2002 - Lim, Steele - A Three-dimensional Nonlinear Active Cochlear Model Analyzed by the WKB-numeric...

8/16/2019 2002 - Lim, Steele - A Three-dimensional Nonlinear Active Cochlear Model Analyzed by the WKB-numeric Method

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A three-dimensional nonlinear active cochlear model analyzed by

the WKB-numeric method

Kian-Meng Lim a;, Charles R. Steele ba Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260

b 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 fluid

effects, an orthotropic cochlear partition with dimensional and material property variation along its length, and a nonlinear active

feed-forward mechanism of the organ of Corti. A hybrid asymptotic and numerical method combined with Fourier series

expansions is used to provide a fast and efficient iterative procedure for modeling and simulation of the nonlinear responses in the

active cochlea. The simulation results for the chinchilla cochlea compare very well with experimental measurements, capturing

several 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 rights

reserved.

Key words: Nonlinear cochlear model; Response suppression; Distortion

1. Introduction

Numerous models have been proposed and used to

simulate the activity in the cochlea. Many are exten-

sions of passive cochlear models with the inclusion of

the micro-mechanics of the organ of Corti, in particular

the 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 nonlinearity

in the activity and obtained both frequency and time

domain solutions using a quasi-linear method. They

also 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: [email protected] (K.-M. Lim).

Abbreviations: A, harmonic coe⁄cients of K ; B , bandwidth of harmonic frequency coupling; b, width of plate; C 1, C 2, constants of proportionality; D11, D12, D22, £exural sti¡nesses of plate; E 11, E 12, E 22, orthotropic Young’s moduli of plate; F BM, F f , F cilia, forces acting onbasilar membrane, £uid and cilia; F c, force exerted by outer hair cell; F BM, F f , F c, harmonic coe⁄cients of F BM, F f , F c ; F Lc , vector of F c in thelinear case; F NLc , nonlinear part of the vector of F c ; F̂ F c, slowly varying amplitude of F c ; f , ¢ber density of plate; f 1, f 2, 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; K p, equivalent sti¡ness of plate; L2, L3, width and height of

£uid chamber; l c, length of outer hair cell; M p, M f , 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; p f , pp, pressures acting on £uid and plate; Rm, Fourier coe⁄cient for the mth component of P ; 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; K 0, constant feed-forward gain factor (linear case); L m, decay coe⁄cient for mth component of P ; y , feed-forward transfercoe⁄cient for F c ; Q m, 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;i 1, i 2, i 3, components of vector i !; 8 m1, 8 m2, Fourier coe⁄cients for mth component of i 1, i 2 ; 6 , matrix of angular frequencies; g , angularfrequency; b f , b p, densities of £uid and plate; a , angle of tilt of outer hair cell; j , local coordinate along the length

Hearing Research 170 (2002) 190^205

www.elsevier.com/locate/heares

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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 passive

transmission 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 been

constructed. Diependaal and Viergever (1989) extended

the negative damping model to two dimensions using a

combination of integral equation and ¢nite di¡erence

method. Two-dimensional ¢nite di¡erence models con-

structed by Neely (1985, 1993) used a feedback law to

model 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 the

activity 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 linear

lumped parameter model with the feed-forward mecha-

nism, and Fukazawa and Tanaka (1996) further in-

cluded the nonlinear behavior in the one-dimensional

model. Two-dimensional (Geisler and Sang, 1995) and

three-dimensional models (Steele et al., 1993; Steele and

Lim, 1999) with the active feed-forward mechanism

were also constructed. But these higher dimensional

models are normally restricted to a linear analysis due

to the formidable computations needed for the iterative

solution procedure of a nonlinear problem.

