Proc. Natl. Acad. Sci. USA Vol. 95, pp. 14594–14599, December 1998 Applied Mathematics, Biophysics
Compression, gain, and nonlinear distortion in an active cochlear model with subpartitions (multiple scale asymptotics͞outer hair cell saturation)
R. S. CHADWICK†
Auditory Mechanics Section, Laboratory of Cellular Biology, National Institute on Deafness and other Communicative Disorders, National Institutes of Health, Bethesda, MD 20892
Communicated by Julian D. Cole, Rensselaer Polytechnic Institute, Troy, NY, September 18, 1998 (received for review March 19, 1998)
ABSTRACT The propagation of inhomogeneous, weakly that nonlinear distortion is small and can be calculated as a nonlinear waves is considered in a cochlear model having two correction to linear theory. Quite surprisingly, large compres- degrees of freedom that represent the transverse motions of sion and gain follow as a natural consequence of this assump- the tectorial and basilar membranes within the organ of Corti. tion. This approach differs significantly from previous com- It is assumed that nonlinearity arises from the saturation of putations in the frequency domain (10, 11), which were based outer hair cell active force generation. I use multiple scale on recursive numerical schemes that utilized an initially asymptotics and treat nonlinearity as a correction to a linear guessed waveform shape. First, the present work treats non- hydroelastic wave. The resulting theory is used to explain linearity in the cochlea in a system with more than one experimentally observed features of the response of the co- transverse degree of freedom. To some extent, the role of the chlear partition to a pure tone, including: the amplification of extra degree of freedom with regard to the nonlinear effects the response in a healthy cochlea vs a dead one; the less than can be established. Second, the method developed here is linear growth rate of the response to increasing sound pres- computationally much simpler. Also, the methods presented sure level; and the amount of distortion to be expected at high allow analytical estimates for distortion and gain in terms of and low frequencies at basal and apical locations, respectively. OHC force production and cochlear geometry. I also show that the outer hair cell nonlinearity generates retrograde waves. Mathematical Formulation and Solution
The mechanical response of the cochlea to sound is rich in The Model. The linear model previously proposed and nonlinear phenomena that are fundamental to the hearing analyzed (8, 9) consists of a straightened duct with a rectan- process in mammals (1). One of the more important effects is gular cross section bounded by rigid walls. The organ of Corti called compression. This term is used to signify that the (OC) separates two fluid-filled spaces. The coordinates (X, Y, displacement of the cochlear partition (CP) at a fixed location Z), respectively, denote the radial, transverse (perpendicular along the cochlea does not increase in proportion to the input to the partition), and axial (along the length of the duct) sound pressure level (SPL) of a pure tone at the stapes, as a directions. Different parts of each OC cross section are free to linear system would, but at a smaller rate. The prevailing oscillate with different amplitudes and phases. Such a repre- hypothesis concerning the source of compression is the satu- sentation is supported by recent experiments demonstrating ration of the active process of outer hair cell (OHC) force significant relative motions within the OC (12, 13). An OHC production (2, 3). OHC active force production is also thought transmits its force through its attachment at its two ends to a by some researchers to be the mechanism of producing gain of Deiter’s cell and to the reticular lamina (RL). The transverse the putative cochlear amplifier (CA) (4). The term, CA gain, component of motion of the RL must be followed by the is utilized to describe the fact that to produce given displace- tectorial membrane (TM) because we have shown that there is ments of the CP in response to a pure tone, a healthy, active no squeezing motion between TM and RL (8). This prediction cochlea requires a smaller SPL than does a damaged or dead is consistent with experimental observations (12, 13). Thus, the cochlea. The effects of nonlinearity are not small, if assessed CP model has to allow at least explicitly for independent from the amount of compression or gain (5). One might expect transverse motions of the basilar membrane (BM) and the a significant nonlinearity to modify various phases of a wave- RL-TM complex. The simplest structural model of the OC is form differently, thus producing a significant harmonic dis- a lumped-parameter model with two degrees of freedom tortion of a pure tone. However, this is not the case for shown in Fig. 1. The observation of both transverse and radial measurements at the base at high frequency (5), although some displacements in the OC (12, 13) are treated as being kine- distortion has been reported at the apex at low frequency (6, matically coupled in the present model, and are not modeled 7). as distinct degrees of freedom. Here it is assumed that OHC The primary aim of this study is to account for large stereocilia sense motion at the TM-RL subpartition, and compression, large gain, and small distortion in an active somatic motility of the OHC acts between the TM-RL complex cochlear model when the source of the nonlinearity is the and the BM (as occurs in vivo). Mechanical axial coupling is saturation of OHC force production. Thus, this work is an included through the fluid–structure interaction. Two fluid extension of our previous work that considered linear, active channels representing the scalae vestibuli and tympani, each models having multiple transverse degrees of freedom, and filled with an incompressible liquid provide the fluid loading which we studied by use of the multiple scale approximation in to the subpartitions. All properties such as mass, stiffness, and the frequency domain (8, 9). Here, I introduce from the outset damping are allowed to vary slowly along the axial coordinate
The publication costs of this article were defrayed in part by page charge Abbreviations: CP, cochlear partition; SPL, sound pressure level; OHC, outer hair cell; OC, organ of Corti; RL, reticular lamina; TM, payment. This article must therefore be hereby marked ‘‘advertisement’’ in tectorial membrane; BM, basilar membrane. accordance with 18 U.S.C. §1734 solely to indicate this fact. †To whom reprint requests should be addressed at: NIH͞NIDCD, 0027-8424͞98͞9514594-6$0.00͞0 Building 9, Room 1E-116, 9 Center Drive, MSC 0922, Bethesda, MD PNAS is available online at www.pnas.org. 20892. e-mail: [email protected].
