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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
J Mech Mater Struct. Author manuscript; available in PMC 2010 May 18.
Published in final edited form as:
J Mech Mater Struct. 2009 September 5; 4(5): 977–986.
doi:  10.2140/jomms.2009.4.977
PMCID: PMC2872487
NIHMSID: NIHMS165527

A COCHLEAR MODEL USING THE TIME-AVERAGED LAGRANGIAN AND THE PUSH-PULL MECHANISM IN THE ORGAN OF CORTI

YONGJIN YOON
Stanford University, Mechanical Engineering, Durand Building, Room 262, Stanford, CA 94305, United States
SUNIL PURIA
Stanford University, Mechanical Engineering, Durand Building, Room 262, Stanford, CA 94305, United States
Stanford University, Otolaryngology — Head and Neck Surgery, Stanford, CA 94305, United States

Abstract

In our previous work, the basilar membrane velocity VBM for a gerbil cochlea was calculated and compared with physiological measurements. The calculated VBM showed excessive phase excursion and, in the active case, a best-frequency place shift of approximately two fifths of an octave higher. Here we introduce a refined model that uses the time-averaged Lagrangian for the conservative system to resolve the phase excursion issues. To improve the overestimated best-frequency place found in the previous feed-forward active model, we implement in the new model a push-pull mechanism from the outer hair cells and phalangeal process. Using this new model, the VBM for the gerbil cochlea was calculated and compared with animal measurements, The results show excellent agreement for mapping the location of the maximum response to frequency, while the agreement for the response at a fixed point as a function of frequency is excellent for the amplitude and good for the phase.

Keywords: cochlear active model, basilar membrane velocity, outer hair cell, push-pull mechanism, gerbil

1. Introduction

The cochlea of the inner ear transforms the input sound into output neural excitation. The basic model consists of a long, straight box of fluid with a flat lengthwise partition. The spiral coiling of the cochlea is neglected, as it has been shown to have little effect on the model response [Loh 1983; Steele and Zais 1985]. The partition has an elastic portion called the basilar membrane (BM). Generally, this membrane is narrow and thick, and therefore stiff, at the input end, where it connects with stapes, and wide and thin, and consequently flexible, at the far end, the apex. From a sinusoidal pressure input, a traveling fluid-elastic wave is generated on the BM.

For a fixed frequency of input, experiments show a traveling wave on the BM that builds to a peak at a certain point and then rapidly decays as it moves further. The location of the peak depends on frequency; thus each point has a “best frequency” (BF) that causes the maximum amplitude of the BM response. In the actual cochlea, a sensory epithelium, called the organ of Corti, is attached to the BM. The organ of Corti contains the sensory hair cells, which respond to the BM motion and initiate neural excitation. However, three rows of outer hair cells (OHC) are piezoelectric and most likely provide an amplification of the wave for low input amplitude; this is known as the active mechanism. Because of the geometry of the outer hair cells, a feed-forward approximation can be used in the cochlear model. Two-dimensional [Geisler and Sang 1995] and three-dimensional models with the active feed-forward mechanism have been employed [Steele et al. 1993; Lim and Steele 2002; Yoon et al. 2007].

In the present work, we improve on this previous feed-forward active model with further consideration of the actual geometry. For the low amplitude of sound intensity, there is significant energy feedback that enhances the wave. Located on the elastic region is the organ of Corti, which contains the sensory hair cells in an interesting arrangement of structural elements. A possible explanation for the form is derived from an analysis of the effect of the electromotile (piezoelectric) expansion and contraction of the outer hair cells which is known as OHC motility. This motility causes a positive spatial “feed-forward” of force on the basilar membrane (the “push”) as well as a negative “feed-backward” of force (the “pull”) from the phalangeal process. These push and pull mechanism work together to provide a two order of magnitude increase in the amplitude of the wave for the short wavelengths, but cancel each other out for the long wavelengths. Thus, the significant enhancement occurs in a narrow band of spatial wavelengths, without the need for special filtering or tuning. The calculations for the simple box model of the cochlea with the push-pull mechanism and the “time-averaged Lagrangian” method [Steele and Taber 1979] yield results very similar to gerbil measurements [Ren and Nuttall 2001].

