the origin of periodicity in the spectrum of evoked...

The origin of periodicity in the spectrum of evoked otoacoustic emissions

George Zweig and Christopher A. Shera a) Hearing Research Laboratory, Signition, Inc., P.O. Box 1020, Los Alamos, New Mexico 87544 and Theoretical Division, Los Alamos National Laboratory, Los Alamo& New Mexico 87545

(Received 15 October 1992; accepted for publication 26 April 1995)

Current models of evoked otoacoustic emissions explain the striking periodicity in their frequency spectra by suggesting that it originates through the reflection of forward-traveling waves by a corresponding spatial corrugation in the mechanics of the cochlea. Although measurements of primate cochlear anatomy find no such corrugation, they do indicate a considerable irregularity in the arrangement of outer hair cells. It is suggested that evoked emissions originate through a novel reflection mechanism, representing an analogue of Bragg scattering in nonuniform, disordered media. Forward-traveling waves reflect off random irregularities in the micromechanics of the organ of Corti. The tall, broad peak of the traveling wave defines a localized region of coherent reflection that sweeps along the organ of Corti as the frequency is varied monotonically. Coherent scattering occurs off irregularities within the peak with spatial period equal to half the wavelength of the traveling wave. The phase of the net reflected wave rotates uniformly with frequency at a rate determined by the wavelength of the traveling wave in the region of its peak. Interference between the backward-traveling wave and the stimulus tone creates the observed spectral periodicity. Ear-canal measurements are related to cochlear mechanics by assuming that the transfer characteristics of the middle ear vary slowly with frequency compared to oscillations in the emission spectrum. The relationship between cochlear mechanics at low sound levels and the frequency dependence of evoked emissions is made precise for one-dimensional models of cochlear mechanics. Measurements of basilar-membrane motion in the squirrel monkey are used to predict the spectral characteristics of their emissions. And conversely, noninvasive measurements of evoked otoacoustic emissions are used to predict the width and wavelength of the peak of the traveling wave in humans. ̧ 1995 Acoustical Society of America.

PACS numbers: 43.64.Bt, 43.64.Jb, 43.64.Kc

INTRODUCTION

Otoacoustic emissions evoked from the human ear often

exhibit a remarkably simple form, their frequency spectra consisting of a series of regularly spaced, almost periodic peaks and valleys. An approximate local periodicity also characterizes the frequency spacing between adjacent spontaneous emissions, the fine structure of acoustic-distortion-

product level curves, and the microstructure of the threshold heating curve (e.g., Kemp, 1981; Schloth, 1983; Horst et al., 1983; Zwicker and Schloth, 1984; Long, 1984; Dallmayr, 1985; Gaskill and Brown, 1990). Local periodicity implies that the emission spectrum remains largely invariant when shifted a small whole number of spectral periods along the frequency axis. In other words, evoked otoacoustic emissions exhibit what might be termed a "discrete spectral symme-

Stimulus-frequency and transiently evoked otoacoustic emissions are believed to arise from the reflection of

forward-traveling waves by localized "impedance irregularities" in the mechanics of the cochlea (e.g., Kemp, 1978; Kemp and Brown, 1983; Ruggero et al., 1983; Sutton and

•)Present address: Eaton-Peabody Laboratory of Auditory Physiology, Mas- sachusetts Eye and Ear Infirmary, 243 Charles St., Boston, Massachusetts 02114.

Wilson, 1983; Zwicker, 1986; Antonelli and Grandori, 1986;

Furst and Lapid, 1988). Pursuing the suggestion by Manley (1983) that the oscillations in emission amplitude originate in quasiperiodic dislocations in the hair-cell lattice induced by the spiral curvature of the organ of Corti, many argue that spectral periodicity can only be explained by supposing that it mirrors an underlying spatial corrugation in the mechanics of the cochlea (e.g., Strube, 1985; Peisl, 1988; Strobe, 1989; Fukazawa, 1992; van Hengel and Maat, 1993). To account for the observed spectral symmetry, these authors have conjectured that the organ of Corti manifests a corresponding translational symmetry somewhat like that of a crystal: Since the cochlea maps frequency into position, increased scattering at certain regularly spaced positions in the cochlea leads to increased emission amplitudes at corresponding frequencies in the ear canal.

Anatomical studies, however, provide no evidence for such periodicity in the mechanics, noting instead a "gener- alized irregularity" and "cellular disorganization" characterizing the arrangement of outer hair cells in the apical turus of the primate cochlea (Wright, 1984; Lonsbury-Martin et al., 1988). In regions where some regularity in the mechanics is found (e.g., in the "scalloping" patterns produced by the sporadic appearance and disappearance of a fourth row of outer hair cells), its spatial period appears incommensurate with the spectral period measured in emissions at corre-

2018 J. Acoust. Soc. Am. 98 (4), October 1995 0001-4966/95/98(4)/2018/30/$6.00 ¸ 1995 Acoustical Society of America 2018

sponding frequencies in the same ear (Lonsbury.-Martin et al., 1988; Martin et al., 1988). Anatonfically, the organ of Corti appears more chaotic than crystalline.

What, then, produces the observed spectral periodicity of evoked emissions? How does spectral symmetry arise when the traveling-wave reflectors are randomly distributed and manifest no corresponding spatial regularity? Under- standing the origin of periodicity in evoked emissions is fa- cilitated by working with the cochlear traveling-wave reflectance, • a quantity more directly related to cochlear mechanics than ear-canal pressure. The cochlear reflectance, clenoted R, is defined as the complex, frequency-dependent ratio of the emitted (or backward-traveling) to the stimulus (or forward-traveling) wave at the stapes:

P stapes (1)

Analysis of emission measurements shows that the regular oscillations apparent in their spectra are equivalent to a cochlear reflectance with a slowly varying amplitude and a phase locally linear in frequency (Shera and Zweig, ][993a).

The nonperiodic frequency dependence of IRI suggests that the wave reflectors that initiate the emissions are distrib-

uted along the organ of Corti in a way uncorrelated with the periodicities observed in the microstructure of the threshold hearing curve. As conjectured by Kemp (1980), oscillations in the ear-canal emission spectra result not from periodic variations in reflectance magnitude, but rather from an alter- nating constructive and destructive interference between the incident and reflected waves that arises from the cyclic cir- cling of their relative phase. The peaks and valleys in ear- canal pressure therefore represent an acoustic interference pattern that arises because the phase of the cochlear reflectance rotates uniformly with frequency (Shera and Zweig, 1993a; Allen et al., 1995). Although these measurements have established the nature of the interference pattern, they leave the "paradox of periodicity" unresolved: How does the simple, empirical form for R--in particular, its smoothly rotating phase--arise when the scattering medium is i•regular or random?

This paper answers this question and shows how the simple periotic periodicily observed in the spectra of evoked emissions results from the coherent scattering of cochlear traveling waves by small, random irregularities in the organ of Corti. The relationship between cochlear mechanics at low sound-pressure levels, as manifest in the traveling wave, and the frequency dependence of evoked emissions, characterized by R, is made precise for one-dimensional models of cochlear mechanics. Since middle-ear transfer functions generally vary slowly compared to the spectra of evoked emissions, emission measurements in the ear canal can be related

to the envelope and phase of the cochlear traveling wave. Measurements of one can be used to predict the other: Not only is the origin of spectral periodicity established in characteristics of the traveling wave, but the traveling wave itself is determined from the frequency dependence of evoked emissions.

A. Overview

Se. ction I reviews the empirical form of R and its rela-. tion to the periodicity ]measured in the spectrum of ear-canal pressure. Before pursuing a more technical analysis, Sec. II presents a heuristic discussion of the scattering mechanism proposed here. Evoked emissions originate through reflection of the forward-traveling wave by mechanical irregularities located in a broad region about the peak of the wave envelope. The resulting backward-traveling wavelets interfere with one another to create a region of coherent reflection whose center sweeps along the organ of Corti as the frequency is varied monotonically. By perturbing a one- dimensional, scaling-symmetric model of cochlear mechanics, we obtain in Sec. III an approximate theoretical expression for R in the form of a scattering integral in which wavelets reflected throughout the cochlea are summed to yield the net reflected wave. The expression relates R directly to the envelope and phase of the cochlear traveling wave. 'The empirical and theoretical forms for R are recon- ciled in Sec. IV, where we determine the nature of the me-

chanical irregularities responsible for evoked emission. Spectral periodicity indicates that emissions result from coherent wave scattering by "place-fixed" irregularities. When the irregularities are r•mdom, the coherence of the reflected wave arises as a consequence of the tall, broad peak and slowly varying wavelength of the traveling wave. The traveling wave acts, in effect, as a narrow-band "spatial- frequency filter" that selects irregularities arrayed within a narrow range of spatial frequencies as the dominant source of scattering. Section V uses the model of cochlear mechanics obtained by solving the cochlear inverse problem (Zweig, 1991) to predict the spectral characteristics of otoacoustic emissions in the squirrel monkey. Numerical simulations confirm that realistic emissions result from small, random irregul•ties in the mechanics. Finally, in Sec. VI we reverse the analysis and use measurements of otoacoustic emissions to predict characteristics of the traveling wave in humans. Appendix A summarizes frequently used symbols and their meaning. Appendix B discusses the middle-ear parameters necessary for understanding retrograde transmission of sound through the middle ear. Appendices C and D derive formulae for R in one-dimensional models of cochlear mechanics.

This paper complements an earlier paper (Shera and Zweig, 1991 a), which explored the implications for cochlear mechanics of measurements of the cochlear input impedance (Lynch et al., 1982). Both papers combine measurements and analysis to deduce constraints on the form of the traveling wave. But whereas the earlier paper discussed properties of the wavelength in the region where the wavelength is predorninantly real (i.e., in the basal, stiffness-dominated region well below "resonance"), this paper explores the region about the peak of the wave, identifying characteristics of the traveling wave necessary for the generation of evoked emissions with the amplitude and spectral periodicity observed experimentally. And whereas the earlier paper focused on cochlear mechanics at sound-pressure levels above 60 dB, deducing the existence of a tapering symmetry that guarantees that waves traveling through the basal turn suffer little

2019 d. Acoust. Soc. Am., Vol. 98, No. 4, October 1995 G. Zweig and C. Shera: Origin of spectral periodicity 2019

lO

5 30

• 0

•- -60

-90

' ' ' l

1000 1500 2000 2500

Frequency (Hz)

FIG. l. Spectral periodicity characteristic of primate evoked emissions. The figure shows a typical stimulus-frequency-emission curve in subject JEM-R at 10 dB above threshold. The triangular markers along the horizontal axes indicate the frequencies of known spontaneous emissions. Roughly periodic oscillations appear superposed on a more slowly varying background. Note that the frequency axis is logarithmic: peaks and valleys appear at regular intervals characterized by the spacing Af/f•. Although Kemp's (1980) "cochlear reflection" mechanism accounts for the existence of evoked emis-

sions, it leaves unexplained the remarkable many-cycle regularity apparent in the interference pattern. Current models require that the observed spectral periodicity mirror a corresponding corrugation in the mechanics of the cochlea. The mechanism proposed here predicts that spectral periodicity emerges spontaneously from coherent scattering off random irregularities in the mechanics.

reflection, here the analysis centers on the low-level linear regime and reveals the existence of a "spontaneously emer- gent" spectral symmetry that arises from coherent wave scattering in a disordered medium. Preliminary accounts of this work have appeared elsewhere (Shera, 1992; Shera and Zweig, 1993b).

I. SPECTRAL PERIODICITY AND THE COCHLEAR TRAVELING-WAVE REFLECTANCE

Figure l shows a typical measurement of human stimulus-frequency emissions, in which the amplitude of the ear- canal pressure is recorded as the frequency of a quiet stimulus tone of constant amplitude is varied monotonically. The measured function contains prominent oscillations superposed on a slowly varying background. On a logarithmic frequency scale, the oscillations appear roughly periodic, with a mean frequency spacing

Af/f• ,4, (2)

or approximately 0.4 Bark in units of critical-band rate (Zwicker and Terhardt, 1980). Whereas the background reflects the acoustic properties of the middle ear and the mea-

suring apparatus, the oscillatory component represents an interference pattern created by the addition to the stimulus tone of a backward-traveling wave originating within the cochlea (Kemp, 1978).

Ear-canal measurements can be related to cochlear me-

chanics by representing the intervening middle ear using scattering coefficients (Appendix B; Shera and Zweig, 1992; Keefe et al., 1993; Voss and Allen, 1994). The measured pressure takes the form of a power series in the cochlear reflectance:

Pec(o•;R) qR - 1 + -- (3)

Pe•(to;0) I -rR

=l+qR(l+rR+r2R2+ '" ) (Irtel<l), (4)

where the complex-valued coefficients q(to) and r(w) summarize the acoustic effects of the residual ear-canal space and middle ear. The cochlear reflectance varies with both the

frequency and the amplitude of the stimulus tone. Although R decreases toward zero at higher stimulus amplitudes, in the linear regime near threshold R remains independent of level. For simplicity, we focus here on otoacoustic emissions evoked in this low-level linear regime; the stimulus amplitude therefore does not appear in the list of independent variables?

The pressure Pec(to;O)=limR_•oPec(to;R) represents the slowly vdrying background upon which the multiple oscillatory components, corresponding to the sum over powers of R in Eq. (4), are superposed. Since the middle-ear response increases' linearly with stimulus amplitude while R diminishes, an estimate of the background Pet(to;0) can be obtained by measuring Pec(o•;R) at high sound-pressure levels (where R is essentially zero) and appropriately rescaling the result. Alternatively, since the oscillatory components vary rapidly with frequency compared to the background, Pec(w;0) can be estimated by locally averaging Pec(w;R) over an appropriate frequency interval f•; that is, Pec(to;O)•(Pec(w;R)) n. These two methods for estimating the background yield essentially identical results (Shera and Zweig, 1993a).

Appendix B shows that the coefficient q characterizes roundtrip sound transmission through the middle ear and sets the overall amplitude of emissions in the ear canal (note that emissions would be unmeasurable if q were zero). The function r represents the middle-ear "reflection coefficient" for backward-traveling cochlear waves reflected at the stapes; the function rR therefore constitutes a product of two reflection coefficients, both evaluated at the base of the cochlear

spiral but measured by driving the ear in opposite directions from the stapes. Note that terms in the series proportional to R 2 or higher vanish in the limit that the stapes represents a perfectly reflectionless boundary (i.e., in the limit [r[-•0). The higher-order terms thus arise from multiple internal reflections within the cochlea.

A. The empirical form of the cochlear reflectance

Measurements of stimulus-frequency emissions in human ears have been used to determine the frequency &pen-


dence of R, which appears as the product of a slowly varying complex function R 0 and a more rapidly varying phasor e (Shera and Zweig, 1993a):

R=Ro[eO(0))]e iø(ø•) (e•l). (5)

We have written R0[e•0)) ] to indicate that changes in R o occur over frequency intervals typically of order l/e the size of those characterizing changes in the phase 0{0)). The value of e can be estimated from the data (see below) and, for the subject of Fig. 1, is approximately •. Since the oscillations in P½c(O•; R) due to the phase variations of R are roughly equally spaced on a logarithmic frequency scale, the phase has the form

O(0))•-• 1,(0)/%o), (6)

where roo0/2rr denotes the maximum frequency of hearing and the dimensionless parameter • varies slowly with frequency. 3 It therefore proves convenient to consider R as a function of the dimensionless frequency variable

;•-ln(to/0)%) (0)<o%o), (7) where the minus sign has been chosen to make •(>0. As shown in Sec. III, the variable •((to) is proportional to the position of the peak of the traveling-wave envelope, about which the largest contributions to the backward-traveling wave are believed to originate. The presence of the logarithm is related to the exponential form of the cochlear frequency- position map and the existence of scaling symmetry (Zweig, 1976). Expressing the phase in terms of $( yields,

= e {8) By regarding R as a function of •(, one effectively regards emissions as a function of their primary site of generation within the cochlea.