The present work extends the linear three-dimension-

al feed-forward model to include the nonlinear behavior

of the outer hair cells. This model is based on the

phase-integral method, commonly known as the WKB

(Wentzel^Kramers^Brillouin) method, which provides

signi¢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 the

scalae and the variation of dimensions and material

properties along the length are included. The frequency

domain formulation in the linear model is retained, but

all the harmonic components need to be considered

together 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 for

comparison with previous linear models and experimen-

tal measurements of frequency responses. The simula-

tion results obtained from this active nonlinear model

successfully 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 of

the outer hair cells for the micro-mechanics in the organ

of Corti is formulated. Combining both formulations

for the macro- and micro-mechanics gives a coupled

system of equations governing the responses for the

various frequency components. An iterative procedure

needs to be used to solve this set of nonlinear equa-

tions.

2.1. Macroscopic model

A schematic drawing of the macroscopic model

showing the top, side, and end views is given in Fig.

1. This model consists of a straight tapered chamber

with rigid walls ¢lled with viscous £uid. The chamber

is divided longitudinally by a cochlear partition intotwo equal rectangular ducts representing the scala ves-

tibuli and scala tympani. The two £uid ducts are joined

at the apical end via a hole representing the helicotre-

ma. The cochlear partition represents a collapsed scala

media with its structural properties dominated by the

pectinate zone of the basilar membrane. This partition

is modeled as an elastic orthotropic plate which is sim-

ply supported by rigid shelves on both sides along its

length. The variations in width, thickness, and ¢ber

density along the length of the basilar membrane are

included, resulting in a partition with varying sti¡ness

along its length.

The £exible cochlear partition is set into motion

when the stapes pushes against the £uid in the scala

vestibuli. The anti-symmetric pressure wave in the

chamber is considered, since only this gives a signi¢cant

displacement to the partition (Peterson and Bogert,

1950; Steele and Lim, 1999). Due to symmetry present

in the model, only one £uid duct needs to be considered

in the simulation. The £uid displacement ¢eld in the

ducts is represented by a sum of the divergence of a

scalar ¢eld P and the curl of a vector ¢eld i !

u!

¼9 P

þ9 Ui

!

ð1

Þwhere P and i ! take the following forms:

P ðx; y; z; tÞ ¼ e3i g tx ðxÞXH m¼0

RmðxÞ cos mZ yL2

cosh ðL mðxÞðL33zÞÞ ð2Þ

K.-M. Lim, C.R. Steele / Hearing Research 170 (2002) 190^205 191


3/16

i !ðx; y; z; tÞ ¼

i 1

i 2

i 3

0BB@

1CCA ¼

e3i g t XH

m¼0

eQ mðxÞz

8 1mðxÞ sin

mZ y

L2

8 2m

ðx

Þ cos

mZ y

L2

0

0

BBBBB@

1

CCCCCA ð3Þ

Here, a harmonic excitation with frequency g is appliedat the stapes. The displacement ¢eld satis¢es the rigid

wall boundary conditions at y = 0, y = L2, and z = L3where the normal £uid displacements are zero. The co-

e⁄cients 8 m1 and 8 m

2 are related to the amplitude x and Rm through no-slip boundary conditions on the

cochlear partition where the tangent displacements are

also zero. The coe⁄cients Rm are determined by match-

ing the normal displacement of the £uid with that of the

partition at their interface. Incidentally, the pressure inthe £uid is given by

pf ¼ 3 b f €P P ð4Þ

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

n2ðxÞ ¼ 3x ;xxx

ð5Þ

the continuity equation for the £uid

v P ¼ 0 ð6Þ

reduces to the algebraic form

L mðxÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n2 þ mZL2

2

s ð7Þ

The WKB approximation is applied here with the as-

sumption of large n. Similarly, the vorticity equation

given by

ð8Þ

can be expressed as

Q mðxÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L 2m3i g b f W

r ð9Þ

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€ww þ D 2

D x2 D11

D 2w

D x2

þ 2 D

2

D xD y D12

D 2w

D xD y

þ

D 2

D y2 D22

D 2w

D y2

¼ pp ð10Þ

where pp is the pressure acting on the pectinate zone, b pis 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.

