1973 - billone, raynor - transmission of radial shear forces to cochlear hair cells

8/16/2019 1973 - Billone, Raynor - Transmission of Radial Shear Forces to Cochlear Hair Cells

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Transmission of radial shear forces to cochlear hair cells

M. Billone

Argonne National Laboratory, Argonne, Illinois

S. Raynor

Northwestern University, Evanston, llinois

(Received 26 June 1973)

The radial shear orces transmitted o the cuticular plates of cochlear sensorycells are investigatedby means

of a mathematical model. The model is based on the fine anatomy of the hair cells and their supporting

structures. t predicts hat the inner hair cells are stimulatedby a viscous orce which is linearly proportional

to, and in phasewith, basilarmembranevelocity. The outer hair cells are stimulatedby a shear force which

is linearly proportional o, and n phasewith, basilarmembranedisplacement t low frequencies e.g., ess han

700 Hz for a midbasal turn cell). At higher frequencies, he shear force transmitted to the outer hair cells is

a function of both displacementand velocity. This outer hair-cell shear force is at least an order of magnitude

greater than the corresponding nner hair cell force for the frequenciesand cochiear positionsstudied. These

results compare favorably to recent cochlear microphonic data from normal and kanamycin-treated guinea

pigs for frequencies ess than 4000 Hz. Shear force amplitude envelopesare presented or 800- and 1600-Hz

pure tones. The results suggest hat the "velocity-sensitive" inner hair cells are candidates for "place-

mechanism" receptors while the outer hail cells are candidates or "frequency-mechanism" eceptors. The

model predicts no mechanical fine tuning in the force transmission.

Subject Classification:4.2.3, 4.2.2.

INTRODUCTION

While the transmission f sound rom pressurewaves

in the atmosphere o traveling displacementwaves

along the basilar membrane s understood easonably

well, much less is known about the mechanismsby

which membrane vibration stimulates the cochlear

sensorycells. The literature contains relatively few

conceptual r quantitative modelswhich describe his

process. elmholtz (1954) and Crane (1966) suggested

that basilar membrane motion causes the sensory

hairs (cilia) to strike against he tectorial membrane,

thereby generating ressurempulseswhich are trans-

mitted to the cell's eceptor ole.Hugginsand Licklider

(1951) hypothesizedhat the cilia experience lternat-

ing tension nd compressionorces ather than mpulses.

Thesemodelsdiffer primarily in their treatment of the

spatial relationship etween he cilia and the tectorial

membrane. After the pioneering work of B•k•sy

(1951, 1953a, 1953b), the focus shifted to shearing

forcesand displacements cting on the receptorpole

of the sensory ells.Khana etal. (1968)useda quantita-

tive model o study ongitudinaP heardisplacement f

the cell surface relative to the basilar membrane.

Johnstonend Johnstone1966)andRhodeand Geisler

(1967)proposed eometrical odelso representadial

displacement f the tectorial membrane elative to the

cell's cuticular surface.

The results of B•k•sy's experimentssuggest hat

both radial and longitudinal shear may be important

mechanical timuli to the hair cells.A workinghypoth-

esis in this article is that the radial shear mode is the

significant one for stimulation--at least as far as

cochlear microphonicgeneration is concerned.This

choices motivatedby the resultsof Wersalland Flock's

(1967)morphologicaltudyof the directional ensitivity

of hair cells and somepreliminary model calculations

(Billone, 1972)which ndicate hat radial displacements

are much arger than longitudinaldisplacements.

The purpose of this investigation s to proposea

mathematical model to describe quantitatively the

radial shear orces which are transmittedby the cilia

to the cuticularplate of a sensory ell. Davis (1965)

hypothesizeshat shear forcesacting on the receptor

pole of the hair cell causea change n the resistance

(to ion flow) of somesensitive egionon the cell surface.

Engstromet al. (1962) speculate hat the "essentially

excitable structure" is in the cuticular-free-region

(CFR). The shear force on the dense cuticular plate

is assumed o move the plate against he softer cuti-

cular-free-region.n this manner, the resistanceof

the region r some lementwithin the region s changed.

Thus, calculating he shear orces ransmitted o the

cuticular late s consistent ith the conceptual odels

of hair cell stimulation suggestedby Davis and

Engstrom. •

Two fundamental uestions hich his modelstudy

attempts o answerare: (1) Does a mechanicaline

tuningoccur n the stimulation f the hair cellswhich

couldaccount or the acutesensitivityof the auditory

system o changesn frequency?2) Are therediffer-

ences n the mechanical nputs to inner versusouter

hair cells which suggest unctional differences?

sensitive ar candetecta 0.2% changen frequencyor

The Journalof the AcousticalSocietyof America 1143

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BILLONE AND RAYNOR

a 2000-Hz tone. The cochlearmechanics nvestigated

thus far (i.e., cochlear artition vibration) for low and

intermediate frequencieshave revealed that a gross

frequencyanalysisby "place" occurs long he cochlear

partition) This analysis annotaccount or the auditory

system's harp requency esolution t these requencies.