2. Mathematical methods

Passive cochlear model

The physical cochlea is modeled as a straight-tapered fluid-filled chamber with rigid walls. The chamber is divided longitudinally by the flexible orthotropic plate into two equal rectangular ducts, which represent the scala vestibuli and scala tympani. The two fluid ducts are joined at the apical end of the chamber by a hole, representing the helicotrema. The flexible plate represents the pectinate zone of the basilar membrane, which bends under the motion of the fluid driven by the stapes motion. A schematic drawing of the model is shown in Figure 1.

Figure 1
Geometric layout of the passive cochlear model: from left to right, side view, cross section (A–A), and top view. Distances are parameterized in cartesian coordinates, x representing the distance from the stapes, y the distance across the scala ...

The equations of the system are formulated in the frequency domain. Harmonic excitation is applied at the stapes and the steady-state response of the system is considered. The equation for wave propagation in the passive model is obtained by equating the time-averaged kinetic and potential energies of the conservative system. The detailed derivations and features for the passive mechanics were described in [Steele and Taber 1979]. Briefly, the three-dimensional fluid equations are integrated over the cross section to obtain the relation between volume flow and fluid pressure, using a Fourier series expansion.

The time-averaged kinetic energy of the fluid (Tf) and of the elastic basilar membrane (Tp) are then obtained, as is the time-averaged potential energy of the plate (Vp). By equating the time-averaged kinetic and potential energies of the system, the eikonal equation is obtained, which can be written as

L=Tf+TpVp=F(n,w)W(x)2(0bη2(ξ)dξ)=0.
(1)

This gives the phase relation for the active case, F(n, ω) = 0, where b is the BM width, n is the wavenumber, and ω is the frequency. Here, L is the time-averaged Lagrangian density of the system, W (x) the amplitude of the wave envelope, and η(ξ) = sin(πξ/b) the shape function for simply-supported edges; the coordinate ξ has its origin at one edge of the BM.

Active cochlear model: push-pull mechanism

For the push-pull active model, the arrangement of the organ of Corti is indicated in Figure 2. If the OHCs expand, the Deiters rod at the horizontal distance x + Δx1 will be pushed down by the OHC with the apex at x, and pulled up by the phalangeal process connected to the OHC with the apex at x + Δx2. The force in an OHC is proportional to the shear force on the cilia at the apex, which is approximately equal to the total force acting on the BM. This push-pull mechanism is quantified by writing the total force on the BM as

FPZ=2FBMf+FBMC.
(2)

in which FBMC is the force due to the OHCs, and FBMf is the force due to the fluid on one side. The force acting through the Deiters rod at x + Δx1 is

FBMC(x+Δx1)=α1(2FBMf(x)+FBMC(x))α2(2FBMf(x+Δx2)+FBMC(x+Δx2)),
(3)

in which the gains from the push and the pull are α1 and α2, respectively, while Δx1 is the longitudinal distance between the apex and the base of the OHC, which depends on the length lOHC of the OHC and its angle θ1 with respect to the longitudinal direction to the reticular lamina (see Figure 2):

Δx1=lOHCcosθ1.
(4)
Figure 2
Schematic longitudinal view of the organ of Corti, showing the tilt of the outer hair cells. For one hair cell with apex at the distance x, the base is at x + Δx1; the phalangeal process connected at the base attaches to the upper surface at ...

Moreover Δx2 – Δx1 is the longitudinal distance between the base and the apex of the phalangeal process, which depends on the length lp of the phalangeal process and its angle θ2 with respect to the longitudinal direction of the reticular lamina:

Δx2Δx1=lpcosθ2.
(5)

Based on the geometric relationship between OHC and phalangeal process, lp can be expressed as

lp=lOHCsinθ1sinθ2.
(6)

By (5) and (6), Δx2 – Δx1 can be reduced without lp:

Δx2Δx1=lOHCsinθ1cosθ2.
(7)

From the phase integral, (3) reduces to

FBMCeinΔx1=α1(2FBMf+FBMC)α2(2FBMf+FBMC)einΔx2.
(8)

The force from the OHC motility is related to the fluid force:

FBMC=2(α1Γ1α2Γ2)FBMf1α1Γ1+α2Γ2,
(9)

where

FPZ=2(1+(α1Γ1α2Γ2)(1α1Γ1+α2Γ2))FBMf=2FBMf1α1Γ1+α2Γ2.
(10)

Therefore, the total force on the BM from (2) is

LW=2F(n,w)W(x)(0bη2(ξ)dξ)=0,
(11)