K.-M. Lim, C.R. Steele/ Hearing Research 170 (2002) 190^205192


4/16

bending sti¡ness components. Taking into account the

sandwiched nature of the pectinate zone with the ¢bers

occupying the lower and upper thirds of the entire

thickness, the bending sti¡ness is determined from

Young’s modulus E ij in the respective directions, Pois-

son ratio X , ¢ber volume density f , and thickness h, asgiven by

Dij ¼ fE ij h3

13X 213

162 ð11Þ

The displacement pro¢le of the partition is assumed to

take the following form

wðx; R ; tÞ ¼ e3i g t WðxÞ sinZR b

ð12Þ

The £uid and partition displacements are matched at

their interface (with W (x) = x (x)) and the coe⁄cientsRm are determined from this assumed shape function

of the displacement.Integrating the pressure across the width and sum-

ming up the Fourier harmonics gives the force per

unit length in the time domain for the partition and

£uid. For the partition,

F BMðx; tÞ ¼X

e3i g j tZ

ppðx; y; g j Þ d y

ð13Þ

¼X

e3i g j t F BMðx; g j Þ ð14Þ

where

F BMðx; g j Þ ¼ ðK pðn; x; g j Þ3g 2 j M pðxÞÞ Wðx; g j Þð15Þ

with the plate’s sti¡ness given by

K p ¼ 2bZ 3 b pg

2h þ D11n4 þ 2D12n2 Zb

2 þ D22 Z

b

4

n oð16Þ

and mass given by

M p ¼

2b

Z b ph

ð17

ÞFor the £uid,

F f ðx; tÞ ¼X

e3i g j tZ

pf ðx; y; g j Þ d y

ð18Þ

¼X

e3i g j t F f ðx; g j Þ ð19Þ

where

F f ðx; g j Þ ¼ b f g 2 j bhf ðn; x; g j Þ Wðx; g j Þ ð20Þ

¼ g 2 j M f ðn; x; g j Þ Wðx; g j Þ ð21Þ

with M f and hf being the e¡ective mass and thickness of the £uid layer over the width of the plate. These quan-

tities are functions of the wave number n(x, g j ), asshown in Lim (2000).

The above provides a physically consistent and sys-

tematic reduction of the three-dimensional model to a

one-dimensional formulation in spatial coordinates.

The sti¡ness and mass quantities are reminiscent of

those used in lumped parameter models, but these are

derived from a physically based three-dimensional mod-

el here.

Fig. 2. Feed-forward mechanism of outer hair cells. Schematic

drawings of the transverse (end) view and longitudinal (side) view

of the organ of Corti are shown in panels a and b respectively. The

free body diagrams for balance of forces are given in panel c.



5/16

2.2. Nonlinear micro-mechanics

In this model, the force exerted by the outer hair cells

on the cochlear partition is assumed to be proportional

to the total force acting on the partition. This can be

seen from the following simplistic arguments. The force

acting on the basilar membrane is transmitted to the

stereocilia by the sti¡ arches of Corti, as depicted in

the transverse view in Fig. 2a. For moment equilibrium

of the arches of Corti about its joint at the spiral liga-

ment,

F ciliaðx; tÞ ¼ C 1ðxÞF BMðx; tÞ2

ð22Þwhere F cilia is the force acting on the stereocilia, F BM is

the resultant force acting on the basilar membrane, and

C 1 is a constant which accounts for the ratio of the

moment arms of each force about the pivot. The resul-

tant force acting on the basilar membrane consists of

those from the £uid in both ducts and the outer haircells

F BMðx; tÞ ¼ 2F f ðx; tÞ þ F cðx; tÞ ð23Þ

The above relations expressing the balance of forces is

illustrated in Fig. 2c.

Due to the longitudinal tilt of the outer hair cells, the

force acting on the stereocilia at x causes the outer hair

cell to push at a point x+v downstream on the basilarmembrane, as shown in the longitudinal view in Fig.

