Correspondence principle in cochlear mechanics

Correspondence principle in cochlear mechanics E. de Boer

Physics Laboratory, Ear, Nose and Throat Department, Wilhelmina Hospital, Amsterdam, The Netherlands (Received 14 January 1981; accepted for publication 25 February 1982)

When only long waves play the most important part in the cochlea, the response can be described by a most simplified model, the one-dimensional model. When short waves are to be included, a more complex model is needed. The response then depends on the dimensionality of the model and is much harder to obtain. This applies especially to the region in the neighborhood of the point where the basilar membrane shows resonance. Both two- and three-dimensional models have been studied to assess the effects of short and long waves. The relative importance of the part played by short waves depends on the damping constant (or loss factor) 6 associated with the resonance of the basilar membrane (BM). For very small 6 a three-dimensional model is really necessary, it cannot be replaced by a model of lower dimensionality. When 6 is small, but not

ß too small, the three-dimensional model can be made equivalent to a two-dimensional one, provided the latter is modified in a specific manner. This paper shows why this is so and which conditions have to be met. The two-dimensional model must undergo two modifications to effect this equivalence. The first modification ensures that the model has the same long-wave behavior. In the second place, a specific additional mass ("added mass") reactante should be added to Z(x ). An expression for the limiting value of 6, above which this correspondence is valid, is given in the paper. A second, larger, limit is presented as well: when 6 is above this limit, the responses of both the three-dimensional and the two-dimensional model are equivalent to that of an appropriately chosen one-dimensional model. In this case too, long-wave behavior must be matched and an "added mass" reactance must be included in Z(x ). This holds true for the entire cochlea including the region of resonance. For both types of transition the amount of "added mass" is given.

PACS numbers: 43.63.Bq, 43.63.Kz

BM

p

w(x) w(•) Wo(•) Q(k)

LIST OF SYMBOLS QJk) basilar membrane q(x) convolution Z(x) 4-21- $o longitudinal coordinate a transversal coordinate M(x) vertical coordinate Mo height of cochlear channel R(x) width of cochlear channel R o width of BM equals ½b 6 effective height: h/½ 6 A 2•r times frequency 6 s density of cochlear fluid C • wavenumber C a BM velocity C s inverse Fourier transform of w(x) k= W(k) for lossless case kA geometry factor ks

Q(k) -Oh Fourier transform of Q(k) specific impedance of BM stiffness component at x = 0 space constant for stiffness mass constant (variable) mass constant (fixed) resistance constant (variable) resistance constant (fixed) damping factor limit of 6

limit of 6

constant subtracted from Q(k)/h one particular value of C one particular value of C limit of k

limit of k

limit of k

INTRODUCTION

Even the simplest possible model of the cochlea is not yet fully understood from all angles. Hence further analysis is still worthwhile to improve our insight. We consider in this paper a two-channel system filled with a linear, incompressible and inviscid fluid: in a general sense a well-studied type of model. Figure 1 shows the geometry of the structure. The channels are separated by a partition of which only a part is flexible: the basilar membrane (BM). The width of the channel (b) and the height (h) are assumed to be constant. A fraction ½ of the width of the partition is occupied by the BM. We assume the BM to be completely specified by its (specific) acoustic impedance g(x), cf. Viergever (1978). Use of this concept implies the model to be

driven sinusoidally; the radial frequency w is, of course, a parameter of Z(x). This type of model en- compasses most of the principal physical phenomena of wave propagation and attenuation in the cochlea. In the form given the model must be solved by a three-dimensional treatment. In this sense it has not yet been studied very thoroughly.