Matching the force to the displacement of the fluid and the OHC to the BM stiffness yields the eikonal (dispersion) relation of the active model. We consider a positive value of α2 to be a negative feedback effect of the phalangeal processes. Because of the flexibility of the upper surface consisting of the tectorial membrane and the reticular lamina in Figure 2, the upward and downward forces tend to cancel each other, so that α1 = α2. Then, for the long wavelength, i.e., the small values of the wave number n, the push-pull effect in (2) are negligible, while for very short wavelengths, the fluid viscosity dominates. Thus, the push-pull is effective only in a band of wavelengths. It is assumed that OHC force production does not roll off with frequency and thus α1 and α2 are constant with frequency. Although this is controversial, there is evidence that suggests that OHCs can generate forces for frequencies up to 50 kHz [Rabbitt 2008].

Wave propagation of BM plate

An energy approach is used involving variational calculus which is applied to the time-averaged Lagrangian density L of the system Equation (1). Independent variation of n and W (x) yields the Euler–Lagrange equations

ddxLn=ddx[F(n,w)nW2(x)(0bη2(ξ)dξ)]=0.
(12)

The eikonal equation is obtained using (11):

F(n,w)=0.
(13)

The amplitude function, W (x), which is the solution of the transport equation (12), is expressed as

W(x)=C(F(n,w)nb)12,
(14)

where C is a constant determined from the boundary condition.

Hamilton's principle is valid only for a conservative system; however, nonconservative terms can be added to the Lagrange equations of motion. Thus, we extend Equation (11) and (12) to the push-pull active model to obtain the eikonal and transport equation.

Ratio of the BM velocity to the stapes velocity

For a given harmonic excitation frequency ω, the eikonal equations for the passive and active case are solved to give the wavenumber (n) by using Newton–Raphson iterative scheme for each cross section along the cochlear duct. Once n is determined, the ratio of the BM velocity VBM to the stapes velocity Vst is obtained. With the stapes motion defined as positive outward, the ratio at x = x* is

(VBMVst)x=x=2nAstbx=0[(bF(n,w)n)x=0(bF(n,w)n)x=x]exp(π2ndx),
(15)

where Ast is the stapes foot-plate area.

Modification for nonconservative system

An extension of the time-averaged Lagrangian method to nonconservative systems is given in [Jimenez and Whitham 1976], in which the adjoint is considered. An alternate approach is to consider the reciprocity principle. For a linear elastic system, consider two independent solutions, denoted by superscript(1) and(2). The stress from (2) multiplied by the displacement of (1) subtracted from the stress from (1) multiplied by the displacement from (2), integrated over the entire surface, must be zero. In standard notation this is

n(σ(1)u(2)σ(2)u(1))dΣ=0.
(16)

This depends on the symmetry of the tensor of elastic constants. The symmetry is preserved when dealing with viscoelastic materials, including a viscous fluid, and working in the frequency domain, for which the coefficients are complex-valued. For a tube such as the cochlea, in which no work is done on the side walls, the integral over one end is the negative of that over the other end. Since the sign change is due to the change in the direction n, the integral itself must be constant with the distance along the tube. So one solution is taken with a positive exponential,

(σ(1),u(1))expλdx,
(17)

while the other is taken with a negative exponential,

(σ(2),u(2))expλdx.
(18)

One is a backward traveling wave, exponentially increasing, while the other is a forward traveling wave, exponentially decreasing. For a conservative system, the condition (16) yields the same result for the transport condition as (14). For a dissipative system with fluid viscosity, we find little difference, since the wave number changes little. In addition, Equation (16) can be used for the case of feedback, when the system is no longer self-adjoint. In this case the second solution in (16) must be taken as a solution of the adjoint problem. Again we find little effect on the solutions for the cochlea, so (14) remains a good approximation.

3. Results and discussion

The cochlear model is used to calculate the response of a gerbil cochlea. The material property values in Table 1 were taken from [Smith 1968; Lim 1980; Steele et al. 1995; Karavitaki 2002], and the dimensions in Table 2 from anatomical measurements for the gerbil cochlea [Sokolich et al. 1976; Greenwood 1990; Dannhof et al. 1991; Cohen et al. 1992; Edge et al. 1998; Thorne et al. 1999; Naidu and Mountain 2007].

Table 1
Gross properties in cochlear model.
Table 2
Property variations in gerbil cochlear model.