2b,

F cðx þ v ; tÞ ¼ C 2ðx; tÞF ciliaðx; tÞ ð24Þ

where C 2 is a transfer function coe⁄cient relating the

two forces and v depends on the outer hair cell length l cand its angle to the x-axis a ,

v ¼ l c cos a ð25Þ

Combining Eqs. 22, 23 and 24 yields

F cðx þ v ; tÞ ¼ C 1ðxÞC 2ðx; tÞ2

ð2F f ðx; tÞ þ F cðx; tÞÞ

ð26Þ

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

where K = (C 1C 2)/2 is referred to as the feed-forwardgain factor. Fig. 3 shows the typical variation of the

cell force F c with the force on the basilar membrane

F BM. Experiments have shown that the outer hair cell

force saturates as the force on the basilar membrane

increases (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 of

the gain factor K with the basilar membrane displace-ment w. The gain factor remains fairly constant for

small 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 F c and the gain factor K can be expressedin terms of their Fourier coe⁄cients, F c(x, g j ) and A(x,g j ), in the frequency domain

F cðx; tÞ ¼X

j

e3i g j t F cðx; g j Þ ð28Þ

K ðx;

tÞ ¼X j e3i g j t Aðx; g j Þ ð29ÞEq. 27 becomes

X j

e3i g j t F cðx þ v ; g j Þ ¼X

k

e3i g k t Aðx; g k Þ !

Xl

e3i g l t ð2 F f ðx; g l Þ þ F cðx; g l ÞÞ !

ð30Þ

and the product on the right hand side can be replaced

by a convolution of the two Fourier series giving

XN j ¼3N

e3i g j t F cðx þ v ; g j Þ ¼

XN j ¼3N

e3i g j tX j þB

k ¼ j 3B Aðx; g j 3k Þð2 F f ðx; g k Þ þ F cðx; g k ÞÞ

ð31Þwhere N is the number of harmonics considered for the

Fig. 3. Saturation of outer hair cell force F c with the force on the

basilar membrane F BM.

K.-M. Lim, C.R. Steele / Hearing Research 170 (2002) 190^205194


6/16

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 F̂ F cðx; g Þei R

xnðj ; g Þdj ð32Þ

where F̂ F c is the slowing varying amplitude, after the rapidly varying part due to the ‘phase’ is being factored out. For

small v, the following approximations are made

F̂ F cðx þ v ; g ÞW F̂ F cðx; g Þ ð33Þ

Z xþvnðj ; g Þdj W

Z xnðj ; 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 numerical

integration. 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 cell

forces at x+v and x are then related by

F c

ðx

þv ; g

Þ ¼ F̂ F c

ðx

þv ; g

Þe

i R xþv nðj ; g Þdj

ð35

ÞW F̂ F cðx; g Þei ð

R xnðj g Þdj þnðx; 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 j Þ F cðx; g j Þ3 X j þB k ¼ j 3B

Aðx; g j 3k Þ F cðx; g k Þ ¼ 2X j þB

k ¼ j 3B Aðx; g j 3k Þ F f ðx; g k Þ ð40Þ

The above equation can be put in a compact matrix form

y ðxÞ F cðxÞ3AðxÞ F cðxÞ ¼ 2AðxÞ F f ðxÞ 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 f ðxÞ ð42Þ

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

F cðxÞ ¼

F cðx; g 3N Þm

F cðx; g 31ÞF cðx; g 0ÞF cðx; g 1Þ

m

F cðx; g N Þ

2666666666664

3777777777775

and F f ðxÞ ¼

F f ðx; g 3N Þm

F f ðx; g ;31ÞF f ðx; g 0ÞF f ðx; g 1Þ

m

F f ðx; g N Þ

2666666666664

3777777777775

ð43Þ



7/16

The o¡-diagonal terms in the matrix A provide the

coupling between various harmonic coe⁄cients of the

forces and the extent of this coupling is determined by

the 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 components

and put in the vector form

F BM

ðx

Þ ¼ 2

F f

ðx

Þ þ F c

ðx

Þ ð46

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

F BMðxÞ ¼ ðK pðxÞ36 2M pðxÞÞ WðxÞ ð47Þ

where 6 , K p(x), and M p(x) are diagonal matrices with

entries g j , K p(x, g j ), and M p(x, g j ) respectively. The

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

F f ðxÞ ¼6 2M f ðxÞ WðxÞ ð48Þ

where the mass matrix M f (x) is also diagonal with en-

tries M f (n; x, g j ).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 whose

solution can be obtained readily (Steele and Lim, 1999).