Expanding the phase 0(0)) in Eq. (6) in a Taylor series about an arbitrary reference frequency demonstrates that over a restricted frequency interval the magnitude o[ R is roughly constant (since • is small) while its phase varies nearly linearly. For example, in a neighborhood about

R• R i e-i(ø•-øq )r•, (9)

where IR,I=IR0[(0)0ll, and

axo ½, 0o) •1 • •R(O'• 1) • • •-d-• • 0) 7 ' •o I •o I

where/_R denotes the phase of R and ½1 =--½(0)0-The function r•(0)) denotes the reflectance group delay; when IRI is approximately constant over the stimulus bandwidth, can be interpreted in the time domain as the emission latency at the stapes for narrow-band tone bursts of center fi'equency 0) (Papoulis, 1962). Empirically, the local appro. ximation (9) holds over several spectral oscillations. As the frequen:y increases, the phasor e -itør• rotates clockwise, passing alter- nately through plus and minus one. The peaks and valleys measured in ear-canal pressure therefore appear with a mean oscillation period equal to 4

Aflf• r/ru•2 rr/½, (11)

where r is the stimulus period. Note that the parameter ½/2rr represents the emission latency expressed in units of the stimulus period. For constant ½, the reflectance group delay ,•(to) varies inversely with frequency, a relation consistent with measurements of the latency of tone-burst-evoked emissions (Wilson, 1980; Norton and Neely, 1987). 5 Typical values near 1300 Hz in the human are (Shera and Zweig, 1993a)

IRIs0.2 and 02)

implying a latency of approximately 15 cycles, or roughly 11.5 ms. These measurements corroborate and extend

Kemp's (1980) heuristic description of evoked emissions, based on a cochlear reflectance with constant amplitude and linear phase.

When combined with power series (4) for P½½(0);R), the approximate local form (9) for R implies that a narrow-band tone burst p(t) of center frequency 0)• results in an ear-canal pressure with the idealized form

Pec(t;R) = p(t) + qR •p(t- r•) + qrR•p(t- 2 r•) +'" , 03)

where R l and the coefficients q and r have been assumed real and constant for simplicity. After the initial stimulus burst, the ear-canal pressure thus consists, in this simplified example, of a series of echoing bursts (indexed by n = 1,2,...) having amplitudes proportional to r • IR• (i.e., occurring in geometric progression) and appearing with latencies n r•. Relaxing the assumptions on q and r and using the approximate global form (8) for R(•) broadens and smears together the returning echoes. Multiple echoes of this sort are often observed experimentally (e.g., van Dijk and Wit, 1987; Norton and Neely, 1987).

B. The distribution of emission spacings

Fourier analysis of ear-canal pressure measurements and simple histograms of emission spacings both indicate that emission spectra are not perfectly periodic, but contain a distribution of frequency intervals Af/f centered about a clear maximum at 0.4 Bark (e.g., Dallmayr, 1987; White- head, 1988; Zwicker, 1989; Lonsbury-Martin etal., 1990; Talmadge et al., 1992). Figure 2 illustrates these distributions for both stimulus-lYequency and spontaneous otoacoustic emissions. Panel (a) shows the results of a Fourier analysis of the ,spectral oscillations apparent in the data of Fig. 1. Since the oscillations appear roughly periodic on a logarithmic frequency scale, the figure gives the spectrum of the function 20 1ogm[Pe,:(0);R)lPe,:(0);O)] with respect to 56 Panel (b) illustrates an analogous distribution assembled from measurements of spontaneous emissions in many subjects. The panel plots a histogram of inleremission spacings (based on binned values offlAf= fx/•dlf.-f•l, where f• and ft, are the frequencies of two adjacent emissions) obtained from an ensemble of human ears, each with at least five emissions

(Talmadge et aL, 1992). Both distributions are strongly peaked and are centered

about the same pronounced maximum at go= •, where go is the Fourier-transform wu'iable conjugate to •. (The aliacriti- cal hat, resembling a peak, is used throughout to indicate a "peak value" in which the underlying variable is evaluated


20

2O

15

5

o[ o

I I I I

I I I I (b)

15 30 4-5 60 75 90

FIG. 2. Distributions of evoked and spontaneous-emission spacings in humans. Panel (a) illustrates the distribution for evoked emissions in a single ear. The panel gives the amplitude of the Fourier transform of the function 201ogm[Pec(to;R)/P½c(to;0)] computed using the measurements illustrated in Fig. 1 (data from the 1-2 kHz region only). Since the spectral oscillations appear roughly periodic on a logarithmic frequency scale, the transform was taken with respect to the the dimensionless frequency variable

,•--- -ln(to/%0 ). The conjugate Fourier-transform variable extends along the horizontal axis. The principal maximum occurs at &/2frei5. The first harmonic, if present, corresponds to the R 2 term in power series (4) and would appear at q/2sr•30. Panel (b) shows an analogous distribution for spontaneous emissions from many ears. The panel gives a histogram of values qv/2zr•-f/Af------f4f-•d[f•-fbl, where f• and fb are the frequencies of two adjacent spontaneous emissions. As in panel (a), the distribution was computed over a limited frequency interval (here, for frequencies 0% ,lb} in the range 1-1.5 kHz). Sampling the distribution in a narrow frequency range minimizes spurious broadening arising from any slow frequency dependence in the middle-ear coefficients q and r, or in ½ itself (compare Fig. 15). Data are taken from the study of Talmadge et al. (1992) and represent 133 emissions recorded from 31 ears (in 22 subjects ranging in age from 7-34 years) with at least five emissions per ear. Note that on the low side of the peak, the histogram includes emissions separated by integer multiples of the fundamental emission spacing Af (i.e., corresponding to values •12,rn, for n =2,3,..). Both distributions manifest a pronounced maximum--indicative of spectral periodicity-•centered about the value •/2•r• 15 and characterized by a half-width Aqo satisfying Aq0/½•0.25--+-0.1.

C. Spectral characteristics of the cochlear reflectance

The distributions of Fig. 2 provide information on the frequency dependence of the traveling-wave reflectance. To see this, note that series expansion (4) for P½c(tO;R) implies that

Pec(tO;R) log Pec(tO;0 )•1og(l+qg+'")•qR (IqR[•l).

(15)

Middle-ear model calculations (Shera and Zweig, 1993a) suggest that the middle-ear coefficient q varies only slightly over the frequency range of the Fourier transform; the illustrated transform is therefore simply proportional to the amplitude of the Fourier transform of R. When evaluated over a limited frequency range, the Fourier transform of R with respect to • therefore has a strongly peaked magnitude, roughly Gaussian in form:

where .37' represents the Fourier-transform operator

• f •d3 e -ivy. (17) Since the cochlear reflectance R has the form

Ro(6•)e i&3 when written as a function of ,•, the Fourier transforms of R and R 0 are but translates of one another along the q• axis. In particular, [ •{R0} I peaks strongly about the origin:

The spectrum of R 0 is therefore a low-pass function of width A½ (i.e., the spectrum is large only for q<Aq). Since Aqo/•<•l, the complex amplitude R 0 typically varies more slowly with frequency than the phasor e •½'•.

An estimate of the value of ß can be obtained by writing the inverse transform of Eq. (18) as a function of •(. Since the Fourier transform of a Gaussian is again a Gaussian,

• - 1 {e - •'2/2 (a qø)2} = e - •2/2(A3)2 • e - (•½•)2 (19) where Af(= 1/Aqo. Solving for e using Eq. (14) yields

ß: a ½ (20)

Emission amplitudes are therefore typically correlated over frequency intervals spanning several spectral peaks.

at the maximum of some function or distribution.) Although intersubject variation contributes to the width of the histogram, 6 the distributions nevertheless appear relatively narrow, with half-widths Aq• given by

A•p/&•0.25_+0.1, with &/2•15. (14)

Other emission data are consistent with these results (e.g., Zwicker, 1989).

II. RESOLVING THE PARADOX OF PERIODICITY

Quasiperiodic spectral oscillations in ear-canal pressure arise because the phase of the cochlear reflectance circles uniformly with frequency (Kemp, 1980; Shera and Zweig, 1993a; Allen et al., 1995). By introducing the cochlear reflectance we have transferred the problem of understanding the origin of spectral periodicity to that of understanding the empirical form of R. Given the apparent absence of periodicity in cochlear micromechanics (Wright, 1984; Lonsbury- Martin et al., 1988), how does the simple, smooth form for R arise when the scattering medium is irregular? To better ap-

2022 J. Acoust. Soc. Am., Vol. 98, No. 4, October 1995 G. Zweig and C. Shera: Origin of spectral periodicity 2022

preciate the dilemma, consider a simple point-reflection model for otoacoustic emissions in which the forward-

traveling wave is reflected by some mechanical pertcrbation located at the peak of the traveling-wave envelope. The total phase difference between the incident and reflected waves at the stapes [i.e., the phase of R] can then be written as a sum of phase shifts arising from wave propagation and from scattering:

O_o.

The subscripts indicate the source of the phase shift: the term O0_.•+Oi_•0=200• • represents the roundtrip phase shift due to propagation between the stapes (at X=0) and the scattering location (at •), and the term O• represents the phase shift due to reflection off the mechanical perturbation.

Recall that the phase of R circles at a nearly uniform rate with frequency. Where does this simple frequency dependence arise.9 To answer this question, consider how the contributions to /_R vary with •(. Perhaps surprisingly, the phase shift 2 O0• i from roundtrip wave propagation remains nearly constant (and is approximately equal to 4•rN, where N is the number of wavelengths of the traveling waw. ß in the cochlea). To see this, note that measurements of basilar- membrane motion made at different positions within the cochlea (Rhode, 1971; Gummet etal., 1987) indicate that basilar-membrane transfer functions manifest a local scaling symmetry (Zweig, 1976), which implies that N, and therefore the phase shift O0_, i, is essentially independent of frequency. Most of the phase shift O0_• • occurs about the peak of the traveling wave envelope where the phase of the traveling wave varies most rapidly. Although waves that peak further from the stapes travel farther, the total number of wavelengths in the wave remains nearly the same. Equation (21) therefore implies that the frequency dependence of/R originates almost entirely in the phase shift Oi due to scattering off the mechanical perturbation. Since varying •he frequency moves the peak of the traveling wave from one perturbation to another along the organ of Corti, the phase of R therefore mirrors the spatial variation of mechanical irregularities. More specifically, if the perturbations vary irregularly with position, then the phase of R varies irregularly with frequency. But, empirically, the phase of R rotates periodically, apparently requiring that the mechanical perturbations somehow vary correspondingly.

This intuitive argument paraphrases the logic that forced Strube (1989) to the conclusion that the cochlea trust be spatially corrugated, with mechanical perturbations •rayed almost sinusoidally along the organ of Corti. This special, periodic pattern of scattering centers (the "cochlear washboard") generates a corresponding periodicity in the phase of the scattered waves as the frequency is varied monotonically and the traveling-wave envelope moves over the spatial washboard. The resulting emission spectrum exhibits locally periodic maxima and minima at frequency intervals Af determined, via the exponential cochlear mapping between position and frequency, by the spatial corrugation period Ax½•,:

Af/f•- Ax½•,/l,

where l represents the distance over which the characteristic frequency changes by a factor of e.

Although the corrugated-cochlea model generates periodicity in the emission spectrum, the required spatial arrangement of irregularities--the quasisinusoidal cochlear washboard---has not been observed. In subsequent sections we resolve this apparent paradox of periodicity by considering the interaction of many wavelets scattered from randomly distributed irregularities. The heuristic discussion pre-

(21) sented in the following section is followed by detailed calculations valid for one-dimensional models of cochlear

mechanics.

A. A heuristic discussion of coherent scattering

Consider first a plane wave of constant wavelength traveling through some uniform background medium sparsely peppered with local inhomogeneities (e.g., a light wave traveling through a block of glass containing localized anomalies in its index of refraction). Whenever encountered, such perturbations scatter the wave, and backward-traveling wavelets subsequently combine to tbrm a net reflected wave. Reflec- tion is greatest when wavelets scattered from different inhomogeneities combine in phase with one another.

When identical point perturbations are arrayed at periodic intervals, as they are in certain simple crystals, scattered wavelets combine in phase whenever the wavelength k of the incident plane wave matches twice the distance Ax between scattering centers; that is, whenever

h = 2 Ax. (23)

When satisfied, condition (23)--tbe so-called "Bragg condition"--implies that the total distance traversed by a wavelet traveling roundtrip between inhomogeneities (there and back again) is precisely one wavelength. Since the phase of the wave changes by one cycle over the course of a wavelength, all scattered wavelets have the same phase; that is, they superpose coherently. Wavelets with wavelengths not approximating the Bragg condition combine out of phase and tend to cancel one another.

Thus, when a superposition of plane waves of different wavelengths (e.g., white light) propagates through a crystal, only those waves satisfying the Bragg condition undergo significant reflection: The net reflected wave returns nearly monochromatic, being dominated by waves of a single wavelength determined by the geometry of the crystal. (For simplicity, we assume here that all plane waves in the incident superposition have wavelengths greater than Ax.) When the point perturbations are arrayed irregularly, however, the scattered wavelets combine with random phases and the net reflected wave is small, representing an incoherent jumble of waves of many wavelengths in which no particular wavelength is favored.

In the ear, irregularities in the organ of Corti scatter cochlear traveling waves. Unlike that in a crystal, however, the background medium in the cochlea is nonuniform (e.g., the stiffness of the basilar membrane decreases gradually from base to apex) and the wavelength of an harmonic traveling wave decreases monotonically with distance from the

(22) stapes. Although the perturbations that scatter forward-


traveling waves are presumably irregular in arrangement, the spectrum of evoked emissions oscillates quasiperiodically with frequency, betraying the existence of an underlying order, analogous to monochromaticity, in the net reflected wave.

This paper proposes that the simple, empirical form for the cochlear reflectance arises from a novel scattering mechanism--an analogue of Bragg scattering in nonuniform, disordered media--that creates spectral order out of spatial irregularity. More specifically, spectral periodicity emerges--given almost any arrangement of densely distributed irregularities--whenever the peak of the traveling wave has a slowly varying wavelength and an envelope that is simultaneously both tall and broad. Empirical basilar- membrane transfer functions indicate that these characteris-

tics are typical of traveling waves in the healthiest prepara- tions at sound levels near threshold. The tall, broad peak in the traveling-wave envelope guarantees that reflected wavelets originating within the peak have much larger amplitudes than those reflected elsewhere. As a result, wave reflection is

largely localized to a relatively broad region containing many reflectors, which are arrayed, in general, at a superposition of many spatial frequencies. Although each of the many reflected wavelets originates from a different location within the peak region, wavelets reflected by a particular spatial frequency, denoted k, undergo a phase change due to scattering by the irregularity that exactly compensates for the phase change due to wave propagation forth and back to the point of reflection. Such wavelets therefore reflect coherently, and their sum dominates the net backward-traveling wave.

Just as in the corrugated-cochlea model, the phase of the net reflected wave rotates periodically with frequency; in this case, however, the spectral period Af characterizing the oscillations in ear-canal pressure is determined not by the length scale of some preexisting spatial washboarding, but by the wavelength of the traveling wave at its peak, where scattering is maximal. More precisely,

• ̂ (24) Ax = $k x ,

where •x is the wavelength at the peak of the traveling wave. The dominant spatial frequency of scattering (expressed in radians using the length scale l) is therefore

k = 2 ,rlt Ax = 2 l/;(x, (25)

where Xx-----kx/2tr. Equation (24) is the cochlear analogue of the Bragg coherence condition given in Eq. (23). The spectral period Af is given by

Af/f•-. Ax/l = Xx / 21. (26)

The phase shift due to scattering off irregularities arrayed at spatial frequency k is kj((to) and accounts for the smooth variation of the phase of R with frequency. The observed spectral periodicity thus emerges as a consequence of cochlear dynamics, rather than of cochlear geometry. Spectral periodicity is spontaneously created.