The model is meshed into 1200 sections along the 12mm length of the cochlea. Forty terms are used in the Fourier expansion across the width of the fluid duct. Calculation with eighty terms for the Fourier expansion shows no difference with forty terms. Running on an Intel Pentium IX (3.40 GHz) processor, the average time taken for a single harmonic excitation calculation is about one second. This method provides a fast and efficient solution compared to a full-scale finite element model. 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 frequency.

The results include the BF-to-place map and the BM velocity frequency response. The modeling results for the gerbil cochlea are compared with recent in vivo experiment measurements in the cochlea.

CF-to-place map

CF versus distance from the stapes along the gerbil cochlear (CF range: 0.3 kHz–50 kHz) is shown in Figure 3. The gerbil CF-to-place map [Sokolich et al. 1976; Greenwood 1990] was measured with cochlear-microphonic recording. The maps from the passive model and measurement are in excellent agreement (Figure 3).

Figure 3
Characteristic frequency (CF) versus position for the passive cochlear model compared to measurements. The present three-dimensional cochlear model represents the gerbil cochlear CF-to-place map from 0.3 kHz to 50 kHz [Sokolich et al. 1976; Greenwood ...

BM velocity frequency response

The gerbil cochlear BM velocity magnitude and phase for 4.2mm from the base (BF = 10 kHz) relative to the stapes displacement were computed up to 20 kHz and are compared with the gerbil experimental data from [Ren and Nuttall 2001] in Figure 4. The passive model shows quantitatively very good agreement with data which are measured at a high stimulus level (100 dB SPL at the ear canal).

Figure 4
Basilar membrane (BM) velocity relative to the stapes: magnitude (left) and phase (right) for the gerbil cochlea at 4.2mm from the base (BF = 10 kHz). For the active model, a value of 0.14 was used for the feed-forward and feedback gain factor. Experimental ...

Karavitaki [2002] gives the angle of tilt of gerbil OHC(θ1) as 84°, closer to perpendicular to the BM (Figure 2). The active model results are calculated with parameters; θ1 = 85° (OHC angle), θ2 = 25° (phalangeal process angle), α1 = α2 = 0.14 (feedforward and feedback gain factor). The active model shows qualitatively and quantitatively good agreement with data at low stimulus level (30 dB SPL at the ear canal) with 27 dB gain.

In the left half of Figure 4, showing the magnitude of the relative BM velocity, the calculated BF place shifts from 10 kHz (passive model) to 13 kHz (active model), providing better agreement with measurements. The BF place shift (two fifths of an octave upwards) observed in the earlier feed-forward active model [Yoon et al. 2007] is resolved by using a push-pull mechanism in the active model. From the parameter study, we find that the BF place shifts to the low frequency by using a small phalangeal process angle in the active case.

The right half of Figure 4 shows that we obtain good agreement in the phase as well, at least in the region below the BF place (that is, up to 10 kHz). Reduced phase excursion is one of the most impressive improvements in the time-averaged Lagrangian model, compared to the 2.5 cycle phase difference near BF region in the earlier model. However, past the BF, the phase of BM velocity from the model shows a larger roll-off than the phase from the data, which represents over-fluctuation of the traveling wave after 10 kHz in the model.

4. Conclusions

We presented a refinement of an earlier physiologically based three-dimensional cochlear model [Yoon et al. 2007] that resolved its two main deficiencies: a relatively large phase excursion near the best frequency region, which was addressed using the time-averaged Lagrangian method, and an overestimation of the best frequency (by about two fifths of an octave) observed in the earlier feed-forward active model, which was ameliorated by the implementation of a push-pull mechanism in the active case. Basilar membrane velocity simulation results for the gerbil cochlea now show excellent agreement in magnitude and good agreement in phase relative to physiological measurements. However, past the best frequency region, the phase in the current model still shows excessive roll-off. We hope that this can be addressed by incorporating into the model a more detailed treatment of the organ of Corti [Steele and Puria 2005].

Acknowledgments

Work supported by the NIDCD of the National Institutes of Health through grant no. R01-DC007910.

List of symbols

BM
basilar membrane
OHC
outer hair cell
BF
best frequency
n
wave number
x, y, z
Cartesian coordinates
ω
angular velocity
L2, L3
width and height of fluid chamber
α1, α2
feed-forward and feedback gain factors
lOHC, lp
length of OHC, length of phalangeal process
θ1, θ2
angle of tilt of OHC, phalangeal process
FPZ, FBMf
forces acting on the pectinate zone and fluid
FBMC
force exerted by OHC

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