Subsequently, the solution to the nonlinear case can be

obtained through successive corrections on the linear

solution. The solution procedure is described as follows.

First, for the linear case with small displacements, the

o¡-diagonal terms of the matrix A are zero and the

diagonal terms are given by the constant K 0. The matrixA can be written as

A ¼ K 0I ð49Þ

where I is the identity matrix. The outer hair cell force

for this linear system, denoted as F Lc for clarity, is thengiven by

F Lc ¼ 2K 0ðy 3K 0I Þ31 F f ð50Þ

The force balance equation (46) becomes

½K p36 2M p32ðI þ K 0ðy 3K 0I Þ31Þ6 2M f W ¼ 0ð51Þ

The dependence on x is implied but omitted in the

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

y ðxÞ ¼

y ðx; g 3N Þ 0 m

X

m 0 y ðx; g 31 0 m

m 0 y ðx; g 0Þ 0 mm 0 y

ðx; g 1

Þ 0 m

X

m 0 y ðx; g N Þ

2666666664

3777777775

ð44Þ

and A(x) is a banded matrix with bandwidth (2B +1) and components A(x, g j 3k )

AðxÞ ¼

Aðx; g 0Þ m Aðx; g 3B Þ 0 mX X

Aðx; g B Þ m Aðx; g 0Þ m Aðx; g 3B Þ 0 mm Aðx; g B Þ m Aðx; g 0Þ m Aðx; g 3B Þ mm 0 Aðx; g B Þ m Aðx; g 0Þ m Aðx; g 3B

X X

m 0 Aðx; g B Þ m Aðx; g 0Þ

2666666664

3777777775

ð45Þ

Fig. 4. Variation of the gain factor K with the basilar membrane

displacement w .



8/16

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 individual

equation gives an eiconal equation from which the

wave numbers can be obtained for that particular fre-

quency. Each equation has multiple complex solutions

for the wave numbers. The complex wave numbers with

a signi¢cant real part correspond to propagating waves,

and the sign of the real part determines the direction of

wave propagation. In the present formulation, positive

and negative real parts correspond to forward and

backward propagating waves, respectively. Normally,

two propagating waves traveling in opposite directions

can be found for each frequency and these provide the

fundamental solutions for the Helmholtz equation (53)

given below.

The amplitudes can be obtained from the transport

equations which are obtained by considering the inte-

gral of the continuity equation over a thin slice of the

duct cross-section N V = L2L3NxZ N V

v P N V ¼ 0 ð52Þ

which gives

F ;xx þ n2F ¼ 0 ð53Þ

where

F ¼ W R0 sinh nL3n

ð54Þ

The amplitude W is obtained by solving for F (x) fromthe Helmholtz equation (53) using a combination of the

asymptotic method in the short wave region (n large)

and the numerical Runge^Kutta method in the long

wave region (n small). The boundary conditions of

matching the volume displacement at the stapes and

zero pressure at the helicotrema are taken into account

for solving the Helmholtz equation.

For the nonlinear case, A is not a diagonal matrix

and the equations in the system (46) are coupled with

each other. In order to derive the nonlinear solution

from 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 cinto a sum of a linear part F Lc as given by Eq. 50and a nonlinear part F NLcF c ¼ F Lc þ F NLc ð55Þ

Fig. 5. Characteristic frequency map of a passive chinchilla cochlea. The frequency^place map obtained using the present 3D model is shown

in comparison with results from Eldredge et al. (1981) and Steele and Zais (1983).