In contrast to conventional Bragg scattering in a crystal, which originates from isolated scattering centers arrayed periodically in space, the coherent reflection responsible for

evoked emissions originates from densely distributed, random irregularities encountered within the peak of the traveling wave. When the forward-traveling wave reflects off irregularities with spatial period Ax within the peak, the resulting wavelets, with wave number /c• k, combine in phase and yield a sizable backward-traveling wave. Wavelets scattered by spatial-frequency components far from k combine incoherently and largely cancel one another. Note that certain very orderly distributions of inhomogeneities [e.g., the almost crystalline regularity characteristic of the arrangement of hair cells in the rodent organ of Corti (Wright, 1984)] have only small spatial-frequency components near k and therefore generate little emission.

III. SCA'I-FERING FROM MECHANICAL IRREGULARITIES

In this section we derive an approximate expression for the cochlear reflectance resulting from small, arbitrarily arrayed irregularities superimposed on the smooth spatial variation of parameters responsible for the frequency- position map. In the simplest case, such perturbations represent preexisting spatial irregularities in the mechanical properties of the organ of Corti. When the cochlear response is nonlinear, however, the forward-traveling wave can itself in- duce transient spatial variations in the mechanics that generate reflections (Kemp, 1979; de Boer, 1983; Allen and Neely, 1992). Subsequent sections reconcile the theoretical expression for R with the empirical form given in Eq. (8).

A. Perturbing the wavelength

In an idealized cochlea the impedance of the organ of Corti varies smoothly with position. In the real ear, however, mechanical properties may change discretely from hair cell to hair cell. In addition, the cochlea manifests mechanical imperfections--arising from natural developmental variabil- ity or from trauma--that perturb the smooth variation of parameters responsible for the frequency-position map. For example, the impedance of the organ of Corti may vary irregularly with position as a result of variations in the number or spatial orientation of the outer hair cells (Wright, 1984; Lonsbury-Martin et al., 1988). 1. Simplifying assumptions

Small local variations in the micromechanics of the co-

chlea perturb the traveling wave, giving rise to minute reflected wavelets. Complete characterization of that scattering requires a detailed, three-dimensional description of cochlear geometry and dynamics, including that of the intricate microstructure of the organ of Corti. Since the necessary measurements of cochlear micromechanics are not currently available, we adopt here a more phenomenological approach based on the simple one-dimensional transmission-line model, in which the scattering of waves within the organ of Corti arises from an effective "scattering potential." Such an approach is adequate for exploring the more qualitative mechanisms underlying evoked emission.

We further simplify the discussion by assuming an approximate local scaling symmetry for the unperturbed cochlea, thereby reducing the number of independent variables from two to one (Zweig, 1976). This approximation is cer-


tainly valid for frequencies greater than 3 kHz (Rhode, 1971; Gumruer et al., 1987). Although scaling symmetry is broken at the frequencies for which human evoked emission appears strongest (i.e., roughly 1-2 kHz), the shape of the transfer function changes only slowly with position, implying that an approximate local scaling symmetry about the peak of the response still holds. Thus, the same qualitative mechanisms for generating evoked emission described in this paper are expected to apply even at these lower frequencies.

The approximate local scaling symmetry implies that rather than depending on position and frequency independently, the unperturbed wavelength depends on those variables only in the combination

/3(x,to) = •o/O•c(X), (27)

where t%(x) is the frequency-position map. In accord with experiment (Liberman, 1982; Greenwood, 1961), the frequency-position map •oc(x) is taken to be exponential in the scaling region:

c%(x) = o•%e-X('O, (28) where

X(x)---x/l, (29)

with i the distance over which the characteristic frequency changes by a factor of e. As a consequence of scaling symmetry, basilar-membrane transfer functions T measured as a function of to at fixed x also describe the lraveling displace- merit wave as a function of x at fixed o•. At fixed position, T is the transfer function; at fixed frequency, the traveling wave.

a. An aside about notation. Before proceeding, we review our notation (see also Appendix A). Three related spatial variables (x, X, and/3 at fixed to) are used throughout this paper. The first, with conventional dimensions of length, represents distance along the organ of Corti. The second, X, represents that length measured in units of the disrenee 1. Finally, the dimensionless variable/3= o•/t%(x) is sometimes employed as a spatial variable when evaluated at fixed frequency. By definition, the transfer function T peaks at/3= 1.

The variable/•o(tO) = ra/to% represents the value of/3 at the basal end of the cochlea. For future reference, note that

Eqs. (27) and (28) imply that X and/3 are related by

X = ln(/3//30)- (30)

At frequency •o the position of the peak of the traveling- wave envelope is denoted •( and is given by

k(to) = in( 1//3o), (31)

an expression obtained by setting /3= 1 in Eq. (30). Recall that •(•o) was introduced in Eq. (7) to characterize the logarithmic frequency variation of R. Since higher frequencies are mapped closer to the stapes, • decreases as m increases. In addition, note that since

•(ro) - X(x) = - In/3(x. o•), (32)

functions of/3 can be written, equivalently, as functions of the difference •(-X.

Finally, we summarize the conjugate Fourier variables: t,-,o•, •O-•qo, X*-•c, and -X•-•k. Note that whereas g represents spatial frequency (e.g., a component of the irregular- ties), k represents wave number (e.g., a component of the traveling wave).

2. The wave equation and a method for its solution

The cochlea is represented as a linear, one-dimensional, hydromechanical transmission line. The differential pressure P satisfies the wave equation (e.g., Zweig, 1991)

d2p 1 -- + -- V = 0, (33) d/32 •2

where the dimensionless function :• represents the complex wavelength of the traveling wave (divided by 2,r). The dia- critical tilde atop the wavelength indicates that :• includes perturbations arising from irregularities in the mechanics of the cochlea. In general, the perturbed wavelength • depends on both position and frequency, or, equivalently, on both X and •. The functional dependence on X and •5 individually, rather than on the difference •(-X, implies a violation of scaling symmetry.

Mechanical irregularities scatter waves traveling along the organ of Corti. To analyze that scattering, first imagine "ironing out" the irregularities to obtain a smoothly varying, scaling-symmetric wavelength, X. Traveling-wave solutions for the smooth cochlea can be obtained by using the WKB approximation (Schroeder, 1973; Zweig et al., 1976). The mechanical irregularities can then be reintroduced as a perturbing potential and their effect on the pressure wave computed. This is effected by writing Eq. (33) in the inhomoge- neous form

d2p I AX 2 I- P = P, (34)

where AX 2--- •2_ X2 vanishes in the absence of any irregularities. An approximate analytic solution to the inhomoge- neous equation, based on WKB solutions to the homoge- neous equation, can be obtained using the Born approximation, in which multiple scattering is neglected (e.g., Shera and Zweig, 1991b).

B. The theoretical form of the cochlear reflectance

The solution to Eq. (34) for P comprises a superposition of incident and reflected waves P-+ traveling in opposite directions along the basilar membrane. We are interested in the ratio R--P-IP • evaluated at the basal end of the cochlea

where its frequency dependence has been determined experimentally (Shera and Zweig, 1993a). Since measurements of the cochlear reflectance typically yield magnitudes of order 0.2, secondary scattering can be neglected. Appendix C shows that the cochlear reflectance can then be approximated as a sum over scattering locations X of all wavelets reflected a single time by the irregularities:


g(•)•p • stap es• floe dx •(X,•)T2(•-X), (35) where

T(,•- X) -• T[13(,•- X)] (36)

is the basilar-membrane transfer function;

i /[2 i

e•- 2•r02 ;•2 AX2•-' 2'-•00 Ax2 (37) is the scattering potential; and T02 is a constant. The transfer function enters as T 2 because in traveling to and from the point of reflection, the scattered wavelet suffers a total phase shift equal to twice the one-way phase shift experienced in propagating from the stapes to X [plus any phase shift due to the irregularities •; compare Eq. (21)]. The integral sums all such wavelets scattered from all points X along the organ of Corti.

Given a description of cochlear mechanics reducible to an equivalent one-dimensional model, Eq. (35) predicts the cochlear reflectance resulting from small perturbations in the mechanics. With the help of Eq. (4) relating R to ear-canal pressure, that prediction can be compared with empirical emission spectra.

C. The place-fixed nature of the irregularities

Note that if q depends solely on the difference •(-X (i.e., if • scales), then the integral (35) for R evaluates to a constant, independent of frequency. When • scales, varying the frequency merely translates the integrand • T 2 along the • axis and leaves the integral unchanged. The strong frequency dependence of R--that is, the rapid clockwise cir- cling of its phase--therefore implies that • must depend on X (and, in general, on •(), and not simply on the difference $(-X. In other words, the spectral symmetry of primate evoked emissions requires a breaking of scaling symmetry.

Thus, if the transfer function scales, the scattering potential cannot (and vice versa). Consequently, in a scaling- symmetric cochlea the irregularities responsible for evoked emission cannot simply move with the wave envelope, as they would, for example, were those irregularities created by the wave itself through some nonlinearity in the mechanics acting near threshold as a "source" of backward-traveling waves (Kemp, 1979; de Boer, 1983; Allen and Neely, 1992). Rather, the irregularities must be fixed in place and exist independently of the traveling wave. This argument recapitu- lates the reasoning underlying the conclusion that "wave- fixed" (or, more precisely, scaling-symmetric) emission mechanisms cannot explain the long latencies characteristic of evoked emission in human ears (Kemp and Brown, 1983; Kemp, 1986; Strube, 1989; Shera and Zweig, 1993b).

In the reflection mechanism proposed here, small irregularities in the mechanics of the organ of Corti provide the requisite breaking of scaling symmetry. Although the region of maximum reflection moves with the wave envelope, the micromechanical irregularities (the "reflectors") remain fixed in place. The irregularities partially reflect the traveling

wave, but when R is small they do not significantly affect either the shape or scaling-symmetric form of the basilar- membrane transfer function.

IV. THE ORIGIN OF SPECTRAL PERIODICITY

In this section we reconcile the empirical and theoretical forms of the cochlear reflectance and indicate what that rec-

onciliation implies for cochlear mechanics. Theory and experiment are combined to obtain the possible forms of the scattering potential •. Two possibilities are discussed, the first corresponding to a corrugated cochlea (Strube, 1989) and the second to a cochlea with coherent backscattering from random irregularities. The scattering mechanism in- volved in this second case is then explored in some detail.

A. The making of spectral symmetry

The origin of spectral periodicity can most easily be understood by assuming that the scattering potential depends only on position, so that •--•(X). Our numerical simulations of scattering from a variety of irregularities suggest that this approximation is generally good to within a few percent. Equation (35) for R then becomes a spatial convolution,

R(k)• f_•o•dx e(x)r2(;•-X)=e©r2; (38) and the transform ,Jr{R}, illustrated in Fig. 2, emerges as the simple product

•½r{R} • •'{ e }•½r{ T2 } . (39)

The transform •'{•} consists of the spatial-frequency components of the irregularities •(X). The function o.•{T 2} multiplying ,3v{•} acts as "spatial-frequency filter" whose form determines the relative contributions to R of the different

spatial-frequency components present in the distribution of irregularities.

At fixed frequency the square of the traveling wave can be represented as a sum of forward-traveling plane waves of wave number k and amplitude •{T2}(k). Since T 2 is a minimum-phase function (Zweig, 1976; Koshigoe and Tubis, 1982), its spectrum is one sided:

d•{r2}(k < 0)=0; (40)

in other words, the plane-wave decomposition of a forward- traveling cochlear wave involves only forward-traveling plane waves. As a result, only non-negative spatial-frequency components in .3r{•} contribute to the reflected wave.

Empirically, the amplitude I:{R}I is a strongly peaked function of q0, roughly Gaussian in form (see Fig. 2). Equa- tion (39) therefore implies that the product I•{•}•r2}l must also be a strongly peaked function. Equivalently, the uniformly rotating phase of R indicates that the net reflected wave is locally "monochromatic" (i.e., comprised of reflected wavelets having a narrow range of wavelengths). The strongly peaked form of I•{g}l can, in principle, arise in either the spatial-frequency transform I.l}l or the filter Iq{r2)l . These two possibilities underlie radically different views of the origin of evoked emissions. In either case, how-


ever, spectral periodicity results from the dominance of wave scattering by a narrow range of spatial frequencies in the scattering potential.

If the peak originates in 13•{0}1, spectral pedodi:ity reflects a corresponding periodicity in the geometric arrangement of irregularities. When a large harmonic corrugation appears in the mechanics, the reflected wave is monochromatic because only a narrow range of spatial frequencies (namely, _AK about k) dominates the scattering potential. As a result, only waves with a corresponding range cf wave numbers (namely, k in the interval k_+AK) undergo significant reflection. Spatial periodicity in, spectral periodicity out: This, in a nutshell, is the corrugated-cochlea model proposed by Sambe (1989). With appropriate choice of the parameters k and AK, the corrugated-cochlea model can be nrtade to approximate the empirical form of the cochlear reflectance; sinusoidal oscillations in the mechanics of the organ of Corti have not, however, been observed (Wright, 1984; Lonsbury- Martin et al., 1988).

If, instead, the peak originates in the filter amplitude •{T2}[, the frequency-dependence of R is determined prin- cipally by the form of the traveling wave, and the observed spectral periodicity originates through a process that we dub "coherent reflection filtering." In this case, the reflected wave is nearly monochromatic because the incident wave is nearly monochromatic. More precisely, a narrow range of spatial-frequency components dominates the scattering because a narrow range of wave numbers (namely, +Ak about •) dominates the incident traveling wave throughout the region of that scattering. Waves with these dominant wave numbers reflect coherently from corresponding spatial frequencies (namely, K in the range/•+Ak) in the distribution of irregularities, giving rise to a large backward-traveling wave. The values k and AK, input parameters in the corrugated-cochlea model, are here determined by the cochlear mechanisms responsible for shaping the traveling wave. In support of this alternate possibility, note that [•'{R}I and I{r2}l are indeed qualitatively similar in shape: The transfer function is a strongly peak function whose transform, like 3•{R}, is also strongly peaked.

In subsequent sections we make these qualitative observations more precise, explore the physical meaning of this alternative model, and discuss its consequences for the origin of otoacoustic emissions and for noninvasive probes of cochlear mechanics.

B. Coherent reflection filtering

In coherent reflection filtering, spectral periodicity arises because the filter .•{T 2} has a strongly peaked form re.•ulting from the dominance of a narrow range of wave numbers in the plane-wave decomposition of the forward-traveling wave. Although the wavelength of the traveling wave changes continuously with position, only the narrow range of wavelengths within the peak contribute substantially to •'{T2}. If the amplitude of the spatial-frequency transform J•{Q} is roughly constant, the approximate proportionality

(41)

determines the spectral characteristics • and Aqo of .•7'{R} through the equations

• and A•p•-•Ak, (42)

where • and Ak are the position and width of the peak of the Fourier transform of T 2 considered as a function of •(-X with • fixed. Note that when the irregularities are random, averaging Eq. (39) over many subjects reduces the fluctuations in R resulting from • and yields

(•R} )or(•{ r•-} }. (43)

The mean frequency spacing Af follows from Eq. (11):

Aflf• 2 •'/•:; (44) the larger the dominant wave number • of the Fourier transform of T 2, the smaller the mean frequency spacing Af between oscillations in ear-canal pressure. The "uncertainty principle" for Fourier-conjugate variables implies that the widths of T 2 and .37{T 2} are reciprocally related:

A xAk• > 1; (45)

the broader the traveling-wave envelope, the narrower the spatial-frequency filter.

