Adler, R. Curi, G. I. Frolenkov and B. Kachar, Expression and localization of prestin and the sugar transporter glut-5 during development of electromotility in cochlear outer hair cells, J. Neurosci., 20 (2000), RC116.

[5]

R. S. Chadwick, Studies in cochlear mechanics, in Mathematical Modeling of the Hearing Process (eds. M. H. Holmes and L. A. Rubenfeld), Lecture Notes in Biomathematics, Springer-Verlag, New York, 1981, 369-386. doi: 10.1007/978-3-642-46445-4_2.

[6]

R. S. Chadwick, Compression, gain, and nonlinear distortion in an active cochlear model with subpartitions, Proc. Nat. Acad. Sci., 95 (1998), 14594-14599. doi: 10.1073/pnas.95.25.14594.

[7]

P. Dallos and B. Fakler, Prestin, a new type of motor protein, Nature Reviews Molecular Cell Biology, 3 (2002), 104-111. doi: 10.1038/nrm730.

[8]

D. Y. Gao, Nonlinear elastic beam theory with application in contact problems and variational approaches, Mech. Res. Commun., 23 (1996), 11-17. doi: 10.1016/0093-6413(95)00071-2.

[9]

R. Glueckert, K. Pfaller, A. Kinnefors, A. Schrott-Fischer and H. Rask-Andersen, High resolution scanning electron microscopy of the human organ of Corti: A study using freshly fixed surgical specimens, Hearing Res., 199 (2005), 40-56. doi: 10.1016/S0378-5955(04)00184-4.

[10]

M. H. Holmes, Frequency discrimination in the mammalian cochlea: Theory vs. experiment, J. Acoust. Soc. Amer., 81 (1987), 103-114.

[11]

M. H. Holmes and J. D. Cole, Cochlear mechanics: Analysis for a pure tone, J. Acoust. Soc. Amer., 76 (1984), 767-778. doi: 10.1121/1.391300.

[12]

A. J. Hudspeth and D. P. Corey, Sensitivity, polarity, and conductance change in the response of vertebrate hair cells to controlled mechanical stimuli, Proc. Nat. Acad. Sci., 74 (1977), 2407-2411. doi: 10.1073/pnas.74.6.2407.

[13]

Z. Liao, S. Feng, A. S. Popel, W. E. Brownell and A. A. Spector, Outer hair cell active force generation in the cochlear environment, J. Acoust. Soc. Amer., 122 (2007), 2215-2225. doi: 10.1121/1.2776154.

[14]

M. C. Liberman, J. Gao, D. Z. He, X. Wu, S. Jia and J. Zuo, Prestin is required for electromotility of the outer hair cell and for the cochlear amplifier, Nature, 419 (2002), 300-304. doi: 10.1038/nature01059.

[15]

J. Lighthill, Energy flow in the cochlea, J. Fluid Mechanics, 106 (1981), 149-213. doi: 10.1017/S0022112081001560.

[16]

K. M. Lim and C. R. Steele, A three-dimensional nonlinear active cochlear model analyzed by the WKB-numeric method, Hearing Res., 170 (2002), 190-205. doi: 10.1016/S0378-5955(02)00491-4.

[17]

J. Meaud and K. Grosh, Response to a pure tone in a nonlinear mechanical-electrical-acoustical model of the cochlea, Biophysical Journal, 102 (1996), 1237-1246. doi: 10.1016/j.bpj.2012.02.026.

[18]

K. E. Nilsen and I. J. Russell, The spatial and temporal representation of a tone on the guinea pig basilar membrane, Proc. Natl. Acad. Sci., 97 (2006), 11751-11758. doi: 10.1073/pnas.97.22.11751.

[19]

J. O. Pickles, An Introduction to the Physiology of Hearing, Emerald Group, Bingley, UK, 2008.

[20]

S. Ramamoorthy, N. V. Deo and K. Grosh, A mechano-electro-acoustical model for the cochlea: Response to acoustic stimuli, JASA, 121 (2007), 2758-2773. doi: 10.1121/1.2713725.

[21]

I. J. Russell, A. R. Cody and G. P. Richarson, The responses of inner and outer hair cells in the basal turn of the guinea-pig cochlea and in the mouse cochlea grown in vitro, Hearing Res., 22 (1986), 199-216. doi: 10.1016/0378-5955(86)90096-1.

[22]

I. J. Russell and K. E. Nilsen, The location of the cochlear amplifier: Spatial representation of a single tone on the guinea pig basilar membrane, Proc. Nat. Acad. Sci., 94 (1997), 2660-2664. doi: 10.1073/pnas.94.6.2660.

[23]

C. R. Steele and L. A. Taber, Comparison of WKB calculations and experimental results for three-dimensional cochlear models, J. Acoust. Soc. Amer., 65 (1979), 1007-1018. doi: 10.1121/1.382570.

[24]

I. U. Teudt and C.-P. Richter, The hemicochlea preparation of the guinea pig and other mammalian cochleae, J. Neurosci. Methods, 162 (2007), 187-197. doi: 10.1016/j.jneumeth.2007.01.012.

[25]

J. A. Tolomeo and M. C. Holley, Mechanics of microtubule bundles in pillar cells from the inner ear, Biophys. J., 73 (1997), 2241-2247. doi: 10.1016/S0006-3495(97)78255-9.

[26]

Y. Yoon, S. Puria and C. R. Steele, Frequency and spatial response of basilar membrane vibration in a three-dimensional gerbil cochlear model, J. Mech. Mater. Struct., 2 (2007), 1449-1458. doi: 10.2140/jomms.2007.2.1449.

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