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Phase of Shear Vibrations within Cochlear Partition

Leads to Activation of the Cochlear Amplifier

Jessica S. Lamb, Richard S. Chadwick*

Section on Auditory Mechanics, National Institute on Deafness and Other Communication Disorders, Bethesda, Maryland, United States of America

Abstract

Since Georg von Bekesy laid out the place theory of the hearing, researchers have been working to understand the

remarkable properties of mammalian hearing. Because access to the cochlea is restricted in live animals, and important

aspects of hearing are destroyed in dead ones, models play a key role in interpreting local measurements. Wentzel-Kramers-

Brillouin (WKB) models are attractive because they are analytically tractable, appropriate to the oblong geometry of the

cochlea, and can predict wave behavior over a large span of the cochlea. Interest in the role the tectorial membrane (TM)

plays in cochlear tuning led us to develop models that directly interface the TM with the cochlear fluid. In this work we add

an angled shear between the TM and reticular lamina (RL), which serves as an input to a nonlinear active force. This feature

plus a novel combination of previous work gives us a model with TM-fluid interaction, TM-RL shear, a nonlinear active force

and a second wave mode. The behavior we get leads to the conclusion the phase between the shear and basilar membrane

(BM) vibration is critical for amplification. We show there is a transition in this phase that occurs at a frequency below the

cutoff, which is strongly influenced by TM stiffness. We describe this mechanism of sharpened BM velocity profile, which

demonstrates the importance of the TM in overall cochlear tuning and offers an explanation for the response characteristics

of the Tectb mutant mouse.

Citation: Lamb JS, Chadwick RS (2014) Phase of Shear Vibrations within Cochlear Partition Leads to Activation of the Cochlear Amplifier. PLoS ONE 9(2): e85969.

doi:10.1371/journal.pone.0085969

Editor: Manuel S. Malmierca, University of Salamanca- Institute for Neuroscience of Castille and Leon and Medical School, Spain

Received September 25, 2013; Accepted December 9, 2013; Published February 14, 2014

This is an open-access article, free of all copyright, and may be freely reproduced, distributed, transmitted, modified, built upon, or otherwise used by anyone for

any lawful purpose. The work is made available under the Creative Commons CC0 public domain dedication.

Funding: This work was supported by the intramural program project DC00033 in the National Institute on Deafness and other Communication Disorders. The

funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing Interests: The authors have declared that no competing interests exist.

* E-mail: chadwick@helix.nih.gov

Introduction

Mammals transduce sound waves to neuronal signals within the

fluid-filled cochlea via traveling waves in the organ of Corti (OC).

This mechanical process is responsible for much of the extraor-

dinary range and sensitivity that is characteristic of mammalian

hearing, yet underlying mechanisms of nonlinear perception that

enable the most interesting aspects of hearing, are still being

determined [1,27]. Investigators probe cochlear mechanics

through a wide variety of methods, including velocimetry of the

basilar membrane (BM) [3], multidirectional measurements of

tectorial membrane (TM) motions [4], and recently measurements

of the sensory epithelia, the reticular lamina (RL) [5]. However,

these methods all offer glimpses of limited sections of the cochlea,

so a well developed traveling wave model can help interpret the

results. Sound manifests in the inner ear as a transverse traveling

wave on the tissue of the OC. Mechanical frequency separation

occurs via gradients in the mass and viscoelastic properties of the

tissue that cause the position of peak vibration to depend on

frequency. The precise mechanical structure of the mammalian

cochlea has inspired many attempts to elucidate the roles of the

different substructures through modeling. A useful way to examine

the whole traveling wave is with a model based on the Wentzel-

Kramers-Brillouin (WKB) approximation, a subset of perturbation

theory ideal for analyzing waves propagating in slowly changing

media. Originally, the approach was developed by using a simple,

flexed basilar membrane (BM) to partition the cochlea fluid [6].

