RETROGRADE WAVES IN THE COCHLEA

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1 August 7, 28 18:2 WSPC - Proceeings Trim Size: 9.75in x 6.5in retro wave 1 RETROGRADE WAVES IN THE COCHLEA S. T. NEELY Boys Town National Research Hospital, Omaha, Nebraska 68131, USA neely@boystown.org J. B. ALLEN University of Illinois at Urbana-Champaign, Urbana, Illinois 6181, USA jontalle@uiuc.eu Retrograe waves in the cochlea are important because they provie information about cochlear mechanics that may be measure in the external ear canal as otoacoustic emissions. By extening traitional wave-variable methos, an explicit equation is erive for the forwar-traveling wave in a one-imensional cochlear moel that is equivalent to the WKB approximation. The corresponing equation for the retrograe waves requires no aitional approximation an provies a simple characterization of cochlear reflectance. Keywors: cochlear moel; wave propagation; reflectance; WKB. 1. Introuction The importance of retrograe waves was not fully appreciate until the iscovery of souns measure in the ear canal that originate within the cochlea [1. These otoacoustic emissions (OAE) provie a noninvasive means to acquire information about cochlear function. Coherent reflection theory is the accepte explanation for stimulus-frequency OAEs (SFOAE) [2. Zweig an Shera use the WKB approximation [3,4 to obtain a solution for the flui pressure in a cochlear transmission line. Their coherent reflection theory provies a theoretical explanation for observe amplitue an phase characteristics of SFOAEs. In this paper, wave-variable methos that have traitionally been applie to uniform transmission lines are extene to obtain the WKB formula escribe by [4. This formulation leas to a simple equation for reflectance looking into the cochlea from the base.

2 August 7, 28 18:2 WSPC - Proceeings Trim Size: 9.75in x 6.5in retro wave 2 2. Cochlear moel The equation for a one-imensional long-wave transmission-line moel of the cochlea[3 is [ [ [ P(x,f) Z = s (x,s) P(x,f), (1) x U(x,f) Y b (x,s) U(x,f) where P(x,f) is the scala flui pressure, U(x,f) is the longituinal volume velocity of the scala flui, Z s (x,s) is the scala impeance (per unit length) an Y b (x,s) is the BM amittance (per unit length). The Laplace frequency s is use for expressions of frequency, such as impeances, that are causal functions in the time omain. The real frequency f (an ω 2πf) are restricte to the Fourier transform of noncausal functions. These moel variables are shown in Fig. 1 in the context of a single section of a transmission line. Fig. 1. Transmission line moel for a section of the cochlea of length x. The seriesimpeance Z s(x, s) x limits flui translational volume velocity U(x, f), while the shuntamittance Y b (x, s) x limits the basilar membrane volume velocity. Scala impeance epens on flui ensity ρ an cross-sectional area A(x). Z s (x,s) = sρ /A s (x). (2) BM velocity is proportional to the graient of the scala volume velocity: where b is the effective BM with. U x (x,f) = Y b(x,s) P(x,f) = b V b (x,f), (3) 3. Wave variables The wave propagation function κ(x,s) an the characteristic impeance z (x,s) are etermine entirely by the meium, an therefore only epen on the per unit length series impeance Z(x,s) an shunt amittance Y b (x,s) [5: κ(x,s) Z s (x,s) Y b (x,s) (4)

3 August 7, 28 18:2 WSPC - Proceeings Trim Size: 9.75in x 6.5in retro wave 3 z (x,s) Z s (x,s)/y b (x,s). (5) The pressure wave variables P + an are efine by [ P+ 1 [ [ 1 z P 2 1 z U We invert Eq. 6 [ [ P 1 1 = U y y [ P+ (6), (7) where y = 1/z, an substitute Eq. 7 into Eq. 1: [ P+ = 1 [ [ y Z s + z Y b + z y y Z s + z Y b z y P+ x 2 y Z s z Y b z y y Z s z Y b + z y. (8) In this equation y = y /x. The equivalence of y Z s = z Y b = κ allows Eq. 8 to be written as [ [ [ P+ κ + ε ε P+ = (9) x ε κ + ε with ε(x,f) 1 2 x lnz (x,s). (1) Note that when z (x,s) is inepenent of x, ε is equal to zero, which ecouples P + (x,f) an (x,f) in Eq. 9. When z (x,s) varies slowly with x, ε/κ is small, which may be exploite to obtain approximate moel solutions. Figure 2 shows examples of P + an that emonstrate the importance of BM roughness in generating retrograe waves. Moel parameters were selecte to maintain nearly uniform characteristic impeance by making the scala area ecrease with x at the same rate as BM stiffness [4. The tall broa peaks of the BM excitation patterns were create by specifying a negative amping region at a location just basal to the tonotopic place. The negative amping was sufficient to boost the tip-to-tail ratio of the BM excitation patterns by about 6 B. The moel results shown in Fig. 2 were obtaine by finite-ifference solution of Eq. 1 an are calle exact to contrast them with approximate results escribe below. 4. Approximate wave variables The top row of Eq. 9 provies a ifferential equation for P + couple to x P + = (ε κ) P + ε. (11) When y (x,s) varies slowly with x, we can obtain an approximate equation for P + that is inepenent of by assuming that (κ + ε) P + ε. With this approximation, Eq. 11 reuces to x P + = (ε κ) P + (12)

