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August  2016, 36(8): 4531-4552. doi: 10.3934/dcds.2016.36.4531

Paradoxical waves and active mechanism in the cochlea

Mohammad T. Manzari 1, and  Charles S. Peskin 1,

1. 

Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, United States, United States

Received  May 2015 Revised  January 2016 Published  March 2016

This paper is dedicated to Peter Lax. We recall happily Lax's interest in the cochlea (and in all things biomedical), culminating in his magical solution of one version of the cochlea problem, as detailed herein. The cochlea is a remarkable organ (more remarkable the more we learn about it) that separates sounds into their frequency components. Two features of the cochlea are the focus of this work. One is the extreme insensitivity of the wave motion that occurs in the cochlea to the manner in which the cochlea is stimulated, so much so that even the direction of wave propagation is independent of the location of the source of the incident sound. The other is that the cochlea is an active system, a distributed amplifier that pumps energy into the cochlear wave as it propagates. Remarkably, this amplification not only boosts the signal but also improves the frequency resolution of the cochlea. The active mechanism is modeled here by a negative damping term in the equations of motion, and the whole system is stable as a result of fluid viscosity despite the negative damping.
Keywords: active mechanism, basilar membrane, Cochlea, paradoxical waves, immersed boundary method..
Mathematics Subject Classification: Primary: 76Z05, 92C35; Secondary: 92C1.
Citation: Mohammad T. Manzari, Charles S. Peskin. Paradoxical waves and active mechanism in the cochlea. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4531-4552. doi: 10.3934/dcds.2016.36.4531
References:
[1]

R. P. Beyer, A computational model of the cochlea using the immersed boundary method, J. Computational Physics, 98 (1992), 145-162. Google Scholar

[2]

P. J. Dallos, The active cochlea, J. Neuroscience, 12 (1992), 4575-85. Google Scholar

[3]

E. Givelberg and J. Bunn, A comprehensive three-dimensional model of the cochlea, J. Computational Physics, 191 (2003), 377-391. doi: 10.1016/S0021-9991(03)00319-X.  Google Scholar

[4]

A. J. Hudspeth, Integrating the active process of hair cells with cochlear function, Nature Reviews Neuroscience, 15 (2014), 600-614. doi: 10.1038/nrn3786.  Google Scholar

[5]

E. Isaacson, A Numerical Method for a Finite-Depth, Two-Dimensional Model of the Inner Ear, Ph.D thesis, New York University, 1979.  Google Scholar

[6]

R. J. LeVeque, C. S. Peskin and P. D. Lax, Asymptotic analysis of a viscous cochlear model, J. Acoustical Society of America, 77 (1985), 2107-2110. doi: 10.1121/1.391735.  Google Scholar

[7]

R. J. LeVeque, C. S. Peskin and P. D. Lax, Solution of a two-dimensional cochlea model using transform techniques, SIAM J. Appl. Math., 45 (1985), 450-464. doi: 10.1137/0145026.  Google Scholar

[8]

R. J. LeVeque, C. S. Peskin and P. D. Lax, Solution of a two-dimensional cochlea model with fluid viscosity, SIAM J. Appl. Math., 48 (1988), 191-213. doi: 10.1137/0148009.  Google Scholar

[9]

C. S. Peskin, Flow patterns around heart valves: A numerical method, J. Computational Physics, 10 (1972), 252-271. Google Scholar

[10]

C. S. Peskin, Lectures on Mathematical Aspects of Physiology (II) The Inner Ear, in Mathematical Aspects of Physiology (eds. F.C. Hoppensteadt), American Mathematical Society, (1981), 38-69. Google Scholar

[11]

C. S. Peskin, The immersed boundary method, Acta Numerica, 11 (2002), 479-517. doi: 10.1017/S0962492902000077.  Google Scholar

[12]

J. J. Stoker, Water Waves, Interscience Publishers Inc, New York, 1957.  Google Scholar

[13]

G. von Bekesy, Experiments in Hearing, Robert E. Krieger Publishing Company, Huntington, New York, 1960. Google Scholar

show all references

References:
[1]

R. P. Beyer, A computational model of the cochlea using the immersed boundary method, J. Computational Physics, 98 (1992), 145-162. Google Scholar

[2]

P. J. Dallos, The active cochlea, J. Neuroscience, 12 (1992), 4575-85. Google Scholar

[3]

E. Givelberg and J. Bunn, A comprehensive three-dimensional model of the cochlea, J. Computational Physics, 191 (2003), 377-391. doi: 10.1016/S0021-9991(03)00319-X.  Google Scholar

[4]

A. J. Hudspeth, Integrating the active process of hair cells with cochlear function, Nature Reviews Neuroscience, 15 (2014), 600-614. doi: 10.1038/nrn3786.  Google Scholar

[5]

E. Isaacson, A Numerical Method for a Finite-Depth, Two-Dimensional Model of the Inner Ear, Ph.D thesis, New York University, 1979.  Google Scholar

[6]

R. J. LeVeque, C. S. Peskin and P. D. Lax, Asymptotic analysis of a viscous cochlear model, J. Acoustical Society of America, 77 (1985), 2107-2110. doi: 10.1121/1.391735.  Google Scholar

[7]

R. J. LeVeque, C. S. Peskin and P. D. Lax, Solution of a two-dimensional cochlea model using transform techniques, SIAM J. Appl. Math., 45 (1985), 450-464. doi: 10.1137/0145026.  Google Scholar

[8]

R. J. LeVeque, C. S. Peskin and P. D. Lax, Solution of a two-dimensional cochlea model with fluid viscosity, SIAM J. Appl. Math., 48 (1988), 191-213. doi: 10.1137/0148009.  Google Scholar

[9]

C. S. Peskin, Flow patterns around heart valves: A numerical method, J. Computational Physics, 10 (1972), 252-271. Google Scholar

[10]

C. S. Peskin, Lectures on Mathematical Aspects of Physiology (II) The Inner Ear, in Mathematical Aspects of Physiology (eds. F.C. Hoppensteadt), American Mathematical Society, (1981), 38-69. Google Scholar

[11]

C. S. Peskin, The immersed boundary method, Acta Numerica, 11 (2002), 479-517. doi: 10.1017/S0962492902000077.  Google Scholar

[12]

J. J. Stoker, Water Waves, Interscience Publishers Inc, New York, 1957.  Google Scholar

[13]

G. von Bekesy, Experiments in Hearing, Robert E. Krieger Publishing Company, Huntington, New York, 1960. Google Scholar

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