While outer hair cells (OHC) and their stereocilia are accepted as

the primary source of nonlinearity, researchers continue to

investigate the role other structural features have on the tuning

curve. The tectorial membrane (TM) has been suggested as a

possible source of secondary filtering or resonance [7,8]. Previ-

ously, we increased the complexity of the cochlea WKB model by

developing it with a modified ‘‘sandwich’’ cross-section [9],

allowing the TM, which has a fluid-facing surface, to vibrate as

a second degree of freedom and found that such a system can

produce a second propagating wave [10]. However, this simple

system did not include shear between the TM and RL, which must

be an input for any amplification mechanisms involving OHCs

[11]. Here, we expand that model and present an active dual wave

model with three self-equilibrating degrees of freedom by

including a shear motion at an angle set in the model. From this

we see that relative phase of vibrating structures strongly affects

active force output, enabling us to offer a novel explanation for

observations of the Tectb mutant mouse [12]. We also consider

longitudinal coupling within the tissue and the affect RL angle has

on activity.

Methods

This model builds on results in [10], the aforementioned model

where differential motion of the TM generates a second mode, and

[13], which added a nonlinear active force onto a WKB-based

model as a perturbation. The complete derivation of this model

comprising aspects of both of these papers as well as the strategies

discussed below, can be found in File S1. Here we focus on

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explaining the novel aspects of this work: the inclusion of the shear

angle h, between the RL and TM, and the active force

perturbation worked out in the two wave system.

Lumped Parameter Model with Hair Bundle Shear

The cochlea is modeled as a long, fluid-filled duct partitioned

into two compartments, the scala vestibule (SV) and scala tympani

(ST), roughly corresponding to the physiology. The lumped

masses, MT, MRand MB(subscripts indicating TM, RL and BM

respectively) are proportional to cochlear width, W(z), where z is

the long dimension. This and other dimensions are based on

observations for the mouse [14]. There are few measurements of

viscoelastic properties in mice, and those that exist are difficult to

interpret as single spring constants, so stiffness coefficients are

based on the observed frequency range of a mouse, and increase

from base to apex by a single exponential gradient. The ratio of

damping to stiffness for the TM was determine from observations

in [15] (rough calculations based on these moduli and the

dimensions of the mouse TM suggest the stiffness values we use are

of the correct order of magnitude.) Damping was otherwise kept

small, increasing by an exponential coefficient that is half that for

the stiffness (Tables S1 and S2 in File S1). We balance the internal

forces of the OC with the fluid by considering the lumped-

parameter cross section illustrated in Fig. 1, which defines most of

the model viscoelastic and dimensional constants.

Some simplifying approximations specific to this model design

are discussed as follows: by introducing the shear between the TM

and RL we also introduce the possibility of two-dimensional

motion the TM. Thin-film viscous adhesion will keep the gap

between the TM and reticular lamina (RL) constant. The direction

of the viscoelastic force due to the TM (VT) varies such that it

always works to restore the mass to its zero position. These

approximations, and those in our previous work [19] lead to a

tractable analytical model.

The partition equation relates the forces from the internal

viscoelastic properties and displacements of the cochlear partition

(CP) to the fluid pressure. These displacements are expressed using

vector notation (~A A(z,t)~½AS(z,t), AU(z,t), AB(z,t)?T) where the

subscripts denote the shear, upper, and basilar components

respectively.

ML2

Lt2

~A A(z,t)~{ DL

LtzS

??

~A A(z,t)zW(z)~P P(x,0,z,t)

z~F FA(AS(z),t)

