# Nonlinear Distortions and Parametric Amplification Generate Otoacoustic Emissions and Increased Hearing Sensitivity

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Partial Differential Equation (Parametric Traveling-Wave Amplifier)

#### 2.2. Finite Difference Scheme (Damped Wave Equation)

#### 2.3. Nonlinear Stiffness Function of Outer Hair Cells

## 3. Results

## 4. Discussion and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Melde, F. Über die Erregung stehender Wellen eines fadenförmigen Körpers [On the excitation of standing waves on a string]. Annalen der Physik und Chemie (Ser. 2)
**1859**, 109, 193–215. [Google Scholar] - Cullen, A.L. Theory of the travelling-wave parametric amplifier. Proc. IEE-Part B
**1959**, 107, 101–107. [Google Scholar] [CrossRef] - Morse, P.M.; Ingard, K.U. Theoretical Acoustics; Princeton University Press: Princeton, NJ, USA, 1986. [Google Scholar]
- Nayfeh, A.H.; Mook, D.T. Nonlinear Oscillations; Pure & Applied Mathematics; John Wiley & Sons: New York, NY, USA, 1979. [Google Scholar]
- Cheng, X.; Blanchard, A.; Tan, C.A.; Lu, H.; Bergman, L.A.; McFarland, D.M.; Vakakis, A.F. Separation of traveling and standing waves in a finite dispersive string with partial or continuous viscoelastic foundation. J. Sound Vibr.
**2017**, 411, 193–209. [Google Scholar] [CrossRef] - Tan, C.A.; Yang, B.; Mote, C.D., Jr. More On the vibration of a translating string coupled to hydrodynamic bearings. J. Vib. Acoust.
**1990**, 112, 337–345. [Google Scholar] [CrossRef] - Huang, F.Y.; Mote, C.D., Jr. On the translating damping caused by a thin viscous fluid layer between a translating string and a translating rigid surface. J. Vib. Acoust.
**1995**, 181, 251–260. [Google Scholar] [CrossRef] - Böhnke, F.; Janssen, T.; Steinhoff, H.-J.; Zimmermann, P. Structural Cochlear Model including the Coupling of Basilar Membrane Fibers in the Longitudinal Direction. Il Valsalva
**1989**, 54 (Suppl. 1), 10–16. [Google Scholar] - Elliott, S.J. Wave propagation in a constrained fluid layer bounded by an elastic half-space and its relevance in cochlear micromechanics. J. Sound Vib.
**2007**, 305, 918–924. [Google Scholar] [CrossRef] - Böhnke, F.; Semmelbauer, S. Acoustic boundary layer attenuation in ducts with rigid and elastic walls applied to cochlear mechanics. J. Fluids Eng.
**2017**, 139, 101202. [Google Scholar] [CrossRef] - Najafi, H.S.; Izadi, F. Comparison of two finite-difference methods for solving the damped wave equation. Int. J. Math. Eng. Sci.
**2014**, 3, 35–49. [Google Scholar] - Preyer, S.; Gummer, A.W. Nonlinearity of mechanoelectrical transduction of outer hair cells as the source of nonlinear basilar-membrane motion and loudness recruitment. Audiol. Neuro-Otol.
**1996**, 1, 3–11. [Google Scholar] [CrossRef] - He, D.Z.; Dallos, P. Somatic stiffness of cochlear outer hair cells is voltage-dependent. Proc. Natl. Acad. Sci. USA
**1999**, 96, 8223–8228. [Google Scholar] [CrossRef] [Green Version] - Deo, N.; Grosh, K. Two-State Model for Outer Hair Cell Stiffness. Biophys. J.
**2004**, 86, 3519–3528. [Google Scholar] [CrossRef] [PubMed] - Rhode, W.S.; Robles, L. Evidence from Mössbauer experiments for nonlinear vibration in the cochlea. J. Acoust. Soc. Am.
**1974**, 55, 588–596. [Google Scholar] [CrossRef] [PubMed] - Ko, W.; Stockie, J.M. An immersed boundary model of the cochlea with parametric forcing. SIAM J. Appl. Math.
**2015**, 75, 1065–1089. [Google Scholar] [CrossRef] - Kielczynski, P. Power amplification and selectivity in the cochlear amplifier. Arch. Acoust.
**2013**, 38, 83–92. [Google Scholar] [CrossRef] - Böhnke, F.; Arnold, W. Mechanics of the Organ of Corti Caused by Deiters Cells. IEEE Trans. Biomed. Eng.
**1998**, 45, 1227–1233. [Google Scholar] [CrossRef] [PubMed] - von Békésy, G. Zur Theorie des Hörens, Die Schwingungsform der Basilarmembran. Physikalische Zeitschrift
**1928**, 29, 793–810. [Google Scholar] - Dong, W. Simultaneous intracochlear pressure measurements from two cochlear locations: Propagation of distortion products in gerbil. J. Assoc. Res. Otolaryngol.
**2017**, 18, 209–225. [Google Scholar] [CrossRef] - Kemp, D.T. Evidence of mechanical nonlinearity and frequency selective wave amplification in the cochlea. Arch. Otorhinolaryngol.
**1979**, 224, 37–45. [Google Scholar] [CrossRef] - Zwicker, E. A hardware cochlear nonlinear preprocessing model with active feedback. J. Acoust. Soc. Am.
**1986**, 80, 146–153. [Google Scholar] [CrossRef] - Verhulst, S.; Dau, T.; Shera, C.A. Nonlinear time-domain cochlear model for transient stimulation and human otoacoustic emission. J. Acoust. Soc. Am.
**2012**, 132, 3842–3848. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**Box model of the human cochlea (

