1973 - Billone, Raynor - Transmission of Radial Shear Forces to Cochlear Hair Cells
Transcript of 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
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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
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TRANSMISSION OF RADIAL SHEAR FORCES
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 / •
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TRANSMISSION OF RADIAL SHEAR FORCES
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).
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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_
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TRANSMISSION OF RADIAL SHEAR FORCES
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_
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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.
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TRANSMISSION OF RADIAL SHEAR FORCES
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
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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)
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TRANSMISSION OF RADIAL SHEAR FORCES
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
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BILLONE AND RAYNOR
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