Where the stiffness component of Z(x) is important, waves can propagate along the length of the basilar membrane. When the wavelength is large compared to the cross section of one cochlear channel, wave propagation can be understood from a still further simplified model, the so-called one-dimensional model. In this model the influence of fluid movements in the y and z directions on the dynamics of the system is ne-

1496 J. Acoust. Soc. Am. 71(6), June 1982 0001-4966/82/061496-06500.80 (D 1982 Acoustical Society of America 1496

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.120.242.61 On: Mon, 24 Nov 2014 14:09:03

FIG. 1. Three-dimensional cochlea model. Shaded area: the

basilar membrane (BM). The origin of the coordinate system is in the center of the BM.

glected. This is allowed for the major part of the cochlea where waves can propagate. In the neighborhood of the point where the BM shows resonance for the frequency under consideration, however, the afore- mentioned condition is no longer fulfilled. Short waves become important and their behavior depends on the geometry of the model studied. In a two-dimensional model the influence of fluid displacement in the transversal direction--the y direction in Fig. 1--is neglected. In a three-dimensional model all three coordinates

x, •, and z are taken into account.

It should be noted that in all three models the fluid

moves in three dimensions. It is the influence of components of fluid movements on the dynamics that is neglected. In the one-dimensional model only the component of fluid velocity in the x direction is taken into account, in the two-dimensional model those in the x and z directions. In the two-dimensional model the BM will

move (in the z direction) over only a fraction (½) of the width of the partition. In this model the variation with y is neglected, hence it is assumed that the fluid in the immediate vicinity of the partition moves (in the z direction) with the average velocity, i.e., with 1/½ of the BM velocity. There is a related point concerning the use of the impedance Z(x). The (vertical) velocity of the B M will not be uniform over its own width. Like-

wise, the transmembrane pressure difference will not be constant over this width, and it will vary in a different way with y. The membrane is described by its impedance as if this were a characteristic of the membrane's substance, hence Z(x) is independent of y. Here, a similar assumption is invoked: it is assumed that Z(x) is the quotient of the averaged pressure difference and the averaged BM velocity. This only applies to the three-dimensional model.

The present paper addresses the question of dimensionality of cochlear waves. In particular, it will be shown under which condition the three-dimensional

model can be described by a two-dimensional model

without loss of detail, and under which condition both models are equivalent to a one-dimensional model. When the former condition is not met, the response of a three-dimensional structure cannot be simulated by that of a structure of lower dimensionality.

I. FORMULATION OF THE COCHLEAR MECHANICS PROBLEM

Especially when one is interested in phenomena related to BM resonance, the following formulation of the cochlear mechanics problem is useful. The velocity of the BM is called w(x) and an integral equation can be derived in terms of the function W(k) which is (apart from a constant factor) the inverse Fourier transform of w(x). Note that both W(k) and w(x• depend on the parameter w. For the three-dimensional model of Fig. 1 and its two- and one-dimensional simplifications this integral equation reads

Z(x) W(k)Q(k)e-•}Xdk = W(k)e- . (la) 2wp

Here w is 27r times the frequency and p is the density of the fluid. As before, Z(x) is the BM impedance and the function Q(k) represents the geometry of the model. See de Boer (1980b, 1981) for the derivation of this equation for two types of three-dimensional models, for simplification and limitations, and for the functional form of Q(k) in such models. The integral equation (la) can be reformulated in the x domain as

[1/Z(x)]w(x) *q(x) = (1/2wp)w(x) , (lb)

where ß denotes a convolution. The kernel ,function q(x) is the Fourier transform of Q(k). A similar equation has been used for the two-dimensional model (e.g., Allen, 1977; Sondhi, 1978; Alien and Sondhi, 1979), our equation is simpler because we have assumed the helicotrema to lie at x = oo and we did not specifically include the excitation of the cochlea by the stapes.

For the solution of Eq. (la) by an analytical method it is convenient to set the lower limits of the integrals

equal to zero. This is allowed because reflections from the helicotrema are neglected (cf. de Boer, 1979) and reflections from the resonance region are insignificant (Lighthill, 1981; de Boer and van Bienema, submitted for publication). In other terms, the spectrum W(k) for the model used here is a one-sided function and the real and

imaginary parts of w (x) form a Hilbert pair. In the later sections of this paper it is sufficient to consider only positive values of k.