**A**). The upper front area (representing the oval window attaching the Scala vestibuli (S.v.)) is excited by a sinusoidal velocity with the frequency 2 kHz and amplitude 1 nm/s. The magnified view of the basilar membrane (BM) (

**B**) illustrates the orthotropic consistency of the BM with collagen containing stiff transverse fibers with Young’s modulus ${E}_{y}$ embedded into softer elastic tissue ${E}_{x}<<{E}_{y}$.

**Figure 2.**Elastic string representing the BM with nonlinear stiffness $k(x,t)$, produced by outer hair cells (OHC) in the cochlea.

**Figure 3.**Displacements along the BM with stimulation frequency 2000 Hz at the base (passive), stimulation amplitude $\widehat{u}=1$ nm.

**Figure 4.**Displacements with stimulation frequency 2000 Hz (parametric amplified), stimulation amplitude $\widehat{u}=1$ nm. Nonlinear distortions are generated by the OHC nonlinear stiffness function.

**Figure 5.**Spectrum of BM displacement at 6.4 mm (n = 40) stimulation frequency 2000 Hz (passive), stimulation amplitude $\widehat{u}=1$ nm.

**Figure 6.**Spectrum of BM displacements at 16 mm (n = 100) stimulation frequency 2000 Hz (active), stimulation amplitude $\widehat{u}=1$ nm.

**Figure 7.**Spectrum of BM displacements at 6.4 mm (n = 40) stimulation frequency 2000 Hz (passive), stimulation amplitude $\widehat{u}=100$ nm. Nonlinear distortions are generated by the cubic nonlinearity of the string.

**Figure 8.**Spectrum of BM displacements at 16 mm (n = 100) stimulation frequency 2000 Hz (active), stimulation amplitude $\widehat{u}=100$ nm. Nonlinear distortions are generated by the OHC stiffness functions and the saturating cubic nonlinearity of the string.

Symbol | Denotation | Value Unit |
---|---|---|

${\rho}_{fl}$ | fluid density per unit area | 1000 kg m${}^{-2}$ |

b(x) | BM width | 0.1–0.5 mm |

u(x,t) | BM displacement | m |

${k}_{d}$ | damping constant | 4000 kg m${}^{-2}$ s${}^{-1}$ |

${k}_{cub}$ | cubic nonlinearity constant | $-{10}^{6}$ kg m${}^{-4}$${s}^{-2}$ |

${E}_{y}$ | transverse Young’s modulus | 100 GPa |

$\Delta {E}_{y}(x,t)$ | variable Young’s modulus | Pa |

h | BM thickness | 10 $\mathsf{\mu}$m |

S | cross section area of canal | 1 mm${}^{2}$ |

L | BM length | 32 mm |

**Table 2.**Electrical and mechanical parameters constituting the parametric amplification by outer hair cells.

Symbol | Denotation | Value Unit |
---|---|---|

${V}_{mp}$ | OHC membrane potential | ${V}_{h}\to \phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{0.166667em}{0ex}}61$ mV |

${V}_{h}$ | OHC membrane potential for maximum hyperpolarization | $-\phantom{\rule{0.166667em}{0ex}}65.9$ mV |

${z}_{1},{z}_{2}$ | force factors | 60 fN, 120 fN |

${x}_{1},{x}_{2}$ | displacements | 56.8 nm, 27.3 nm |

$kT$ | Boltzmann-constant $\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}$ temperature (energy) | $4.11\times {10}^{-21}$ J |

${x}_{st}$ | stereocilia displacement | m |

${k}_{0}$ | OHC stiffness | 0.001 Nm${}^{-1}$ |

$a,b$ | constants | $0.001415,-0.0419$ |

$c,d$ | constants | $2.665,-0.003801$ |

${f}_{act}$ | factor | $8.2\times {10}^{-10}$ |

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**MDPI and ACS Style**

Böhnke, F.
Nonlinear Distortions and Parametric Amplification Generate Otoacoustic Emissions and Increased Hearing Sensitivity. *Acoustics* **2019**, *1*, 608-617.
https://doi.org/10.3390/acoustics1030036

**AMA Style**

Böhnke F.
Nonlinear Distortions and Parametric Amplification Generate Otoacoustic Emissions and Increased Hearing Sensitivity. *Acoustics*. 2019; 1(3):608-617.
https://doi.org/10.3390/acoustics1030036

**Chicago/Turabian Style**

Böhnke, Frank.
2019. "Nonlinear Distortions and Parametric Amplification Generate Otoacoustic Emissions and Increased Hearing Sensitivity" *Acoustics* 1, no. 3: 608-617.
https://doi.org/10.3390/acoustics1030036