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 asymptoticsyouter 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: NIHyNIDCD, 0027-8424y98y9514594-6$0.00y0 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 5 y 5 y 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 e 5 y ,, stapes end. The ratio W0 L 1 is the fundamental small parameter that forms the basis of the multiple scale expansions W ~ ! 5 W ~u ! 1 e W ~u ! 1 Q z, t Q0 , z Q1 , z ... [2a] W ~ ! 5 W ~u ! 1 e W ~u ! 1 P y, z, t P0 , y, z P1 , y, z ..., [2b] u 5 v 2 e21 *z z 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(e0) 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 @v2 ­ 1 v ­ 1 # W ~u ! the TM, which is kinematically coupled to Q(1) in the present model, M(z) uu C(z) u 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 ` 2 W 5 rv ~ ! y ~ ! E ~ u w!­ww ~w ! w are not shown so as to keep figure simple. W z W0 k z G z, ; Q0 ,zd 2` Z. Locally, the masses are assumed to perform piston-like 1 @ 2 #Tc~ ~1!! 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, u; w) is derived in the OHCs is modeled through a wavelength-dependent active Appendix, force mechanism (9). 1 cosh p~u 2 w! Partition Equation. The dynamic equations of motion of the ~ u w! 5 F 2 G G z, ; p log ~ ! ~ !y 1 , [4] partition can be written in matrix form 2 k z HT z W0 @M~Z!­ 1 C~Z!­ 1 K~Z!#QW ~Z t! 5 FW ~Z t! 1 FW ~Z t! u w 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 W 5 y where the structural matrices for mass M, damping C, and introducing the normalized displacement vector q0 Q0 Q*, W 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 c(Q(1)) saturates.