A three-dimensional nonlinear active cochlear model analyzed bythe WKB-numeric method

Kian-Meng Lim a;�, Charles R. Steele b

a Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260b Division of Mechanics and Computation, Department of Mechanical Engineering, Stanford University, Stanford, CA 94305-4035, USA

Received 9 July 2001; accepted 30 April 2002

Abstract

A physiologically based nonlinear active cochlear model is presented. The model includes the three-dimensional viscous fluideffects, an orthotropic cochlear partition with dimensional and material property variation along its length, and a nonlinear activefeed-forward mechanism of the organ of Corti. A hybrid asymptotic and numerical method combined with Fourier seriesexpansions is used to provide a fast and efficient iterative procedure for modeling and simulation of the nonlinear responses in theactive cochlea. The simulation results for the chinchilla cochlea compare very well with experimental measurements, capturingseveral nonlinear features observed in basilar membrane responses. These include compression of response with stimulus level, two-tone suppressions, and generation of harmonic distortion and distortion products. - 2002 Elsevier Science B.V. All rightsreserved.

Key words: Nonlinear cochlear model; Response suppression; Distortion

1. Introduction

Numerous models have been proposed and used tosimulate the activity in the cochlea. Many are exten-sions of passive cochlear models with the inclusion ofthe micro-mechanics of the organ of Corti, in particularthe active behavior of the outer hair cells.

The inclusion of negative damping in one-dimension-

al, lumped parameter models had been used by de Boer(1983) and Diependaal et al. (1987). Kanis and de Boer(1996, 1997) extended this model to include nonlinearityin the activity and obtained both frequency and timedomain solutions using a quasi-linear method. Theyalso demonstrated the phenomena of two-tone suppres-sion and distortion product generation which are com-monly observed in experimental measurements.

0378-5955 / 02 / $ ^ see front matter - 2002 Elsevier Science B.V. All rights reserved.PII: S 0 3 7 8 - 5 9 5 5 ( 0 2 ) 0 0 4 9 1 - 4

* Corresponding author. Tel. : +65 6874 8860; Fax: +65 6779 1459.E-mail address: limkm@nus.edu.sg (K.-M. Lim).

Abbreviations: A, harmonic coe⁄cients of K ; B, bandwidth of harmonic frequency coupling; b, width of plate; C1, C2, constants ofproportionality; D11, D12, D22, £exural sti¡nesses of plate; E11, E12, E22, orthotropic Young’s moduli of plate; FBM, Ff , Fcilia, forces acting onbasilar membrane, £uid and cilia; Fc, force exerted by outer hair cell ; FBM, F f , F c, harmonic coe⁄cients of FBM, Ff , Fc ; FL

c , vector of F c in thelinear case; FNL

c , nonlinear part of the vector of F c ; F̂F c, slowly varying amplitude of F c ; f, ¢ber density of plate; f1, f2, frequencies of tones used indistortion product generation; H, number of Fourier components used across width L2 ; h, thickness of plate; hf , equivalent thickness of £uid; I,identity matrix; i, imaginary number

ffiffiffiffiffiffiffi31

p; j, k, l, indices for harmonic frequencies; Kp, equivalent sti¡ness of plate; L2, L3, width and height of

£uid chamber; lc, length of outer hair cell ; Mp, Mf , equivalent masses of plate and £uid; m, index for the Fourier expansion of u! across width L2 ;N, number of harmonic frequencies; n, wave number; pf , pp, pressures acting on £uid and plate; Rm, Fourier coe⁄cient for the mth component ofP ; t, time; u!, £uid displacement vector; V, volume of integration; w, de£ection of plate; W, coe⁄cient of w ; x, y, z, Cartesian coordinates; K, feed-forward gain factor; K0, constant feed-forward gain factor (linear case); Lm, decay coe⁄cient for mth component of P ; y, feed-forward transfercoe⁄cient for F c ; Qm, decay coe⁄cient for mth component of P ; v, feed-forward distance; R, local coordinate across width of plate; W, dynamicviscosity of £uid; X, Poisson ratio of plate; P, scalar potential for £uid displacement; x, coe⁄cient of P ; i!, vector potential for £uid displacement;i1, i2, i3, components of vector i

!; 8m1, 8m

2, Fourier coe⁄cients for mth component of i1, i2 ; 6, matrix of angular frequencies; g, angularfrequency; bf , bp, densities of £uid and plate; a, angle of tilt of outer hair cell ; j, local coordinate along the length

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Other researchers (Zweig, 1991; Allen and Fahey,1993) used a ‘second ¢lter’ to describe the micro-me-chanics of the organ of Corti in addition to a passivetransmission line model. Hubbard et al. (1996) imple-mented the second ¢lter using a traveling wave ampli-¢er in their active cochlear model.

Higher dimensional active models have also beenconstructed. Diependaal and Viergever (1989) extendedthe negative damping model to two dimensions using acombination of integral equation and ¢nite di¡erencemethod. Two-dimensional ¢nite di¡erence models con-structed by Neely (1985, 1993) used a feedback law tomodel the active mechanism. Full three-dimensional ¢-nite element models including varying details and com-plexities of the organ of Corti were also built by Kol-ston and Ashmore (1996), Bo«hnke and Arnold (1998)and Dodson (2001).