1. A phenomenological model

We illustrate these general remarks and explore the relationship between the shape of the traveling wave and the spectrum of evoked e•nissions by considering an idealized case ia which the traveling wave has a roughly Gaussian envelope and a linear phase. This phenomenological model captures the general features of the traveling wave essential for understanding the origin of spectral periodicity while greatly simplifying the analysis. (A more realistic form for T 2 is analyzed in Appendix D.) The simplified T 2 has the form

r2(,•- X) = •'2e - ('•- x}212(Ax)2e i(•- X)0', (46) with maximum amplitude •- at •, width AX, and phase slope 0'. Measurements of T indicate that 0'>0 (e.g., Rhode, 1971). For future reference, note that 0' can be written in terms of the wavelength of the traveling wave. The wavelength X x is defined as the distance over which the traveling- wave phase changes by exactly one cycle: 7

•x--= •x/2 ,r=-- 2/0'. (47) (The factor of 2 arises because 0' is the slope of /_ T 2 =2/_T.)

The spatial-frequency filter, given by .•{T2}, also has a Gaussian amplitude and a linear phase. In particular,

12jV:[T2}I = •2e -(• '•)2/2{ Ak)2, (48) where

k = O' = 21• x , (49) and

tax. (50)


Note that Eq. (48) agrees with the roughly Gaussian form for I.{g}l observed experimentally [compare Eq. (16), with q0=k], confirming Eq. (41). Evidently, the spectral characteristics of otoacoustic emissions are controlled by the shape of T 2 about its maximum, in particular by the size of its wavelength and the width of its envelope.

Of course, actual traveling-Wave envelopes are less symmetric than the simple Gaussians considered here, and their phases deviate from perfect lineadty. Consequently, actual spatial-frequency fi•lters .½V{T2} are also asymmetric, peaking at wave numbers k only approximated by Eq. (49) and having widths A k somewhat greater than the minimum value 1/A X obtained from the uncertainty relation for Gaussian transforms.

Since scaling symmetry is only an approximate local symmetry, the shape of the traveling wave--and hence the dominant wave number •: and its spread Ak--presumably varies slowly with position in the cochlea. Since traveling waves are minimum-phase functions (Zweig, 1976), their a•mplitude and phase are related. Consequently, variations in k and Ak are not independent. Although analyticity properties reflect global characteristics of the transfer function, an empirical rule of thumb associates narrow peaks (small AX) with steep phase slopes (or, equivalently, small •x) and broad tuning (large AX) with gentle slopes (large •x)' The rule of thumb therefore predicts that narrower traveling-wave envelopes (smaller AX) lead to smaller relative emission spacings (smaller /Xf/f• «•'x)' Data consistent with these predictions are presented in Sec. VIA.

The scaling-symmetric form of T implies an alternate, time-domain interpretation of the slope 0'. Note that /-T(•-X) can be varied by fixing • and varying X, or by fixing X and varying •(. In the first case, T represents the traveling wave and the slope of its phase determines the wavelength [e.g., through Eq. (47)]. In the second case, T represents the transfer function, and slope of its phase represents the transfer-function group delay. Explicit calculation of the derivative yields

d/_T d/_T Orr------ro do• peak-- •'• peak' (51)

Equation (46) for the simplified T 2 implies that

wrr = 0'/2. (52)

By using Eqs. (42) and (49) for • and •:, one obtains

½/2 *r= 2 rr/ r, (53)

where r is the stimulus period. In other words, the parameter ½/2,r defining the emission latency through Eq. (10) equals twice the group delay for wave travel between the stapes and the peak of the transfer function, expressed in units of the stimulus period. Equations (49) and (53) imply that the wavelength of the traveling wave and the group delay of the transfer function, expressed in periods, are reciprocally related:

•xrr/r= 1. (54)

2. A localized region of coherent reflection

To understand the scattering mechanism more fully, it is helpful to imagine decomposing the irregularities into spatial-frequency components and examining the reflection due to each component separately. Using the scattering potential •=2 cos(tcx)=eiKX+e-iKx and substituting the simplified transfer function into the convolution (38) for R yields

R(•) • fpeakdXeiKXe-(•-x)2/2(aX)2e2i(•-x)/•x, (55) where Eq. (40) allows one to ignore the negative spatial- frequency component e i•x in Q. The peak region dominates the scattering because waves reflected near •( have much larger amplitudes than those reflected elsewhere. The cochlear reflectance R(•) is obtained by convolving Q with a function of Gaussian amplitude and linear phase. Over a scattering region of size +/XX about •((ro), the irregularity e i•x dx is (1) Gaussianly weighted according to its distance from the peak of the traveling wave; (2) multiplied by the phasor e2i(•-x)/'(x representing the phase shift for roundtrip wave propagation between the peak and the scattering location; and (3) added to form the integral. Wavelets combine coherently when the phase of the scattering integral is stationary; that is, when

d

• (•cX- 2X/•x): 0. (56) Thus, when many spatial frequencies •: appear in the irregularities, wavelets scattered within the peak region by components arrayed at spatial frequencies

•= 2/• (57) sum coherently and dominate the net reflected wave. Wave- lets scattered about the peak by irregularities arrayed at other spatial frequencies combine incoherently and tend to cancel one another.

Equation (56) implies that wavelets combine coherently when the phase change tc•y due to scattering by the irregu-

larities exactly compensates for the phase change 2 X/ •x due to wave propagation forth and back across the scattering region. The reciprocal relation between A X and Ate follows immediately: The broader the peak (i.e., the larger AX), the greater the number of scattered wavelets combining in the sum and the more complete the cancellation of contributions from spatial frequencies away from ;r (i.e., the smaller Ate). In addition, the degree of coherence (extent of off-frequency cancellation) should increase with the number of wavelengths of the traveling wave that fit within the scattering region. And, indeed, Eqs. (49) and (50) imply that the selec- tivity, or Q, of the spatial-frequency filter varies as the ratio of the size of the scattering region to the wavelength:

Qk- 2Ak - AX/•x. (58) The mechanism of coherent reflection filtering is sche-

matized in Fig. 3, which illustrates the computation of the integral


Incoherent reflection (a)

Coherent reflecti•. .. . .•• .-"'"'-:.: " ..

,ncoherenl reflection • (c)

FIG. 3. Coherent reflection filtering. In each panel, the sinusold ( ..... ) represents a particular spatial-frequency component cos(Kx) in the distribution of irregularities upon which a segment of the function Re{T 2} near its peak ( ) has been superposed (arbitrary vertical scale centered about zero). The spatial frequency •: of the irregularities increases frorn top to bottom. tn panel (a) the spatial period 2*rIK is larger than the wavelength •x/2 of T z at its peak (the wavelength of T 2 is half that of T); in panel (b) the two are approximately equal; and in panel (c) the spatial period is smaller than the wavelength. For each panel, the real part of the corresponding reflectance is proportional to the integral of the product of :he two curves. When the spatial period of the irregularities matches the wavelength of T 2 at its peak (i.e.. when 2rr/K•.•/2), wavelets reflected at differera locations within the peak region combine in phase and the reflec.ance is large (middle panel). Wavelets reflected by spatial frequencies outside this narrow range (e.g., top and bottom panels) combine out of phase and tend to cancel one another. Thus, when many spatial frequencies are pre•nt simultaneously. contributions from spatial frequencies lying outside the passband of the "filter" are suppressed and the net backward-traveling wave is dominated by wavelets reflected from spatial frequencies determined by the wavelength of the traveling wave at its peak.

Re{R}oc •peakdX cos( (59)

at fixed frequency 0 for three different values of K. (A similar diagram involving Im{T 2} could be constructed for Im{R}.) Each panel shows the real part of T 2 about its peak superposed on a sinusold representing a particular spatial- frequency component cos(•c.y) in the distribution of i•regularities. The contribution of each component is obtained by integrating the product of the two functions along the spatial axis. When, as in the middle panel, the spatial period of the irregularity matches the wavelength of T 2 at its peak, wave-

lets reflected ti'om difikrent locations within the peak combine in phase with one another and the integral is a maximum. (Note that the wavelength of T 2 is half the wavelength of T.) But, when the spatial period of the irregularities is either considerably larger (top panel) or smaller (bottom panel) than the peak wavelength, wavelets reflected from different locations largely cancel one another and their total contribution to the backward-traveling wave is small.

3. A narrow-band "irregularity spectrogram"

The scattering mechanism responsible for the simple, smooth form of the cochlear reflectance can also be viewed

from a perspective analogous to "time-frequency" analysis. Equation (38) for R can be written in the form

f (60) After substituting the simplified traveling wave from Eqs. (46) and (,47), one obtains

fdx ..•-'{.•{elTl2}}e 2i(i' x)/•x, (61) where

] T] 2 = •2 e - (k-,Y)2/'2{AX)2. (62) Expanding the inverse transform and switching the order of integration yields

d K ei•Xe2il,• - x)l• x R(,ff) = .el tl

=.½qelrl}l_ae '• (with k=2/•). (63) The cochlear reflectance thus has the form

R(,• )• S( ;•,k )e i•, (64) where

$(.•, g) --.-•{ e (x) ITI 2(.•_ g)} (65)

is the •noving Fourier transform (e.g., Allen and Rabiner, i 977) of the irregularities • computed using the narrow-band windowing function IT[ • centered at •. The function S(X,g) represents a complex "irregularity spectrogram" in which the signal under analysis is the spatial distribution of irregularities and the roles conventionally played by time and frequency are played, respectively, by position X and spatial frequency •c.

When compared with the empirical form (8) for R (with •= •:), Eq. (64) indicates that

R0--sc•,k). (66)

At frequency 60, the complex reflectance amplitude R 0 is given by the value of the spectrogram at the point (•o),k) in the "position-spatial frequency" plane. Varying the frequency moves the analysis window (i.e., the square of the traveling-wave envelope) along the organ of Corti. Equations

2029 J. Acoust. Soc. Am., Vol, 98, No, 4, October 1995 G. Zweig and C. Shera: Origin of spectral periodicity 2029

(64) and (66) imply that the cochlear reflectance R($() is equal to the product, evaluated at the peak of the traveling wave, of (1) the scattering potential e ikx for which scattering within the peak is coherent and (2) the spatial-frequency component at k present in the surrounding irregularities.

Since k = 21• x, the phase shift k• is equivalent to that experienced by a plane wave of wavelength •x traveling the roundtrip distance 2• between the stapes and the scattering location.

V. PREDICTING EMISSIONS IN THE SQUIRREL MONKEY

This section uses the model of cochlear mechanics de-

duced from measurements of basilar-membrane transfer

functions (Zweig, 199 l) to compute the filter •q{T 2} and predict the characteristics of otoacoustic emissions at 8 kHz in

the squirrel monkey. A numerical simulation, based on the addition of random perturbations to the model mechanics, illustrates the origin of spectral periodicity and verifies the analysis culminating in Eq. (39).

A. The estimated wavelength

Rhode's measurements of basilar-membrane motion in

the squirrel monkey (Rhode, 1978; Rhode, 1974) have been used to obtain an empirical estimate of the wavelength X by solving the cochlear inverse problem in the absence of the small perturbations considered here (Zweig, 1991). That empirical estimate--valid in the squirrel monkey at frequencies in the local scaling region above approximately 3 kHz-- accurately describes the motion of the basilar membrane at stimulus levels near threshold where the mechanics of the

cochlea are linear and evoked emissions are proportionately the largest.

The estimated wavelength, &, was shown to have the form s

(4N)[e) 2•' 1 -/32 + i Bt3+ pe- 2•rit•tt, (67)

implying that a section of the organ of Corti responds as though it were a negatively damped harmonic oscillator (since, empirically, 8<:0) stabilized by a time-delayed feedback force of strength p>0. (The feedback strength p should not be confused with the scattering potential •.) Whereas the stabilizing force acts with a time delay of approximately it•<l• cycles of the local oscillator period, the negative damping originates through the action of a fast-acting positive-feedback force with a delay small compared to that period. The parameter N denotes the approximate number of wavelengths of the traveling wave present in the cochlea in response to sinusoidal stimulation (Zweig et al., 1976). Al- though its value presumably varies with the species (and slowly with characteristic frequency within an individual), N•5 near 8 kHz in the squirrel monkey.

In the absence of irregularities, the behavior of the solutions to Eq. (34) for P is determined by the location of the poles of l/X 2 in the complex/3 plane. The parameter values found by Zweig (1991) imply that the estimated wavelength has, among an infinite series of poles, two closely spaced poles just above the real axis near/3= 1. By making both the

90

80

70

60

50 40

30 20 l0

0

-10 1

-7

I I

0 1 2 3 4 5 6 7 8

Frequency (kHz)

FIG. 4. Theoretical and empirical transfer functions in the squirrel monkey. The figure shows the theoretical basilar-membrane transfer function ( ..... ) computed by using the estimated wavelength •t½ with parameter values con- strained by requiring that its two zeroes near/3= 1 coincide. The transfer function is compared with extrapolated measurements (0) of basilar- membrane motion and with the empirical transfer function derived from the wavelength obtained by solving the eochlear inverse problem ( ); both are taken from Fig. 10 of Zweig (1991). Parameter values are 8=-0.1223; p=0.1309;/x=1.746; N=5.24; and o•r/2•r = 7.75 kHz.

wavelength and its derivative small, the series of poles creates the tall, broad peak of the transfer function (Zweig, 1990). By choosing slightly different parameter values, the locations of the two adjacent poles can be made to coincide without significantly altering the corresponding transfer function (see Fig. 4). Indeed, requiring that the two poles coincide at a given distance from the real axis uniquely determines the parameters •, p, and it (for/z in a neighborhood of 12). 9 For ease of analysis, we adopt this simpler "double- pole" form of the estimated wavelength Xe; the locations of its poles are plotted in Fig. 5.

B. The squirrel-monkey filter

The form of the filter •7•T2} predicts the characteristics of squirrel-monkey emissions near 8 kHz. Figure 6 plots the model I•{T2}l computed numerically. As suggested by the foregoing analysis, the filter is roughly •Gaussian in form (compare Fig. 2). Peaked about the value k/2•r•56, the filter predicts an emission latency •/• of approximately 56 cycles and a spectral spacing

Aflf• (squirrel monkey), (68)

a quantity not yet measured. Although the predicted latency differs by more than a factor of three from the value (•15) typical of human emissions near I kHz, Sec. VIA presents evidence suggesting that the value is consistent with the

2030 d. Acoust. Soc. Am., VoL 98, No. 4, October 1995 G. Zweig and C. Shera: Origin of spectral periodicity 2030

0.½

0.3

0.0 0.5

I I

ß {oo I I

1.0 1.5

I I __ 2.0 2.5 •.0

FIG. 5. Poles of 1/A• in the squirrel monkey. The figures shows Ihe pole locations (O) for the double-pole form of IlAr(0 in the • plane near •=1 computed as in Zweig (1991, Fig. 14). The complex variable • equals fl along the real axis. The pole closest to the real axis, shown with a large dot and denoted •0o, is double (i.e., two poles coincide). Parameter values are the same as those listed in the caption to Fig. 4 and were determined by firing the experimental transfer function (yielding lm{•0o}•O.03)..•q infinite string of poles lies on the dotted line outside the range of the: figure. Since all poles appear in the upper half of the complex plane, the model is stable. In the simulations described below, perturbations in the mechanics were introduced by randomly varying lm{•0} with position.

spacing of human spontaneous emissions at higher frequencies.

The filter has bandwidth Ak/l•}, implying that A•0/• « (squirrel monkey), (69)

in approximate agreement with the width of the distribution of frequency spacings (A•0/•0.25_+0.1) characterizing hu-

1.0

0.4

0.2

0.0 0

I I I

I•.•l 100 200 300 400 500

FIG. 6. The bandpass spatiai-fi'equency filter for the squirrel monkey. The figure plots the function [:•{T•}l, normalized to a maximum value of one, computed by using the estimated wavelength. The horizontal axis shows wave numbers up to spatial frequencies characteristic of variations on the scale of a hair cell: •d2•'=l/h•500, where l•5 mm and h•-10/am. The vertical dashed line (---) indicates the principle wave number ,•/2a' com- p•uted from Eq. (49). Here. for characteristic frequencies near 8 kHz, k/2•56, corresponding to the width of roughly nine hair cells. Approxi- mately Gaussian in form• Io•{TZ[I represents a bandpass "spatial-frequency filter" satisfying Ak/fc• (compare Fig. 2 for humans).