For the one-dimensional model, Q(k) has the form

Qx(k) = 1/h.,,k :• , (2)

where her f is the "effective height," i.e., the ratio of channel area and BM width (cf. de Boer, 1980a), in fact, heff= h/½. For a "strict" two-dimensional model in which the membrane is moving uniformly over the entire width:

Q•.a(k) = 1/k tanh kh . (3a) This function does not have the same long-wave behavior as Eq. (2). By averaging the velocity over the

1497 J. Acoust. Soc. Am., Vol. 71, No. 6, June 1982 E. de Boer: Correspondence principle in cochlear mechanics 1497


width of the partition (as mentioned in the preceding section) a "modified" two-dimensional model is con- structed with Q(k) given by

Q•.b(k) = ½/k tanh kh. (3b) This model has the proper long-wave behavior. The modified two-dimensional model can be derived from

the strict two-dimensional model by scaling of the magnitude of the impedance Z(x). It is only this form of the two-dimensional model that is used in this paper.

The Q(k) function Qa(k) for the three-dimensional model of Fig. 1 is given in de Boer (1981), the precise mathematical form of it is not important at the moment. For small k, Qa(k) behaves as Qx(k) of Eq. (2) and as Q2t•(k) from Eq. (3b). It is to be remembered that an alternative derivation of Qa(k) is possible (Steele and Taber, 1979). In that case the derivation starts not from the differential equations describing the dynamics but from an asymptotic solution to the problem. Al- though that method is not exact it has undeniable ad- vantages, for instance, in the handling of minor modifications of the model.

For every function Z(x) with resonance the nature of the solution to Eq. (1) depends critically on the function Q(k). Physically, icopQ(k) denotes the pressure at the surface of the BM in a fluid wave with wavenumber k

and unity velocity. In an alternative formulation: Q(k) represents the way BM pressure and BM velocity are related as due to the geometry of the fluid columns and their modes of movement. The character of the Q(k) function for the three types of model is shown by Fig. 2. The abscissa is kh where h is the height of the cochlear channel, the ordinate is the dimensionless quantity Q(k)/h. The width b of the channel is assumed to be equal to h. Curve 1 applies to the one-dimensional model, and follows Eq. (2). The BM is assumed to occupy 0.1 of the width b (½ = 0.1). Curve 2 applies to the two-dimensional model and follows Eq. (3b). Curve 3 is for the rectangular block model of Fig. 1, again with ½ = 0.1 (de Boer, 1981). The curve for a "cylindrical" three-dimensional model is similar.

Curve 1 in Fig. 2 is very simple indeed and does not need much comment; in curve 2 two regions can be distinguished: "long waves" for kh < 1 and "short waves" for kh > 1. Curve 3 has three regions; the behavior in the region of the highest values of k is not too relevant, however, since here the wavelength tends to be smaller than the width of the BM. Hence, for curve 3 there are also two principal regions, long and short waves and curves 2 and 3 are found to differ mainly in the transition from long to short waves and in the nature of the function Q(k) in the short-wave region. This feature allows us to understand the correspondence between the various types of model. It also illustrates the basic physical features of the problem.

II. THE "SHORT-WAVE LIMIT"

In general, the following functional form is used for the BM impedance Z(x):

Z(x) -- [S O exp(-ax) /ico]+ icoM(x)+ R(x) . (4)

Q(k) / h

100.00

1O. O0

1.00

0.10

0.01

O. 1 1.0 10.0 100.0

kh

FIG. 2. Curves 1, 2, and 3: Q(k) functions for one-, two-, and three-dimensional models, respectively. Abscissa: kh; ordinate: Q(k )/h.

For further simplification the mass function M(x) is often assumed to be constant (Mo). The resistance R(x) is only important in the region of resonance, i.e., where the imaginary part of Z(x) is Small or zero. In this region R(x) will vary only little and it makes hardly a difference when it is replaced by a constant (Ro).

