A three-dimensional model of the hydro-elastic waves in the mammalian cochlea is presented along with numerical simulations. The cochlear fluid is treated as linear, incompressible, and inviscid. The cochlear partition is treated as a massless thin plate loaded by the fluid. This model is then reformulated by analytically removing the fluid variable with the use of a Dirichlet-to-Neumann operator. The resulting fifth-order nonlocal PDE for the motion of the partition is simulated using a novel implicit numerical scheme. Simulations demonstrate that this model exhibits traveling wave characteristics and a clear place principle. Asymptotic analysis in the small aspect ratio of the cochlea is performed on the given model equations with energetic concerns in mind. The results of simulations along with these asymptotic arguments suggest a relationship between the form and function of the cochlea which we compare to physiological data.
@article{CML_2011__3_3_523_0,
author = {Holmes, William R. and Jolly, Michael and Rubinstein, Jacob},
title = {Hydro-elastic waves in a cochlear model: {Numerical} simulations and an analytically reduced model},
journal = {Confluentes Mathematici},
pages = {523--541},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {3},
number = {3},
year = {2011},
doi = {10.1142/S1793744211000382},
language = {en},
url = {http://archive.numdam.org/articles/10.1142/S1793744211000382/}
}
TY - JOUR AU - Holmes, William R. AU - Jolly, Michael AU - Rubinstein, Jacob TI - Hydro-elastic waves in a cochlear model: Numerical simulations and an analytically reduced model JO - Confluentes Mathematici PY - 2011 SP - 523 EP - 541 VL - 3 IS - 3 PB - World Scientific Publishing Co Pte Ltd UR - http://archive.numdam.org/articles/10.1142/S1793744211000382/ DO - 10.1142/S1793744211000382 LA - en ID - CML_2011__3_3_523_0 ER -
%0 Journal Article %A Holmes, William R. %A Jolly, Michael %A Rubinstein, Jacob %T Hydro-elastic waves in a cochlear model: Numerical simulations and an analytically reduced model %J Confluentes Mathematici %D 2011 %P 523-541 %V 3 %N 3 %I World Scientific Publishing Co Pte Ltd %U http://archive.numdam.org/articles/10.1142/S1793744211000382/ %R 10.1142/S1793744211000382 %G en %F CML_2011__3_3_523_0
Holmes, William R.; Jolly, Michael; Rubinstein, Jacob. Hydro-elastic waves in a cochlear model: Numerical simulations and an analytically reduced model. Confluentes Mathematici, Tome 3 (2011) no. 3, pp. 523-541. doi : 10.1142/S1793744211000382. http://archive.numdam.org/articles/10.1142/S1793744211000382/
[1] J. B. Allen and M. M. Sondhi, Cochlear macromechanics: Time domain solutions, J. Acoust. Soc. Am. 66 (1979) 123–132.
[2] G. V. Bekesy, Experiments in Hearing (McGraw-Hill, 1960).
[3] J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd edn. (Dover, 2001).
[4] E. Givelberg and J. Bunn, A comprehensive three-dimensional model of the cochlea, J. Comput. Phys. 191 (2003) 377–391.
[5] J. Keener and J. Sneyd, Mathematical Physiology (Springer, 2001).
[6] J. B. Keller and J. C. Neu, Asymptotic analysis of a viscous cochlear model, J. Acoust. Soc. Am. 77 (1985) 2107–2110.
[7] Y. Kim and J. Xin, A two-dimensional nonlinear nonlocal feed-forward cochlear mode and time domain computation of multitone interactions, Multiscale Model. Simul. 4 (2005) 664–690.
[8] L. Landau and E. Lifshitz, Theory of Elasticity, Course of Theoretical Physics, Vol. 7, 3rd edn. (Elsevier, 1986).
[9] J. Lighthill, Energy flow in the cochlea, J. Fluid Mech. 106 (1981) 149–213.
[10] 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.
[11] D. Manoussaki, R. S. Chadwick, D. R. Ketten, J. Arruda, E. K. Dimitriadis and J. T. O’Malley, The influence of cochlear shape on low-frequency hearing, PNAS 105 (2008) 6162–6166.
[12] S. T. Neely, Finite difference solution of a two-dimensional mathematical model of the cochlea, J. Acoust. Soc. Am. 69 (1981) 1386–1393.
[13] C. S. Peskin, Partial Differential Equations in Biology. Courant Inst. Lecture Notes. (Courant Institute of Mathematical Sciences, 1975–76).
[14] L. C. Peterson and B. P. Bogert, A dynamical theory of the cochlea, J. Acoust. Soc. Am. 22 (1950) 369–381.
[15] J. O. Pickles, An Introduction to the Physiology of Hearing, 3rd edn. (Emerald, 2008).
[16] J. Rubinstein and M. Schatzman, Variational problems in multiply connected thin strips I: Basic estimates and convergence of the Laplacian spectrum, Arch. Rational Mech. Anal. 160 (2001) 271–308.
[17] C. R. Steele, Behavior of the basilar membrane with pure-tone excitation, J. Acoust. Soc. Am. 55 (1974) 148–162.
[18] J. W. Stephenson, Single cell discretizations of order two and four for biharmonic problems, J. Comput. Phys. 55 (1984) 65–80.
[19] J. Xin, Y. Qi and L. Deng, Time domain computation of a nonlinear nonlocal cochlear model with applications to multitone interaction in hearing, Comm. Math. Sci. 1 (2003) 211–227.
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