The motivation for the secondquestion s the broad

dynamic range of the auditory system. t seems ikely

that more than one mechanism is involved in transmit-

ting frequency nformation o the brain.

I. MODEL DESCRIPTION

The basic features of the model for mechanical

stimulation of inner and outer hair cells are shown in

Fig. 1. The input to the system s assumed o be the

radial shear motion (s) of a point on the tectorial

membrane TM) relative to an opposing oint on the

reticular membrane (RM). The response f interest

is the shear orce (f) actingon the cuticularplate of

the cell. The plate is treated as a rigid body which is

tightly held in place by its stiff attachment to the

reticularmembrane Spoendlin,.966). n quantitative

x

s

[///////////////////////,dr,

11 la -tLn II II ENDOLYMPH

II Ii

....__H•__•-CFR

x

Fro. 1. Radial viewof shear orcemodel or inner (a) andouter

(b) hair cells. Vibration of the tectorial membrane shears the

embeddedOHC cilia directly and the free OHC and IHC cilia

throughhe medium f the viscousndolymph.he cilia ransmit

these orces o the cuticularplate (CP).

terms, his assumptionmplies hat the motionof the

cuticularlate elativeo RM is small omparedo s.

The cilia in the modelare cylindrical antilevered

beamsfuniformrossectionndmodulusfelasticity.

The ciliabeams mergerom heirbuilt-insupport t

the cuticular late into the viscous ndolymphluid

which fills the spadebetweenTM and RM. None of

the inner hair cell (IHC) cilia make contact with the

tectorialmembrane. hesecilia experience viscous

drag inducedby the movement f the endolymph

fluid. The outerhair cell (OHC) modeldiffers rom this

in that the tallest row of cilia are embedded in shallow

groovesn the tectorialmembrane Engstrom t al.,

1962 and Kimura, 1965). This TM contact s suitable

for transmittingshear orces o the tall cilia without

exerting ny axial forces r moments. he remaining

rows of OHC cilia are not connected to the tectorial

membrane.

The IHC cilia are arrangedn three ongparallel

rowswith 10 to 20 cilia per row. While all IHC cilia

are assumedo have the samediameter, he height

of each ow increasesrogressivelys the position f

the row approacheshe cuticular-free-regionCFR).

The center-to-center distance between cilia in a row

is approximatelyhree radii. This value s alsoused or

the center-to-center distance between rows of cilia.

The average eightof cilia per cell ncreasesrogres-

sivelywith the cellspositionalong he cochlearom

stapes o helicotrema.

The model or OHC ciliary arrangements similar o

the one for IHC cilia. Three exceptionsre that (1)

the number of rows per cell varies from three to six

with 20 to 40 ciliaper row dependingpon he species

and the locationalong he cochlea; 2) the numberof

cilia per cell decreasesrom stapes o helicotrema;

and (3) the center-to-centerpacings less han three

OHC radii. A more quantitative treatment of the

modelparameterss presentedn Sec. II.

Although a cilium is analyzed as a beam with

constant eometricalndmaterial roperties,t actually

tapers down to about half of its maximum diameter

(seeKimura),andbecomes oredense s t approaches

the cuticularplate. The stiff rootlet which anchorshe

cilium o the cuticular late continuesp the centerof

the ciliuma shortdistance dding o both the density

and the stiffnessf the neckportion.The stiffnesser

unit lengthof a beam n bendings proportionalo the

productof the fourth powerof the radius a4) and the

modulus f elasticity E). While the decreasingadius

tends o weakenhe ciliumneck, he ncreasing odulus

of elasticity ends o strengthenhe neck egion. o a

first approximation,he productof these wo effectss

assumedo be constantalong the axis of the cilium

(i.e., Ea4= constant).

Anothersimplification hich has been ncorporated

into the model s the assumptionhat OHC cilia are

arrangedn straight ows. t is well establishedhat the

1144 Volume 54 Number 5 1973



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TRANSMISSION OF RADIAL SHEAR FORCES

OHC cilia form a "W" pattern. While the shapeof the

row may effect the transmission f viscous orces, t

hasno effecton the shear orces ctingon the embedded

tips of the tall OHC cilia. Thus, it is a goodassumption

at low frequencies here the viscous orces ransmitted

to the outerhair cellsare negligible seeSec. V-B).

II. SHEAR FORCE ANALYSIS

For pure-tonestimulation applied at the eardrum,

the radialvibrationof a point on the tectorialmembrane

relative to an opposing oint on the reticularmembrane

may be expressed s

s=S cosCt+•), (1)

where S=S(x;o•), •=•(x;o•), x is the longitudinal

positionalong the cochlea, is the time, and o• s the

stimulus requency. he phase •) is measured elative

to the displacement f the stapes.This radial shear

oscillation nducesmotion in both the endolymph

fluid and the ends of the tall OHC cilia which are in

direct contactwith the TM. A free cilium experiences

a viscous rag per unit lengthwhich s proportional o

the product of the viscosity (t•) and the velocity

difference between itself and the fluid. An embedded

cilium is stimulated primarily by the shear force

acting on its TM contact point. In each case, the

cilium transmits he shear orce to the cuticularplate

in which it is rooted.

