Mechanics and nonlinearity of hair cell stimulation

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The finding that the integrity ¢,¢ hab ceils is e,.~enual for th~ norm~ generation of aural distortton (e.g. [16]), however, argue~ against site 1, where th~ m~chanical load of the o~gan of Corti ~s in fact neglected. In that case the state of the orga,~ of Corti would play no role in the generation of distortion product~. So we are left with option~; 2 and 3. In the absence of hard evidence we will rely for the time being on ~ mc :[rated guess. In order t', account for the data the i~onlineafity Ires to be followed by a second fdter. Option 3 locates the second filter m the ele,:trochemistry of Ihe ~ ceLl~. It reqaires h,dr cell filters, the tuning frequencies of which roughly match the basilar n~embrune tonotopy. At this point we tend to believe this to be a little far.fetched. WL t.ave speculated for some time [1,2] on the possibility that the second fdter is linked t.o ~he di~ecttonal sensiuv~ty of the hair ceils, If this turns out to be the case, then the norlin.arity should be vt or before that stage. Therefore, we propose to use option 2 as our working hypothesis, I.EASI~ILITY STUDY Of THE 'HAIR CELL BPNL MODEL' Fig. 3 gives a block dtagram of the 'hair cell BPNL m jdel'. We have two linear filters determtr:ing the longitudinal and the radial components of the driving force on the hair cell at positron x. The magnitude of the vector sum of the two is determined in the next block. The vector sum r(x, t) has the sign of the radial component. The nonlinearity R =[(r) operates on r(x, t). The multiplic~ttion by rr(x, t)/r(x, t) models the directtona! sensitivtty, the factor equ',ds the cosine of the angle betwe~*n vector sum and radial direction. Thi.~ ~s the second filter operation in terms of the BPNL model. These basic assumpttons are described m detail elsewhere [1,2], albert lr~ a different form. The main points may be summarized as follows. (I) covhlear hair cells are sensmve to, iadial shear between tectorial membrane .and cutlcular plate~ 12) longitudinal shear affects the hair :airs sensttivity l o radlai shear tf the mechamcal resi:onse to the sheanng force is nonhnear, (3) nonhn~Jr interaction between radial and longitudinal shear produces substanttal

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F~g 3. The ~hatr cell BPNL model'. At location x the r~dlal and longitudinal driving forces on the cflta of the h~it cell are dc',cnbed as responses of the radial end longitudinal fillers Hr(x. )9 and/-/l(X, J). The resultant force m the radial-longltudinal phne is the vector ~,,umr(x, t). The mechamcal response lo the force foUowsfrom the nonlinear transfer ftmctien R = f(r) The hau cell is sensitive to the ladial component of R, Rr, ~blch ts obtal-ted by mu,~tlplyingR by the cosine of the angle between v~ctor sum and radii dtrection, which is equal to rr/r.

497

sharpening and lateral (two-tone) sup)ress~on provided that the nonlinearity ts compr~s. sire and that the rat'-~ between radial ~nd longitudinal sh6ar follows a certain pattern. The present feas;bflity study a~ms at answering the following questions. What ~at~o between radial .',~d longitudinal shear is required to account for sharpening and supplesstun, and secondly, ~s that ratio likely to occur in tke cochlea" Stared otherwise: Can cochlear (rmero.) mechamcs explain sharpening and lateral supressionq In our previous studies [I ~] we assumed, rather arbitrarily, tl~t the stimulation dnectmn in the radialqongitudinal plan,-, depended on place and fiequtncy in a very s~mple way. At a given place the d~rectmn of the resultant vector sum was a:sumed to change monotonically with frequency. In terms of the block dmgram m F~g, 3 that assumption can be restated as follows, radial tuning is of the bandpass type, longitudinal ~s bandstop, and differences between phase characteristics were not taken into account. We will now follow the more general approach..This states that there are radtal and longitudinal forces, Fr(x,.D and F~(x,f) respectively, acting on the hair cell at longitudinal pos~t~o~ x (we are not at present interested in the transversal force; the ~adml, longitudinal and transversal directions are defined tn F~g. 8). For the sthnulusAcos (2n'ft)the |o~ces take the form