Table 1

Gross properties in chinchilla cochlear model

Basilar membrane b p = 1.0U103 kg/m3

E 11 =1.0U1032 GPa

E 22 = 1.0 GPa

E 12 = 0.0 GPa

X =0.0

Scala £uid b f = 1.0U103 kg/m3

W =0.7U1033 Pa s

Stapes area Ast =0.7 mm2

Outer hair cell a =30‡

K 0 = 0.35



9/16

where

F NLc ¼ 2 ððy 3AÞ31 A3K 0ðy 3K 0I Þ31Þ F f ð56Þ

The nonlinear system of equations for force balance

then becomes

½K p36 2M p32ðI þ K 0ðy 3K 0I Þ31Þ6 2M f W ¼ F NLcð57Þ

The left hand side of the above equation is the same as

the homogeneous system in the linear case (Eq. 51). The

extra term F NLc on the right can be treated as a forcing

function to the linear system. The solution of the linear

system provides the homogeneous part of the nonlinear

solution. The particular solution due to the driving

term on the right hand side of the nonlinear system

can be found from the homogeneous solution using

the method of variation of parameters. Since the driv-

ing term F NLc contains A which depends on the dis-placement W , the particular solution obtained also de-pends on the ¢nal displacement. This leads to an

iterative procedure in which the ¢nal displacement

and the driving term are repeatedly calculated until

the system of equations (57) is satis¢ed within a given

tolerance.

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 2

Property variations in chinchilla cochlear model

x (mm) b (mm) h (mm) f L2, L3 (mm) l c (Wm)

0.0 0.130 0.0150 0.030 0.750 25.0

2.2 0.165

4.3 0.187 0.708

5.3 0.0100

7.5 0.596

8.8 0.200

10.8 0.0055 0.562

12.6 0.226

13.5 0.531

15.3 0.240 0.0030

18.0 0.287 0.007 0.470 45.0



10/16

Fig. 7. Nonlinear compression of response with input level. The basilar membrane response at 5 mm is plotted against the input sound pressure

level. The response gets compressed nonlinearly as the input level increases, due to suppression of the activity in the cochlea. Experimental

data (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 is

reduced 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.

K.-M. Lim, C.R. Steele/ Hearing Research 170 (2002) 190^205 199


11/16

3. Results

The nonlinear active model is used to simulate the

response of a chinchilla cochlea. The input parameters

used 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 gives

the properties whose variations along the length of the

cochlea have been taken into account, such as basilar

membrane width and thickness, and outer hair cell

length.

The cochlear model is discretized into 1600 sections

along its length of 18 mm, and H = 40 terms are used in

the Fourier expansion across the width of the model.Results for the nonlinear coupling of frequencies are

obtained using a coupling bandwidth of B = 3 in order

to maintain a manageable level of computational oper-

ations without signi¢cant compromise to the accuracy

of results. The average time taken for a single frequency

calculation within an iterative step is about 12 s on a

600 MHz Pentium-III desktop computer. A distortion

product generation problem involving about 30 fre-

quencies and requiring about 100 iterations to achieve

a convergence tolerance of 5% would take about 10 h

for solution. This method provides a fast and e⁄cient

solution compared to a full-scale ¢nite element model

taking into account the complexities and strong non-

linearity present in the model. Therefore results are ob-

tained without the need of computational resources

such as high performance super-computers and work-

stations. We note that the computation time indicated

by Parthasarathi et al. (2000) is measured in hours of

computing time for the linear solution for a single fre-

quency.

The model produces a frequency^place map with

characteristic frequency range of 20 kHz to 20 Hz

from the basal to the apical end of the cochlea, in

line with animal measurements made by Eldredge et

al. (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 several

mammals, and given by

g 2 ¼ D222 b f

Z

b

5 ð58Þ

The local ‘resonance’ involves the plate sti¡ness and the

£uid inertia, and marks a point near the actual peak

amplitude in Fig. 5. Thus from this simple formula, the

variation 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 the

stimulus frequency (fundamental), the responses at the higher harmonics are also shown. These higher harmonic responses become signi¢cant

with high input levels.



12/16

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 present

model (with pectinate zone only) is the same near the

base but much more compliant at the apex. However,

the calculation of the complete organ of Corti of the

water 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 the

present study correspond to the actual chinchilla co-

chlea.