A. Shear Force Transmitted by a Free Cilium

The problem of computing he viscousdrag trans-

mitted to the cuticularplate by a free cilium s divided

into three parts:

(1) The calculation f the fluid velocityprofile (V)

between the tectorial and reticular membranes far

away rom the cilia [seeFig. 2(a)-].

(2) The computation f the dragper unit lengthon a

rigid ciliumwhich s a memberof an array of cilia and

is exposedo a freestream elocity V) [seeFig. 2(b)•.

(3) The evaluation of the shear force transmitted

to the cuticularplate by a flexiblecilium beam which

is in a viscouslow ield [-see ig. 3(a)•.

The flow is assumedo be incompressiblend laminar.

Also, because he cilia are long compared o their

diameter nd the relativeverticaldisplacementetween

the two membraness small, the fluid velocity n the

verticaldirection z) is neglected.

1. Fluid Flow Between the Tectorial and

Reticular Membranes

Far away from the cilia, the flow between TM and

RM is assumed o be an oscillatingCouette flow

[seeFig. 2(a)•. The governingquation ndboundary

v L

y

•y

0 0 0.•_2. : :

•uo00 : _-

000 _- :

000 : ,

000

000 : :

(b)

Fro. 2(a). Viscous flow between the tectorial and reticular

members. b) Flow in the xy plane for the IHC shear orcemodel.

Far away from the cilia, the velocity (V) is uniform n spaceand

oscillating n time. In the neighborhood f the cilia, the flow has

componentsu,v).

conditions for this flow field are

pO / Ot= t•OV/ Oz , (2a)

V(O,t)=0, (2b)

V(L,t) =•, (2c)

where

•=So• cosCt+•+«•r), (2d)

and p is the density of endolymph. The solution to

Eq. 2a is given by Lamb (1945). In the caseof slow

viscous low (i.e., po•L2/t•


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BILLONE AND RAYNOR

(u,v). As the cilia are treated as infinitely long n this

analysis, o motion n the z direction s induced.

The number of cilia in a row is considered to be

infinite. The drag analysissimplifiesgreatly in this

casebecause he fluid streamlines re symmetricabout

the y axis of each cilium and the drag is the sameon

all of the cilia in a row. This assumptions particularly

reasonable for IHC cilia. While each cell carries about

20 cilia per row, the cellsare lined up so close ogether

that a row is effectively as long as the cochleaand

includes thousands of cilia.

The continuity equation or an incompressiblelow

with no motion n the z direction s givenby

Ou/Oxq-Ov/Oy=O. (4)

Stokesequations seeLandauand Lifshitz, 1959) for

slow viscous low are a good approximation o the

momentum quationsor flow through owsof cilia5'

• (02u/Ox2q-O2u/Oy2)op/Ox, (5)

• (O2v/Ox•+O%/Oy) = Op/Oy, (6)

where p is the pressure.The boundaryconditions n

the xy plane for this problemare

u=v=0, on all cylindersurfaces, (7a)

0, as lyl (7b)

v, lyl (7c)

Miyagi (1958) solvedEqs. 4-7 for the velocity ield

througha single ow of infinitely ongcircularcylinders.

He found that the drag per unit length on a cylinder

in a single ow is given by

Dv= S•rcl• , (8a)

where

c=[1-2 log(2r)+ (2/3)r •-- (1/9)r4+ (8/135)r6

-- (53/1350)rS+ (1112/42 525)r1ø

--(241 643/13 395 375)r1=

+(18 776/1488 375)r14... •-1. (8b)

The expressionor c in Eq. 8b is applicableor 0 < a/q


5/14


where

k4= 8rct•oo/(EI). (17b)

The general olution o Eq. 7 is

•=S Real{•z/L-t-B•sin(kilz)-t-B2os(kilz)

+B3 sinh(kilz)+B4cosh(kilz)eg•t},

(18)

where i = V'- 1.

The constants (B•,B2,Ba,B4) are determined from

the four boundaryconditionsor this problem.At the

built-in end of the cilium, the deflectionand slope are

zero while the moment and shear are zero at the free

end. Mathematically, theseconditions re

r/(0,t)= 0, (19a)

0r/

--(0,t) =0, (19b)

Oz

EI•(l,t) =0, (19c)

Oar/

--EI•(l,t)=O. (19d)

Oza

Solving or the constantsn Eq. 18 and substituting

these esults nto Eqs. 12 and 14 yieldsan expression

for the viscous hear orce (fv) transmittedby a free

cilium'

fv= F•S coso•t+•+ •), (20a)

where

F•= 4rc•ooL F•* [

qv= Phase{F•*},

2il2(kli•)2 sinh(klil)sin(klil)

L2•l +cosh(klil) cos(kill)

(20b)

(20c)

(20d)

For kl


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BILLONE AND RAYNOR

ment (S)'

f,= (3EI/la)S cos(wt-+-•).