Finally, there is a class of models that include theactivity in the organ of Corti as a feed-forward mech-anism, taking into account the longitudinal tilt of theouter hair cells. Kolston et al. (1989) developed a linearlumped parameter model with the feed-forward mecha-nism, and Fukazawa and Tanaka (1996) further in-cluded the nonlinear behavior in the one-dimensionalmodel. Two-dimensional (Geisler and Sang, 1995) andthree-dimensional models (Steele et al., 1993; Steele andLim, 1999) with the active feed-forward mechanismwere also constructed. But these higher dimensionalmodels are normally restricted to a linear analysis dueto the formidable computations needed for the iterativesolution procedure of a nonlinear problem.

The present work extends the linear three-dimension-al feed-forward model to include the nonlinear behaviorof the outer hair cells. This model is based on thephase-integral method, commonly known as the WKB(Wentzel^Kramers^Brillouin) method, which providessigni¢cantly faster computations than the ¢nite di¡er-ence or ¢nite element methods, especially for high fre-quency responses. The viscous e¡ects of the £uid in thescalae and the variation of dimensions and materialproperties along the length are included. The frequencydomain formulation in the linear model is retained, butall the harmonic components need to be consideredtogether since they are coupled in the nonlinear prob-lem. The time domain solution can readily be obtainedby synthesis of these harmonic components. The fre-quency domain results also prove to be convenient forcomparison with previous linear models and experimen-tal measurements of frequency responses. The simula-tion results obtained from this active nonlinear modelsuccessfully demonstrate various features of the re-sponse in a live cochlea, as observed in in vivo experi-ments.

2. Formulation

First, a macroscopic model based on the WKB meth-od subjected to harmonic excitation is brie£y described.Next, the nonlinear feed-forward active mechanism ofthe outer hair cells for the micro-mechanics in the organof Corti is formulated. Combining both formulationsfor the macro- and micro-mechanics gives a coupledsystem of equations governing the responses for thevarious frequency components. An iterative procedureneeds to be used to solve this set of nonlinear equa-tions.

2.1. Macroscopic model

A schematic drawing of the macroscopic modelshowing the top, side, and end views is given in Fig.1. This model consists of a straight tapered chamberwith rigid walls ¢lled with viscous £uid. The chamberis divided longitudinally by a cochlear partition intotwo equal rectangular ducts representing the scala ves-tibuli and scala tympani. The two £uid ducts are joinedat the apical end via a hole representing the helicotre-ma. The cochlear partition represents a collapsed scalamedia with its structural properties dominated by thepectinate zone of the basilar membrane. This partitionis modeled as an elastic orthotropic plate which is sim-ply supported by rigid shelves on both sides along itslength. The variations in width, thickness, and ¢berdensity along the length of the basilar membrane areincluded, resulting in a partition with varying sti¡nessalong its length.

The £exible cochlear partition is set into motionwhen the stapes pushes against the £uid in the scalavestibuli. The anti-symmetric pressure wave in thechamber is considered, since only this gives a signi¢cantdisplacement to the partition (Peterson and Bogert,1950; Steele and Lim, 1999). Due to symmetry presentin the model, only one £uid duct needs to be consideredin the simulation. The £uid displacement ¢eld in theducts is represented by a sum of the divergence of ascalar ¢eld P and the curl of a vector ¢eld i

!

u! ¼ 9 P þ 9Ui

! ð1Þ

where P and i! take the following forms:

P ðx; y; z; tÞ ¼ e3ig tx ðxÞ

XHm¼0

RmðxÞ cosmZyL2

� �

cosh ðL mðxÞðL33zÞÞ ð2Þ

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i

!ðx; y; z; tÞ ¼i

1

i2

i3

0BB@

1CCA ¼

e3ig tXHm¼0

eQ mðxÞz

8 1mðxÞ sin

mZyL2

� �

8 2mðxÞ cos

mZyL2

� �0

0BBBBB@

1CCCCCA ð3Þ

Here, a harmonic excitation with frequency g is appliedat the stapes. The displacement ¢eld satis¢es the rigidwall boundary conditions at y=0, y=L2, and z=L3

where the normal £uid displacements are zero. The co-e⁄cients 8m

1 and 8m2 are related to the amplitude x

and Rm through no-slip boundary conditions on thecochlear partition where the tangent displacements arealso zero. The coe⁄cients Rm are determined by match-ing the normal displacement of the £uid with that of thepartition at their interface. Incidentally, the pressure inthe £uid is given by

pf ¼ 3b f €PP ð4Þ

where bf is the density of the £uid. Using the followingde¢nition of the wave number n(x)

n2ðxÞ ¼ 3x ;xx

x

ð5Þ

the continuity equation for the £uid

v P ¼ 0 ð6Þ

reduces to the algebraic form

L mðxÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 þ mZ

L2

� �2

sð7Þ

The WKB approximation is applied here with the as-sumption of large n. Similarly, the vorticity equationgiven by

ð8Þ

can be expressed as

Q mðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL

2m3ig

b f

W

rð9Þ

where W is the dynamic viscosity of the £uid. The equa-tion governing the bending motion of the cochlear par-tition, modeled as an orthotropic tapered plate withwidth b(x) and thickness h(x), is

b ph€wwþ D2

D x2D11

D2wD x2

� �þ 2

D2

D xD yD12

D2w

D xD y

� �þ

D2

D y2D22

D2wD y2

� �¼ pp ð10Þ

where pp is the pressure acting on the pectinate zone, bpis the density of the plate, and D11, D12, and D22 are the

Fig. 1. Schematic drawing of the macroscopic model: (a) side view of chamber, (b) end view of chamber, and (c) top view of cochlear parti-tion. A typical basilar membrane vibration response due to harmonic excitation is shown.