Several response features commonly observed in in

vivo experimental measurements are captured by this

model, including compression of the response with

stimulus level, harmonic distortion, two-tone suppres-

sion, and distortion generation. For quantitative com-parison with experimental measurements on sound

pressure levels, a simple model of the middle ear with

20 dB gain in pressure at the stapes, accounting for the

ratio of areas of the eardrum and stapes and the lever

e¡ects of the ossicles, is used.

3.1. Response compression with stimulus level

Fig. 6 shows the basilar membrane responses when

the stapes is excited at 10 kHz at various input levels.

The ¢gure gives a plot of the velocity amplitude against

the distance along the cochlea duct measured from the

stapes. The response of the basilar membrane increases

with increase in the input level. This occurs linearly

over most of the region basal to the characteristic place

which is located at about 5 mm for the frequency of 10

kHz. For the region around the characteristic place, the

response of the membrane is nonlinear. The response

gets suppressed with increasing input level, as the re-

sponse curves get ‘compressed’ close to each other. The

peak 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 plotted

against the input level in Fig. 7. The response increases

linearly for low input level (up to 20 dB SPL), but turns

nonlinear as activity in the model gets suppressed with

higher stimulus level. At high input level, the response

is suppressed by about 30 dB. This nonlinear compres-sion in the response with input stimulus level is also

observed in the experimental measurements by Ruggero

et al. (1997), as shown by the points in the plot. The

computation and experimental data agree quite well

with each other.

The frequency response over a range of excitation

frequencies 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 the

stapes 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 of

the basilar membrane by about 10 dB. The model calculations (calc) are compared with the experimental data (expt) of Ruggero et al. (1992).



13/16

For a low input level, a large ampli¢cation is present

due to the active process in the cochlea. With increasing

input 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 quite

well with the model computation results. The phase of

the response obtained from the model shows a much

larger roll-o¡ with frequency than the experimental

measurements. In the model, the phase is normalized

to the volume £ow at x = 0, as the stapes is assumed

to be a piston at the end of the £uid chamber. The

actual position of the stapes in the cochlea extends

over a small portion of the basal end of the scala which

may result in this discrepancy in the phase. Neverthe-

less, the phase calculated from the model shows little

dependence 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). A

study on the transient response for the linear case of

the feed-forward model also shows the invariance of the

phase 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 due

to 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 plots

show the amplitudes of the basilar membrane response

normalized by the stapes input plotted against the dis-

tance along the cochlea. For low input level (up to 50

dB 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 basilar

membrane begins to respond with the higher harmonics

of the excitation frequency (second harmonic 10 kHz,

third harmonic 15 kHz, and fourth harmonic 20 kHz).

The higher harmonic responses also increase in relative

amplitude, as the nonlinearity in the cochlea increases

with the input level. At the high input level of 80 dB

SPL, the higher harmonic responses become almost

comparable to the response at the fundamental fre-

quency. The presence of harmonic distortions in thebasilar membrane response at high stimulus levels had

been measured by Cooper and Rhode (1992) in the cat

cochlea.

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 two

tones 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 basilar

membrane by about 10 dB. The model calculations (calc) are compared with the experimental data (expt) of Ruggero et al. (1992).



14/16

Fig. 10 shows the case of two-tone suppression by the

high frequency. A probe tone of frequency 8 kHz is

presented to the cochlea and monitored at its character-

istic place. This probe tone is presented alone, or with a

higher frequency suppressor tone of 10 kHz at various

levels. The response of the basilar membrane at the

location with characteristic frequency of the probe

tone (8 kHz) is plotted against the input level of the

probe tone. The solid line gives the response with the

probe tone alone, and the dashed and dotted line, the

response with the suppressor tone at 40 and 60 dB

SPL. In all cases, the curves are similar to that of non-

linear compression shown in Fig. 7. The response is

linear for low input levels of the probe tone and slowly

gets nonlinear and suppressed with increase in input

level. 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 probe

tone gets reduced by about 10 dB in the presence of

a 10 kHz suppressor tone at 60 dB SPL. The experi-

mental measurements by Ruggero et al. (1992) are

also included in the ¢gure (plotted as points) for com-

parison with the model computation. It can be seen that

both set of data are in very good agreement with each

other.