10

5

0

-5

9.00 8.75 8.50 8.25 8.00 7.75 7.50

Chorocteristic Frequency (kHz) 7.25

FIG. 7. Simulated random irregularities in the squirrel-monkey organ of Corfi. The figure shows the spatial variation of the irregularities used in the simulation shown in Figs. 8 and 9. The lower panel plots deviations in lm{•0a}, where •00 is the location of the double pole of I/X• found near the real axis close to t= 1 (compare Fig. 5). The pole location. expressed as percent deviation {x 100) from its unperturbed value is plotted as a function of the local characteristic frequency. Note that the characteristic-frequency axis is reversed; the distance from the basal end of the organ of Corti increases along the axis. The upper panel indicates the approximate size of the scaRering region by plotting a temporal snapshot of the square of the traveling wave (i.e., Re•T•'ei'•t[ and its envelope) computed by using the estimated wavelength at a stimulus frequency of 7.75 kHz. lm{•0o} was varied every 10/zm by picking random values from a Gaussian distribution with a width corresponding to a variation of 0.01% of the mean. The width of the distribution was chosen to yield a cochlear reflectance of mean amplitude IRIs0.2 similar to that measured experimentally in humans (Shem and Zweig, 1993a).

man spontaneous and evoked otoacoustic emissions. Esti- mates of the width of the transfer function (obtained from Fig. 4) indicale that A,yAk•l.4, and hence

AXA,p • 1.4 (squirrel monkey). (70)

Measurement of emissions around 8 kHz in the squirrel monkey would provide an important test of these predictions.

C. Simulating squirrel-monkey emissions

This section describes a simulation based on the esti-

mated wavelength in which irregularities in the mechanics of the organ of Corti are introduced by randomly varying the strengths of the fast and slow feedback forces. Those two forces are responsible, respectively, for creating a net negative dmnping and for stabilizing the resulting oscillator (Zweig, 1990). The form of the irregularities is motivated by the conjecture that the feedback strengths manifest small spatial variations. Parameter values were varied in such a way that the distance between the real axis and the double pole in 1/•[• changed randomly and discontinuously on the scale of a hair cell. The corresponding fluctuations in the location of the double pole are illustrated in Fig. 7. Other types of random mechanical perturbations yield qualitatively similar results. Although the analysis presented above neglects higher-

2031 J. Acoust. Soc. Am., Vol. 9B, No. 4, October 1995 G. Zweig and C. Shera: Origin of spectral periodicity 2031

1.0

0.8

--• 0.6

Q.

E 0.4

0.2

0.0 0

I I I I

-9 7.50 7.75 8.00 8.25 8.50 8.75

Frequency (kHz)

-5

30

I I I I

I I I I

I I 7.50 7.75 8.00 8.25 8.50 8.75

Frequency (kHz)

FIG. 8. Simulated cochlear reflectance for the squirrel monkey. The figure plots R computed from a simulation based on the random irregularities illustrated in Fig. 7. As predicted, the phase of R noodles about •((to) and appears almost linear over several spectral oscillation periods. The amplitude of R varies more slowly; its fractional change over intervals of size A f---corresponding to a full rotation of the phase--is typically small. Both amplitude and phase are in qualitative agreement with emission measurements in humans.

order scattering, the numerical simulations discussed here represent calculations performed to all orders in the scattering potential.

Figures 8 and 9 illustrate, in two equivalent but comple- mentary ways, the stimulus-frequency emissions produced by the model. The first figure plots the cochlear reflectance R(to); the second shows the resulting quasiperiodic oscillations that appear in the cochlear input impedance. ]ø The cochlear impedance has been normalized to its value in the absence of apical reflections (i.e., to its value at high stimulus amplitudes or in the corresponding unperturbed cochlea):

z(•o;R) l+R(•o)

(71) Equation (71) assumes that the tapering symmetry deduced from measurements of the cochlear input impedance in cats (Shera and Zweig, 1991a; Puria and Allen, 1991) applies to squirrel monkeys as well.

The phase of R in Fig. 8 varies almost linearly, changing by about nine cycles over the range of the figure for an average frequency spacing Aft-55 Hz, in agreement with the periodicity predicted by Eq. (68). Since the scattering region is much larger than the width of a single hair cell, the fine structure present in the irregularities has been "smoothed out" in IRI. The smoothing obscures the relation between the local pattern of irregularities and the value of I R[ at corresponding frequencies (compare Figs. 7 and 8).

FIG. 9. Spectral oscillations in the simulated cochlear input impedance of the squirrel monkey. The figure plots the normalized cochlear input impedance E(w) corresponding to the reflectance shown in Fig. 8. Despite the random nature of the irregularities, the amplitude and phase of the cochlear input impedance manifest a pronounced periodicity, with peaks and valleys appearing at intervals predicted by Eq. (68). The irregularity apparent near 8.25 kHz resembles the "anomalous regions" found in human stimulus- frequency emission curves (Shera and Zweig, 1993a). Consistent with its definition as a ratio of driving-point impedances, E(to) is a minimum-phase function, in agreement with the measured analyticity properties of empirical emission spectra.

Unlike actual emissions measured in the ear canal--

which have been filtered by the middle ear and so appear superposed on a slowly varying background (compare Fig. 1)--the oscillations in E(to) appearing in Fig. 9 were computed near the stapes at frequencies to <• •Oc0 and therefore oscillate about an almost constant baseline. Although the perturbations responsible for the wave scattering are disordered on the scale of a hair cell (i.e., the parameters change discontinuously every 10/.•m), the resulting oscillations in •,(to) show considerable spectral regularity, with peaks and valleys arrayed nearly periodically, as in human emission spectra. Since the irregularities are small, dense, and randomly distributed, coherent backscattering generates sizable emissions without undue effect on basilar-membrane tuning or an ob- vious breaking of local scaling symmetry in the transfer functions.

The periodicity apparent in the spectra can be exhibited more explicitly by performing a Fourier analysis of the cochlear reflectance. Figure 10, analogous to distributions assembled from human data in Fig. 2, gives I.•½'•{R}I computed from Fig. 8, together with the results of averaging the Fou- rier transforms of 1000 similar simulated reflectances, each

corresponding to a different random arrangement of irregularities. A pronounced maximum at the value ½/2,r•52 is readily apparent. By contrast, the spectral density computed


A 20

10

I I I I

I I lOO

I o 200 ,oo

1.0

0.0

{: (•)

0 100 200 •00 •00 500

FIG. !0. Spectral periodicity without cochlear corrugation. Panel (a) plots •{R}I computed using the simulated reflectance from Fig. 8 ( ..... ) and the mean value (].•-•"{R}[) ( ) computed from 1000 simulations performed by varying the pattern of randomly generated irregularities. To minimize end effects, the reflectances were Hamming-windowed before the transform was computed. Fourier transforms were taken with respect to the logarithmic frequency variable ,• = -in(osl•o%). The conjugate Fouricr-uansform variable extends along the horizontal axis. The distribution shows a pronounced peak of width A•p centered about ½ {compare Figs. 2 and 6). Panel (b) gives the corresponding distribution of spatial frequencies t• present in the irregularities used to generate the emissions. The dotted cup, e ( ..... ) corresponds to the irregularities shown in Fig. 7, the solid curve (- ) to the average of 1000. In both cases, the fluctuations in Ira{stoa} were Ham- ming windowed and Fourier transformed with respect to the spatial variable X- The conjugate Fourier variable extends along the horizontal axis. The distribution shows no pronounced maxima or minima corresponding to the peak seen in the top panel. The fact that the spectrum of the irregularities is not white, but rolls off at high spatial frequencies reflects the existence of an underlying "discretizafion scale" over which the mechanical parameters remain roughly constant. In the simulations described here, the discretization scale corresponds to the width of a single hair cell.

for the underlying irregularities has no such corresponding structure: The spectral periodicity apparent in the eatssion curves has no spatial correlate in the mechanics.

Note that the slight difference in peak locations between .½"{T 2} illustrated in Fig. 6 (which peaks at fd2rr•56) and the averaged spectra ( ,•'{R}l ) shown in Fig. 10 (which peaks at ½/2•-•52) arises because the mechanical irregulafitie.• used in the simulation vary with frequency at fixed position (i.e., the simulated Q depends on both X and •). Using irregularities in which Q varies only with position (i.e., by perturbing the damping) n eliminates this small discrepancy.

In Fig. 8 the amplitude of R noodles slowly about an average value of roughly 0.2; corresponding variations appear in the phase. These variations, which reflect the

-lO

-2o

-3o

I I

7.5 8.0 8.5 9.0 9.5 10.0

Frequency (kHz)

FIG. 11. Spectral "grouping" arising from amplitude fluctuations in the cooblear reflectance. The figure plots IRI from Fig. 8 over a wider frequency range and using a logarithmic amplitude scale. Fluctuations in ]R] give rise to the appearance of regions of relatively prominent emissions separated by narrower regions in which emissions become small and unmeasurable. The spectrum is reminiscent of those observed experimentally (e.g., compare with Fig. 6 of Wilson, 1980).

frequency-dependence of the complex amplitude R 0, arise from random spatial fluctuations in the irregularities initiating the reflection. Large fluctuations in R o can give rise to the phenomenon of "spectral grouping" in which regions of relatively large evoked emissions are separated by narrower regions in which emissions become small or unmeasurable (e.g., Wilson, 1980). For example, Fig. 11 plots IRI from Fig. 8 over a wider frequency range and on a logarithmic amplitude scale (to facilitate comparison with experiment). The simulation yields a spectrum qualitatively similar to amplitude spectra measured experimentally (for example, compare with Fig. 6 of Wilson, 1980).

1. Changing the shape of the transfer function

We now examine how the cochlear reflectance depends on the shape of the transfer function. Moving the unperturbed value of the double pole in l//i• further from the real axis changes the shape of the transfer function by lowering its height and increasing its width. With the broadening of the transfer function comes a reduction in the slope of its phase (i.e., an increase in ;•) and a corresponding decrease in the dominant wave number,/• (see the discussion in Sec. IV B 1). This downward shift in the peak of the spatial- frequency filter predicts an increase in the spectral period and a corresponding decrease in the emission latency.

These effects are illustrated in Figs. 12-14, which plot, respectively, the functions IT21 and corresponding filters 1.7{T•}I, the oscillations in the cochlear input impedance E(co) due to emissions, and the cochlear reflectances R for a series of transfer functions obtained by moving the double pole successively further from the real axis. Along with the amplitude and phase of R, Fig. 14 plots the reflectance group delay rn measured in units of the stimulus period r.

Figure 14 illustrates that as the transfer functions broaden, the dominant wave number • shifts to lower values. The changes in T •' are reflected in the emission spectra: The decrease in height produces a corresponding decrease in the reflectance magnitude IRI; and with the broadening of the wave envelope (downward shift in •), comes a decrease in


1.0

(•) 0.8

0.8

-- 0.4

0.2

0.0 -0.•5 -0.10 -0.05 0.00 0.05

1.0

0.8

0.$

0.4

0.2

0.0 0 50 100

I 150

I (b)

200

FIG. 12. Peak shifting in the spatial-frequency filter due to shape changes in the transfer function. Panel (a) plots IT2I on a linear vertical scale for a sequence of transfer functions obtained by moving the double pole in l/X• further from the real axis. The parameter lm{•00} varies between 0.03 to 0.04 in equal steps. The transfer functions are normalized so that the tallest has a maximum value of one. As the poles move up. the transfer-function peaks decrease and broaden. Associated with these increases in width, which are

camouflaged here by the large changes in height, are corresponding decreases in the slope of the phase in the peak region (not shown). Panel (b) plots the corresponding filters •.•[T2}[, normalized similarly. As the transfer functions broaden, the filter center frequencies, located at •/21r, shift to lower values.

I/-R1 that decreases the reflectance group delay r R and increases the average spectral period A f, all in accordance with Eqs. (42)-(45).

Although the linearty of the model precludes any quantitative exploration of the level dependence of evoked emissions, many of the qualitative features of that dependence may originate in the mechanisms described here. Note, for example, that raising the stimulus level out of the linear regime near threshold produces changes in the shape of the "describing function "tz qualitatively similar to those effected in the transfer functions above (e.g., Rhode, 1978; Ruggero and Rich, 1991). And, indeed, the emission spectra illustrated in Fig. 13 are strongly reminiscent of level series measured in humans. For example, whereas empirical emission amplitudes decrease rapidly with stimulus level, corresponding increases in their spectral period, while apparent, are proportionally much smaller (Shera and Zweig, unpublished observation).

co

o

-5

3o

-30

7.50

I I I I

7.75 8.00 8.25 8.50 8.75

Frequency (kHz)

FIG. 13. Changes in the cochlear input impedance with transfer-function shape. The figure plots E(o•) as in Fig. 9 for the sequence of shorter, broader transfer functions illustrated in Fig. 12(a). As expected, the oscillation amplitudes decrease as the transfer function height diminishes. Less apparent in this representation is the concomitant increase in the mean spacing Af pre- (ticted by the shifts in ,• illusu:ated in Fig. 12(b). Such changes are more readily observed in the cochlear reflectance (see Fig. 14).

D. Simultaneous ear-canal and basilar-membrane measurements

Basilar-membrane transfer functions can be measured

near threshold by recording the Doppler shift of laser light reflected from tiny microbeads placed on the basilar membrane or atop the organ of Corti (e.g., Ruggero and Rich, 1991). But such beads reflect more than the light of the laser: By acting as artificial impedance anomalies, they reflect sound-induced cochlear traveling waves as well, and their introduction should therefore produce changes in the evoked- emission spectrum.

Equation (35) predicts that the change AR in the cochlear reflectance resulting from the placement of a micro- bead at position Xb,ad is given by

AR[•( •o) ]o•e_•,T2(•,_ Xb•aa)= •o rZ(•,_X•.•a) ' o)c 0

(72)

where the bead has been approximated as a point mass? In other words, the emission change AR due to scattering off the bead depends on the square of the transfer function mea-

sured at the bead. Because of the factor of •o/•oc0, the bead should be placed as close to the stapes as possible, subject to the constraint that the emission be largely unattenuated when passing through the middle ear.

Equation (72) assumes that placement of the bead af- fects only the pattern of irregularities and not the transfer


-o

o

1.0

0.8

0.6

0.4

0.2

0.0 0

-3

-6

-9

100

75

50

25

0 7.50

I I I I

ß , •.- . :

;• -- I'.

7.75

/:,

Frequency (kHz)

FIG. 14. Changes in the cochlear reflectance with transfer-function shape. The figure plots the amplitude, phase, and group delay of the sequence of cochlear reflectances corresponding to the input impedances shown in Fig. 13. The group delay r R is given in units of the stimulus period. The dashed horizontal line corresponds to the value •12vr=fcl2• indicated by the dashed vertical line in Fig. 12(b). As the transfer function height diminishes, the amplitude IRI decreases; as the peak broadens, both the average and--in regions of approximately constant [Rl--the local slope of Z.R decnmse in magnitude. This change in phase slope decreases the corresponding group delay and implies that the mean spectral period increases, as predicted by the shifts in k apparent in Fig. 12(b).

function the bead was inserted to measure. This assumption implies that emissions observed prior to placing the bead remain unchanged and subtract out in the difference, AR. Spurious contamination of AR by changes in the emission background can be minimized by perfornfing the experiment in rodents or other animals with small evoked emissions.

Equation (72) illustrates how simultaneous measurement of basilar-membrane transfer functions and otoacousfic

emissions---experimental quantities related by Eq. (35)---can test the model.