14594 Downloaded by guest on September 25, 2021 Applied Mathematics, Biophysics: Chadwick Proc. Natl. Acad. Sci. USA 95 (1998) 14595
Multiple Scale Expansion. To exploit the slenderness of the uncoiled cochlear geometry, the following scaled coordinates ϭ ͞ ϭ ͞ are introduced: z Z L; y Y W0, where L is the length of the cochlea and W0 is the width of the cochlear partition at the ⑀ ϭ ͞ ϽϽ stapes end. The ratio W0 L 1 is the fundamental small parameter that forms the basis of the multiple scale expansions ជ ͑ ͒ ϭ ជ ͑ ͒ ϩ ⑀ ជ ͑ ͒ ϩ Q z, t Q0 , z Q1 , z ... [2a] ជ ͑ ͒ ϭ ជ ͑ ͒ ϩ ⑀ ជ ͑ ͒ ϩ P y, z, t P0 , y, z P1 , y, z ..., [2b]
ϭ Ϫ ⑀Ϫ1 ͐z where t 0 k( )d is the wave phase variable, with k(z) a dimensionless nonuniform wavenumber scaled by the characteristic width W0. Substituting Eq. 2 into Eq. 1, and grouping terms together of O(⑀0) yields, to the zeroth order, the coupled nonlinear homogeneous integro-differential equa- FIG. 1. Structural elements in the transverse (X, Y) plane. Q(1) and Q(2) denote the vertical displacements of the two masses TM and BM. tion system. The horizontal displacement Q(s) denotes the shearing displacement of ͓2 Ѩ ϩ Ѩ ϩ ͔ ជ ͑ ͒ the TM, which is kinematically coupled to Q(1) in the present model, M(z) C(z) K(z) Q0 , z thus resulting in two independent degrees of freedom. The TM is constrained by a torsional spring and a rigid rod. Viscous components ϱ 2 ជ ϭ ͑ ͒ ͞ ͑ ͒ ͵ ͑ ͒Ѩ ͑ ͒ are not shown so as to keep figure simple. W z W0 k z G z, ; Q0 ,zd Ϫϱ Z. Locally, the masses are assumed to perform piston-like ϩ ͓ Ϫ ͔T͑ ͑1͒͒ oscillations, independent of X, displacing liquid transversely 1, 1 Q0 , [3] and axially in the adjacent channels. Elastic coupling of adjacent sections is neglected, but electrical axial coupling of where the Green’s function G(z, ; ) is derived in the OHCs is modeled through a wavelength-dependent active Appendix, force mechanism (9). 1 cosh ͑ Ϫ ͒ Partition Equation. The dynamic equations of motion of the ͑ ͒ ϭ ͫ Ϫ ͬ G z, ; log ͑ ͒ ͑ ͒͞ 1 , [4] partition can be written in matrix form 2 k z HT z W0 ͓M͑Z͒Ѩ ϩ C͑Z͒Ѩ ϩ K͑Z͔͒Qជ ͑Z t͒ ϭ Fជ ͑Z t͒ ϩ Fជ ͑Z t͒ tt t , f , c , , where is the observation point, is the source point, and HT [1] is the scala height. Subsidiary Small Distortion Expansion. Rescale Eq. 3 by ជ ϭ ͞ where the structural matrices for mass M, damping C, and introducing the normalized displacement vector q0 Q0 Q*, ជ stiffness K are detailed in the Appendix. Q(Z, t) is a vector where Q* is a characteristic value of the TM displacement at having components (Q(1), Q(2)) that represent the transverse which the active force function (Q(1)) saturates. Also, rewrite (1) Ϫ1 ⌿ (1) displacements of the TM and BM (note that the TM radial (Q ) in the form ik (z)Fa (Q ), where Fa is a the (S) (1) ជ ϭ ⌿ motion Q is proportional to Q in the model); Ff(Z, t) maximal active force per unit length of CP and is the ជ Ϫ1 W(Z)P(Z, t) is a fluid forcing vector having components W(Z) normalized active force function shown in Fig. 2. The ik (z) (ϪP(1), P(2)), where W(Z) is the local width of the partition, factor is introduced to show the dependence of the active force and P(1) and P(2) are, respectively, the pressures, averaged over on the local wavenumber.‡ Also, divide each term of Eq. 3 by 2 ϭ 2 the transverse coordinate X and evaluated at the upper surface the inertia of the BM per unit length M2 HB(z)W(z) ជ of the TM and lower surface of the BM. Fc(Z, t) is the active to obtain control vector having the components ((Q(1)), Ϫ(Q(1))), ͓ Ѩ ϩ Ѩ ϩ 2 ͑ ͔͒ជ ͑ ͒ where (Q(1)) is a saturating function of the TM transverse m(z) c(z) K z q0 , z