By way of the following reasoning the relative importance of short and long waves can be assessed. First, the resistance term is omitted from Eq. (4) so that a purely imaginary impedance Z(x) remains. Equation (la) can be thought to be solved for this imaginary impedance function, let the solution for W(k) be called Wo(k). Next the damping term is reintroduced in Eq. (4) but by a special trick: in the lossless impedance function the real variable x is replaced by a complex variable: x-i6/a with 6 positive and smaller than unity. In this way Z(x) acquires a positive real part. For small 5 the real part is 6S o exp(-ax)/co and the imaginary part is hardly changed (this holds true when the mass is constant). Gver the region of resonance the resistance introduced varies somewhat but it

can safely be replaced by its value at the resonance point. Clearly, the parameter 6 then has the meaning of a loss factor, it is (at resonance) the ratio of the resistance and one of the constituent components of the teacrance. Substitution of x-i6/a for x in Eq. (la) yields immediately that the solution W(k) with losses relates to the lossless solution Wo(k) as follows:

W(k) = Wo(k) exp(-6k/ot). (5)

Since, as said, the spectrum is one-sided, this equation needs only to be evaluated for positive k. Equation (5) suggests already that regions with higher k become progressively less important when k increases. From previous papers (de Boer, 1979, 1980b) it is seen that for large k the lossless solution Wo(k) tends to be a function with a constant magnitude for both the "straight-line" and the "hyperbolic" approximations of



the impedance function Z(x). more general solutions.

The same holds true for

Consequently, for large k the ma•nitude of W(k) de- cays exponentially with increasing k and this indicates that the solution to the integral Eq. (la) will not change noticeably when the upper limit of the integrals is replaced by a certain limiting value k,•. It is clear from Eq. (5) that the limit k= should be related to From experience it is known that a good value for the limit is (for the models treated in the papers men- rioned):

k•: 2a/6. (6)

When this limit is used as the upper limit in the integrals of Eq. (la), the errors in the amplitude of w(x) are not larger than i dB and features associated with B M resonance are well-preserved. It is recalled that a is the "space constant" involved in the exponential dependence of the stiffness term in Z(x) on •. A typical value for a is 3.0 (cm -z) and for h- 0.1 (cm) so we find as a typical limit for k=h (the abscissa in Fig. 2) the value 3.0 for 6:0.2 and 12.0 for 6: 0.05. The value

given for the limit k•, Eq. (6), depends on the magnitude level of the BM reactance components at resonance and hence upon the mass constant (the local value, represented by Mo). The limit k• will, in effect, be proportional to (Mo) -u•'. In the papers referred to Mo was taken as 0.05 (g cm-•'), a value that almost cer- tainly seems to be on the high side and that probably will have to be modified in the future. The mass com-

ponent cannot be omitted in Eq. (4) because resonance behavior in the model is essential for simulating char- acteristics of experimental data (Alien, 1977; Light- hill, 1981).

III. REDUCTION OF THE O(k) FUNCTION

In the interest of brevity as well as clarity, we will concentrate on the Q(k) function and not on response forms obtained by solving Eq. (la). In Fig. 3 the three Q(k) functions of Fig. 2 are redrawn. We observe in curve 3 a certain tendency to approach a constant value for kh larger than 1. To illustrate this, two further curves are added. Curve A shows the three-dimension-

al function with a constant Ca subtracted from Q(k)/h. In this case C A equals 0.050. The resulting curve is seen to be a good approximation of the two-dimensional function up to the point kA, where the product kh is about 8.0. For simplicity, the part of curve A far above that limit is omitted from Fig. 3. In the next section it will be shown that subtracting a constant from Q(k) can be simulated by addition of a mass teaclance to the impedance function Z(x).

We can use k A as the limit kin; from this we see that for values of the loss factor 6 higher than 6A= 2a/kA = 0.25ah, the response of the three-dimensional model will be almost indistinguishable from that of a two-dimensional model provided with the proper modification. For values of 6 below this limit it will not be possible to find a two-dimensional model that has the same or

almost the same response. In other words, for 6 below this limit three- and two-dimensional models are fun-

C•Ck] / h

10.00

1.00

0.10

0.01

O. 001

0. 1 1.0 10.0 100.0

kh

FIG. 3. Curves 1, 2, and 3: Q (k) functions for one-, two-, and three-dimensional models, respectively. Curve A: Curve B: @(k)?h-Cs, both for the three-dimensional model. See text for the meaning of C,• and Cs. Of curves A and B only the part where the function depicted is positive is shown.

damentally different. It is noted that with a = 3.0 and h = 0.1 the limit amounts to 6A = 0.075. TO explain de- tails of experimental findings a value of 6 on the order of 0.050 is needed (cf. de Boer, 1980b); apparently this is somewhat below the limit where two- and three-

dimensional models can be made equivalent.