(23)

C. Total Shear Transmitted to a Hair

Cell Cuticular Plate

1. Inner Hair Cells

The shear force (f•) transmittedby cilia to the

cuticularplate of an inner hair cell is calculated y

addingup the forcecontribution f eachcilium.As all

of the IHC cilia are free, the force transmitted by

each HC cilium is given by Eq. 20. Summing hese

forcesgives

fl=F•S1 cos(wt+•+•l), (24a)

where

3

Fi= (4•rclucoLiT1/3)E F•n*[, (24b)

3

Phase{ F•n*}, (24c)

2il•'(klll•il) -•' sinh(klll•i ) sin(klll•il)

Fin= , (24d)

Ll•'[-1-}-cosh(kd•il) os(kll•il) ]

k 1= [8•rcluCo/(EI1)']-:, (24e)

and n is the cilia row number with the shortest row

labeled 1), the middle ow (2), and the tallestrow (3);

l• is the length of a cilium in the nth row; T• is the

total number of IHC cilia per cell. At frequenciesow

enougho justify the rigid ciliumassumptionk / •


7/14


position x) in AppendixA for a modelsimilar o the

one usedby Rhode and Geisler.

B•k•sy (1949) measuredvolumedisplacement n-

velopes A) in the human or severalow frequencies.

These envelopes re redrawn n Figs. 4 and 5 as a

functionof normalized osition x•) for 800 and 1600

Hz. While no envelopesor the guineapig have been

measured,B•k•sy (1960) did determine the position

of maximum cochlearpartition motion in the guinea

pig as a functionof frequency. is data showshat the

peak positionsor 800- and 1600-Hz onesare approx-

imately the same ractionaldistance long he cochlea

for the guinea ig andthe human.As a first approxima-

tion, it is assumedhat the human envelopesor these

two frequencies ay be used or the guineapig with

position xpresseds a fraction x•) of partition ength.

To convert volumedisplacement nvelopeso mid-

point displacementnvelopes,he following elation-

ship basedon Allaire's 1972a,b)beam modelof the

basilarmembranemay be used'

D= 1.67A/w (29)

whereA is the volumedisplacement/unitengthand

w is the width of basilarmembrane.The displacement

envelopesD) plotted n Figs.4 and 5 are calculated

from Eq. 29 by using B•k•sy's data for A and the

guinea ig data or w (seeAppendix ).

The radial shearenvelopesor 800 and 1600 Hz are

shownn Figs.4 and 5. It is interestingo note hat the

envelopesor S and D are flatter than the ones or A.

A sharpeningf these urveswouldbe moreconsistent

with the PlaceTheory of frequency nalysis.

Rossi (1914) and Vilstrup et al. (1955) measured

valuesof viscosity rom 2.9 cp at 20øC to 1.7 cp at

25øC or the shark'sendolymph.A reasonable stimate

for mammalianendolymph t 37øC s

ju= cp=0.0 dyn-sec/cm'.

Engstrom t al., observehat the cilia "standupright

like stiff bristles or fine rods." As there are no measure-

ments reported n the literatureof the mechanical

propertiesf cochlearilia, HC andOHC shear orces

are calculatedor a rangeof Young'smoduli

108 E < 1011 yn/cm '.

Thisrangencludes iological aterials uch scartilage

(• 108)motilecilia n nonauditoryystems109 o 1011)

and bone (1011).Shear force amplitude and phase

results re comparedo cochlearmicrophonic ata in

Sec. V to determinehe approximatealueof E which

leads to shear force response onsistentwith micro-

phonic esponse.

The geometricalarametersa,q,N,J) are assumed

to be constantalong he length of the cochleawhile

the parameters T,L,1,) are allowed o vary linearly

with position.Numericalvalues or theseparameters

are listed n Table I alongwith values or the material

o 20 40 60 80 IOO

COCHLEAR POSiTiON (Xp), ø/o

Fro. 4. Normalized volume displacementenvelope (A/Am,,O

adapted from Bdkdsy (1949) for an 800-Hz tone. Calculated

displacement nvelopesor midpointbasilarmembranedisplace-

ment (D) and radial sheardisplacementS).

properties. They are based on the qualitative and

quantitative anatomical work of Engstrom et al.,

Kimura, and Spoendlin.

IV. RESULTS

A. Shear Force Transmitted by a Single Cilium

The degree o which the free cilium force deviates

from a simple velocity-dependent orce can be seen

by comparinghe solutionsor a flexible ree cilium and

a rigid free cilium. Let Rv be the ratio of flexiblecilium

force (Eqs. 20a-20d) to rigid cilium force (Eq. 13) and

let 0vbe the phasedifference.Then

I g,*l, (30a)

Phase{ ,* } -- -}•r, (30b)

where

2i(klil)-•. sinh(klil)sin(klii)

R,* = . (30c)

l +cosh(klil) cos(kilt)

Allowing or the variationsn the dynamicand geomet-

ric parameters, long with the large uncertainty n

I'- 1.0

z

• o.$

• 0.4.