displacement (Fig. 2), and the signs of the components reflect ϱ that an equal and opposite effect acts on the TM and BM. ϭ ͑͞ ͑ ͒ ͵ ͑ ͒Ѩជ ͑ ͒ W0 k z HB(z)) G , ,z q0 ,zd Ϫϱ
ϩ ␦ Ϫ1͑ ͓͒ Ϫ ͔T⌿͑ ͑1͒͒ i k z 1, 1 q0 . [5]
␦ ϭ ͞ 2 The ratio Fa ( M2) of maximal active control force to BM inertia is evidently small (see Discussion for estimates) and can serve as an auxiliary expansion parameter for the following
‡The dependence of the active force on wavenumber was found previ- ously by solving a linear inverse problem (9). There the central idea was to determine the active wavenumber k(z, ) from the approximation k(z, (2) (2) ) Ӎ i⑀Ѩz log Q , where measured values of Q at low sound pressure levels were used. When that k(z, ) is used in Eq. 7, with a Z(z, ) including a contribution from a linearized active force, we found that the active force happened to be approximately inversely proportional to k(z, ). We interpreted this result as an indication of axial electrotonic FIG. 2. Saturation of OHC active force production. The curve is coupling between OHCs. The factor i is introduced to provide a ͞2 drawn with the hypebolic tangent functions: tanh(x) for x Ͼ 0; 1͞4 phase lead of the active force with respect to the TM displacement. tanh(4x) for x Ͻ 0 to represent data obtained for OHC receptor Without this factor the model cannot produce enough gain with a potential vs. stereocilia bending (16). realistic value of maximal OHC active force. Downloaded by guest on September 25, 2021 14596 Applied Mathematics, Biophysics: Chadwick Proc. Natl. Acad. Sci. USA 95 (1998)
small distortion expansions§: into the O(␦) partition equation. The O(␦) partition equation for the nth harmonic can be written as a forced linear system ͑ ͒ ϭ ͑ ͒ i ϩ ␦ ͑ ͒ ϩ u1 ,z A1 z e u11 ,z ... [6a] T ជ ͓Z ͑z, ͔͓͒a , b ͔ ϭ f , [12] ͑ ͒ ϭ ͑ ͒ i ϩ ␦ ͑ ͒ ϩ n n n n u2 ,z A2 z e u21 ,z ... [6b] where [Z (z, )] is given by ͑ ͒ ϭ ͑ ͒ ϩ ␦ ͑ ͒ ϩ n k z k0 z k1 z ..., [6c] W0n ជ ͫϪn2m͑z͒ Ϫ I coth͑nk ͑z͒H ͑z͒͞W ͒ where (u1, u2) are the components of q0. Substitution of Eq. k ͑z͒H ͑z͒ 0 T 0 6 into Eq. 5 and letting ␦ 3 0 (no active force) yields, to zeroth 0 B order O(␦0), a coupled linear homogenous system for the ᠬ ϩ ͑ ͒ ϩ 2 ͑ ͒ͬ passive amplitude vector A with components (A1, A2): in c z K z [13] ͓Z͑z, ͔͓͒A͑z, ͔͒ ϭ 0, [7] ជ and fn is given by where [Z(z, )] is given by ic W k ͑z͒ n ͓ Ϫ ͔T ϩ 0 1 ͑ ͑ ͒ ͑ ͒͞ ͓͒ ͔T 1, 1 2͑ ͒ ͑ ͒ coth k0 z HT z W0 A1, A2 . W k0 k0 z HB z ͫϪ ͑ ͒ Ϫ 0 ͑ ͒ ͑ ͒͞ m z I ͑ ͒ ͑ ͒ coth(k0 z HT z W0) [14] k0 z HB z The linear system defined by Eqs. 12–14 is straightforward to ϭ ϩ ͑ ͒ ϩ 2 ͑ ͒ͬ solve, except the case n 1 which needs special attention. i c z K z . [8] ϭ ϭ When n 1, Z1 Z (compare Eqs. 8 and 13). Because Det[Z] ϭ 0, Eq. 12 cannot be solved unless its right-hand side is A nontivial solution of Eq. 7 can exist only if the determinant orthogonal to the eigenvector of the adjoint of Z (14). There- ជ ϭ of Z vanishes. We thus look for solutions k0(z) of Det[Z(z, )] fore fn must satisfy f1,1Z12 f1,2Z11. This condition leads to the ϭ ϭ 0. Notice that if k0 is a solution of Det[Z(z, )] 0, then amplitude-dependent wavenumber correction Ϫ so is k0, because [Z(z, )] is an even function of k0. So for ik ͑z͒H ͑z͒c ͑z, A ͒͞A r͑z͓͒1 Ϫ r͑z͔͒ every forward wave there exists the possibility of a retrograde ͑ ͒ ϭ 0 B 1 1 1 k1 z, A1 ͑ ͑ ͒ ͑ ͒͞ ͒ ϩ 2͑ ͒ . [15] wave having the same wavenumber. This retrograde wave is W0 coth k0 z HT z W0 1 r z not needed in the present study because the forward wave ϭ becomes sufficiently small at the helicotrema (z ϭ 1) to satisfy