When the loss factor 6 is substantially larger than 6A, it is possible to show equivalence of three-dimensional and one-dimensional models. Curve B in Fig. 3 shows Q(k)/h for the three-dimensional model but now with a constant C s equaling 0.083 subtracted from it. Up to the limit ksh-- 2 this function is similar to curve 1. Hence when 6 is so large that only values of k up to the limit k s are relevant, the three-dimensional model is equivalent to an appropriately modified one-dimensional model. The same holds obviously true for the two-dimensional model. Quantitatively,' for the same parameters as mentioned earlier, the limit for 6 is 6s= c•h= 0.3.

In conclusion, for 6 larger than hA-- 0.25ah the three- dimensional model can be replaced by a suitably modified two-dimensional model. When 5 is larger than bs = ah , the three-dimensional and two-dimensional models are equivalent to a suitably modified one-dimensional model. This constitutes the first part of the correspondence principle we wish to show in this paper. In its general form it holds true for al! models described by Eq. (la) and impedance functions of the type given by Eq. (4). The numerical values quoted are somewhat dependent on the size of the mass constant as mentioned in connec- tion with Eq. (6).

The proper constants to be subtracted from Q(k)/h depend, of course, on (; the reader is referred to the cited papers for analytical expressions for Qs(k), from these it is possible to derive estimates for the more general case. In particular, for small ( the constant C s for the best fit to the one-dimensional case turns



out to be equal to ½. This value follows from a second- order expansion of k•'Q(k)/h- ½ in terms of k. In actual practice a somewhat smaller value (e.g., 0.083 instead of 0.1) allows a better approximation over a wide range of k values to be obtained (curve B in Fig. 3).

IV. EQUIVALENCE: THE CONCEPT OF "ADDED MASS"

We now study the effect of replacing Q(k)/h by Q(k)/h -C on the integral equation (la) and its solution. Substitution of Qc(k) - Q(k) -Ch and using k,, as the upper limit of integration yields

Z(x) Qc(k) W(k)e-'•xdk

= 2cop - Z(x) W(k)e-i•xdk' (7) A further reduction leads to

Z(x) + 2icopCh Qc(k)W(k)e-i•Xdk

= 2•op '• W(k)e-•'xdk (8)

and this is of the same form as the original Eq. (la) but with Z(x) replaced by Z(x)+ 2ic•pCh. The additional term has the character of a mass teaclance. Hence:

the effect of replacing Q(k) by Qc(k) is equivalent to adding a corresponding mass teaclance to Z(x). The magnitude of the "added mass" is 2pCh; for the ex- amples considered in the previous section, the values are 0.10ph and 0.166ph, respectively. The statement in italics above constitutes the second part of the principle of correspondence. It is valid without restric- tions for all models described by Eq. (la).

In various forms the concept of added mass has ap- peared in earlier publications, notably in those con- sidering two-dimensional models (e.g., Oetinger and Hauser, 1961; Viergever and Kalker, 1975; Sondhi, 1978; Alien and Sondhi, 1979; Viergever, 1980). In the derivation above the concept is shown to be quite a general one when one attempts to approximate the response by way of that of a model of lower dimension-, ality. It is, therefore, not surprising to find it in a similar form in asymptotic expansions that relate two- dimensional to one-dimensional models. Note that the

difference between the constants C s and C,• amounts to 0.033, signifying an added•mass of 0.066ph when going from a two-dimensional to a one-dimensional struc-

ture. The theoretical value for the (modified) two-dimensional model is (2/3) ½ph, our estimate agrees well with this value.