Vl. PREDICTING TRANSFER FUNCTIONS IN THE HUMAN

In the previous section, knowledge of the empirical transfer function was used to predict the spectral character-

istics of squirrel-monkey emissions resulting from random irregularities in the mechanics. But the theory of coherent reflection filtering can also be used to solve an inverse scattering problem. For example, given measurements of the amplitude and phase of scattered waves (i.e., measurements of R), one can use Eq. (39) to find the irregularities • responsible for that scattering if the transfer function T is known. Alternatively, one can determine the transfer function given knowledge of the scattering potential. Here, we discuss this second alternative and demonstrate how, with the help of Eq. (3), measurements in the ear canal can be used to determine characteristics of basilar-membrane transfer functions in humans.

Uncertainty in •{q} prevents using Eq. (39) to solve directly for the transfer function (i.e., by computing .3z'-•{,9•{R}/.'27'{•}}). Three possibilities suggest themselves: First, one can smooth an initial estimate of T 2, obtained by assuming that .)-'•{•} is constant, by locally averaging over an appropriate frequency interval

( T2).•o• - • {(•;[R} )a}. (73) Thus, a smoothed form of Fig. 2(a) approximates the shape of 3•'{T2} for the right ear of subject JEM. Second, one can average 5Z{R} across subjects to obtain the estimate

( T 2 ) oc.• - 1 {(•?'{R} )}. (74) Figure 10 was generated by performing an analogous average over many different simulations. Third, one can model the transfer function parametrically by assuming a particular form for T 2 and fitting the resulting expression for •{T2•I to the measurements of I/•,z{R}l . For example, combining the Gaussian approximation introduced in Sec. IV B with the empirical values from Eq. (14) for emissions near 1300 Hz yields the value

•'r•l/•/•7.5 cycles (human), (75) •' 2\2•r1

for the transfer-function group delay (i.e., the slope of the phase). The group delay can be reexpressed as an estimate of the wavelength using Eq. (54):

•,•--•2/(27) •-960 /am (human), (76) for /•7.2 mm (Greenwood, 1961; Greenwood, 1990); the wavelength thus spans roughly 96 hair cells at 1300 Hz. Finally, the Q of the transfer function becomes

•< -- • 8 (human), (77) Q- 2Aco 2v• where the relations Ato/c%=A/•=v•X, together with Eqs. (42) and (45) have been used to compute the bandwidth Am. This estimated upper bound for the Q of the transfer function is consistent with the Q's of the narrowest psychophysical tuning curves measured near 2 kHz (Zwicker, 1974).

These values are expected to vary with characteristic frequency. The data presented in the following section suggest that •'r/•' increases with increasing characteristic frequency (,and, correspondingly, that the wavelength • decreases). Measurement of the frequency dependence of

2035 J. Acoust. Soc. Am., VoL 98, No. 4, October 1995 G. Zweig and C. Shem: Origin of spectral periodicity 2035

would enable one to predict the frequency dependence of the width of the transfer function based solely on ear-canal emission measurements (as measured psychophysica!ly, the transfer-function Q increases with frequency).

A. Frequency dependence of the distribution of emission spacings

The theory of coherent reflection filtering predicts that spatial changes in transfer-function shape should be reflected in the spectral characteristics of evoked emissions. Recall, for example, that the tips of neural and psychophysical tuning curves broaden at low characteris. tic frequencies (e.g., Liberman, 1978; Zwicker, 1974). If human transfer functions manifest similar increases in width and corresponding decreases in peak phase slopes (see Sec. IV B)--the model predicts a concomitant decrease in the ratio f/Af at low frequencies. Measurements of the distribution of emission spacings provide evidence for such variations with frequency.

Figure 15 shows a scatter plot of frequency spacings f/Af between adjacent spontaneous emissions obtained from the data of Talinurge et al. (1992) and Bums (1993, personal communication). Data representing frequency spacings between adjacent spectral maxima in Fig. 1 are also included. When the data in the scatter plot are pooled across frequency and projected onto the vertical axis one recovers a conventional histogram of emission spacings evincing a strong maximum at f/Af•-15, corresponding to a spacing of approximately 0.4 Bark (Dallmayr, 1987; Zwicker, 1989; Tal- madge et al., 1992).

Unlike the pooled histogram, however, the scatter plot reveals an interesting trend. Although the data at higher frequencies are limited, the peak of the distribution appears to increase with frequency. At frequencies above 2 kHz, the peak value off/Af lies considerably above the value (• 15) typical of lower frequencies? Although interpretation of the frequency dependence of f/Af is complicated by possible variations in the middle-ear coefficients q and r, the data suggest that f/A f--related to the slope of the transfer function phase through Eqs. (44) and (49)--increases slowly with frequency, perhaps approaching the value (-•56) obtained by using the squirrel-monkey wavelength at high frequencies (see Fig. 6). ]5 Additional data are needed to corroborate these trends and, as discussed above, determine whether the

width of the distribution also varies with frequency. The observed variation in f/Af implies a gradual deviation from scaling symmetry qualitatively consistent with a general broadening of mechanical tuning at low characteristic frequencies.

The relation between the peak value of f/Af and the parameter & suggests that variations in f/Af with frequency should be apparent in measurements of emission latency. And, indeed, the trend apparent in Fig. 15 is corroborated by measurements of tone-burst-evoked emissions in human ears

(Wilson, 1980), which indicate that their latency, when expressed in cycles of the stimulus period, increases slightly with frequency above 1.5 kHz.

60

30

20

12

lO

60

50

20

15

12

10

2 3 4 5 6 7 B

Geometric-Mean Frequency f (kHz)

FIG. 15. Frequency dependence of the distribution of emission spacings. The figure shows scatter plots off/Af versus f for frequency separations Af between adjacent spontaneous emissions (O) and stimulus-frequency- emission spectral maxima (,). As in Fig. 2, the frequency f rapresents the geometric-mean frequency of the pair of corresponding emissions. In panel (a), the spontaneous-emission data are taken from die study of Talmadge et aL (1992} and represent 487 emissions recorded from 67 ears (in 44 subjects ranging in age from 7-49 years) with at least two emissions per ear. Emissions with frequencies less than 1 kHz are subject to breathing artifacts (C. Talmadge, 1992, personal communication) and have not been included. For clarity, frequency separations Af far from the peak of the distribution (i.e., greater than approximately twice the value 0.4 Bark characteristic of emissions near I kHz} are not shown. Slimulus-frecluency-emission data are taken from Fig. I. Panel (b) shows data for 56 spontaneous emissions in three children (age 2) collected by Bums (E. Bums, 1993, personal communication). In young children the transfer characteristics of the middle ear permit measurement of emissions at frequencies higher than in adults. In each panel, the dashed line (---) indicates the valueflAf= 15 locating the peak of the distribution when the adult data are pooled and projected onto the vertical axis (compare Fig. 2). Above 2 kHz the peak falls substantially above the value projected by the dashed line, which is based predominantly on data near 1 kHz where the majority of emissions are found. Although the dala are relatively sparse at higher frequencies (especially in adults), the scatter plots indicale that the peak of the distribution (½) increases slowly with frequency. Note that at higher frequencies, many of the data points located far from the peak of the distribution (e.g., below the dashed line) may correspond to emissions separated by mulliples of the fundamental spacing. Despite interspecies differences that complicate the comparison, the distributions are not inconsistent with the value f/Aft56 obtained in Eq. (68) from the estimated wavelength near 8 kHz in the squirrel monkey (0).

VII. DISCUSSION

Coherent cochlear scattering from random impedance irregularities represents a novel analogue of Bragg scattering in nonuniform, disordered media (see Table I). Bragg reflection in a crystal occurs when multiple reflected plane waves--scattered throughout the crystal by a large number of identical, discrete scattering centers arrayed at periodic intervals---combine in phase (Bfillouin, 1946). In contrast to the constant amplitudes and wavelengths of plane waves, the


TABLE I. Comparison of conventional Bragg scattering in a crysial with coherent cochlear scattering.

Bragg scaltering in a crystal Evoked otoacoustic emission

uniform background medium

incident plane waves have constant amplitude and wavelength

all scattering centers identical

scattering centers sparsely distributed

scattering centers arrayed at discrete, periodic intervals Ax

scattering occurs throughout the medium

scattering coherent at a single frequency (and its integer multiples)

wavelength of scattered plane waves determined geometrically via the Bragg condition: h=2Ar

peri9dic interference pattern produced by fixing the observation point and translating the cqtstal along the beam ax•s

nonunilbrm background medium

incident traveling waves have spattally varying amplitude and wavelength

scattering centers have strengths that vary randomly with position

scattering cenlers den•ly dislributed

scattering centers randomly distributed

scattering localized about the peak of the traveling-wave envelope

scauering coherent at all frequencies

spatial period of dominant scattering irregularities determined dynamically: Ax = •.•/2

periodic interference pattern produced by varying the stimulus frequency (i.e., by moving the scattering region along the organ of Corfi)

amplitudes and wavelengths of cochlear traveling waves both vary dramatically with position. The dynamical action of the cochlea produces a tall, broad peak in the envelope of the traveling wave, which localizes the region of seatiering. Unlike the reticular structure found in a crystal, cochlear irregularities appear to be rather randomly distributed along the organ of Corti. Nevertheless, an analogue of the Bragg coherence condition [i.e., Eq. (24)] guarantees that many scattered wavelets combine coherently to form a large reflected wave whose phase rotates periodically with frequency. Spectral regularity emerges dynamically from spatial disorder.

The theory of coherent reflection filtering resolves the paradox of periodicity by considering the interaction of many wavelets reflected over a broad region of the cochlea. In contrast to the predictions of the point-reflection :nodel described in Sec. II, the phase of the reflectance represented by Eq. (35) is no simple sum of phase shifts resulting from wave propagation and from reflection [compare Eq. (21)]. Unlike the point-reflection model, which represents the backward-traveling wave as arising from a single reflection at the peak of the traveling wave (and which leads, inexora- bly, to the conclusion that the cochlea must be mechanically corrugated), the integral in Eq. (35) represents the reflectance as a sum of backward-traveling wavelets originating throughout the cochlea. The theory described here indicates how those multiple reflected wavelets interfere with one another--both constructively and destructively---to generate a large backward-traveling wave whose phase rotates periodically with frequency. That interference between wavelets originating over an extended region plays a crucial role in generating the spectral characteristics of evoked emissions.

By representing the cochlear reflectance as the integral of a spatial, or "place-tixed," component (namely •) with a scaling, or "wave-fixed," component (namely T2), Eq. (35) highlights the respective roles of geometry and dynamics in the generation of evoked otoacoustic emissions. The mechanism described here implies that both components. geometric and d. vnamic, contribute essentially to the production of spectral periodici.ty: The geometric because it provides the necessary breaking of scaling symmetry without which emissions could not depend strongly on frequency, and the dy- namic because it provides the spatial-frequency filtering necessary to make the emission spectrum quasiperiodic.

Although one-dimensional models of cochlear mechanics predict that R and T • are simply related via Eq. (35), their precise relationship depends on details of the mechanics not known with any certainty. Empirically, however, the Fourier transforms of R and T 2 are similar in shape, suggesting that the two functions share a common origin in the cochlear mechanisms responsible for mechanical tuning at low levels. The more qualitative mechanisms described here for the origin of otoacoustic emissions and their spectral periodicity presum,qbly remain valid in more realistic models as well.

The reflection mechanism proposed here modifies Kemp's (1980; 1986) heuristic description of evoked emissions in a surprising way. Kemp proposed that the structure apparent in emission spectra represents an interference between the stimulus tone and a reverse traveling wave originating by reflection off an impedance discontinuity located along the basilar membrane. Spectral maxima and minima occur at frequency intervals inversely related to the travel


time to and from the punctuate anomaly. In the picture presented here, however, well-defined roundtrip travel times re- fiect not so much any discrete localization of the anomalies•which, on the contrary, may actually be densely distributed along the organ of Corti--but arise through the dynamical action of the cochlea, which generates a tall, broad peak in the traveling wave and thereby creates an extended region of coherent reflection that sweeps over the organ of Corti as the frequency is varied monotonically.

In simulations performed using analogue and computer models of cochlear mechanics, Zwicker observed a relation-

ship between the spacing between adjacent maxima in the simulated emission spectra and the slope of the phase of the model traveling wave (e.g., Zwicker, 1989; Zwicker and Peisl, 1990; Zwicker, 1990a). Conjecturing that the slope of the transfer-function phase was somehow responsible for the regular emission spacing observed experimentally, Zwicker suggested that measurements of emission spacings could provide estimates of human traveling-wave characteristics. This paper elucidates the scattering mechanism responsible for Zwicker's model results and verifies the spirit, if not the letter, of Zwicker's phase conjecture. (For example, the cor- relation with the slope of the phase is only approximate and the underlying mechanism involves the coherent backscattering of the forward-traveling wave rather than the local "feedback enhancement" he postulated.)

A. Constraints on the arrangement of irregularities

Because the cochlea maps frequency into position, it is natural to associate a length scale Ax/ with the observed spectral period Af. The two intervals are related by

Ax//l = Aflf, (78)

an equation obtained by differentiating the exponential form of the frequency-position map (l is the distance over which the characteristic frequency changes by a factor of e). Near 1300 Hz in humans, Ax/corresponds to the distance spanned by roughly 48 hair cells. Contrary to recent arguments (e.g., ManIcy, 1983; Strobe, 1985; Peisl, 1988; Strobe, 1989; Fuka- zawa, 1992), which associate the length scale Ax! with an ostensible regularity in the spacing of impedance disconti- nuities along the organ of Corti, this paper has shown that Ax/need have no geometric correlate in the structure of the cochlea.

Although a narrow range of spatial frequencies does dominate the scattering, those spatial frequencies need not dominate the scattering potential. Indeed, the irregularities can be completely random, representing a superposition of many different spatial frequencies. For although every spatial frequency scatters the incident wave, only wavelets scattered

by spatial frequencies near • • 2/• x contribute significantly to the net reflected wave. In this view, the distance Axf=2•rl/k has no objective correlate independent of the traveling wave. The dominant spatial frequency, determined by the wavelength of the traveling wave at its peak, emerges through an analogue of the Bragg coherence condition. The only constraint on the arrangement of irregularities is that their spatial-frequency spectrum contain components near • a requirement met when the arrangement is random.

Although the impedance irregularities used in the simulation vary from hair cell to hair cell, qualitatively similar results are obtained from other spatial distributions so long as the irregularities are "densely distributed." A rough upper bound on the spacing between irregularities can be obtained by requiring at least one within any region of width A X. This condition guarantees that the scattering potential contains local spatial-frequency components near k. Expressing the maximum spacing between irregularities as a multiple, n, of the hair-cell width, h, yields the relation nh<Ax!. Rewrit- ing the inequality using Eqs. (42), (49), and (50) yields the estimate

For humans near 1300 Hz, the inequality implies that the spacing between irregularities must be less than roughly 30 hair cells [see Eqs. (14) and (76)].

The narrow-band tBrm of the spatial-frequency filter has unexpected consequences. For example, the irregularities that make the most effective "reflectors" are not necessarily those in which the impedance changes most rapidly (e.g., discontinuously) with position. Because the high spatial frequencies associated with sharp transitions fall outside the passband of the filter, they generate wavelets that contribute little to the net reflected wave. Indeed, even irregularities that vary smoothly and continuously with position make effective reflectors so long as their spectrum contains contributions near •:. Sizable reflection then results even though the standard conditions for validity of the WKB approximation (e.g., that the spatial derivative of the perturbed wavelength be small relative to one) may be everywhere well satisfied. Those conditions, however, are all local conditions (e.g., Bender and Orszag, 1978). The reflection found here constitutes a global phenomenon resulting from the coherent sum- marion of small, reflected wavelets.