ð1Þ

Here M, D, and S are slowly varying matrices describing

respectively the internal mass, damping, and stiffness per unit

length of the OC (Fig. 1 gives the subscript notation for each

structure) and~P P(x,0,z,t) is a vector of fluid pressure at the

partition (located at y~0). ~F FA represents the active force,

discussed below. Because all viscoelastic elements have mass and

damping terms in parallel, the matrices D and S have the same

form, and we represent them both with V. The cross section

matrices and form of the pressure vector can be derived by

balancing forces due to the elements of the CP and the forcing

fluid pressure. The matrices are

M~

MT

MTsinh

MTzMR

0

0

MTsinh

0

0

MB

2

64

3

75

ð2Þ

V~

VTMzVGap

VTMsinh

0

VTMsinh

VTMzVOC

{VOC

0

{VOC

VOCzVBM

2

64

3

75

ð3Þ

To derive an analytical expression relating~P P(x,0,z,t) to~A A(z,t),

we recognize that the problem is one of multiple scales, where the

cross section and wavelength dimensions are much shorter than

the overall length of the cochlea. We thus use a WKB method to

expand both quantities around the small number E~W(0)=L,

where L is the length of the cochlea, the ratio of the short to the

long scales. We proceed to solve Laplace’s equation in both fluid

chambers, using AB or the total displacement of the TM,

ASsinhzAU, to describe the partition boundaries [10]. Com-

bining this with the force balance in the cross section, the pressure

term on the RHS of Eq. (1) becomes

W(0)~P P(x,0,z)v2mf

sin2h

sinh

0

sinh

10

001

2

64

3

75~A A(z)

0k(g)dg is the

ð4Þ

where v is the driving frequency, w(k,v)~vt{Ðz

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Figure 1. Illustration (not to scale) of the lumped-parameter

cross section of the cochlea. This figure illustrates the scala height

(H), OC width (W(z)), tissue masses (M), and viscoelastic elements (V).

The direction of VTMis drawn at an arbitrary angle - the force always

acts in a direction that restores the TM to its resting position.

doi:10.1371/journal.pone.0085969.g001

Phase of Shear Vibrations in Cochlear Partition

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Page 3

accumulated phase and k is the wavenumber. The matrix is

brackets will be shown as Mf and

mf(k)~rW(z)

k(z)

cothHk(z)

ð5Þ

with r being the fluid density. All of the solutions will have the

form C(z)eiw(k,v), where C(z) is A(z),P(z),… as appropriate.

Active Force

Somatic electromotility in OHCs [16] implies the possibility of

force applied between the RL and BM in vivo. The active force in

this model is developed with this force profile and a standard

sigmoid function, without further specifying the origin of the

dominant nonlinearity. A Fourier coefficient is generated for the

fundamental frequency and harmonics are neglected in the

solution. The active force term can then be written as.

~F FA(AS(z),v)~fAY(AS(z))eiw(k,v)

0

{1

1

2

64

3

75

ð6Þ

where Y is the Fourier coefficient defined as

Y(AS)~1

2p

ðp

{p

S cos DASD

zarg(AS)

ðÞ½?

fge{id

ð7Þ

and S is the sigmoid function

S(x)~

2

1z0:005

x

ls

{1:

ð8Þ

The range of the function is scaled by the input saturation length,

defined by the inverse of the sigmoid function ls~S{1(0:99). The

output is normalized with the maximum force given explicitly by

fA. To use the perturbation method in [13] we recognize that fAis

much smaller than the scale of the inertial forces, i.e. v2MB(0)ls.

The ratio of the two gives a small number, d, about which we

expand~A A and k. We also introduce the subscript notation

~A A0,~A A1,... to indicate the order the displacement after treated with

the perturbation expansion. This gives us the homogeneous and

perturbed equations

v2MzmfMf

??zivDzS

??~A A0(z)~0

ð9Þ

v2MzmfMf

~pfHk1mfMf~A A0(z)z1

??zivDzS

??~A A1(z)

~F FA(AS(z))

d

ð10Þ

where the enumerated subscripts indicate expansion orders and pf

is a coefficient dependent on k0 that arises through the small

number expansion and is expressed in File S1.

Solution

Eq. (9) represents the passive problem, which we solve as in [10]

and then use the results to determine the active contribution.

Briefly, this is a generalized eigenvalue problem for mfthat can be

solved computationally, using the WKB transport equation and

the boundary conditions at the base to scale the eigenvectors.

Although this model has three coupled equations, because Mf is

singular there are only two eigenmodes. The passive response of

one mode is strongly influenced by the properties of the TM, and

has larger amplitude on that structure, while the same can be said

of the other and the BM, motivating the use of(B,T)to denote each

mode. Given this, the active system can be described via the

expansions

~A A~(AB

0zdAB

1)eiwBz(AT

0zdAT

1)eiwTz...