When n 1, Eq. 12 has a particular solution and a homoge- the boundary condition of zero pressure difference across the neous solution with an arbitrary multiplicative factor that we are free to specify. Choose the homogeneous solution such that CP. There is an infinite set of roots for the wavenumber. ϭ However, computations show only one root with a positive real a1 0, then part has a small negative imaginary part. This root determines ic ͑z, A ͒ 1 ϩ r͑z͒ ͑ ͒ ϭ Ϫ 1 1 the dominant forward propagating wave. All other roots b1 z, A1 ͑ ͒ ͑ ͒ ͑ ͓͒ ϩ 2͑ ͔͒ . [16] represent evanescent waves that rapidly decay from the stapes k0 z Z11 z r z 1 r z (z ϭ 0), and these roots are not used in the present study. ϭ ͞ The theory thus far indicates that the nonlinear character- Corresponding to each k0, an amplitude ratio r A1 A2 can istic shown in Fig. 2 produces a full set of harmonics that be calculated. Calculation of the individual amplitudes is propagate apically (n ϭ 1, 2, 3, . . .) and basally (n ϭϪ1, Ϫ2, discussed in the section Determination of Partition Amplitudes. Ϫ3, . . .), as well as producing a dc-shift (n ϭ 0). The same can Harmonic distortion of the partition and the correction of be said even for a purely antisymmetric characteristic with ⌿ ϭϪ⌿Ϫ the wavenumber due to the active OHC force are determined (A1) ( A1), except that a dc-shift is not produced. from the O(␦) terms of Eq. 5. To this end introduce the Fourier The n ϭ 1 harmonic produced by the nonlinearity has a special series expansions: importance in that it determines both the gain and compressive ϱ ϱ characteristics of the CA (see Discussion). ͑ ͒ ϭ ͑ ͒ in ͑ ͒ ϭ ͑ ͒ in Determination of Partition Amplitudes. Consideration of u11 ,z an z e , u21 , z bn z e , nϭϪϱ nϭϪϱ the O(⑀) equations of the multiple scale expansion leads to an [9a,b] energy integral that determines the partition amplitudes (8). When the structural matrices are symmetric, as they are here, where the an and bn are to be found. Also consider the Fourier the energy integral states that the integral of the product of series representation of the normalized nonlinear active force axial velocity and pressure over a cochlear cross section is an ⌿ function : axial invariant. Hence ϱ ␣͑ ͒ ⌿͑ ͒ ϭ ͑ ͒ in z d 0 d , z cn z e , [10] ͑ ͒ ͵ 2͑ ͒ ϩ ͵ 2͑ ͒ ϭ nϭϪϱ w z k0(z) ͭ P1 ,z P2 ,z ͮ E0, ͑ ͒ ͑ ͒ k0 z k0 z 0 Ϫ␣͑z͒ where the coefficients are calculated from [17]
1 Ϫ where c ͑z, A ͒ ϭ ͵ ⌿͑A ͑z͒ei ͒e in d. [11] n 1 2 1 Ϫ cosh͓␣͑z͒ Ϫ ͔ ͑ ͒ ϭ Ϫ2 ͑ ͒ P1 ,z W0Q*A1 z ͑ ͒ ␣͑ ͒ [18] k0 z sinh z Note that it is mathematically consistent to neglect the O(␦) correction term of Eq. 6a in the argument of the integrand of and a similar expression exists for P2. Integration over the cross Eq. 11 because it would generate a higher order term of O(␦2) section yields ␣͑z͒ ϩ 1͞2 sinh 2␣͑z͒ §Expansions in previous linearized versions of the model (8, 9) 2 2 ͑ ͓͒ ϩ ͑ ͒Ϫ2͔ ϭ A1 (z)Q* w z 1 r z 2͑ ͒ 2 ␣͑ ͒ constant. correspond to Eq. 6 with ␦ ϭ 0, but with the k0 term including the k0 z sinh z contribution from the linearized active force. [19] Downloaded by guest on September 25, 2021 Applied Mathematics, Biophysics: Chadwick Proc. Natl. Acad. Sci. USA 95 (1998) 14597