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The precise course of A~o as a function of f Is not crucml in the sub;equent ana~y~lg of asymptotic behaviour; we only use the abo~e values. With thege, we calculated the responses to two-tone sttmuk, which are transformed bac,k mto eqtu~,~lef~t radial input level units. "llats implies that we take as respome r~easure the level of a smgk. rad,,,l component winch gk,e~ the same rms re;porlse as ~he ~vvo4oae stimulus, Pe,,,h~ ~'e shown in F~g. 4, where tw~e 1 is at CF and tovie 2 ofi C'F, so that &~o, -~ O. ,.3L~ ~r~d :-J~z

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! ig. 4. Two-tore suppression predicted b y the model Tone I Is at CF, with ~¢q = ~r/2; for z~Li (the level dffferente between longitudinal and n,dlal components of ~ove 1) we used two values" - 1 8 , avd 6, d B; the level o~" tone I wag fixed at L ! --0 dB. For the seco,~d tc no, off CF, we assumed z ~ 2 = 0. ALz i.,, the parameter, and L 2 ~s the independent variable. A F-th law device was used for the nonlinean~y, w~th p = 0 3. The response ~s e"pressed m terms of a single radial component giving the same response, or L R ~s R r I/p expressed m dB. Fig. 5 in ord~.r to ma~cll measured frequency behavmur in two-tone supprersion, the Iongitud~ml component off CF has t,~ be stronger (relative to the radml compf,nent) than the one at CF. This mapll,.~s that the radial ct mponent should be tuned more tharply than the lor.gRudinal component. This is indicated schematically (For frequencies well abo~e CF, suppression ~s found ,o diminish. In that area the slopes may converge again.)

are parameters and L2, the level of tone 2, is the independent variable. The nonhneadty is ~,pproxlmat ed wRh an odd-order power law rectifier with power 0.3. It is seen that, although the depth of the ~uppression notch depends on ALl, for any value of ALt we can find a ALz that will produce suppression. This shows the superfluity of the original assumption that at CF rite longitudinal component Ft should be zero, The presence of a longitudinal component at CF does produce self-suppression, as follows from the reduction m ~esl:onse for small L~ with increasing AL~. However, as long as the suppre~sor, tone 2, has a stronger longitudinal component, It appears to be able to produce additional suppresuon. P~ych~phys~cal and hemal data show that rate suppression ~s a lateral effect, i.e. it requires a c~rtam dlstanc~ ~ , ~en suppressor and suppressee frequencies (e.g. [3,20]). Although the threshold for the effect tends to increase as the suppressor frequency moves away from CF, the range and depth of the suppression area also increase so that ~uppressm3 appears lo be more prominent. This requires that the longitudinal cornpoaent becomes stronger as f~ moves away from CF. In other wo~:ls, we arnve at the

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F:g. 6. Schematic cross-sect]on of ~he organ of Cortr radial cut m panel a, longttu6mal m b Panel a sl-.ows that fcr small travelling wave amplitudes the radml force on the cdta of a hair cell Is propo,tgmal to the tramversal deflectmn of the me'nbrane. In panel b it is dlustrat¢d that lcng-~udmal forces may arise if the hair cells follow the longitudinal wave mouon. Maximum torte ~s found where tli¢ f~:rst spatial derivative is maxunum. (tin. rectorial membrane, dae tuner hatr cell; tl rzt~cv~ar h mina, bin: basilar membrane, bex): width of the bastlar membrane at longitudinal p,~stt~onx .y(v. t) t ansversal bm wave: cr and q" geometrteal constants).

interesting coneluraoa that the radial tuning must be sharper than the longJtut~i~al t u m n g in order [or the model in Fig. 3 to predict lateral suppression properly (Fig. 5). The next step is to relate the tun;ng o f the ra6ial and longttudina! forces on f~e hair c~ll to the cochlear travelling wave. First we consider the classlcal approach. This assumes t~la~ the ~adial force is proportional to the transversal basilar m e m b r a n e motaon y(x, t), and the l o n g i t u d i n ~ force to its spatial derivatiue (see Fig. 6).