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bending sti¡ness components. Taking into account thesandwiched nature of the pectinate zone with the ¢bersoccupying the lower and upper thirds of the entirethickness, the bending sti¡ness is determined fromYoung’s modulus Eij in the respective directions, Pois-son ratio X, ¢ber volume density f, and thickness h, asgiven by

Dij ¼fEijh3

13X2

13162

ð11Þ

The displacement pro¢le of the partition is assumed totake the following form

wðx; R ; tÞ ¼ e3ig t WðxÞ sinZRb

ð12Þ

The £uid and partition displacements are matched attheir interface (with W (x) =x(x)) and the coe⁄cientsRm are determined from this assumed shape functionof the displacement.

Integrating the pressure across the width and sum-ming up the Fourier harmonics gives the force perunit length in the time domain for the partition and£uid. For the partition,

FBMðx; tÞ ¼X

e3ig j tZ

ppðx; y; g jÞ dy� �

ð13Þ

¼X

e3ig j t FBMðx; g jÞ ð14Þ

where

FBMðx; g jÞ ¼ ðKpðn; x; g jÞ3g2j MpðxÞÞ Wðx; g jÞ

ð15Þ

with the plate’s sti¡ness given by

Kp ¼ 2bZ

3b pg2hþD11n4 þ 2D12n2

Z

b

� �2 þD22

Z

b

� �4

n oð16Þ

and mass given by

Mp ¼ 2bZ

b ph ð17Þ

For the £uid,

F fðx; tÞ ¼X

e3ig j tZ

pfðx; y; g jÞ dy� �

ð18Þ

¼X

e3ig j t F fðx; g jÞ ð19Þ

where

F fðx; g jÞ ¼ b f g2j bhfðn; x; g jÞ Wðx; g jÞ ð20Þ

¼ g2j Mfðn; x; g jÞ Wðx; g jÞ ð21Þ

with Mf and hf being the e¡ective mass and thickness ofthe £uid layer over the width of the plate. These quan-tities are functions of the wave number n(x, gj), asshown in Lim (2000).

The above provides a physically consistent and sys-tematic reduction of the three-dimensional model to aone-dimensional formulation in spatial coordinates.The sti¡ness and mass quantities are reminiscent ofthose used in lumped parameter models, but these arederived from a physically based three-dimensional mod-el here.

Fig. 2. Feed-forward mechanism of outer hair cells. Schematicdrawings of the transverse (end) view and longitudinal (side) viewof the organ of Corti are shown in panels a and b respectively. Thefree body diagrams for balance of forces are given in panel c.

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2.2. Nonlinear micro-mechanics

In this model, the force exerted by the outer hair cellson the cochlear partition is assumed to be proportionalto the total force acting on the partition. This can beseen from the following simplistic arguments. The forceacting on the basilar membrane is transmitted to thestereocilia by the sti¡ arches of Corti, as depicted inthe transverse view in Fig. 2a. For moment equilibriumof the arches of Corti about its joint at the spiral liga-ment,

F ciliaðx; tÞ ¼ C1ðxÞFBMðx; tÞ

2ð22Þ

where Fcilia is the force acting on the stereocilia, FBM isthe resultant force acting on the basilar membrane, andC1 is a constant which accounts for the ratio of themoment arms of each force about the pivot. The resul-tant force acting on the basilar membrane consists ofthose from the £uid in both ducts and the outer haircells

FBMðx; tÞ ¼ 2F fðx; tÞ þ F cðx; tÞ ð23Þ

The above relations expressing the balance of forces isillustrated in Fig. 2c.

Due to the longitudinal tilt of the outer hair cells, theforce acting on the stereocilia at x causes the outer haircell to push at a point x+v downstream on the basilarmembrane, as shown in the longitudinal view in Fig.2b,

F cðxþ v ; tÞ ¼ C2ðx; tÞF ciliaðx; tÞ ð24Þ

where C2 is a transfer function coe⁄cient relating thetwo forces and v depends on the outer hair cell length lcand its angle to the x-axis a,

v ¼ lc cos a ð25Þ

Combining Eqs. 22, 23 and 24 yields

F cðxþ v ; tÞ ¼ C1ðxÞC2ðx; tÞ2

ð2F fðx; tÞ þ F cðx; tÞÞ

ð26Þ

¼ K ðx; tÞð2F fðx; tÞ þ F cðx; tÞÞ ð27Þ

where K= (C1C2)/2 is referred to as the feed-forwardgain factor. Fig. 3 shows the typical variation of thecell force Fc with the force on the basilar membraneFBM. Experiments have shown that the outer hair cellforce saturates as the force on the basilar membraneincreases (Pickles, 1988). From the relation in Eq. 27,

the gain factor K is given by the chord joining a pointon the curve to the origin. Fig. 4 shows the pro¢le ofthe gain factor K with the basilar membrane displace-ment w. The gain factor remains fairly constant forsmall basilar membrane displacement and tapers o¡to zero as the basilar membrane displacement increases,due to the saturation in the outer hair cell force.