Z 0.2

I I I i I ' I

IGO0 Hz

_ D/DMAx--•

_s's,,,x-,,//

//

/'";• • I , I "•,

0 40 60 80

COCHLEAR POSITION (Xp), ø/o

Fro. 5. Normalized olumedisplacementnvelope A/Am,,•)

adapted rom Bdk•sy (1949) or a 1600-Hz one. Calculated

displacementnvelopesor midpoint asilarmembraneisplace-

ment (D) and radialsheardisplacementS).

The Journalof the AcousticalSocietyof America 1149



8/14

BILLONE AND RAYNOR

TA,.v.I. Valuesof parameters sed n shear orcecalculationsor the guineapig.

Parameter IHC (1) OHC (3)

SheardisplacementS) SeeFigs.4 and5 SeeFigs.4 and5

Endolymphiscosityu) 0.01dyn-sec/cm" 0.0 dyn-sez/cm"

Young'smodulusE) 108 E < l0 n dyn/cm" 108 E < 10 dyn/cm"

Number of cilia per cell (T) 40 120-60Xp

Number of rows of cilia (N) 3 3

Number of free rows 3 2

Cilium radius (a) 0.15X 10 4 cm 0.12X 10 4 cm

Center-to-center 0.5X 10 4 cm 0.3X 10 4 cm

cilia spacing q)

TM-RM spacingL) SeeAppendix (4Xpq-l.6)X 0 4 cm

Cilium lengths l) /n=0.5/•a /a•--0.5/aa

l•.= 0.75/•a la•.= 0.75/3a

/•a= (4xv-}-2)X 10 4 cm laa= (4xvq-2)X 10 4 cm

Source

oo.

Rossiand Vilstrup et al., for Shark

To be chosen to match CM data

Engstromet al., Kimura, and Spoendlin

Engstromet al., Spoendlin

Estimated

Kimura's micrographs f guinea pig cilia

Kimura's micrographs f guineapig cilia

Kimura's squirrel monkey data

Kimura's squirrel monkey data and

Spoendlin's at data

Young'smodulusor cilia, the rangeof possiblealues

for kl is

0.01 < kl< 10.

Figure6 showshe changen amplitude atio andphase

deviationas kl varies. The rigid cilium approximation

for free cilia is a very goodone for kl_


9/14


IO

'40(•.01

[ • [

0.02 0.5 I

k•

o I ' ' I ' [ ''l I ' [ I ' ' ''l

[ [ [ iO -tRANGEFMMPLITUDEATAORUINEAI65

OT INDICATES AVERA6E OF OF :5 SETS OF DATA (DALLO•, 1972)

-20 - --SHEAR FORCE ODELREDICTIONOREn 5x 909dyltel/cm

\

\\

-I-O

•vA._/I/

FIG. 6. Forceamplitude atio (Rv) and phasedifference0v) or

flexible ree cilium as compared o a rigid-freecilium.

As thereare approximatelyour OHC to everyone HC,

CM may be written as

Cm = H (4 3+ f O. (33)

The microphonic mplitude atio and phasedifference

recordedby differential electrodesn Dallos's second

set of measurements are related to the calculated

shear orcesby

I CMll/I CMI--Ifll/14fa+f11, (34)

Phase{CM •} --Phase{CM} = Phase{ •}

--Phase{4faq-f•}. (35)

The shear orces f•,fa) are evaluated s a functionof

frequencyor Young'smoduli n the range10a< E < l0 n

dyn/cmh The value which yields shear force results

which behavevery similarly to microphonic ata is

E = 3 X 10ødyn/cmh

The solid ines n Figs. 8 and 9 representhe calculated

results or this value of Young'smodulus seeSec.V).

4O

'1%.Ol 0.02

/ "7

,'/

I I I I I-•o

0.5 I 2 5 IO

k.•

Fro. 7. Forceamplitude atio (Re) and phasedifference 0e) or

an embedded ilium in a viscousmediumas compared o one n

an inviscid medium.

FIG. 8. Comparison between amplitude ratio of cochlear

microphonicdata from kanamycin-treated o normal guinea pigs

and the calculated ratio based on the shear force model.

In the previous subsection, t is shown that a free

cilium acts like a rigid rod in a viscous low field for

kl_


10/14

BILLONE AND RAYNOR

at x-4 mm. The above limiting frequencieswill be

smaller for positions arther from the stapes. The

limiting requency aries nversely s the fourthpower

of the cilium length (see Eq. 36) which increases

with distance rom the stapes.

C. Shear Force Envelopes

The distribution of shear forces on the IHC cuticular

plates s shownn Fig. 10 for 800 and 1600Hz. As the

simplevelocityrelationship Eq. 14) is valid for the

frequencies nd positionsplotted, the results are

independent f Young'smodulus.The shapeof these

curves is approximately the same as the volume

displacementnvelopes easured y B•k•sy (seeFigs.

4 and 5). Thus, while the geometric actorscontained

in G(x) tend to amplify displacements n the stapes

sideof the maximum, the IHC shear orce ransmission

counters this effect.