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I tg 7, Schem,JtlC top view el the tec'on.d membrane aad radial cross-section of the organ of CorU. (he te~torial r~cmbra.e has J ttbre structure oriented a~ ~ ,,mall angle wtth the radial direction. We assume that the *ectortal membrane ~tdfne~s is oriented m the ~ame d~rection. The tectorial membrane point !. above ~he tuner hatr cell at x t. ts htr, ged m S at Xo, and attached to the reticular h m i n a m C at ¢~,

to produce sharpening. At tlus petal we must conzlude either that the interaction between longitudinal and radial driving forces does not provide the mechanism for suppressaon, or that the est,nale of radial ;,,nd longitudinal threes in Eqn. 5 Is mcorre:t. Ao a poss~bihty we mall cons~de' the following modlfic,~tion of Eqn. 5 Recent morphological studies of lhe tectori.'d membrane have revealed a fibre-like ~tracture [12,17.I8]. The structure has a clear, dominant orientation wluch ss at a small angle with the radial dtrect~on This i~ depicted schematically m Fig. 7. The slant appears to vary shghtly with posttton along the membrane i!8] If we assume that a tectorial membrane stiffness is assocmted wit;~ this slant structure (lor the ttme being we take no account of tectonal memt~rane mass and damptag), and secondly that stretch along the structure is nommtform, e.g. because .~f the va~ying thick,tess of .he te~.torial membrane, then the model :an work. The tectonal membrane above the inner hatr cell at xt is driven at posit~on xz (Figs. 7 and 8). The transversal wave at x2, .~,(x:, t), causes a radmi motion at point C. Because of the nonuniform stretch, this causes a radial and a longitudinal compo,ent ,n the dtsplacem,:nt of T (Fig. q~. This displacement is proportt')nal to the radial dlsphcement of C, and tht~s to ylx:, t). The displacement of the cuticular plate of the ha~ cell at xt is determined '3y the travelling wave at x~. The major terms determining the radial and longRudmal force;are nDw *: ~" At ~hl~ point It is i o t of into' est whether the cilia of the inner hair cells are actually attached to the to,tarsal membrane (F~ensen's strmpe) ~x not. ~n the h t t e r case the endings will be m the close proximity of the tectorial n, embrane, The -tlatl~e motion of the rectorial membrane above the c~ha, and the cllttcuhr plato .~t the~ feet, will in both cases determine the force exerted on the elba. The dynamical bei~aviou~ ot the elba, y motmn, however, ~s fikely to be different.

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7 wzth the tectonai rier]l~rane cut along the shnt tibre

F,(x~, t)= ay(x,, t)- by(x2, o

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Fbg. 9. | f t h e stretch Mogg *he fibre 1. net ,anb,Jrm, bcraose ~hln patt~ ~treteh more t~_o :b~,P, t)xt~, then l ra:7.tJ pull at C wtl! _-e~lr in ~ rx dta~ ~s wci[ a'~L~ngl~,Kd,~Hfd~:l?["cc~ e~" ~f T

502 the more F~ follows the spatial derivative. In Fi, on the other "hand, the first term appears to be several orders of magnit,.=de smaller than the ~cond, so thai F~ behaves as the tr~.velhng wave. Therefore, Fr is now ttmed more sharply than Fb and the model predicts ~hacpenmg and lateral suppre~;ion. DISCUSSION

