# American Institute of Mathematical Sciences

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

## Paradoxical waves and active mechanism in the cochlea

 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.
Citation: Mohammad T. Manzari, Charles S. Peskin. Paradoxical waves and active mechanism in the cochlea. Discrete and 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. [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.

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. [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.
 [1] Qiang Du, Manlin Li. On the stochastic immersed boundary method with an implicit interface formulation. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 373-389. doi: 10.3934/dcdsb.2011.15.373 [2] Harvey A. R. Williams, Lisa J. Fauci, Donald P. Gaver III. Evaluation of interfacial fluid dynamical stresses using the immersed boundary method. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 519-540. doi: 10.3934/dcdsb.2009.11.519 [3] Robert H. Dillon, Jingxuan Zhuo. Using the immersed boundary method to model complex fluids-structure interaction in sperm motility. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 343-355. doi: 10.3934/dcdsb.2011.15.343 [4] Pavel Eichler, Radek Fučík, Robert Straka. Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 819-833. doi: 10.3934/dcdss.2020349 [5] Daniele Boffi, Lucia Gastaldi. Discrete models for fluid-structure interactions: The finite element Immersed Boundary Method. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 89-107. doi: 10.3934/dcdss.2016.9.89 [6] Shunfu Jin, Wuyi Yue, Chao Meng, Zsolt Saffer. A novel active DRX mechanism in LTE technology and its performance evaluation. Journal of Industrial and Management Optimization, 2015, 11 (3) : 849-866. doi: 10.3934/jimo.2015.11.849 [7] Giovanni Alessandrini, Elio Cabib. Determining the anisotropic traction state in a membrane by boundary measurements. Inverse Problems and Imaging, 2007, 1 (3) : 437-442. doi: 10.3934/ipi.2007.1.437 [8] Champike Attanayake, So-Hsiang Chou. An immersed interface method for Pennes bioheat transfer equation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 323-337. doi: 10.3934/dcdsb.2015.20.323 [9] Jian Hao, Zhilin Li, Sharon R. Lubkin. An augmented immersed interface method for moving structures with mass. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1175-1184. doi: 10.3934/dcdsb.2012.17.1175 [10] Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184 [11] Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196 [12] Thomas Y. Hou, Pingwen Zhang. Convergence of a boundary integral method for 3-D water waves. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 1-34. doi: 10.3934/dcdsb.2002.2.1 [13] So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343 [14] Scott W. Hansen. Controllability of a basic cochlea model. Evolution Equations and Control Theory, 2016, 5 (4) : 475-487. doi: 10.3934/eect.2016015 [15] Paolo Piersanti. On the improved interior regularity of a boundary value problem modelling the displacement of a linearly elastic elliptic membrane shell subject to an obstacle. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 1011-1037. doi: 10.3934/dcds.2021145 [16] Sheng Xu. Derivation of principal jump conditions for the immersed interface method in two-fluid flow simulation. Conference Publications, 2009, 2009 (Special) : 838-845. doi: 10.3934/proc.2009.2009.838 [17] Derrick Jones, Xu Zhang. A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces. Electronic Research Archive, 2021, 29 (5) : 3171-3191. doi: 10.3934/era.2021032 [18] Youngmok Jeon, Dongwook Shin. Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions. Electronic Research Archive, 2021, 29 (5) : 3361-3382. doi: 10.3934/era.2021043 [19] Mostafa Ghelichi, A. M. Goltabar, H. R. Tavakoli, A. Karamodin. Neuro-fuzzy active control optimized by Tug of war optimization method for seismically excited benchmark highway bridge. Numerical Algebra, Control and Optimization, 2021, 11 (3) : 333-351. doi: 10.3934/naco.2020029 [20] Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems and Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577

2020 Impact Factor: 1.392