Figure 11 shows he distributionof OHC shear

forces or 800 and 1600 Hz. The curvesare relatively

flat between he positionof maximumbasilarmembrane

displacement nd a positionabout 10% from the

stapes.The 800-Hz curve appears o have two peaks:

a largerpeakat a position f about10% anda slightly

smaller peak at about 55%. The 1600-Hz envelope

has a peak at about 50% which correspondso the

positionof maximum membranedisplacement. ow-

ever, in both cases he envelopes ppear to lack a

distinct sharppeak. The OHC shear orce eads he

basilarmembrane isplacementy 10ø to 50ø in the

regionbetween •-10% and the position f maximum

membrane isplacement.his result ndicateshat the

viscous orces ransmitted to the OHC are significant

at these requencies.

The force transmittedby an embedded ilium is

inversely roportionalo the third powerof the cilium

length. Because he cilia decreasen height as they

approachhe stapes, here s a largegain due to this

geometricactor. f largervaluesof Young'smodulus

are used n the calculations E•10 n dyn/cm2), the

,oo,

ß . o.6-

0 \\.• l

0 2o 4o 60 8o

COCHLEAR POSITION (Xp), '/.

Fro. 10. Calculated IHC shear force envelopes or 800- and

1600-Hz tones. Results demonstrate that inner hair cells may

functionas place eceptors.Arrows ndicatepositions f maximum

basilar membranedisplacement.

Fro. 11. Calculated OHC shear force envelopesor 800- and

1600-Hz tones. Results demonstrate that outer hair cells are

poor place receptors,but that they may function as "frequency"

receptors.Arrows indicate positionsof maximum basilar mem-

brane displacement.

resultingcurveshave a singlepeak near the stapes.

Smaller Young's moduli (E•108 dyn/cm ) tend to

maintain the peak at its original position but to

flatten out the stapedalsideof the envelope.

V. DISCUSSION

Two fundamental modeling assumptions re that

(1) the IHC cilia are free from contactwith the tectorial

membrane nd (2) the radial shear orce s the signif-

icant mechanicalnput to the hair cells.The CM results

cited in Sec. V-B support he assumption oncerning

IHC cilia. Also,becausehe radial shear orcespredict

IHC and OHC microphonics hich are consistentwith

the experimentalresults, it appears that the radial

mode is significant or CM generation.This doesnot

negate he possibility hat the longitudinalmode s im-

portant n the generation f otherelectrical ignals e.g.,

the summating otentialas suggestedy Davis, 1960).

A Young's modulusof 3X100 dyn/cm ' is used to

bring the shear force amplitude and phasevariations

into agreementwith the cochlearmicrophonic ata for

the guinea pig. This value appears o be reasonable

for a fibrous biological material. Young's modulus

estimatesbasedon the performanceof motile cilia in

nonauditory systems ange from 10ø to l0 n (Sleigh,

1962). For the Young'smodulus hosen,he predicted

ratio of IHC microphonic o total microphonic s

within 3 dB of the range of measuredvalues for fre-

quenciesess han 8 kHz (seeFig. 8). The corresponding

phasedifferences re in reasonably oodagreement or

frequenciesess than 4 kHz (see Fig. 9). At higher

frequencies,he microphonic hasedifference ecreases

rapidly from a positive to a negativephasedifference.

One possibleexplanation for the high-frequency

discrepancy between calculated shear forces and

measuredmicrophonicsas to do with the experimental

approach. At low frequencies, he basilar membrane

phase distribution in the region of the electrodes s

reasonablylat. This means hat the microphonic hase

is not strongly ependent nposition t low frequencies.




11/14


However, at higher frequencies,he position of max-

imum vibrationapproaches mm and the phasechanges

very rapidly with position.Because hase data from

one set of animals is subtracted from the data from

another set, large errors could be introduceddue to

slightvariations n the positions f the electrodes.

The amplitude envelopesor the IHC shear force

(seeFig. 10) are very similar o the basilarmembrane

volume displacementenvelopes or the frequencies

investigated.The positionsof the maxima and the

shapesof the curves are approximately the same.

This result implies that the inner hair cells are can-

didates or receptorswhich analyze requencyaccording

to the Place Theory. Unfortunately, the processof

convertingmembrane displacementso shear forces

does nothing to strengthen he Place Theory. The

envelopesbecome increasingly latter at lower fre-

quencies s they do for basilarmembrane isplacement.

The results n Fig. 10 ndicate hat it wouldbe relatively

simple to distinguishbetween an 800-Hz tone and a

1600-Hz tone. However, a good ear can discriminate

between 1600 and 1604 Hz. On the basisof _thisstudy

it appearsunlikely that the sharp requency esolution

propertiesof the auditory system at low and inter-

mediate frequencies re causedby mechanicalevents

within the cochlear duct.

The envelopesor OHC shear orces (see Fig. 11)

are significantlydifferent from the basilar membrane

envelopes.The OHC shear forces are distributed in

such a way that thousandsof cells receive approx-

imately the same large amplitude signal. This result

suggests hat the outer hair cells are inappropriate

frequency receptors according o the Place Theory.