The integration constant can be evaluated at z ϭ 0 in terms of behavior at the apex and base, respectively, is determined by ϭ͗ ͘ the averaged stapes pressure PS P1( ,0). passive dynamics of the model. Active properties of the model affect only the higher harmonics. This seems to be consistent Computational Results with experimental observations that phase is relatively unaf- fected by SPL (5) or efferent stimulation (17). All computations are carried with parameter values given in Why Is 6 the Dominant Distortion Harmonic at the Apex the Appendix, which were selected with the chinchilla cochlea (cf. Fig. 4)? The amplitude and harmonic number of the in mind. The input͞output relation shown in Fig. 3 clearly dominant distortion harmonic (n ϭ 6) is in qualitative agree- shows the compression and gain effects recently measured in ment with recent observations at the apex of the guinea-pig the chinchilla using laser velocimetry (ref. 5, cf. their figure 6). cochlea using laser heterodyne interferometry (S. M. Khanna, Linearity of the response at low SPL and its reemergence at personal communication). Irrespective of the symmetry of the high SPL as seen in our computation as well as others’ (10, 11), active force nonlinearity, a resonance effect in the model has been recently observed (5). Fig. 4 (Left) demonstrates predicts that Ϯ2, Ϯ3, Ϯ4, Ϯ5, and Ϯ6 are the respec- negligible distortion in computed displacement waveforms of tive dominant distortion harmonics going from the base to the both the TM and BM at the base for a 10 kHz sinusoidal input apex at the characteristic frequency of each location, with signal in agreement with observations in ref. 5. Fig. 4 (Right) corresponding increases in magnitude. This effect should not indicates observable distortion of a 1 kHz sinusoid at an apical be confused with the dominance of cubic distortion products location. Fig. 4 (Middle and Bottom) show the amplitudes of the when two tones are input. harmonics relative to the fundamental. Tuning curves are How Can a Nonlinear Active Force Mechanism That Pro- computed at a basal location for various input sound pressure duces a Small Distortion at a Basal Location Produce a Large levels in Fig. 5. The phenomena of broadening and leftward- Gain at the Same Location (cf. Fig. 3)? The origin of the gain shifting of the peak with increasing SPL have been reported in can be interpreted as a large ‘‘distortion’’ of the fundamental ϭ the chinchilla (ref. 5, cf. their figure 8). (n 1) harmonic. In fact, for this case, the BM amplitude is ␦͞⑀ ͐z augmented by the multiplicative factor exp[ 0 Im(k1)dz], Discussion where k1 is the correction term for the wavenumber given by Eq. 15. The analysis requires that the phase of the active force ͞ Why Is Harmonic Distortion Observable at the Apex but leads that of the TM displacement by 2. The biophysical Not at the Base of the Cochlea (cf. Fig. 4)? The relative origin of this phase lead is as yet unclear; it may reside in more magnitude of distortion is controlled by the parameter ␦ intricate micromechanics than contained in the present model, ͞ introduced in Eq. 5. This parameter is the ratio of the local and or it may be a result of electrotonic axial coupling of OHC active control force to the inertial force of the BM at a OHCs. Because the Im(k1) integral is an O(1) quantity at the base, setting it equal to one gives a simple estimate of the gain saturating amplitude. Estimates of these forces per unit length ␦͞⑀ ϳ ␦͞⑀ ␦ ϳ ͞ of partition are: for the control force, (0.1 nN)͞(mV-cell) (0.25 in dB: 20( )log10e 9( ). Taking 1 80 at the base and ͞ ͞ ͞ ⑀ ϳ 1͞360 for the chinchilla cochlea, the basal gain is estimated mV) (3 cells row) (15 m row), and for the inertial force, ϳ ⑀ 2HWQ*, where is the density of the BM (ϳ1g͞cm3), H is to be 40 dB. The presence of the slenderness ratio in the the effective BM thickness (ϳ50 m), is the best frequency gain estimate is interesting because it implies that the CA (rad͞sec) at the observation location, W is the local partition would depend on cochlear geometry and not just the local width (ϳ0.15 mm at a basal location and ϳ0.5 mm at an apical behavior of OHCs near the peak of the response. The model location), and Q* is the saturation amplitude (ϳ1 nm). The 0.1 predicts there would a cumulative effect of active force gen- nN͞mV sensitivity was determined in isolated guinea-pig eration, suppressing the response basal to the peak, while OHCs (15); the 0.25 mV receptor potential is estimated from amplifying the response closer to the peak. The in-phase data obtained from mouse organotypic cochlear culture (16). behavior of the TM and BM at the apex, and their amplitude ratio near unity contribute to distortion but not to the gain. At Thus, at a basal location at 10 kHz, ␦ ϳ 1͞80 or Ϫ38 dB, which the apex, geometrical factors (e.g., the decreasing scala height would be difficult to observe, while ␦ ϳ 1͞4orϪ12 dB at an and increasing tilt of the OHCs) conspire to reduce the apical location at 1 kHz, which would be observable. Note that magnitude of the Im(k ) integral, and thereby prevent the gain the scaling method to obtain ␦ is not unique; it can also be 1 from increasing in proportion to ␦. interpreted as the ratio of the active force to the BM elastic ␦ ϭ ͞ 2 What Is the Mechanism for the Leftward Shift of the Tuning restoring force at the local characteristic place, Fa ( M2) ϭ ͞ 2 ϭ 2͞ 2 ϳ ͞ ͞ Curve Peak with Increasing SPL (cf. Fig. 5)? It has been FaK2 ( M2K2) Fa 2 ( K2) Fa K2, because 2 ϭ suggested that this phenomena is evidence for softening of the O(1). Because the stiffness is higher at the base than at the BM by lengthening of OHCs (18). Here, it is demonstrated that apex, the same arguments follow. the same effect can be produced without this softening mech- What Determines the Relative Phase Between the TM and anism. In the present model, the leftward shift is related to the BM in the Model (cf. Fig. 4)? The in-phase and anti-phase compression shown in Fig. 3. Because compression cannot reduce the response at a given axial location with increasing SPL, but merely results in a less than linear increase, it is clear the peak of the spatial waveform at a given frequency must more toward the apex, which is equivalent to a shift towards lower frequencies in the tuning curve. What Happens to the Gain Mechanism in the Present Model If It Is Constrained to Have One Transverse Degree of Freedom? The gain is proportional to the k1 integral as previously discussed, and because k1 is proportional to the factor r(1 Ϫ r), where r is the TM͞BM amplitude ratio (cf. Eq. 15), the gain will diminish as either r 3 1orr 3 0. The former would occur if parallel supporting cells were made very stiff, while the latter would occur if the TM were made very stiff. This is not to say that an active model with one degree of FIG. 3. Computed BM input͞output function. Frequency: 10 kHz; freedom cannot produce a large enough gain. In fact a large normalized axial location: z ϭ 0.18. gain has been demonstrated in such models (10, 11), but Downloaded by guest on September 25, 2021 14598 Applied Mathematics, Biophysics: Chadwick Proc. Natl. Acad. Sci. USA 95 (1998)
FIG. 4. Computed waveform distortions. (Left) Basal location, z ϭ 0.18. (Right) Apical location, z ϭ 0.9. (Top) Waveforms: solid lines, BM response; dashed lines, TM response. (Middle and Bottom) The amplitudes of harmonics n ϭ 2, 3, . . . , 10 relative to the fundamental and expressed in dB. The dc offset from asymmetric saturation of OHC active force has been omitted.