The force Fc and the gain factor K can be expressedin terms of their Fourier coe⁄cients, F c(x, gj) and A(x,gj), in the frequency domain

F cðx; tÞ ¼Xj

e3ig j t F cðx; g jÞ ð28Þ

K ðx; tÞ ¼Xj

e3ig j t Aðx; g jÞ ð29Þ

Eq. 27 becomes

Xj

e3ig j t F cðxþ v ; g jÞ ¼Xk

e3ig kt Aðx; g kÞ !

Xl

e3ig l t ð2 F fðx; g lÞ þ F cðx; g lÞÞ !

ð30Þ

and the product on the right hand side can be replacedby a convolution of the two Fourier series giving

XNj¼3N

e3ig j t F cðxþ v ; g jÞ ¼

XNj¼3N

e3ig j tXjþB

k¼j3B

Aðx; g j3kÞð2 F fðx; g kÞ þ F cðx; g kÞÞ

ð31Þwhere N is the number of harmonics considered for the

Fig. 3. Saturation of outer hair cell force Fc with the force on thebasilar membrane FBM.

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vibration of the basilar membrane, and B+1 gives the number of terms used in the expansion for K. To determineF c(x+v ; g), the functional form of F c is assumed to be similar to F f , which is a phase-integral form

F cðx; g Þ ¼ F̂F cðx; g ÞeiR

xnðj ; g Þdj ð32Þ

where F̂F c is the slowing varying amplitude, after the rapidly varying part due to the ‘phase’ is being factored out. Forsmall v, the following approximations are made

F̂F cðxþ v ; g ÞW F̂F cðx; g Þ ð33Þ

Z xþv

nðj ; g ÞdjWZ x

nðj ; g Þdj þ nðx; g Þv ð34Þ

The approximation in Eq. 34 is ¢rst order accurate, similar to the explicit Euler scheme commonly used in numericalintegration. The error induced depends on the variation of n over the interval v, and this is small in the long-waveregion near the stapes and tends to increase towards the apical short-wave region of the cochlea. The outer hair cellforces at x+v and x are then related by

F cðxþ v ; g Þ ¼ F̂F cðxþ v ; g ÞeiR

xþv nðj ; g Þdj ð35Þ

W F̂F cðx; g ÞeiðR

xnðj g Þdjþnðx; g Þv Þ ð36Þ

¼ F cðx; g Þeinðx; g Þv ð37Þ

¼ y ðx; g Þ F cðx; g Þ ð38Þ

where y is given by

y ðx; g Þ ¼ einðx; g Þv ð39Þ

Equating the Fourier coe⁄cient in Eq. 31 gives

y ðx; g jÞ F cðx; g jÞ3XjþB

k¼j3B

Aðx; g j3kÞ F cðx; g kÞ ¼ 2XjþB

k¼j3B

Aðx; g j3kÞ F fðx; g kÞ ð40Þ

The above equation can be put in a compact matrix form

y ðxÞ F cðxÞ3AðxÞ F cðxÞ ¼ 2AðxÞ F fðxÞ ð41Þ

and its rearrangement gives the following expression for the outer hair cell forces in terms of the £uid forces

F cðxÞ ¼ 2ðy ðxÞ3AðxÞÞ31 AðxÞ F fðxÞ ð42Þ

Here, the vectors F c(x) and F f (x) contain the Fourier coe⁄cients F c(x, gj) and F f (x, gj) respectively,

F cðxÞ ¼

F cðx; g3NÞm

F cðx; g31ÞF cðx; g 0ÞF cðx; g 1Þ

m

F cðx; gNÞ

2666666666664

3777777777775and F fðxÞ ¼

F fðx; g3NÞm

F fðx; g ;31ÞF fðx; g 0ÞF fðx; g 1Þ

m

F fðx; gNÞ

2666666666664

3777777777775

ð43Þ

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The o¡-diagonal terms in the matrix A provide thecoupling between various harmonic coe⁄cients of theforces and the extent of this coupling is determined bythe parameter B.

2.3. Solution of nonlinear system

Eq. 23 expressing the balance of forces on the parti-tion can also be expressed in its Fourier componentsand put in the vector form

FBMðxÞ ¼ 2 F fðxÞ þ F cðxÞ ð46Þ

where FBM is vector of Fourier coe⁄cients of FBM(x, t)similar to F c and F f in Eq. 43. Eq. 15 written in thesimilar matrix-vector form gives

FBMðxÞ ¼ ðKpðxÞ36 2MpðxÞÞ WðxÞ ð47Þ

where 6, Kp(x), and Mp(x) are diagonal matrices withentries gj , Kp(x, gj), and Mp(x, gj) respectively. The

column vector W(x) contains the displacement ampli-tudes at various frequencies W(x, gj). Also, the vectorof Fourier coe⁄cients of the £uid force is given by

F fðxÞ ¼ 6 2M fðxÞ WðxÞ ð48Þ

where the mass matrix Mf (x) is also diagonal with en-tries Mf (n ; x, gj).

