Article Contents
Article Contents

# Paradoxical waves and active mechanism in the cochlea

• 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.
Mathematics Subject Classification: Primary: 76Z05, 92C35; Secondary: 92C10.

 Citation:

•  [1] R. P. Beyer, A computational model of the cochlea using the immersed boundary method, J. Computational Physics, 98 (1992), 145-162. [2] P. J. Dallos, The active cochlea, J. Neuroscience, 12 (1992), 4575-85. [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. [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. [5] E. Isaacson, A Numerical Method for a Finite-Depth, Two-Dimensional Model of the Inner Ear, Ph.D thesis, New York University, 1979. [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. [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. [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. [9] C. S. Peskin, Flow patterns around heart valves: A numerical method, J. Computational Physics, 10 (1972), 252-271. [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. [11] C. S. Peskin, The immersed boundary method, Acta Numerica, 11 (2002), 479-517.doi: 10.1017/S0962492902000077. [12] J. J. Stoker, Water Waves, Interscience Publishers Inc, New York, 1957. [13] G. von Bekesy, Experiments in Hearing, Robert E. Krieger Publishing Company, Huntington, New York, 1960.