4 August 7, 28 18:2 WSPC - Proceeings Trim Size: 9.75in x 6.5in retro wave 4 Fig. 2. Forwar-traveling (soli) an retrograe (ashe) pressure components for the approximate moel. The magnitue an phase are shown relative to pressure at the stapes. The results on the left sie are for smooth BM impeance. The results on the right sie are for rough BM impeance. The arrow inicates the phase at the characteristic place. which can be integrate to obtain [ x P + (x,f) = P + (,f) exp ε(x 1,f) κ(x 1,f)x 1, (13) giving an approximate expression for P +. Surprisingly this expression is the WKB approximation for the forwar-traveling pressure wave [3,6,7. The bottom row of Eq. 9 gives us a ifferential equation for that is couple to P +, x = (κ + ε) εp +. (14) No further approximations are neee to obtain a solution for the retrograe wave. Substitute Eq. 13 into Eq. 14 to obtain x (κ + ε) = εp + (,f) exp [ x (κ ε) x 1. (15) The integration of this equation requires a bounary conition. If we assume that (L,f), we obtain [ x L [ x2 (x,f) = P + (,f) exp (κ + ε) x 1 εexp 2 κx 3 x 2, (16) x where x 2 an x 3 are aitional integration variables for x. The sum of Eqs. 13 an 16 provies an approximate solution for the total pressure P(x,f). The accuracy of this approximation is emonstrate in Fig 3 by comparing exact an approximate solutions of peak pressure (at the characteristic place) relative to stapes pressure for versions of the cochlear moel with smooth an rough BM. The exact solutions require about 2 times longer to compute.

5 August 7, 28 18:2 WSPC - Proceeings Trim Size: 9.75in x 6.5in retro wave 5 Fig. 3. Peak pressure (re stapes pressure) as a function of frequency. The peak pressure is efine as the maximum pressure magnitue over the entire length of the BM at each frequency. The four curves represent ifferent moel conitions: (1) exact-smooth (thick ashe); (2) exact-rough (thick soli); (3) approximate-smooth (thin ashe); (4) approximate-rough (thin soli). The approximate results (thin) are barely visible in this figure because they are covere by the exact results (thick). We use Eqs. 13 an 16 to write an approximate equation for reflectance [ x L [ x2 R(x,f) = exp 2 κ(x 1 )x 1 ε(x 2 )exp 2 κ(x 3 )x 3 x 2, (17) which at the stapes reuces to R(,f) = L x [ x2 ε(x 2 )exp 2 κ(x 3 )x 3 x 2. (18) Comparison between exact an approximate reflectance (not shown) emonstrate excellent agreement. 5. Discussion Figure 2 shows exact moel results for forwar-traveling pressure P +, an retrograe pressure, at one frequency. The results on the left of Fig. 2 are for a smooth BM stiffness. The relatively small level at the stapes inicates that reflection at the tonotopic place is insignificant. The results on the right are for a rough BM stiffness. The rough stiffness ha a small, ranom increase ae to the smooth stiffness at each place along the BM. The ranom stiffness increase was uniformly istribute between an.1%. This small amount of roughness was sufficient to prouce significant reflection of the forwar-traveling wave in the vicinity of the tonotopic place. The positive slope of the phase of the component inicates that retrograe energy is being propagate. Another interesting feature emonstate by the cochlear moel results in Fig. 2 is that the roun trip elay of an SFOAE may be less than the twice forwar elay to the characteristic place. In Fig. 3, we see that the pressure ifference between the smooth an rough moels (i.e., the peak pressure fine structure) is the same for the exact an approx-

6 August 7, 28 18:2 WSPC - Proceeings Trim Size: 9.75in x 6.5in retro wave 6 imate results. In other wors, the fine structure of the approximate pressure is the same as the fine structure of the exact pressure. Moel peak-pressure fine structure represents threshol fine-structure rather than SFOAE fine-structure [8. In Eq. 18, the accumulate phase of the incient wave P + at location x comes primarily from the integral of κ. In Eq. 18, note that the contribution to the stapes reflectance R of the retrograe wave from any location x has twice the phase accumulation of the incient wave to that location. This oubling of phase is consistent with the mechanism of reflection being place-fixe. Experimental observations of OAE phase as a function of frequency suggest that some types of emissions are generate by a place-fixe mechanism [9. 6. Conclusions The wave variable ecomposition traitionally applie to uniform transmission lines may be extene to the case of nonuniform transmission lines, which may be use to represent one-imensional moels of cochlear mechanics. When the characteristic impeance z varies slowly, the equations for P +, P an R provie useful approximations for solutions of cochlear moels with tall broa peaks. Acknowlegments This work was partially supporte by a grant from NIH-NIDCD (R1-DC8318). References 1. Kemp, D., 198. Stimulate acoustic emissions from within the human auitory system. J. Acoust. Soc. Am. 5, Zweig, G., Shera, C.A., The origin of perioicity in the spectrum of evoke otoacoustic emissions. J. Acoust. Soc. Am. 98, Zweig, G., Lipes, R., Pierce, J.R., The cochlear compromise. J. Acoust. Soc. Am. 59, Shera, C.A., Zweig, G., Reflection of retrograe waves within the cochlea. J. Acoust. Soc. Am. 89, Brillouin, L, Wave propagation in perioic structures. Dover, Lonon, secon eition. 6. Durney, C.H. Johnson, C.C., Introuction to moern electromagnetics. McGraw- Hill, New York. 7. Viergever, M.A., e Boer, E., Matching impeance of a nonuniform transmission line: Application to cochlear moeling. J. Acoust. Soc. Am. 81, Choi, Y.-S., Lee, S.-Y., Parham, K., Neely,S., Kim, D., 28. Stimulus-frequency otoacoustic emission: Measurements in humans an simulations with an active cochlear moel. J. Acoust. Soc. Am. 123, Shera, C.A., Guinan, J.J., Evoke otoacoustic emissions arise by two funamentally ifferent mechanisms: A taxonomy for mammalian OAEs. J. Acoust Soc. Am