The p, imary objectv, e of tile above t'~asibility study is to indicate that precise geometrical and phystc,q propert,es of the elements of the organ of Corti can play a crucial role m cochlear tr~nsduction. Progress in the understanding of this transduct~on process can only be made on the basis of more detailed data m this area. The propose,] interpretation of the BPNL model in terms of cochlear processes Is m fact no longer a true EPNL model. Tl',e two important filters, viz. the longitudinal and the radial tilter (Fig. 3), do not correspond in a slmpb, way to the first and second filters. Both filters play a role m the processing before ard after the no~dineanty. Therefore, both the first and the second filter are determined by the mechanical ttining of the basdar membrane and the geo'uetrj of the orga~ of Corti. Nevertheless, this scheme mhnics many of the characteristics of the BP~L model. For that reason wc use the term 'hair cell BPNL model'. In order for the mechamcal model to display successfully two-tone suppression effects we had to speculate aI-out ~.he role of the tec~orial membrane. We conclude that its action a:ounts for more than the transformation from basdar membrane elev~.tion to radial shear at the edna (at least m the workmg model). The idea that the rectorial membrane plays a more actwe role m tl~e cochlear tiansductJon process l~ by no means new (see e.g. [l I ]). However, it is only, ecently that di ect evtdence on this point has become available. In the alligator hzard o~~ly part of the h air cells 3n the basilar papilla is covered with a tec'~orial membrane. Aupr, rentb, these at..= the cells that exhibit sharp tunirg and lateral suppression [1022}. el course this still leaves the question of/tow the tectorial membrane does it. One wotdd aope to find a general mechanism across many species. Data to check our current h)pothesls, expressed in ~qn. 6, are unfortunately lacking. We assumed that the noulinearRy R = j{~) is 1sotropic. except that we aPowed for an even-order compopent, which cause,.; d~ffereat responses for positive and negative radial mp~Jts. In vtew ot the morphology this assumption seems rather ad hoe. We use it only because of its simlJ~icity and economy. It v,,3uid cost an extra parameter to let the nonImeamy depend o : the direction of stimulati ;n. Ihe load wtuca the nonhnearRy imposes on the basdar membrane is not neghgible. ]'hus the ,n~e-way BPNL model of Fig. 1 or 3 i:, an oversimplification. The models of basdar membrane motion that assume; a healing=at damping are closer to correctly describing 1he cechlear filtering One significant effect is the propagation ~f cochlearl~, generated dl~-t~-~r~on Ftodu:ls [8], A second r; the shift o~ tunmg frequem'y with lzvel and ~he asymmetry m h~3t,-freq~ency and low-frequenc.~ suppres,.io~ areas [9.15]. These ~uedels p~edim suppression at the meehan~ca~ ~evel for a ~fign-fr~quency suppressor only. This then would rest It m an a~,ymroetnc behav[our :~ the hair ceLt output, although the octed~recfl~nal~y coupled mo d~ p~edicts qu,dita-tivcly equal b~ha',lour above and below supp-essse frequet~cy. The dz=ta [3,201 clearly shvw such an. asymmetry. In hne with the

503

theorizing on stimulated acoustic emission, e.g. [13,14,1 ~], the propagation of d~ztor~;on products cap be conceived of as resulting from outputs of active sources in the cochlea. The nonlinear damping at point x can be thought of as th~ combmauon of hne,~ d, roping and a nonlinear source of distoruon p:oducts. The exten~ to which the products couple back into the mechanics depends on the cochiear impedat,ce a~ seen by this source at x. The retrograde effect will m general differ from the forw.'rd effect because of the different impedances. The forward channel also contains addmonal active sources. The study of the quantitative relation between ,~e.l,al and itl6dta~t~cal distortion products will therefore, rely on a precise description of the nonlineai transducer. One true -: phy:,ological experiment suggested by our theory ~s to che~k whethec longitudinal sv":a.,~atlon of the haL- cell reduces the response to ~adiaI stimulaiton. This requires h~,r co" r, -,:.n~ measurement under controlkd mechanical stimulation m two dtrect~,'ms. The re~uit of tim expen,nent will answer the quesUon of vA~ether directions; .~ensauvity can act as the second fiher. We would ,very much like hair cell phys~,~teg.,:*'~Io take up this issue. ACKNOWLEDGMLNT