The considerations put forward in the foregoing can be generalized. From Eqs. (7) and (8) it is shown that the addition of a constant to Q(k) can be simulated by the addition of a corresponding constant to Z(x). In- versely, a constant component of Z(x) can be left out and its effect replaced by a proper modification to Q(k). This can be done, for instance, with the resistance: Z(x) •s considered as a pure reactance and Q(k) is replaced by Q(k)- iRo/2Cop, where R o is the resistance

1500 J. Acoust. Soc. Am., Vol. 71, No. 6, June 1982

that ought to be included in Z(x). In terms of the damping constant 5, Q(k) is to be replaced by Q(k)- iSMo/2P where Mo is, as before, the mass parameter. Use of this method to account for losses instead of the one im-

plicit in Eq. (5) reduces computing time for the response considerably.

v. CONCLUSION

The following reviews the results of the considerations put forward in this paper. The conclusions are pertinent to all cochlea models that can be described in the form of Eq. (la). The coefficients cited apply to the case studied here where the BM occupies 0.1 of the width of the partition that separates the channels and the mass constant M o equals 0.05. It is recalled that the loss factor 5 is the ratio of the resistance term in

Z(x) to any of the reactive terms at the resonance lo- cation.

When the loss factor 5 is larger than 5A= 0.25ah, the response of the three-dimensional model is almost indistinguishable from that of an appropriately modified two-dimensional model.

As the loss factor becomes smaller than the afore-

mentioned limit, this equivalence degenerates: a three-dimensional model can no longer be simulated by .a two-dimensional model.

When the loss factor 5 is larger than 5r= ah, the response of the three-dimensional model is almost indistinguishable from that of an appropriately modified one-dimensional model.

Subtracting a constant C from Q(k)/h is equivalent to adding the reactance of a mass 2pCh to the BM impedance Z(x), where h is the height of a cochlear channel.

Two types of modification of the BM impedance Z(x) are sufficient to effect the correspondence between models: (a)the magnitude [Z(x)[ must, if necessary, be scaled to obtain the correct behavior for very small values of k (long waves), and (b) an appropriate added- mass reactance must be added to Z(x).

Allen, J. B. (1977). "Two-dimensional cochlear fluid model: New results," J. Acoust. Soc. Am. 61, 110-119.

Allen, J. B., and Sondhi, M. M. (1_979). "Cochlear macromech- anics-Time domain solutions," J. Acoust. Soc. Am. 66, 123-!32.

de Boer, E. (1979). "Short-wave world revisited: Resonance in a two-dimensional cochlear model," Hear. Res. 1, 253- 281.

de Boer, E. (1980a). "Auditory physics. Physical principles in hearing theory. Part 1," Phys. Rep. 62, 87-174.

de Boer, E. (1980b). "A cylindrical cochlea model--The bridge between two and three dimensions," Hear. Res. 3, 109-131.

de Boer, E. (1981). "Short waves in three-dimensional cochlea models. Solution for a 'Block' model," Hear. Res. 4, 53 -77.

E. de Boer: Correspondence principle in cochlear mechanics 1500


de Boer, E., and van Bienema, E. "Solving cochlear mechanics problems with higher-order differential equations," J. Acoust. Soc. Am. (submitted for publication).

Lighthill, J. (1981). "Energy flow in the cochlea," J. Fluid. Mech. 106,149-213.

Oetinger, R., and Hauser, H. (1961). "Ein elektrischer Ketten- leiter zur Untersuchung der mechanischen Schwingungsvor- gaenge im Innenohr," Acustica 11, 161-177.

Sondhi, M.M. (1978). "Method for computing motion in a two-dimensional cochlear model," J. Acoust. Soc. Am. 63, 1468-1477.

Steele, C. R., and Taber, L. A. (1979). "Comparison of 'WKB' calculations and experimental results for three-dimensional cochlear models," J. Acoust. Soc. Am. 65, 1007-1018.

Viergever, M. A., and Kalker, J. J. (1975). "A two-dimensional model for the cochlea. The exact approach," J. Eng. Math. 9,353-365.

Viergever, M. A. (1978). "On the physical background of the point-impedance characterization of the basilar membrane in cocldear mechanics," Acustica 39,292-297.

Viergever, M. A. (1980). Mechanics of the Inner Ear, a Mathematical Approach, Academic thesis (Delft U. P., Delft).



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Correspondence principle in cochlear mechanics