ð11Þ

w(B,T)~vtz

ðz

0

k(B,T)

0

(g)zdk(B,T)

1

(g)

hi

dgz...

ð12Þ

Once the passive model is solved, we can use the resulting A(B,T)

to calculate Y(B,T)for each eigenmode. Eq. (10) is only solvable if

the RHS is orthogonal to the eigenvector of the homogeneous

equation, and from this solvability condition we can derive the

expression for k1, the active wavenumber correction

S

k1~

MB(0)lsY(AS0(z))(AB0(z){AU0(z))

h

mfpfHAS0(z)sinhzAU0(z)

ðÞ2A2

B0(z)

i:

ð13Þ

The LHS of Eq. (10) requires choosing an eigenmode to set mf.

Obviously the homogeneous solution will be the eigenvector for

this mode; less obviously the particular solution will be the

eigenvector of the other mode, which we need to scale. By

assuming that the contribution from the homogeneous solution is

entirely accounted for in the passive problem and setting it to 0, we

can scale~A A1(z) by solving Eq. (10). This involved expression is

given in File S1.

Additional Terms

We wished to consider the effect of longitudinal coupling in the

TM, both to test the effect of a longitudinal coupling source

besides the fluid on our model. We developed a term to be added

to a two degree of freedom model, without any shear motion. SLT

represents longitudinal spring constants. A ‘‘waves on a string’’

problem typically involves taking two derivatives in the direction of

wave propagation, z in our model. However, in the WKB

approximation we have employed, these derivatives are replaced

by the wavenumber, we add the term

k2SLT

0

00

??

~A A(z)

ð14Þ

to the passive equation of motion. The active expansion derived in

[13] cannot be worked with this term in place, but the passive

model can still provide some insight.

Phase of Shear Vibrations in Cochlear Partition

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Results and Discussion

Passive Model Comparison

Table 1 shows the zeroth order wavenumbers for the original

passive model, that with added longitudinal coupling in the TM,

and that with shear. The wavenumbers with large real and small

imaginary components indicate propagating modes, and there are

clearly two for each model. This is significant because neither the

extra degree of freedom introduced by shearing, nor the additional

longitudinal coupling added propagating waves. Previous works

have dealt with evanescent wave modes, showing they make a

perceptible contribution near the frequency cutoff [17,18], or than

they emerge with added components such as longitudinal coupling

[19] and fluid chambers [20]. However, to our knowledge only

another sandwich model [9] suggested an additional propagating

mode is possible. We conclude the fluid-partition interfaces carry

propagating modes like surface waves. This analysis demonstrates

that only independent motions of the fluid-facing surfaces of the

CP, the TM and BM create propagating waves. Since the CP is a

viscoelastic structure, such independent motions could certainly

arise in vivo, making the second wave an important consideration.

Also of note is the extreme similarity of wavenumbers between

the system with and without shear. Adding this component lets us

extract more information about the system, but the waves are

essentially the same. Thus conclusions about the two degree of

freedom system [10], including those regarding relative phases of

the two modes and mode conversion, are valid as we increase

model complexity.

Varying TM Stiffness

The most significant result we found was that by increasing the

stiffness of TM, the selectivity (as defined by the quality factor) of

the BM increased, but the sensitivity (as defined by the maximum

velocity) decreased, as shown in Table 2. Fig. 2 shows how the

frequency responses of BM velocity and other quantities change

with the TM stiffness. To maximize the impact of the active force

we use an input SPL of 20 dB (referenced to 20 mPa, as is

standard, and without modeling the middle ear transfer function,

which likely adds some further gain [21].) We can see the decrease

in quality factor corresponds to a shift in the region of significant

gain. After testing the frequency of this transition against many

model parameters, such as BM stiffness, OC stiffness, and TM

damping (see File S1), we find it is strongly dependent on the TM

stiffness and fairly insensitive to any other parameter examined.