The system of equations in Eq. 46 is generally non-linear because the gain K depends on the displacement.However, for small displacements, K is constant asshown in Fig. 4, and this gives a linear system whosesolution can be obtained readily (Steele and Lim, 1999).Subsequently, the solution to the nonlinear case can beobtained through successive corrections on the linearsolution. The solution procedure is described as follows.

First, for the linear case with small displacements, theo¡-diagonal terms of the matrix A are zero and thediagonal terms are given by the constant K0. The matrixA can be written as

A ¼ K 0I ð49Þ

where I is the identity matrix. The outer hair cell forcefor this linear system, denoted as FL

c for clarity, is thengiven by

FLc ¼ 2K 0ðy3K 0IÞ31 F f ð50Þ

The force balance equation (46) becomes

½Kp36 2Mp32ðI þ K 0ðy3K 0IÞ31Þ6 2Mf � W ¼ 0

ð51Þ

The dependence on x is implied but omitted in the

y is a diagonal matrix with the elements given by y(gj)

y ðxÞ ¼

y ðx; g3NÞ 0 m

X

m 0 y ðx; g31 0 m

m 0 y ðx; g 0Þ 0 m

m 0 y ðx; g 1Þ 0 m

X

m 0 y ðx; gNÞ

2666666664

3777777775

ð44Þ

and A(x) is a banded matrix with bandwidth (2B+1) and components A(x, gj3k)

AðxÞ ¼

Aðx; g 0Þ m Aðx; g3BÞ 0 m

X X

Aðx; gBÞ m Aðx; g 0Þ m Aðx; g3BÞ 0 m

m Aðx; gBÞ m Aðx; g 0Þ m Aðx; g3BÞ m

m 0 Aðx; gBÞ m Aðx; g 0Þ m Aðx; g3B

X X

m 0 Aðx; gBÞ m Aðx; g 0Þ

2666666664

3777777775

ð45Þ

Fig. 4. Variation of the gain factor K with the basilar membranedisplacement w.

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above equations for clarity. Since all the coe⁄cient ma-trices in this case have a diagonal structure, the equa-tions in the system are uncoupled. Each individualequation gives an eiconal equation from which thewave numbers can be obtained for that particular fre-quency. Each equation has multiple complex solutionsfor the wave numbers. The complex wave numbers witha signi¢cant real part correspond to propagating waves,and the sign of the real part determines the direction ofwave propagation. In the present formulation, positiveand negative real parts correspond to forward andbackward propagating waves, respectively. Normally,two propagating waves traveling in opposite directionscan be found for each frequency and these provide thefundamental solutions for the Helmholtz equation (53)given below.

The amplitudes can be obtained from the transportequations which are obtained by considering the inte-gral of the continuity equation over a thin slice of theduct cross-section NV =L2L3NxZN V

v P N V ¼ 0 ð52Þ

which gives

F ;xx þ n2F ¼ 0 ð53Þ

where

F ¼ WR0 sinh nL3

nð54Þ

The amplitude W is obtained by solving for F(x) fromthe Helmholtz equation (53) using a combination of theasymptotic method in the short wave region (n large)and the numerical Runge^Kutta method in the longwave region (n small). The boundary conditions ofmatching the volume displacement at the stapes andzero pressure at the helicotrema are taken into accountfor solving the Helmholtz equation.

For the nonlinear case, A is not a diagonal matrixand the equations in the system (46) are coupled witheach other. In order to derive the nonlinear solutionfrom the linear solution, the linear system of equations(51) needs to be isolated from the nonlinear system.This is achieved by decomposing the cell force F c

into a sum of a linear part FLc as given by Eq. 50

and a nonlinear part FNLc

F c ¼ FLc þ FNL

c ð55Þ

Fig. 5. Characteristic frequency map of a passive chinchilla cochlea. The frequency^place map obtained using the present 3D model is shownin comparison with results from Eldredge et al. (1981) and Steele and Zais (1983).

Table 1Gross properties in chinchilla cochlear model

Basilar membrane bp = 1.0U103 kg/m3

E11 = 1.0U1032 GPaE22 = 1.0 GPaE12 = 0.0 GPaX=0.0

Scala £uid bf = 1.0U103 kg/m3

W=0.7U1033 Pa sStapes area Ast = 0.7 mm2

Outer hair cell a=30‡K0 = 0.35

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where

FNLc ¼ 2 ððy3AÞ31 A3K 0ðy3K 0IÞ31Þ F f ð56Þ

The nonlinear system of equations for force balancethen becomes

½Kp36 2Mp32ðI þ K 0ðy3K 0IÞ31Þ6 2M f � W ¼ FNLc

ð57Þ

The left hand side of the above equation is the same asthe homogeneous system in the linear case (Eq. 51). Theextra term FNL

c on the right can be treated as a forcing

function to the linear system. The solution of the linearsystem provides the homogeneous part of the nonlinearsolution. The particular solution due to the drivingterm on the right hand side of the nonlinear systemcan be found from the homogeneous solution usingthe method of variation of parameters. Since the driv-ing term FNL

c contains A which depends on the dis-placement W, the particular solution obtained also de-pends on the ¢nal displacement. This leads to aniterative procedure in which the ¢nal displacementand the driving term are repeatedly calculated untilthe system of equations (57) is satis¢ed within a giventolerance.