Of course, not all arrangements of irregularities contain spectral components near •. For example, irregularities arrayed in an orderly, crystalline fashion (e.g., the "staircase" variation of parameters produced by spatial discretization in numerical simulations of cochlear models) only contain large components at spatial frequencies t%12,r=llw and higher (where w is the distance over which the mechanical properties remain roughly constant). If w is set by the width of a hair cell, then •:,.,16 and spatial frequencies associated with the cellular discretization scatter only incoherently (i.e., lie outside the passband of the filter). Such orderly patterns therefore generate little emission compared to more irregular variations at the same scale. As an empirical correlate of this observation, note that in contrast to the cellular disorder

characteristic of the primate organ of Corti, Wright (1984) reports that outside the most apical tums anatomical regularity constitutes an "impressive feature" of the guinea pig cochlea. Such spatial order may therefore underlie the observation that emissions evoked from rodent ears are considerably smaller in amplitude (Zwicker and Manley, 1981; Schmiedt and Adams, 1981) than those measured in primates I? (Ander- son, 1980; Wit and Kahmann, 1982; Zurek, 1985; Brown and Gaskill, 1990; Probst et al., 1991).

2038 J. Acoust. Soc. Am., Vol. 98, No. 4, October 1995 G. Zweig and C. Shem: Origin of spectral periodicity 2038

The theory of coherent reflection filtering does not, of course, preclude the possibility that the cochlea is, in fact, mechanically corrugated, but demonstrates that such corrugation, although sufficient, is unnecessary for the generation of spectral periodicity. Indeed, unless the ostensible corruga- tions occur at a spatial frequency that falls within the passband of the filter, the mechanism proposed here will filter them out in favor of spatial-frequency components close to

In contrast to the corrugated-cochlea model, the theory of coherent reflection filtering predicts that spectral periodicity hinges on the physiologically active elements of cochlear mechanics responsible for mechanical tuning near threshold. In principle, one can test this prediction by measuring evoked emissions after those elements have been largely dis- abled (e.g., by application of furosemide, salicylate, or other ototoxic drugs): One should observe (in addition, of course, to a decrease in emission amplitude) a degradation in the periodicity of their spectrum. ]8

Although the irregularities used in the simulations cannot be directly related to corresponding micromechanical perturbations in the organ of Corti, the small size of the irregularities necessary to produce realistic emission amplitudes (i.e., only a few hundredths of a percent variation in pole location) is nevertheless somewhat surprising. Equation (35) for R implies that the size of the necessary irregularities scales inversely with the square of the amplitude of the transfer function (assuming that all else is held fixed). In active models, the enormous gains typically required to match empirical basilar-membrane transfer functions (I ]'l--30-40 dB) imply correspondingly small irregularities (lelc, l1-2 lelo.m-O.l%).

B. Emissions from a passive model

The requirement that the basilar-membrane transfer function have a tall, broad peak to define a localized scattering region containing many scattering centers impose., pow- erful constraints on cochlear mechanics. For example, recent analyses show that conventional passive cochlear models cannot match the peaks of measured transfer functions (Zweig, 1991; Brass and Kemp, 1993; de Boer, 1993). The failure of passive models to generate realistic transfer functions is echoed in the spectrum of their emissions. Figure 16 shows the oscillations in the cochlear input impedance computed by using a typical passive model supplemented with random irregularities. Although the underlying irregularities are more than three orders of magnitude larger than those shown in Fig. 7, the resulting emissions are minute: The magnitude of the cochlear reflectance is only of order i•- Even more stalking, however, is the complete absence of spectral periodicity.

These problems originate in the unrealistic shapes of passive-model transfer functions, which cannot simultaneously be made both tall and broad. Unlike the active model, which creates the transfer-function peak pri•nafily through the spatially distributed amplification and absorption of traveling-wave energy, passive models rely entirely on the action of a "resonant denominator" (Zweig et al., 1976). Fit- ting the width does not provide the necessary gain, and such

o

0.50

0.25

0.00

-0.25

-0.50

2

0

-2

7.50

I I I I

7.75 8.00 8.25 8.50 8.75

Frequency (kHz)

FIG. 16. Irregular emission spectrum characteristic of passive models. The figure plots the cochlear input i•npedance computed as in Fig. 9, but by using a passive model (obtained from X, by setting p=O and 8=0.01) in which the traveling-wave envelope lacks the tall, broad peak observed experimentally. The underlying irregularities have the same spatial variation as those shown in Fig. 7 but were made 2500 times larger (the width of the Gaussian distribution corresponds to a 25% deviation in pole location) in order to compensate for the unrealistic transfer-function peak and generate measurable emissions. Despile the large irregularities, emissions are small. with IRI•T[•. Unlike emission spectra observed experimentally--or generated by using the estimated wavelength--the spectrum here appears irregu- lax and contains •to dominant spectral period.

models therefore require substantially larger irregularities ([•l•100%) to generate measurable emissions. One can at- tempt to compensate tbr the small gain by lowering the damping and moving the zero in the denominator closer to the real axis. In this manner, the passive transfer function can be made arbitrarily tall, but only at the cost of decreasing its width. A passive transfer function of sufficient height becomes far too narrow to match the widths of empirical transfer functions measured near threshold. Since the uncertainty principle tums narrow, unrealistic transfer functions into broad, unrealistic spatial-frequency filters, the randomness of the scattering medium survives unfiltered in the emission spectrum.

C. The origin of spontaneous emissions

Adding random perturbations to the mechanics of the active model only somewhat larger than those employed in the simulation often results in the generation of sharp, regularly spaced peaks in the cochlear input impedance Iz(•o)]. Concomitant values of Re{Z} <0 indicate that the cochlear model is "emitting" energy at these frequencies. In other words, the magnitude IRI of the cochlear reflectance, given by the product of the magnitudes of the local reflection and roundtrip amplification factors, exceeds one. This insta- bility--a "spontaneous emission"---occurs despite the fact

2039 J. Acoust. Soc. Am., Vol. 98, No. 4, October 1995 G. Zweig and C. $hera: Origin of spectral periodicity 2039

that the oscillators composing the organ of Corti are individually stable (i.e., all spectral zeroes of its impedance lie above the real frequency axis). Retrograde traveling waves are reflected at the cochlear boundary with the middle ear with a net reflection factor gst (see Appendix B). As the product RRst of forward and reverse reflection factors approaches unity, the emission amplitude continues to grow without bound. In the real ear, or an appropriate nonlinear model, emission amplitudes are stabilized by saturating nonlinearities that limit the energy produced (e.g., Talmadge and Tubis, 1993). Once initiated, the emission therefore requires no external sound for its maintenance: The stapes acts as a partially reflecting "mirror" that feeds energy back to the cochlea as a forward-traveling wave (Kemp, 1980; Zweig, 1991). Unlike the case in which a localized section of the organ of Corti acquires a net negative damping, the spontaneous emissions described here would not be self-sustaining were the stapes a perfectly reflectionless boundary (i.e., if Rst were zero). As a consequence, manipulation of the stapes reflection factor (e.g., by changing the acoustical probe impedance or the static pressure in the ear canal) can easily alter emission amplitude and frequency, effects that are observed experimentally (e.g., Kemp, 1978; Zwicker, 1990b; Allen et al., 1995). (Appendix B provides a formula for gst in terms of the middle-ear scattering coefficients and the reflectance of the acoustic source assembly.) In this view, spontaneous emissions--and their regular spacing with frequency--arise as a collective response to random perturbations in the mechanics.

VIII. SUMMARY

The following list summarizes the major results of the paper.

(1) At sound-pressure levels in the linear regime near the threshold of hearing, ear-canal measurements of evoked otoacoustic emissions take the form of a power series in the cochlear reflectance, R (Shera and Zweig, 1993a):

Pcc(O;)

Pec(tO;0 ) - 1 +qR(1 +rR+r2R2+ '"), for [rRl< 1, (80)

where the coefficients q and r characterize the mechanics of the middle ear. The reflectance R--defined as the ratio of the

emitted to the stimulus wave at the stapes--has the empirical form

R[,•( to) ]•-Ro[ e•;•( to) ]e i'•3('•), (81) where the logarithmic frequency variable

•:(to) --- - In(to / to%}, (82)

and tOCo/2•r denotes the maximum frequency of hearing. Over much of the frequency range where emissions are measured, the middle-ear coefficients q and r vary slowly with frequency compared to the phase ½•((to). The complex amplitude R0, typically of magnitude 0.2, also varies slowly with frequency (i.e., e•l). Almost periodic spectral oscillations in ear-canal pressure arise because the phase of R var-

ies nearly linearly over frequency intervals for which R 0 is roughly constant. The parameter •/2•r, which sets the spacing Af between spectral peaks through the equation Af/f•2rr/•, equals the mean emission group delay at the stapes expressed in units of the stimulus period. At 1300 Hz, ½/2•r is approximately 15 cycles in humans. Since •(m) locates the pe• of •e traveling-wave envelope, Eq. (81) re- g•ds evoked emissions as a function of their pfim• site of generation within the cochlea.

(2) Em-c•al measurements and histograms of emission spacings were used to estimate the Fourier •sform of R with respect m •(m) (Sec. I C). When t•en over a limited •equency inte•al, the •ansfom .•R} has a roughly Gauss- i• peak of wid• A• centered about the value •. Ne• 1300 Hz in humans, A•/•0.25, implying that e• •.

(3) A •eoretical expression for R, based on the wave equation describing linear, one-dimensional, scaling-sym- meffic models of cochle• mechanics, was given in the form of a scattering integral in which wavelets reflected •rough- out •e cochlea me summed to yield the net reflected wave (Sec. III B):

(83) where • is the dis•bution of mechanical perturbations and T is the basil•-membrane •sfer function (or, equivalently, the traveling-wave displacement nomalized to that of •e stapes). Given a description of cochlear mechanics reducible to an equivalent one-dimensional model, R(•) can be computed from Eq. (83), compmed wi• the empirical fore (81), •d used in conjunction with power series (80) to predict •e e•-c•al emission spectrum resulting from small perturbations in the mechanics.

(4) The scattering integral was used to demons•ate that the many-cycle latencies ch•acteristic of primate evoked emissions requke •at the mechanical perturbations initiating ß e reflection remain fixed in space (Sec. III C). Unlike e•s- sion sources induced by nonlinear distortion, the perturbations responsible for evoked e•ssion cannot move wi• •e wave envelope. •en •e iaegul•ties • depend we•y on frequency, the place-fixed nature of the perturbations allows an approximate factofization of the scattering integral into a product of place- and wave-fixed components. In that case, R t•es the fore of a convolution (Sec. IV A):

R(•)• f f•d X e(x)T2(•- X); (84) or, equivalently,

•g}•e}•r2}. (85) (5) Since, empirically, both T 2 and •T 2} manifest tall,

broad pe•s, the approximate proportionality

Ig}llr}l (86) was postulated to explain •e roughly Gaussian fore of Ig}l (Sec. IV B). The proponionality requires that •e spatial-frequency spec•m [•}l be roughly constant, as expected if the i•egul•ties •e essentially random in •- rangement. Contributions to •R} from random fluctuations in •} c• be reduced by averaging across subjects:


(87)

(6) The function •{T 2} acts as a "spatial-frequency filter" whose center frequency • and bandwidth Ak determine both the dominant spatial frequencies of scattering (i.e., and Ag•Ak) and the spectral characteristics of evoked emissions (i.e., ½•/• and Aq0•-Ak). The reflectance R(•() is approximately equal to the product, evaluated at the peak of the traveling wave, of (1) the scattering potential e i•X for which scattering within the peak is coherent and (2) the spatial- frequency component at k present in the surrounding irregularities. Thus, spectral periodicity can originam in the shape of the traveling wave, rather than in some putative geometric regularity in the distribution of impedance anomalies.

Taken together, Eqs. (80) and (84) enable ear-canal measurements of otoacoustic emissions to be interpreted dtrectly in terms of cochlear mechanics: Measurements of one can be

used to predict the other (Sees. V and VI). By assuming that the transfer characteristics of the middle ear vary .,;lowly compared to oscillations in the emission spectrum, noninvasive measurements of otoacoustic emissions can be related to

cochleax mechanics, even in the absence of complete knowledge of the middle ear.

ACKNOWLEDGMENTS

We thank Carrick Talmadge and his colleagues at Pur- due University and Ed Bums at the University of Washington for generously making their data available for analysis.. Jont Allen, Paul Fahey, John Guinan, Stephen Neely, Sunil Puria, Wendy Schaffer, Hans Werner Strobe, Carrick Talmadge, Ar- nold Tubis, and Martin Whitehead provided helpful critical readings of and comments on the manuscript. This work was supported by DARPA and AFOSR contract N00014-86- C0399, the DOE Office of Health and Environmental Re- search and Applied Mathematics Program, NIH-NIDCD Na- tional Research Service Award F32-DC00108 to CAS, and

the Theoretical Division of Los Alamos National Laboratory.

APPENDIX A: FREQUENTLY USED SYMBOLS AND THEIR MEANING

In the following summary of frequently used symbols, the double-headed arrow (•) indicates a pair of Fourier- conjugate variables.

Operators

Re{R} real part of R Im{R} imaginary part of R /_R phase of R 3Z{R} Fourier transform of R

Independent variables

t time (with t•to) f stimulus frequency r stimulus period 1/f ro angular frequency 2•'f (with t•-•o) x distance from the stapes X dimensionless distance x/l (with X•K) K dimensionless spatial frequency (with X•-•K) •( frequency-dependent position of the peak of travel-

ing wave, - ln(•o/o•c0) (with qo variable conjugate to •, interpreted as a tone-pip

latency measured in units of the stimulus period (with

/• dimensionless scaling variable to It%(x)

• ratio to/to% (value of •t at x=0) •-X dimensionless scaling variable -In/•1, interpreted as

the distance from the peak of the traveling wave k dimensionless wave number (with -x•k at fixed

k)

Dependent variables

Pec ear-canal pressure q function characterizing roundtrip transmission

through the middle ear r middle-ear reflection coefficient for retrograde co-

chlear waves

P pressure difference across the cochlear partition (with P=P+ +P-)

P+ forward-traveling cochlear pressure wave P- backward-traveling cochlear pressure wave R cochlear reflectance (the ratio P-/P+ at the stapes) R 0 slowly varying complex amplitude of R T at fixed frequency, the forward-traveling cochlear

displacement wave (normalized to stapes displace- merit); at fixed position, the basilar-membrane transfer function

r n reflectance group delay •r r transfer-function group delay h• wavelength of the traveling wave at its peak (mea-

sured in the same units as x)

•x dimensionless wavelength of the traveling wave at its peak (measured in the same units as X)

X unperturbed, scaling symmetric wavelength/2,r (measured in the same units as

• perturbed wavelength/2• (dimensionless) X e estimated X for the squirrel monkey (Zweig, 1991) • distribution of mechanical perturbations (scattering

potential)

Cochlear parameters

t%(x) 2,rXposition-frequency map for the cochlea Derived •o% to•(x=0) (i.e., 2rrXmaximum frequency of hear- Af

ing) I distance over which toe(x) decreases by a thctor of

e in the basal tums Aqo N approximate number of wavelengths of an harmonic

traveling wave in the cochlea h width of a hair cell (taken to be •10

values

frequency spacing between adjacent spectral peaks (or spontaneous emissions) peak of 5•{R} transformed with respect to •( half-width of the peak of .SZlR} transformed with respect to •( peak of .5•{T 2} transformed with respect to -X at fixed •(


Ak

AK

half-width of •{T 2} transformed with respect to -X at fixed • dominant spatial frequency of scattering half-width of the distribution of spatial frequencies K contributing to the reflected wave dimensionless parameter characterizing relative rates of change of IR[ and /_R

APPENDIX B: RELATING EAR-CANAL AND COCHLEAR REFLECTANCES

We relate ear-canal measurements to cochlear mechanics

by representing the middle ear as a linear, two-port network characterized in the frequency domain using reflectance and transmittance coefficients (Shera and Zweig, 1992). The scattering matrix describing the residual ear-canal space and middle ear is defined by the equation

(Bi)

where P•o are the incident and reflected pressure waves measured at the microphone probe tip in the ear canal and P -+ the traveling waves at the basal end of the cochlea near the stapes.