The wavelength correction defined in Eq. (13) is responsible for

most of the active force gain. Examining the equation analytically

and looking at predictions for the frequency dependent behavior of

the quantities on which k1depends do not reveal any dominating

transitions that account for the sudden rise in gain, although

several of them exhibit small bumps around the start of the gain.

The model does predict a strong transition in the phase difference

between the shearing motion and the basilar membrane vibration.

Although the formula for k1does not explicitly involve this term, it

is quite reasonable to expect the amplifier needs to work at a

specific phase to the BM vibration to be effective. Indeed, Dong

and Olson recently observed a transition the phase difference

between the extracellular voltage of the scala vestibuli, which is

related to the active force, and the fluid pressure near the BM,

which indicates its motion [22]. They also suggest this phase shift

optimizes the timing of the cochlear amplifier, and if one considers

the phase of the active force should mimic the phase of the input

shear, their result supports our prediction of a phase transition.

We can think of this transition as restricting the frequency range

of the active region, somewhat akin to a high-pass filter. The

abrupt loss of passive amplitude at the cutoff frequency restricts on

the high-frequency end, together making a narrow band region of

gain. Shifting the frequency of the ‘‘high-pass filter’’, by changing

the TM stiffness, but maintaining the frequency cutoff alters the Q

of the frequency response. For stiffness values that allow the

filtering effects to overlap, the frequency of maximum vibration is

impacted by the gain-reducing qualities and is diminished,

lowering sensitivity, trading sensitivity for selectivity. The high-

frequency end of the band can be altered by varying BM stiffness,

giving the same sensitivity-selectivity trade off (see File S1) but the

frequency of the phase transition remains constant.

This result is particularly interesting given that velocity

threshold measurements on the BM of the high-frequency (basal)

region of the Tectb mutant mouse displays increased selectivity

and somewhat decreased sensitivity compared to the wildtype

[12]. The mutated protein is found extensively in the TM of this

animal and the membrane, which at the base lacks an organized

striated-sheet matrix and Hensen’s stripe, and is further disrupted

at the apex. This TM phenotype and experiments [23] have led

researchers to attribute the altered tuning curve to a decrease in

longitudinal coupling associated with the loss of the striated-sheet

matrix. It is suggested that OHCs on the periphery of the

frequency place will be not be engaged in the mutant, leading to a

decrease in overall amplification as well as finer tuning [24].

Modeling bears this out as a possible cause of fine tuning, but does

not demonstrate decreased sensitivity [25]. Our result offers an

alternate explanation, which can achieve the same phenomena

based solely on changes in local resonance properties of the TM,

without implicating longitudinal coupling.

As part of this interpretation, we must consider how the changes

observed in the TM of the Tectb mutant may manifest themselves

as the increase in stiffness required by our model. Our lumped

stiffness parameter simultaneously represents TM attachment,

elastic modulus, and any bending effects which are left out of our

model. For modeling purposes we assumed a fairly homogeneous

cross section, but the actual cochlea is quite complex, and any

inhomogeneity due to fluid pressure, geometry or differential force

from the cells versus the numerous fluid spaces in the OC may

Table 1. Wavenumbers with imaginary parts ordered from least to most negative for three models.

Independent TM only Longitudinal TM coupling Shearing between TM and RL

69.00–00.39i 70.65–0.30i 68.96–0.39i

91.37–01.64i 27.04–25.26i91.64–1.66i

0.32–888.52i0.05–892.11i0.32–889.46i

0.06–892.37i 0.02–1792.46i 0.06–892.37i

Values are presented in units of mm21and calculated at z~0 where the waves are longest and least likely to be evanescent.

doi:10.1371/journal.pone.0085969.t001

Phase of Shear Vibrations in Cochlear Partition

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make the bending characteristics quite important. In this case, a

loosely coupled system might in effect be stiffer by requiring a

greater fluid pressure at some local point to produce the same

amount of force transmitted at another local point (such as directly

over the hair cells). Thus the observed disorganization of the TM

structure that led other researchers to think about a decrease in

longitudinal coupling might increase the lumped parameter spring

constant.