Fig. 6. Active basilar membrane response at various input levels. The basilar membrane response of the nonlinear active model is given for var-ious stimulus levels. Most parts of the membrane behave linearly, except for the region around the characteristic place (5 mm) where the re-sponse curves are ‘compressed’, i.e., close to each other.

Table 2Property variations in chinchilla cochlear model

x (mm) b (mm) h (mm) f L2, L3 (mm) lc (Wm)

0.0 0.130 0.0150 0.030 0.750 25.02.2 0.1654.3 0.187 0.7085.3 0.01007.5 0.5968.8 0.20010.8 0.0055 0.56212.6 0.22613.5 0.53115.3 0.240 0.003018.0 0.287 0.007 0.470 45.0

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Fig. 7. Nonlinear compression of response with input level. The basilar membrane response at 5 mm is plotted against the input sound pressurelevel. The response gets compressed nonlinearly as the input level increases, due to suppression of the activity in the cochlea. Experimentaldata (expt) from Ruggero et al. (1997) are included for comparison.

Fig. 8. Nonlinear frequency response of basilar membrane. The frequency response at 5 mm from the base on the basilar membrane with char-acteristic frequency of 10 kHz is shown. The input level is varied from 20 dB SPL to 80 dB SPL. The amplitude of the normalized response isreduced with increasing input level, as the activity in the cochlea is suppressed. Experimental data (expt) from Ruggero et al. (1997) are in-cluded for comparison.

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3. Results

The nonlinear active model is used to simulate theresponse of a chinchilla cochlea. The input parametersused in the model are based on anatomical data (Smith,1968; Dallos, 1970; Cabezudo, 1978; Bohne and Carr,1979; Lim, 1980). Table 1 gives the gross properties,such as density of £uid, and modulus basilar mem-brane, and angle of tilt of outer hair cells. Table 2 givesthe properties whose variations along the length of thecochlea have been taken into account, such as basilarmembrane width and thickness, and outer hair celllength.

The cochlear model is discretized into 1600 sectionsalong its length of 18 mm, and H=40 terms are used inthe Fourier expansion across the width of the model.Results for the nonlinear coupling of frequencies areobtained using a coupling bandwidth of B=3 in orderto maintain a manageable level of computational oper-ations without signi¢cant compromise to the accuracyof results. The average time taken for a single frequencycalculation within an iterative step is about 12 s on a600 MHz Pentium-III desktop computer. A distortionproduct generation problem involving about 30 fre-quencies and requiring about 100 iterations to achievea convergence tolerance of 5% would take about 10 h

for solution. This method provides a fast and e⁄cientsolution compared to a full-scale ¢nite element modeltaking into account the complexities and strong non-linearity present in the model. Therefore results are ob-tained without the need of computational resourcessuch as high performance super-computers and work-stations. We note that the computation time indicatedby Parthasarathi et al. (2000) is measured in hours ofcomputing time for the linear solution for a single fre-quency.

The model produces a frequency^place map withcharacteristic frequency range of 20 kHz to 20 Hzfrom the basal to the apical end of the cochlea, inline with animal measurements made by Eldredge etal. (1981), as shown in Fig. 5. Also shown is the loca-tion of the local resonance of plate strip and in¢nite£uid, discussed in Steele and Zais (1983) for severalmammals, and given by

g2 ¼ D22

2b f

Z

b

� �5 ð58Þ

The local ‘resonance’ involves the plate sti¡ness and the£uid inertia, and marks a point near the actual peakamplitude in Fig. 5. Thus from this simple formula, thevariation in physical properties of the pectinate zone of

Fig. 9. Harmonic distortion generation. The basilar membrane responses for various input levels are shown. In addition to the response at thestimulus frequency (fundamental), the responses at the higher harmonics are also shown. These higher harmonic responses become signi¢cantwith high input levels.

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the basilar membrane can explain the frequency map-ping. Naidu and Mountain (1998) measured the distri-bution of point load sti¡ness along the gerbil cochlea.Compared to their results, the sti¡ness of the presentmodel (with pectinate zone only) is the same near thebase but much more compliant at the apex. However,the calculation of the complete organ of Corti of thewater bu¡alo by Steele (1999) gives a distribution re-markably close to that measured by Naidu and Moun-tain (1998). So to the best of our knowledge, the pecti-nate zone dimensions and properties used for thepresent study correspond to the actual chinchilla co-chlea.