apparently at the expense of requiring a very large active force distortions to the dominant wave propagating from base to to be generated by OHCs. apex that have a negative harmonic index n. These harmonics Does a Nonlinear Active Force Mechanism Contribute to correspond to retrograde waves, traveling backward toward the the Generation of Otoacoustic Emissions? According to the base. Whether or not these waves retain sufficient amplitude present theory, the nonlinear active feedback force produces at the oval window to contribute to otoacoustic emissions still needs to be investigated. Another proposed mechanism for emissions is scattering from random inhomogeneities, which is thought to be the dominant mechanism at low SPL (19, 20).
Appendix
Structural Matrices. The mass, damping, and stiffness matrices corresponding to the two degree of freedom system shown in Fig. 1 are ϩ Ϫ ϭ M1 0 ϭ C1 C12 C12 M ͩ ͪ , C ͩ Ϫ ϩ ͪ , 0 M2 C12 C2 C12 ϩ ϩ Ϫ͑ ϩ ͒ ϭ K1 K12 Kc K12 Kc K ͩ Ϫ͑ ϩ ͒ ϩ ϩ ͪ , K12 Kc K2 K12 Kc FIG. 5. Computed BM tuning curves for various SPLs (dB). Normalized axial location: z ϭ 0.18. Note at high SPL the system where the elements are slowly varying functions defined per appears to respond linearly as in Fig. 3. unit length of the cochlear partition. Downloaded by guest on September 25, 2021 Applied Mathematics, Biophysics: Chadwick Proc. Natl. Acad. Sci. USA 95 (1998) 14599
Parameter Values. Computation were carried out with the Integration with respect to gives the pressure field following parameters, many of which have yet to be measured in the chinchilla but seem reasonable to the author. Geomet- 2 ϱ ͑1͒ W0 ͑1͒ ϭ P ͑, , z͒ ϭϪ ͵ ѨQ ͑, z͒ rical quantities: uncoiled length L 20 mm; scala height HT 0 2k(z) 0 ϭ 0.44 mm; BM effective thickness H ϭ 55 m; TM effective Ϫϱ H ϭ W z ϭ ϩ thickness 40 m; BM width function ( ) 0.055(1 ͑ Ϫ ͒ ͑␣ Ϫ ͒ z 5 ) mm. Elements of structural matrices are normalized by the ϫ logͫcosh ϩ cos ͬd. BM mass͞length and are expressed in terms of frequency (Hz): ␣ ␣ ϭ ϭ Ϫ ϭ ϭ K1 4,000; K2(z) 30,000 exp[ 4z]; K12 10,500; Kc Ϫ ϭ ϭ ϭ ͞ ϩ ϭ 3,000 exp[ 0.25z]; C1 C2 C12 200 (1 5z). Viscous Evaluating this expression on the surface 0 gives the damping has been neglected because this value of partition Green’s function Eq. 4 in the text. damping substantially exceeds viscous damping by three orders of magnitude even at 1 kHz. The maximal active force of an Note Added in Proof. The predicted dominance of the 2nd harmonic OHC is taken as 25 pN at a saturation value of TM displace- and its relative magnitude at the base (Fig. 4) has recently been experimentally confirmed (21). ment of 1 nm. Green’s Function for Surface Pressure. The fluid flow in the I am grateful for useful discussions and criticisms from Daphne fluid-filled spaces above the TM and below the BM is assumed Manoussaki, Kuni Iwasa, and Emilios Dimitriadis. to be described by solution of the linearized Navier Stokes equations for an incompressible fluid having density . The 1. Patuzzi, R. (1996) in The Cochlea, eds. Dallos, P., Popper, A. N. pressure