Phase of the Cochlear Amplifier

To amplify BM vibrations, OHC electromotility must be

properly timed in the vibration cycle [26]. Nilsen and Russel

[27] discuss this and conclude amplification occurs at the

maximum velocity of the OHC cycle, which occurs a quarter

cycle after the maximum displacement. Chadwick [13] found a

factor of i was needed for correct timing in his model. Examining

Fig. 2, reveals the maximum gain in this model occurs when the

RL displacement is nearly a quarter cycle different from the BM,

corresponding to these experimental and model results.

In general, we assumed the phase of action of the cochlear

amplifier was zero. That is, a positive shear of OHC stereocilia

causes a contraction at the same point in the phase cycle without

any sort of delay [28]. However, since in vivo factors including local

damping from surrounding tissues and slow electrical response

times may introduce delay, we investigated altering the phase of

the input-output relation. The maximum velocity in these

scenarios, shown in Fig. 3 is smaller, but interestingly the best

frequency is also shifted. Even with no phase shift, our model

shows peak activity at higher frequency than the passive velocity

Figure 2. Demonstration of the frequency response on TM

stiffness. Vibrations calculated at 1.4 mm and stiffness are given per

length in units of N/mm2. Fig 2a: As stiffness increases, velocity peaks

decrease and narrow. Fig 2b: Gain (in the BM mode) calculated as the

quotient of the active displacement over the passive displacement

peaks at progressively higher frequencies with increasing stiffness.

Vertical lines correspond to maximum gains. Fig 2c: The phase

difference between the shear vibrations and the BM vibrations in the

BM mode undergoes a dramatic shift from out-of-phase to in-phase at a

frequency that is strongly dependent on TM stiffness. This shift seems

to correspond well with best gain the beginning of the active force

peak.

doi:10.1371/journal.pone.0085969.g002

Table 2. Quality factors and sensitivities for different TM

stiffnesses.

TM stiffness (N/mm2)Q Maximum Velocity (mm/s)

0.052.4 7.261024

0.072.85.061024

0.093.42.961024

0.11 3.71.561024

All values calculated at z~1:4 mm.

doi:10.1371/journal.pone.0085969.t002

Figure 3. Illustration of the shift in best frequency caused by a

shift in the phase of the IO function. The vertical dotted line

indicates the best frequency of the passive response.

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Phase of Shear Vibrations in Cochlear Partition

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maximum, consistent with experiments that show the observed the

best frequency shifts for very high input sound pressure levels

(SPL) [29]. This suggests that phase differences in the amplifier

could affect tuning as well as sensitivity.

Reticular Lamina Angle

By introducing the shear angle into the model, we gained a

means to examine the role of this quantity which is known to grow

from base to apex [14,30]. While the angle has little effect on the

passive BM vibration, a greater angle leads to much greater

amplification as shown in Fig. 4. This is due to the larger shearing

motion on the stereocilia, which is the input to the active force. Eq.

(13) shows the wavenumber correction is directly proportional to

this number. The larger angle couples more of the fluid force

directed perpendicular to the partition into the shear direction,

giving rise to a larger active force. Since this angle increases from

base to apex, it may help aid amplification of low frequencies that

must be carried far into the cochlea. This echoes work that

suggests low frequency sound perception is enhanced at the apex

by mechanics that selectively promote shear in this region [31].

Conclusions

Our model offers an intriguing possibility to explain the

simultaneous increase in selectivity with loss of sensitivity of the

Tectb mutant - a band pass filtering mechanism, with the low

corner being determined by the TM-BM interaction. This idea

lends further support to previous studies that suggest the properties

of the TM are important to overall cochlear tuning [8,32,33] and

specifically echoes work that suggests the TM-RL movement can

act as a filtering mechanism [7,11]. This work provides an in

depth description of the mechanics, and adds nonlinear cochlear

activity.