Several response features commonly observed in invivo experimental measurements are captured by thismodel, including compression of the response withstimulus level, harmonic distortion, two-tone suppres-sion, and distortion generation. For quantitative com-parison with experimental measurements on soundpressure levels, a simple model of the middle ear with20 dB gain in pressure at the stapes, accounting for theratio of areas of the eardrum and stapes and the levere¡ects of the ossicles, is used.

3.1. Response compression with stimulus level

Fig. 6 shows the basilar membrane responses whenthe stapes is excited at 10 kHz at various input levels.

The ¢gure gives a plot of the velocity amplitude againstthe distance along the cochlea duct measured from thestapes. The response of the basilar membrane increaseswith increase in the input level. This occurs linearlyover most of the region basal to the characteristic placewhich is located at about 5 mm for the frequency of 10kHz. For the region around the characteristic place, theresponse of the membrane is nonlinear. The responsegets suppressed with increasing input level, as the re-sponse curves get ‘compressed’ close to each other. Thepeak of the response is also shifted in the basal direc-tion, towards the stapes, with increase in the input level.

The response at the characteristic place is plottedagainst the input level in Fig. 7. The response increaseslinearly for low input level (up to 20 dB SPL), but turnsnonlinear as activity in the model gets suppressed withhigher stimulus level. At high input level, the responseis suppressed by about 30 dB. This nonlinear compres-sion in the response with input stimulus level is alsoobserved in the experimental measurements by Ruggeroet al. (1997), as shown by the points in the plot. Thecomputation and experimental data agree quite wellwith each other.

The frequency response over a range of excitationfrequencies for the same location is shown in Fig. 8.Each curve gives the amplitude and phase of the re-sponse on the basilar membrane normalized by thestapes input displacement, for various input levels.

Fig. 10. Two-tone suppression with a higher frequency. The basilar membrane response at the location with a characteristic frequency of 8 kHz(probe frequency) is plotted against the probe input level. Introduction of a suppressor of frequency 10 kHz at 60 dB reduces the response ofthe basilar membrane by about 10 dB. The model calculations (calc) are compared with the experimental data (expt) of Ruggero et al. (1992).

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For a low input level, a large ampli¢cation is presentdue to the active process in the cochlea. With increasinginput level, the response is suppressed. The experimen-tal measurements of Ruggero et al. (1997) are also in-cluded in the amplitude plot, and they compare quitewell with the model computation results. The phase ofthe response obtained from the model shows a muchlarger roll-o¡ with frequency than the experimentalmeasurements. In the model, the phase is normalizedto the volume £ow at x=0, as the stapes is assumedto be a piston at the end of the £uid chamber. Theactual position of the stapes in the cochlea extendsover a small portion of the basal end of the scala whichmay result in this discrepancy in the phase. Neverthe-less, the phase calculated from the model shows littledependence on the stimulus level, similar to the exper-imental data. This is also in agreement with the invar-iance of ¢ne time structure in the basilar membranewith variation of stimulus intensity (Shera, 2001). Astudy on the transient response for the linear case ofthe feed-forward model also shows the invariance of thephase of time oscillations with variation of the feed-forward gain (Lim and Steele, 1999).

3.2. Harmonic distortion

Fig. 9 shows the generation of higher harmonics dueto the nonlinearity in the model. The model is excited

with a single fundamental frequency of 5 kHz, at var-ious input levels (50, 60, 70, and 80 dB SPL). The plotsshow the amplitudes of the basilar membrane responsenormalized by the stapes input plotted against the dis-tance along the cochlea. For low input level (up to 50dB SPL), there is almost no harmonic distortion gen-eration. At 50 dB SPL input level, the basilar mem-brane only has a response at the fundamental fre-quency. As the input level is raised, the basilarmembrane begins to respond with the higher harmonicsof the excitation frequency (second harmonic 10 kHz,third harmonic 15 kHz, and fourth harmonic 20 kHz).The higher harmonic responses also increase in relativeamplitude, as the nonlinearity in the cochlea increaseswith the input level. At the high input level of 80 dBSPL, the higher harmonic responses become almostcomparable to the response at the fundamental fre-quency. The presence of harmonic distortions in thebasilar membrane response at high stimulus levels hadbeen measured by Cooper and Rhode (1992) in the catcochlea.

3.3. Two-tone suppression

When more than one tone is presented to the ear,inter-modulation occurs between the various tones.The following demonstrate the interplay between twotones that are presented simultaneously to the cochlea.

Fig. 11. Two-tone suppression with a lower frequency. The basilar membrane response at the location with a characteristic frequency of 7 kHz(probe frequency) is plotted against the probe input level. Introduction of a suppressor of frequency 1 kHz reduces the response of the basilarmembrane by about 10 dB. The model calculations (calc) are compared with the experimental data (expt) of Ruggero et al. (1992).