However, the OHC envelope s consistentwith the

Frequency Theory of pitch preception. In the Fre-

quencyTheory, the positionof a cell along he cochlea

is not important. Frequencynformation s assumedo

be transmitted to the brain directly by meansof the

phase-lockediring pattern of neural mpulses.

While the Place Theory is weak at low frequencies,

the FrequencyTheory encounters ifficultiesat high

frequenciesecause f the imitation on he phase-locked

firing pattern of nerve fibers.Weaver (1949) hypoth-

esized that both mechanism's are needed to allow the

ear to operate over such a broad frequency range. If

this is the case, hen the outer hair cellsmay function

as low-frequency eceptorswhile the inner hair cells

functionas high-frequencyeceptors.

VI. SUMMARY AND CONCLUSIONS

(1) The radial shear force transmitted to an IHC

cuticularplate is linearly proportional o, and in phase

with, basilar membranevelocity for low and interme-

diate frequenciese.g., ess han 7000Hz for a midbasal

turn cell). At higher requencies,he flexibilityof IHC

cilia cause decreasen force ransmission nd a phase

lag.

(2) The OHC shear orce s linearlyproportionalo,

and in phasewith, basilarmembrane isplacementt

low frequenciese.g., less han 700 Hz for a midbasal

turn cell). At higher frequencies,he viscous orces

acting on OHC cilia causean increasen amplitude

and a phase ead.

(3) The OHC shear force is at least an order of

magnitude larger than a correspondingHC force.

This leads to a total OHC microphonicwhich is more

than 30 timesas large as the IHC microphonic.

(4) The model redictionsre n good greement ith

cochlearmicrophonicdata for frequenciesess than

4000 Hz.

(5) The shear orceenvelopesuggesthat the inner

hair cells may function as high-frequency place"

receptorswhile the outer hair cells may function as

low-frequencyeceptors. he modelpredictsno spatial

sharpening f the mechanical ignal ransmitted rom

the basalmembrane o the cell'scuticularplate.

ACKNOWLEDGMENTS

The authorswish to thank Prof. Peter J. Dallos for

his cooperationand encouragement. he funds for

this research ereprovidedby Bioengineeringraining

Grant (5T01GM00874-10).

APPENDIX A: RADIAL SHEAR DISPLACEMENT

Rhode and Geisler (1967) and Billone (1972) have

proposed geometrical models for calculating the

amplitude of the opposingpoint radial displacement

(S) as a function of the midpoint basilar membrane

displacement D). Both modelsassume igid bodies

for the organ of Corti and the tectorial membrane.

The primary difference etween he two models s that

the Rhode and Geislermodelassumeshat all opposing

points on the tectorial and reticular membranes are

in contact n the rest positionwhile the Billone model

assumesan initial separation of the two membranes

exceptat the outer tip of the TM whereslidingcontact

is maintained seeFig. A-i). Anotherdifferences that

Allaire's (1972) beam model is used to calculate the

shapeof the deflectedbasilarmembrane or the model

in Fig. A-1 while Rhode and Geisler use a piecewise

linear shape.

The quantitative results for the two models are

similar. The shear displacement per unit basilar

membranedisplacement ecreases onotonically long

the cochlea rom base to apex (3 to « in the Rhode

and Geisler model and 2 to • in the Billone model).

Both modelspredict that the relationshipbetween $

and D is linear and frequency ndependentwithin the

human auditory range.Lastly, the sheardisplacement

above an inner hair cell is approximately he same as

that above an outer hair cell (less han 40% difference

for the Rhode and Geisler model and less than 1%

difference or the Billone model).

The Journal of the Acoustical Societyof America 1153



12/14

BILLONE AND R•kYNOR

,w/2 •

$LG

Fro. A-1. Radial shear displacementmodel. The tectorialmembrane TM), the reticularmembrane RM), the pillarsof Corti

(IP, OP), the bony spiral amina (BSL), the spiral imbus (SL), and the spiral igament SLG)are treatedasrigid bodies. he

basilar membrane BM) is assumed o be a beam.

The model in Fig. A-1 is chosen or this study for

two reasons:

(1) It allows or a separation etween he tectorial

and reticular membranes above the hair cells. This is

consistent with anatomical observations,and it is a

critical factor in the shear orce analysis.

(2) It incorporates beam model for basilar mem-

brane deflection which is based on both the structure

and the performance f the basilarmembrane.

The responsef the model o an upwarddisplacement

of the basilar membrane s shown n Fig. A-2. The

radial sheardisplacementSy) s defined s the compo-

nent of relative motion between opposingpoints

(Mi,Qi) which s resolved long he reticularmembrane.