The authors gratefully ackncwledge the contributions of Professor ~'gbert de Boer from the University of Amsterdam, through many stimulating d)seusstons on the top'c REFERENCES I l l Daft'hub, .q. ~197~[) An alternative approach [o the second filter. In Facts and 'doo¢!s m Heating, pp. 100-! 03. Editors E. Zwjcker and E. Terhardt, SpLulger-Verlag, Berl:n [2] Duifhm% H (1976) Cocl'd~ar nonlinearity and second fll~er ~,osslble mecham~m ant Dmrhcatzoas J. At.oust Soc. A~. '~9, '~08-423 [3] Dmfhu~, H. (198C)" L~el effects m psychophysJcal two-ton~ su:pression J Acou~t. $oc Am 67 Cm prez~) [4J Engeb~t,~n, A.M. aqd Eldredge, D.H. g1~68) Mod~l Foc nonhneag chzr~c[e~l~:ic~ o! c~dd~.a~ potemiah, J. Aeoust. See. Am. 44,548-554. [5] Fettiplace, R. and Crawford, A.C, (1978): The coding of ~o~nd pressure and freqt~enc~' m cochlear ha~ ceils of the terrapin. Ptoe R See. Lonci B 203,209-218. [6] Gold~ein, l.L. (1967) Auditory nunL~earity. J. Aeoust. Soc Am, 41. 676-689. [7] Hail, LL (t972) Auditory distorhon product~ f~ - f j and 21~ - h . J Acoust. So~ Am 51, 1863-1871. [8] Hail, J L. (I974) Two-tone distortion producb. ~n a nonlmL,ar mode[ of the basflar membrane L hcoost. See. Am. 56, 1 8 l o - i $28. [9] Hail, J.L. (1977)" Two-tonL ~uppte~ion m a nonl~l~r model of the t~asit~r merabtane. J ~_cou,t Sac. Am. 6 1 , 8 0 2 - M 0. [101 Holton, T. and We,~,,g,T.F. (1978) Two-tone rate sappres;Jo.J m hzard cocide~'r nerve fibers, relation to receplor organ ra ~rphology. Brim Res 159, 2i9--222. [11 j Hoggins, W.H. and Ltckhdet, J C R ~f!95i ~, P|ace me¢|,aq,qms of asd~o.,v r~equcr~.'y - ~Jy~s J. Acous~. Soc. Am. 23~ 290-299 [12] Hunter-Duva~, 1 M. (1978). Electron r~,cxoscomc a,~ses~me,t of the co~ht~a, ~ome fc~l,ntqdes and results. Aeta OIo-tar.~,ngoi. Suppt. 35 t [I3j Kemp, D L (1978)" Stimufiated go~osnc em~s=to~ from w~lhh; lb," b~,x,ar a~taaor:r ,v~tem L Aeoust. She. Am. 65, { 38~-~ 391. [14] Kemp, D.T. d980)" Towards a n~od¢l tot the orig~n ef .ochlear ech~t~ !I~a~,L-- be,. 2. 533548.

50,* [ ~5| Elm, D.O., M~lnar, C.E. and Pfe~fcr R R. (1973): A syste,~ of noaki~e~ differential equations m~ell~tg b a ~ t membrane moUon..L A ~ u ~ . Soc. Am. 54. 1517-1529. [l~l Ktm, D.O., Molnaz, C.E. and Mztth~ws, J.W. (1979): Cochlea~ x~ec1~mics" nonSneaz behavior m •wa-tone responses as reflected i~ ~chlezr nezre tibet ~e~,~on~ and m ear-c~.nal soun~ press'~,te. J AcousL Soc. Am. (in press). [ l ? i l~tone~er-Frei, A. (1979): The e f f ~ I of cl-.~nges in eado~rmphatic ion concentratiom on the te~onal membrane. Heating Res 1 , 8 1 - 9 4 . [18] Lun, D J . (1979)" Cochlear anatom) related ta ~ c ~ ~aecha~Jcs. J. Aeoust. Sac. Am. 65, $27(A). [ ! 9 ] Pfefffez, R.R. (1970): A mo:lei for two-tone inhibition of single ¢ochleax nerve hbers. J. Aooust. Soc. Am.4B, 1373-1378. [2r)I bachs, M.B. and Kmng, N.Y.-S. (1965)" T w o ~ n e inhibition in auditory-nerve f i b , s. J. Acoust. Soc. Am. 45, 1025-1036. [21 | Sellick, P.M ~nd Russell, [.J. (1979) Two-tone ~appre~lon in cochlear hair celIs. Hearing Res. !, 227 -239. |221 Weiss. T.F., Muffoy, M J , Turner, R.G. and P~ke, C.L. (1976): Tuning of ~JIgle fibers in ~he cochlear netw of the all~ator hzatd ~elation to receptor raorphology. Brain Res. 115, 71 - 90.