Suppression can also occur with the suppressor at a

lower 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 of

the basilar membrane at the location with characteristic

frequency of the probe tone (7 kHz) is plotted against

the probe input level. The response due to the probe

tone alone is given by the solid line, and that in the

presence of the probe and suppressor tones is given by

the dashed line. Again, the response is reduced (by

about 10 dB) by the suppressor which is applied at a

very high level of 80 dB SPL. Also, the computation

results 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 f 1 and f 2 are pre-

sented simultaneously to the cochlea, distortion prod-

ucts of frequencies mf 1+nf 2 (where m and n are integers)

are generated due to the nonlinearity in the cochlea.

The strongest distortion product that is measured by

researchers is the 2 f 13 f 2 distortion. The active model

is hereby used to compute the distortion product gen-

eration. Two tones with frequency f 1 and f 2 of equal

Fig. 12. Distortion products generation for closely spaced inputs f 2/ f 1 =1.05. The basilar membrane velocity spectra at the location with charac-

teristic frequency 2 f 13 f 2 are given for various input levels. The model calculations (calc) are compared with the experimental data (expt) of Ro-

bles et al. (1997).



15/16

intensity are presented to the cochlea, and the basilar

membrane response at the location with characteristic

frequency 2 f 13 f 2 is monitored. Fig. 12 gives the case

where f 1 and f 2 are close together with f 2/ f 1 = 1.05, and

Fig. 13 gives the case where f 1 and f 2 are far apart with

f 2/ f 1 = 1.15. In each case, the plot of the basilar mem-

brane velocity spectrum is given. The input levels of the

two tones are prescribed at four di¡erent values ranging

from 30 to 80 dB SPL as shown in the four plots in

each ¢gure. It can be seen that for a high input level, a

broad frequency spectrum of the response is obtained,

due to the nonlinear distortion generation. As the input

level is reduced, the spectrum narrows since the distor-

tions get trimmed down and disappear. The experiment

measurements 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 each

other, but they show a similar trend. The discrepancy in

distortion generation results is probably due to the sim-

pli¢ed middle ear model that assumed a constant gain

of pressure over the range of frequencies. A complete

system consisting of realistic models of the cochlea, the

middle ear and the ear canal would certainly provide

better simulation results for comparison with experi-

mental data.

4. Conclusions

The current nonlinear active cochlear model is an

extension 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-mechanics

in the organ of Corti. The hybrid asymptotic and nu-

merical method combined with the use of Fourier series

expansions provides a fast and e⁄cient iterative proce-

dure for modeling and simulating the strong nonlinear

activity in the cochlea. Using a single set of physiolog-

ically based parameters, the model is able to reproduce

several characteristic nonlinear behaviors of the active

cochlea commonly observed in experimental measure-ments on the chinchilla.

It is remarkable that a simple feed-forward model

gives simulation results with reasonably good agree-

ment with experimental data. No speci¢c material ‘sec-

ond ¢ltering’ is needed, although the gain factor of the

feed-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 main

contribution to the characteristic active behavior ob-

Fig. 13. Distortion products generation for widely spaced inputs f 2/ f 1 = 1.15. The basilar membrane velocity spectra at the location with charac-

teristic frequency 2 f 13 f 2 are given for various input levels. The model calculations (calc) are compared with the experimental data (expt) of Ro-

bles et al. (1990).



16/16

tained in the simulations. For further improvement, a

physically based piezo-electric model of the outer hair

cells should be used to provide a more realistic material

description 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 frequency

of excitation, longitudinal position along the cochlea,

and displacement of the basilar membrane. Finally,

the present cochlear model can be coupled with realistic

middle and outer ear models, providing a complete sys-

tem model of the ear. More complicated hearing-related

phenomena, such as otoacoustic emissions, can be sim-

ulated using this complete model.

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