For BM displacementsD) small compared o the

membrane width (w), the following relationship

between i and D applies for a detailedderivation f

these elationships,eeBillone,1972):

Si=Gi(x)D, j= 1, 2, 3, 4,

G•(x)= (4Cq/w)E2(P/w)a-3(P/w)•+ 1•,

C4i= 2 csc(2•) (1--Ca)•Liq-ri sin(•--?) •-t-Ca(gsin?-t-h os•) /(tan-•q-ctn-¾),

Ca= (C2--C• tan•) cos2•/(ycoq-g),

C•=yco sec•-t-(Cx-t-rxsin•x--rx cos•x an•) tan•,

(h--z,o sec•'•)y,o+(h--rxsin•x+rx cos•x an•)z,o tan•

C1 •

g+y,o sec• -- (h - rx sin•x+rx cos•x an•) tan•

y•o=•W,

Z,o=hx+(•w--gx+g) tan%

t'= [(y,o+g)•+(Z,o--h)•,

•= arctan[(Z,o--h)/(y,o+g) ,

rx=[(gx--g)•+h• •,

•= arctan[hx/(gx--g)

ry=•{[(4--j)gx+jg4--4g]•+[4hx+j(g4--gx) tan,]•} •, j•l,

•= arctan{[4hx+j(g4--gO an•]/[(4--j)g+jg4--4g•}, j• l,

ti = [(g+ri sin•/--Li sin•)•+(h--ri sin•i--Li cos•)•] ,

•= --arctan[(h--ri sin•i--Li cos•)/(g+ri sin•i--Li sin•)•,

Ly= {[(Z,o--r• sin•y)•+(y,o--ri cos•i)•[(Z,o--ra sin•a)•+(y,o--racos•a)•} •.

(A1)

(A2)

(A3)

(A4)

(AS)

(A6)

(A7)

(A8)

(A9)

(A10)

(All)

(A12)

(A13)

(A14)

(A15)

(A16)

(A17)




13/14


y#

g

+a)

BSL Y

z,

•2 Lj- $j

Yj Mj yj

(P,O)

(•)

lb)

'o - • SLG

Fro.A-2.Responsef he adial hearisplacementodeloa positiveM displacementD). (a)TheorganfCorti otateshrough

ananglea)about SL. he TM otateshroughnangle/•)about L. b)Opposingoint Qi)on heTM movesdistanceSj)

in theshear irectionelative o reticularmembraneoint Mi).

The nine model parameters w,h,h•,g,g•,g4,'y,L3,p)

must be specified efore Eqs. A1-A17 can be used to

calculateG(x). Each of theseparameters arieswith

position.Let xv be the normalized ositionalong he

cochleai.e., distancerom stapes/totalengthof the

basilar membrane). Based on Fernandezs (1952)

guinea ig data, the function w) is approximatedy

w= (1.52xv-3-1.2) 10 2 cm. (A18)

The nextsixparameters avebeenmeasured y Rhode

and Geisler in the cochlear duct of the cat. As a first

approximation,heir data is assumedo apply to the

guineapig as well:

h= (--0.359x•q-1.07)X 10 2 cm, (A19)

h•= (0.0033x•q-0.459) 10 2 cm, (A20)

g= (1.01xv+0.242) 10 • cm, (A21)

g•- (1.04xv+0.324)X 10 2 cm, (A22)

g4= (1.34xv+0.755) 10 2 cm, (A23)

• = 0.103xvq-0.265. (A24)

The parameter L3) is given n Sec. II as

La= (0.04xv-3-0.016)10 2 cm. (A25)

The authorshave measured he ratio (p/w) for the

cat. Using his data for the guineapig gives

P= (0.608xv+0.48)X 10 2 cm. (A26)

• The longitudinaldirection (x) is measured long the basilar

membranerom stapedal nd (x=0) to the apicalend (x--BML).

•'The radial direction is measured across the cochlear from

spiral imbus o spiral igament long lineparallel o thereticular

membrane.

a A shear orce s defined o be a forcewhich s applied o a given

surfacen a direction arallel o that surface.n this paper, he

cuticular late s the reference urface. he radial component f

the shear orce s onewhich s resolved long he axispointing n

the radial direction (i.e., the y axis n Fig. 1).

4Recent studies of basilar membrane vibration by Rhode

(1971) and Johnstone, aylor, and Boyle (1970) demonstrate

that the frequencyresponseof positions n the first cochlear

turn are tuned quite sharply o high frequencies e.g., 5000-20 000

Hz). As this paper s concerned ith the spatial distributionof

mechanical timuli along the cochlear artition (i.e., amplitude

envelopes)ather than the frequency esponse urves or individ-

ual positions, he data of theseauthors s not appropriate or this

modelstudy. Thus, resultsare presented nly for B•k•sy's wave

envelopedata which is in the low- and intermediate-frequency

range.

5 The Stokes quationsor slowviscouslow are goodapproxi-

mations or an incompressionlow field when the inertia terms n

the momentum equations are small compared to the viscous

terms (Landau and Lifshitz, 1959). The Reynolds number

(R=pScoq/•) gives an estimate of the ratio of the convective

inertia terms to the viscous erms. The parameter (B=pcoq2/u)

representsan estimate of the linear inertia terms relative to the

viscouserms.Using he valuesp--•l g/cma, u•10 -2 dyn-sec/cm,

q•-•8X10 -5 cm, and S•5X10 -* cm when co•10 * rad/sec

gives R


14/14

BILLONE AND RAYNOR

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1973 - billone, raynor - transmission of radial shear forces to cochlear hair cells

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