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Fig. 10 shows the case of two-tone suppression by thehigh frequency. A probe tone of frequency 8 kHz ispresented to the cochlea and monitored at its character-istic place. This probe tone is presented alone, or with ahigher frequency suppressor tone of 10 kHz at variouslevels. The response of the basilar membrane at thelocation with characteristic frequency of the probetone (8 kHz) is plotted against the input level of theprobe tone. The solid line gives the response with theprobe tone alone, and the dashed and dotted line, theresponse with the suppressor tone at 40 and 60 dBSPL. In all cases, the curves are similar to that of non-linear compression shown in Fig. 7. The response islinear for low input levels of the probe tone and slowlygets nonlinear and suppressed with increase in inputlevel. However, the response gets reduced in the pres-ence of the suppressor tone as depicted by the down-ward shift of the curves. Over its linear response region(less than 60 dB SPL), the response to the 8 kHz probetone gets reduced by about 10 dB in the presence ofa 10 kHz suppressor tone at 60 dB SPL. The experi-mental measurements by Ruggero et al. (1992) arealso included in the ¢gure (plotted as points) for com-parison with the model computation. It can be seen thatboth set of data are in very good agreement with eachother.

Suppression can also occur with the suppressor at alower frequency than the probe. Fig. 11 shows the re-sults for this case. Here, the probe frequency is 7 kHz,and the suppressor frequency is 1 kHz. The response ofthe basilar membrane at the location with characteristicfrequency of the probe tone (7 kHz) is plotted againstthe probe input level. The response due to the probetone alone is given by the solid line, and that in thepresence of the probe and suppressor tones is given bythe dashed line. Again, the response is reduced (byabout 10 dB) by the suppressor which is applied at avery high level of 80 dB SPL. Also, the computationresults agree quite well with the experimental measure-ments by Ruggero et al. (1992), plotted as points in the¢gure.

3.4. Distortion products

When two tones of frequencies f1 and f2 are pre-sented simultaneously to the cochlea, distortion prod-ucts of frequencies mf1+nf2 (where m and n are integers)are generated due to the nonlinearity in the cochlea.The strongest distortion product that is measured byresearchers is the 2f13f2 distortion. The active modelis hereby used to compute the distortion product gen-eration. Two tones with frequency f1 and f2 of equal

Fig. 12. Distortion products generation for closely spaced inputs f2/f1 = 1.05. The basilar membrane velocity spectra at the location with charac-teristic frequency 2f13f2 are given for various input levels. The model calculations (calc) are compared with the experimental data (expt) of Ro-bles et al. (1997).

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intensity are presented to the cochlea, and the basilarmembrane response at the location with characteristicfrequency 2f13f2 is monitored. Fig. 12 gives the casewhere f1 and f2 are close together with f2/f1 = 1.05, andFig. 13 gives the case where f1 and f2 are far apart withf2/f1 = 1.15. In each case, the plot of the basilar mem-brane velocity spectrum is given. The input levels of thetwo tones are prescribed at four di¡erent values rangingfrom 30 to 80 dB SPL as shown in the four plots ineach ¢gure. It can be seen that for a high input level, abroad frequency spectrum of the response is obtained,due to the nonlinear distortion generation. As the inputlevel is reduced, the spectrum narrows since the distor-tions get trimmed down and disappear. The experimentmeasurements of Robles et al. (1990, 1997) are alsogiven in the plots as points. The experimental and com-putational data are not in excellent agreement with eachother, but they show a similar trend. The discrepancy indistortion generation results is probably due to the sim-pli¢ed middle ear model that assumed a constant gainof pressure over the range of frequencies. A completesystem consisting of realistic models of the cochlea, themiddle ear and the ear canal would certainly providebetter simulation results for comparison with experi-mental data.

4. Conclusions

The current nonlinear active cochlear model is anextension of the linear model based on the WKB meth-od, with the inclusion of the three-dimensional viscous£uid e¡ects, an orthotropic tapered cochlear partition,and the active nonlinear feed-forward micro-mechanicsin the organ of Corti. The hybrid asymptotic and nu-merical method combined with the use of Fourier seriesexpansions provides a fast and e⁄cient iterative proce-dure for modeling and simulating the strong nonlinearactivity in the cochlea. Using a single set of physiolog-ically based parameters, the model is able to reproduceseveral characteristic nonlinear behaviors of the activecochlea commonly observed in experimental measure-ments on the chinchilla.

It is remarkable that a simple feed-forward modelgives simulation results with reasonably good agree-ment with experimental data. No speci¢c material ‘sec-ond ¢ltering’ is needed, although the gain factor of thefeed-forward mechanism may be interpreted as an ab-stract but simpli¢ed representation of the ‘¢lter’. Never-theless, the interaction between the feed-forward mech-anism and the propagating wave mechanics is the maincontribution to the characteristic active behavior ob-

Fig. 13. Distortion products generation for widely spaced inputs f2/f1 = 1.15. The basilar membrane velocity spectra at the location with charac-teristic frequency 2f13f2 are given for various input levels. The model calculations (calc) are compared with the experimental data (expt) of Ro-bles et al. (1990).

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tained in the simulations. For further improvement, aphysically based piezo-electric model of the outer haircells should be used to provide a more realistic materialdescription of the gain factor (Baker, 2000). In general,this gain factor may be represented by a complex trans-fer function with explicit dependence on the frequencyof excitation, longitudinal position along the cochlea,and displacement of the basilar membrane. Finally,the present cochlear model can be coupled with realisticmiddle and outer ear models, providing a complete sys-tem model of the ear. More complicated hearing-relatedphenomena, such as otoacoustic emissions, can be sim-ulated using this complete model.

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