There is a the growing body of evidence that the TM is a critical

structure in the precise nature of mammalian hearing [34],

especially frequency filtering. Working with the TM in isolation

can provide valuable insight into its mechanical properties, though

we emphasize that the results we get with this model require the

components be assembled as a system. We have demonstrated that

TM properties affect a transition in the phase of the hair bundle

shear that limits the effectiveness of OHCs in amplifying BM

vibrations at low frequencies, increasing selectivity.

Supporting Information

File S1

TM damping. More TM damping leads to a smoother curve,

but does not significantly shift the peak or the frequencies of the

phase transition. Sensitivity increases with selectivity. Figure S2,

Frequency responses due to change in OC stiffness. In

this case, increased coupling between the TM and BM leads to a

sharper, bumpier curve, and sensitivity increases with selectivity.

Figure S3, Frequency responses due to change in BM

stiffness. Changing the BM stiffness leads to a trade in sensitivity

for selectivity. In this case, we can observe that the peak frequency

increases with BM stiffness, as expected, while the phase difference

remains the same. This is compatible with our band pass filter

description of the tuning mechanism, but in this case we are

moving the high frequency corner. Figure S4, Diagram of the

three masses of the cochlear partition and the forces on

them per unit length. Arrows are intended to depict direction

of force due to positive displacements. The direction of VT

depends on TM position. Table S1, Viscoelastic Parame-

ters. Table S2, Structural Dimensions.

(PDF)

Figure S1, Frequency responses due to change in

Acknowledgments

We would like to thank D. Manoussaki for her advice on certain

mathematical points, and for critical comments. We also thank S. Smith for

critical comments.

Author Contributions

Conceived and designed the experiments: JL RC. Performed the

experiments: JL. Analyzed the data: JL RC. Contributed reagents/

materials/analysis tools: JL RC. Wrote the paper: JL RC.

References

1. Ashmore J (2008) Cochlear outer hair cell motility. Physiol Rev 88: 173–210.

2. Hudspeth AJ (2008) Making an effort to listen: mechanical amplification in the

ear. Neuron 59: 530–45.

3. Ruggero MA, Rich NC, Recio A, Narayan SS, Robles L (1997) Basilar-

membrane responses to tones at the base of the chinchilla cochlea. J Acoust Soc

Am 101: 2151–2163.

4. Hemmert W, Zenner HP, Gummer AW (2000) Three-dimensional motion of

the organ of corti. Biophys J 78: 2285–97.

5. Chen F, Zha D, Fridberger A, Zheng J, Choudhury N, et al. (2011) A

differentially amplified motion in the ear for near-threshold sound detection. Nat

Neurosci 14: 770–4.

6. Steele CR, Taber LA (1979) Comparison of wkb and finite difference

calculations for a two-dimensional cochlear model. J Acoust Soc Am 65:

1001–6.

7. Allen JB (1980) Cochlear micromechanics-a physical model of transduction.

J Acoust Soc Am 68: 1660–70.

Figure 4. Dependence of vibration on RL angle. Green lines

indicate passive shear velocity, blue passive BM velocity, and red active

BM velocity.

doi:10.1371/journal.pone.0085969.g004

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8. Ghaffari R, Aranyosi AJ, Freeman DM (2007) Longitudinally propagating

traveling waves of the mammalian tectorial membrane. Proc Natl Acad Sci USA

104: 16510–5.

9. de Boer E (1990) Wave-propagation modes and boundary conditions for the

Ulfendahl-Flock-Khanna preparation. In: Dallos P, Geisler CD, Matthews JW,

Ruggero MA, Steele CR, editors, The Mechanics and Biophysics of Hearing,

Berlin: Springer-Verlag. 333–339.

10. Lamb JS, Chadwick RS (2011) Dual traveling waves in an inner ear model with

two degrees of freedom. Phys Rev Lett 107: 088101.

11. Zwislocki JJ, Kletsky EJ (1979) Tectorial membrane: a possible effect on

frequency analysis in the cochlea. Science 204: 639–41.

12. Russell IJ, Legan PK, Lukashkina VA, Lukashkin AN, Goodyear RJ, et al.

(2007) Sharpened cochlear tuning in a mouse with a genetically modified

tectorial membrane. Nat Neurosci 10: 215–23.