fields then satisfy Laplace’s equation. Consider the & Fay, R. R. (Springer, New York), pp. 186–257. space above the TM, and introduce the stretched vertical 2. Rhode, W. S. (1971) J. Acoust. Soc. Am. 49, 1218–1231. coordinate ϭ ky. The leading pressure term of the expansion 3. Ruggero, M. A., Robles, L. & Rich, N. C. (1990) J. Acoust. Soc. (1) Ѩ ϩ Eq. 2b, P0 ( , , z) then satisfies Laplace’s equation ( Am. 87, 1612–1629. Ѩ (1) ϭ 4. Ruggero, M. A. & Rich, N. C. (1991) Neuroscience 11, 1057– )P0 ( , , z) 0 in an infinite strip with the boundary Ϫ 1067. conditions: ѨP (1) ϭϪ2 W k 1(z)ѨQ (1) on the TM ϭ 0 0 0 5. Ruggero, M. A., Rich, N. C., Recio, A., Narayan, S. S. & Robles, Ѩ (1) ϭ ϭ ͞ ϭ ␣ 0, and P0 0 on the top k(z)HT(z) W0 . To solve L. (1997) J. Acoust. Soc. Am. 101, 2151–2163. this boundary value problem introduce the Fourier transform 6. Cooper, N. P. & Rhode, W. S. (1995) Hearing Res. 82, 225–243. pairs 7. Khanna, S. M., Ulfendahl, M. & Flock, A. (1989) Acta Otolaryng. Suppl. 467, 195–203. 1 ϱ 1 ϱ 8. Chadwick, R. S., Dimitriadis, E. K. & Iwasa, K. H. (1996) Proc. ͑͒ ϭ ͵ ͑͒ Ϫi ͑͒ ϭ ͵ ͑͒ i f ͱ f e d , f ͱ f e d . Natl. Acad. Sci. USA 93, 2564–2569. 2 Ϫϱ 2 Ϫϱ 9. Chadwick, R. S. & Dimitriadis, E. K. (1997) in Diversity in Cochlear Mechanics, eds. Lewis, E. R., Long, G. R., Lyon, R. F., The normal derivative of the transformed problem has the Narins, P. M., Steele, C. R. & Hecht-Poinar, E. (World Scientific, form River Edge, NJ), pp. 409–415. 10. Kanis, L. J. & de Boer, E. (1993) J. Acoust. Soc. Am. 94, 2 Ϫ1͑ ͒2 ͑1͒͑ ͒ ͑␣ Ϫ ͒ 3199–3206. k z Q0 ,zsinh Ѩ ͑1͒͑ ͒ ϭ 11. Nobili, R. & Mammano, F. (1996) J. Acoust. Soc. Am. 99, P0 , , z ␣ . sinh 2244–2255. 12. Gummer, A. W., Hemmert, W. & Zenner, H. P. (1996) Proc. By the convolution theorem, the normal derivative can be Natl. Acad. Sci. USA 93, 8727–8732. written 13. Ulfendahl, M., Flock, A. & Khanna, S. M. (1989) Hearing Res. 40, 55–64. Ϫ2 Ϫ1͑ ͒ ϱ 14. Boyce, W. E. & DiPrima, R. C. (1977) Elementary Differential ͑1͒ k z ͑1͒ Ѩ ͑ ͒ ϭ ͵ Ѩ ͑ ͒ P0 , , z ͱ Q0 ,z Equations and Boundary Value Problems (Wiley, New York), p. 2 Ϫϱ 288. 15. Iwasa, K. H. & Adachi, M. (1997) Biophys. J. 73, 546–555. ϫ g͑ Ϫ , ͒d, 16. Russel, I. J., Cody, A. R. & Richardson, G. P. (1986) Hearing Res. 22, 199–216. where 17. Murugasu, E. & Russel, I. J. (1996) J. Neurosci. 16, 325–332. 18. Allen, J. B. (1997) in Diversity in Cochlear Mechanics, eds. Lewis, ϱ ͑␣ Ϫ ͒ E. R., Long, G. R., Lyon, R. F., Narins, P. M., Steele, C. R. & 1 sinh Ϫ g͑, ͒ ϭ ͵ e i d Hecht-Poinar, E. (World Scientific, River Edge, NJ), pp. 167– ͱ sinh ␣ 175. 2 Ϫϱ 19. Zweig, G. & Shera, C. A. (1995) J. Acoust. Soc. Am. 98, 2018– 2047. (␣ Ϫ ) sin 20. Talmage, C., Tubis, A., Pikorski, P. & Long, G. R. (1997) in ␣ Diversity in Cochlear Mechanics, eds. Lewis, E. R., Long, G. R., ϭ ͱ . Lyon, R. F., Narins, P. M., Steele, C. R. & Hecht-Poinar, E. 2␣2 (␣ Ϫ ) cos ϩ cosh (World Scientific, River Edge, NJ), pp. 462–471. ␣ ␣ 21. Cooper, N. P. (1998) J. Physiol (London) 509, 277–288. Downloaded by guest on September 25, 2021