13. Chadwick RS (1998) Compression, gain, and nonlinear distortion in an active

cochlear model with subpartitions. Proc Natl Acad Sci USA 95: 14594–9.

14. Keiler S, Richter CP (2001) Cochlear dimensions obtained in hemicochleae of

four different strains of mice: Cba/caj, 129/cd1, 129/svev and c57bl/6j. Hear

Res 162: 91–104.

15. Gavara N, Chadwick R (2010) Noncontact microrheology at acoustic

frequencies using frequency-modulated atomic force microscopy. Nature

Methods 8: 650–4.

16. Brownell WE, Bader CR, Bertrand D, de Ribaupierre Y (1985) Evoked

mechanical responses of isolated cochlear outer hair cells. Science 227: 194–6.

17. Elliott SJ, Ni G, Mace BR, Lineton B (2013) A wave finite element analysis of the

passive cochlea. The Journal of the Acoustical Society of America 133: 1535–

1545.

18. Watts L (2000) The mode-coupling liouville-green approximation for a two-

dimensional cochlear model. J Acoust Soc Am 108: 2266–71.

19. Elliott S, Ni G, Mace B, Lineton B (2011) How many waves propagate in the

cochlea? In: AIP Conference Proceedings. volume 1403, p. 563.

20. Chadwick RS, Dimitriadis EK, Iwasa KH (1996) Active control of waves in a

cochlear model with subpartitions. Proc Natl Acad Sci USA 93: 2564–9.

21. Saunders JC, Summers RM (1982) Auditory structure and function in mouse

middle ear: an evaluation by sem and capacitive probe. J Comp Physiol 146:

517–25.

22. Dong W, Olson ES (2013) Detection of cochlear amplification and its activation.

Biophys J 105: 1067–78.

23. Ghaffari R, Aranyosi AJ, Richardson GP, Freeman DM (2010) Tectorial

membrane travelling waves underlie abnormal hearing in tectb mutant mice.

Nat Commun 1: 96.

24. Lukashkin AN, Richardson GP, Russell IJ (2010) Multiple roles for the tectorial

membrane in the active cochlea. Hear Res 266: 26–35.

25. Meaud J, Grosh K (2010) The effect of tectorial membrane and basilar

membrane longitudinal coupling in cochlear mechanics. J Acoust Soc Am 127:

1411–21.

26. Russell IJ, Kossl M (1992) Sensory transduction and frequency selectivity int the

basal turn of the guinea-pig cochlea. Phil Trans R Soc Lond B 336: 317–24.

27. Nilsen KE, Russell IJ (2000) The spatial and temporal representation of a tone

on the guinea pig basilar membrane. Proc Natl Acad Sci USA 97: 11751–8.

28. Santos-Sacchi J (1992) On the frequency limit and phase of outer hair cell

motility: effects of the membrane filter. J Neurosci 12: 1906–16.

29. Robles L, Ruggero MA (2001) Mechanics of the mammalian cochlea. Physiol

Rev 81: 1305–52.

30. Dallos P, Popper AN, Fay RR (1996) The Cochlea, volume 8 of Springer

handbook of auditory research. New York: Springer.

31. Manoussaki D, Chadwick RS, Ketten DR, Arruda J, Dimitriadis EK, et al.

(2008) The inuence of cochlear shape on low-frequency hearing. Proc Natl Acad

Sci USA 105: 6162–6.

32. Allen JB, Fahey PF (1993) A second cochlear-frequency map that correlates

distortion product and neural tuning measurements. J Acoust Soc Am 94: 809–

16.

33. Zwislocki JJ (1986) Analysis of cochlear mechanics. Hear Res 22: 155–69.

34. Richardson GP, Lukashkin AN, Russell IJ (2008) The tectorial membrane: one

slice of a complex cochlear sandwich. Curr Opin Otolaryngol Head Neck Surg

16: 458–64.

Phase of Shear Vibrations in Cochlear Partition

PLOS ONE | www.plosone.org7 February 2014 | Volume 9 | Issue 2 | e85969