Equation (B1) implies that the ear-canal reflectance takes the form

P•c r+ t+t-R (B2) Rec----- P-•c = + 1-r I• ' where R is the cochlear reflectance defined by Eq. (1). At stimulus amplitudes above approximately 60 dB SPL, the cochlear reflectance R becomes negligible and the ear-canal reflectance reduces to the value

lim Reo = r +. (B3) R•0

Careful measurements of r + have been reported for both human infants and adults (Keefe et al., 1993; Voss and Allen, 1994). As the stimulus level is lowered, IRI increases and nonlinear contributions to Ree originating in the cochlea become significant (Allen et al., 1995). At the lowest levels near threshold, R becomes independent of level and cochlear contributions to the ear-canal reflectance are maximal.

The coefficients q and r in power series (4) can be obtained by noting that

eeo(R) = P•+c( 1 +Rec(R)), (B4) so that the ratio

Pe½(R) t+t-R Pec(0•- 1 + (1 +r+)(1-r-R)' (B5)

Comparison with Eq. (3) yields

q--=t+t-/(l+r +) and r--r-. (B6)

The function t+t - appearing in q characterizes roundtrip transmission through the middle ear. The coefficient r represents the middle-ear reflection coefficient at the stapes for backward-traveling cochlear waves. It can be shown that energy conservation implies that q goes to zero as r approaches

one. Note that the pressure ratio is independent of the transducer characteristics.

When emissions are measured with a transducer charac-

terized by the "Thevenin-equivalent" soume reflectance R s, the reflection coefficient Rst for retrograde traveling waves at the stapes has the value

Rst • P•__ t + t-Rs (B7) stapes = r- q- 1 - r+R• ' An earlier paper (Shera and Zweig, 1991b) gives estimates of Rst for the cat under both natural and simulated recording conditions.

APPENDIX C: THE SCATrERING INTEGRAL DERIVED

By applying the WKB approximation to the inhomoge- neous wave equation (34), one can develop the solution P to Eq. (34) as a cochlear scattering series. When secondary scattering is neglected one obtains the expression (Shera and Zweig, 1991b),

p-or-W- W+ o-W+d]•' +O(o-2), (C1)

for the net backward-traveling wave P-. Here, the functions

W -+-= 4• expWif; ø dfl' (C2) represent the first-order WKB solutions for the traveling pressure waves, and

(c3) 2i•2/{2 '

The upper limit of integration in the expression for P-, shown as "m" for convenience, indicates that the integration extends to the helicotrema. Because of the rapid fall in IW+I as/• increases above one, the limit "o•" means, in practice, any convenient value of/5 somewhat greater than one.

By dividing Eq. (C1) by the value of the forward- traveling wave, W +, substituting the value of • from Eq. (37), and evaluating the integral at fi=fl0•0 near the stapes, one obtains the cochlear reflectance at the basal end of the

cochlea:

e- fo ø at• (C4) R(/•0)=•-•stapes •' •)(/•,/•0)T2(] •) ]•, where

r(/•) • ro•l- 3/2e -if•odlV/X• rolJX- 3/2e -iJ'o•dlV/X (c5)

is the basilar-membrane transfer function with T o a constant (Zweig, 1991). When the sound frequencies are much less than the maximum frequency of hearing, the integrarids are small in the interval [0,/•0]. Therefore, the lower limits of integration have been decreased from/•0 to 0. Ignoring the /•0 dependence of the transfer function in this manner neglects an overall phase factor of the form

e,J'o d•/x• e,pO/So, (C6)


where

X0---- lim X---- 1/4N, (.C7)

and N represents the approximate number of wavelengths of the wave along the organ of Corti) 9

Finally, rewriting R using the variables X and • yields Eq. (35):

R• f•e(X,,•)T2(,•- x)dx, (C8) where

X) -= r[fi(- X) ]. (C9)

APPENDIX D: SADDLE-POINT CALCULATION OF THE SPATIAL-FREQUENCY FILTER

This Appendix uses the saddle-point method to find the first term of an asymptotic expansion in 4N of.57{T2} for the model of Zweig 0991). The resulting expression is further approximated to obtain simple formulae, accurate to within roughly 20%, for the center frequency and bandwidth of the spatial-frequency filter.

The WKB expression for T is given by Eq. (C5); when expressed in terms of/3, 20 its Fourier transform takes the form

rg fxg%ik'na-2fg ct. (DI) To put the integral in standard form, note that 1//• e is proportional to the large parameter 4•N. Anticipating the result that the integral peaks at a value k also proportional to 4N, we adopt the notation

K----kl4N and A----4NX•, (D2)

so that K and A are independent of N. Regarding the integrand as a function of the complex variable • and rewriting the integral to exhibit the dependence on 4N yields

where

a(•,K)--=iK In •-2i f: d•' (D4) A'

The contour • extends along the real axis from fl0•-0 to m but can be deformed to pass over a saddle in the function le4•t• I = e4N Re{a} along a line of constant phase.

For large 4N, the integral is dominated by a limited range of g' about the saddle illustrated in Fig. D l, which plots, in the first panel, the real parts of the two terms appearing in Eq. (D4) for a, and, in the second panel, their sum. In this restricted region, the two contributions to Re{a} appear as intersecting surfaces in the complex plane. The first term produces an approximately planar surface whose inclination depends on the value of K; increasing K steepens the tilt by rotating the plane counterclockwise about the real axis. The second surface, independent of K, generally slopes

1.05

1

0.03

0

1.05

-2O

0.03

0

]m •

FIG. DI. Existence of a saddle point. The two surfaces shown in the top panel represent the two terms in Fx I. (D4) for the function 4N Re{a(![,K) } in the region about •= 1 for the value of K at which the saddle point lies upon the real axis. The sum of the two terms is shown in the bottom panel. The approximately planar surface (large grid) corresponds to the first term. In- creasing the value of K rotates the plane counterclockwise about the real axis. The curved surlhce (small grid) represenls the second term, which slopes downward from a peak region located at a position determined by the double pole of ll•. As N im reases, the saddle becomes taller and steeper.

downward from a peak region located at a position determined by the double pole of I/X• 2 . Summing the two surfaces yields the function 4N Re{a}. Equation (40) allows us to restrict attention to posilive K. For K>0, the opposing incli- nations guarantee the existence of a saddle point, generally located in the vicinity of •1. By varying K one varies the tilt of the planar surface and thereby moves the saddle point around in the complex plane.

The Cauchy-Riemann equations imply that the saddle point occurs when dodd•=O. Evaluating the derivative yields the relation

2•.p h•.(•.) = K, (DS)


0.05 [__ I I I I -0.05 --

-O. lO I I I I 0.6 0.7 0.8 0.9 1.0

FIG. D2. Location of the saddle points. The figure plots •sp(K) as K is varied monotonically. The curved trajectory corresponds to the estimated wavelength; the straight line to a generic passive model obtained from X c by setting p=0 and 8=0.01. To indicate the rate at which the trajectory is traversed, small dots appear at intervals AK= 2•/4N. Both trajectories be- gin at the left near the value K= 10•r/4N and approach the lowest zero(es) of the corresponding wavelength (marked by large bullets) as K increases. In the passive case the trajectory remains entirely above the real axis ( ); in the active case the trajectory lies largely below the axis, crossing the real line in a positive-going direction near ;• I.

whose solution •rsp(K ) locates the saddle point as a function of K. Figure D2 illustrates the function •sp(K) obtained by using the estimated wavelength and, for comparison, a conventional passive model. Re{•p} is a monotonically increasing function of K and the curves •rsp(K ) asymptotically approach the poles of the reciprocal of the corresponding wavelength as K-•. Since K is a real number, Eq. (DS) implies that

Z_•sp=/_h((sp ) . (D6) In passive models, for which Im{A} is everywhere positive, the curve •sp(K) remains above the real axis. The estimated wavelength, however, has a negative imaginary part over an extended region basal to the "resonance" location: the corresponding saddle-point trajectory therefore lies largely below the real axis.

For each value of K, the analyficity of the integrand justifies deforming the contour • so that it passes over the corresponding saddle along a line of constant phase. Figure D3 illustrates this deformed contour in the active model for

the value of K at which the •sp(K) crosses the real axis near s r 1. Although the corresponding saddle point lies on the real line, the path of steepest ascent/descent does not lie along the real axis.

Expanding the function a in a Taylor series about the saddle point and integrating along the constant-phase contour yields an asymptotic expansion whose first term is given by (e.g., Bender and Orszag, 1978)

y---{ T2}(K) • 2x/•(4N)3r0 2

•½ 4Not(•,K) eiO• x (4N large).

•rsp(K)

(D7)

Asymptotically, the integral is proportional to the value of the integrand at the saddle point times the width of the

0.04

0.02

0.00

-0.02

-0.04

0.90 0.92 0.94 0.96 0.98 1.00

/g = Re)•' I

FIG. D3. Traversing the saddle along the deformed contour. The figure shows a contour plot of the saddle in the function 4N Re{a(sr, K)} evaluated at that value of K for which •sp(K) crosses the real axis in the vicinity of (= 1. Contour intervals represent steps of approximately -+ 1. The dashed line (---) represents the constant-phase contour traversing the saddle in the direction (arrow) of increasing Re{•} along the line of steepest ascent/ descent. Except in the immediate vicinity of the saddle point (marked with a +), the integrand is small.

saddle, approximated by 1/4,• a"[, where the primes de- note derivatives with respect to •. The angle 0•[•sp(K)] represents the inclination of the constant-phase contour and has the value

0• = -/_ a"/2 + •r/2 •- -/_A'/2 + •r/4. (D8)

For the estimated wavelength, 0• •-0.5 radians.

Center frequency and bandwidth

For large 4N, the filter characteristics of .SrlT 2} can be obtained by noting that the K dependence of Eq. (D7) is dominated by the exponential factor e 4Na(10, where a(K) •a[•p(K),K]. Ignoring the K dependence outside the exponent and expanding 4Na(K) in a Taylor series permits one to obtain simple formulae relating the filter center frequency and bandwidth to the wavelength and its derivative.

The wave number K at which .•-{T 2} reaches its maximum defines the value •. At large values of 4N,. the ampli~ tude IoqlT2}l is controlled by the exponential factor e4•V a•{o•m}, which has a local maximum at that wave number Ksp satisfying 2!

a Re{o4 0 -- (D9) d K I •qp

= Re{/ In •p(K•p)} = -/_ •p(K•p), (DI0)

that is, at that wave number for which the phase of •sp(K) vanishes and the saddle point lies on the real axis. The supplemental condition

2044 J. Acoust. Soc./•,;'n., Vol. 98, No. 4, October 1995 G. Zweig and C. Shera: Origin of spectral periodicity 2044

d 2 Re{a} d/•sp] .--:•.• <0 (DII)

guarantees that the stationary point at K•p represents a local maximum in Re{a} and requires that the zero crossing in Im{•sp(K)} • positive-going (i.e., that Im{•p} change sign from negative to positive as K increases through KspL

•e doanant wave number •Ksp can be related to the positive-going zero-crossing of the imaginary p• of the wavelength using Eq. (D6). The filter •T 2} peaks when the saddle point •p(K) falls at that location along the real axis where the imaginary pa• of the wavelength passes t[xough zero in a positive-going direction. This location, denoted flsp, is close m, but is not exactly at, the pe• of the basilar- merebrae tr•sfer function. 22 BasM to fl•p the ima.gin•y pan of the wavelength is negative and the •aveling wave is amplified as it propagates. Apic• to flsp the imagin• pa• of the wavelength is positive and power flows out of the traveling wave into the organ of CoHi (Zweig, 1991).

Solving Eq. (DI0) for Ksp using •e saddle-point equation (D5) yields

•Xsp_ 2•,. (D12) or, equivalently,

(m3) •is approximation yields a value •/2•46, a value within 20% of that obtained by eva•uating .•T 2} numerically.

The width AkI•=AKIK dete•ines the selectivi W of the nter. Since [T2}I primarily controlled by the exponential e 4•Re{•}, the width AK can be estimated fi'om a %ylor-sefies expansion of the exponent 4N Rely} about Ksp:

I Kse = 1 Ksp ' AK• q4Nld2 Re{M/dK2 ] . •4N(d•Csp/dK) •uations (DS) and (D6) can be used to express Ak/• in te•s of the derivative of the wavelength: 23

where t•e•d&ld•.

•In an earlier paper (Shem and Zweig, 1993a) R was called the "cochlear traveling-wave ratio." To prevent possible confusion with the standing- wave ratio familiar from transmission-line theory (and to emphasize that, unlike the standing-wave ratio, R is complex), we adopt here the more descriptive term "cochlear reflectance."

2At sound levels near threshold, the amplitude of evoked otoacoustic emissions varies linearly with the level of the stimulus (e.g., Kemp and Chum, 1980: Wit and Ritsma, 1979; Wilson, 1980), and the principle of superposition applies (Zwicker, 1983). Although this paper focuses on emission characteristics in the frequency domain (i.e., on stimulus-frequency emissions), so long as the evoking stimulus is quiet enough to elicit a linear response. the characteristics of transiently evoked cochlear echoes follow immediately from the corresponding stimulus-frequency-emission function with the aid of stan 'dard Feuder analysis.

•In the paper describing the measurements on which this paper is based

(Shem and Zweig, 1993a} the parameter O is denoted &. 4Relation (ll } between the frequency spacing Af and the reflectance group delay r• is only approximate because the coefficients q and r appearing in power series (3) are not constant but vary slowly with frequency.

Sin Sec. Vl A we discuss experimental evidence suggesting that ½ increases slowly with frequency in humans.

6The agreement between the two distributions illustrated in Fig. 2 provides support tbr an analog of the ergodie principle: To the extent that local scaling •ymmetry is applicable in the unperturbed cochlea, averages over an ensemble of irregularities explored at fixed frequency (i.e., over an ensemble of subjects) yield roughly the same results as averages over frequency land hence position} assembled from measurements on a single individual.

*The wavelength Xa defined in Eq. (47} is real. To account for euergy loss or gain during wave propagation, models of cochlear mechanics typically assume that the wavelength is complex and that Ihe traveling wave has Ihe form • e ,$a•t•. At the peak of the traveling wave, the Iwo wavelengths are related by

1//{r'•Re{ I//•.•}. Sin this paper the scaling variable fi is defined a•s the model-independent ratio • I•oe{.t }, where oc(x)i2•r is the characteristic frequency defined by the peak of the transfer function at threshold. Note, however, that in the model of cochlear mechanics deft ned by Eq. (67) (Zweig, 1991 ),/51 refers to the ratio m/to,{x), where •odx)/2rr represents the resonant frequency of a section of the organ of Corti in the limit in which the damping 8 and feedback strength p are both negligible. For the model parameters given in the caption to Fig. 4, these two frequencies are related by o c I•or•-0.97. For the purposes of this paper, the smMI difference between these two frequencies (and their respective/•s) is unimportant and has been ignored.

*rhree equations determine three unknowns: Specifying the imaginary part of one of the two closely spaced poles and that the real and imaginary parts of the other pole coincide with those of the first fixes the model parameters 8, p, and /z. More preeizly, given y•lm{•0o}>0, where • denotes the location of the double pole in the complex-,8 plane, one determines the three parameters {•,p,/.,}, the real part of the double pole (denoted x=Re{•o,}), and the auxiliary variable a by solving the system of five simultaneous equations:

y=JlZ+a;

2*ra#= 1;

alx= tan[2 rr (hi2 + 3/4-/.•x}];

x 2= 1 --(812)2--a2; and

p= 2a[ ! _( f12)•jm e 1•. with n = I (K•r/• 1-34). mStimulus-tYequency emissions measured in the ear canal reflect oscillations

in the middle-ear input impedance caused by retrograde waves originating within the cochlea. Since the reverse-transmission characteristics of the

squirrel-monkey middle eas are not known, we plot the corresponding oscillations in the cochlear input impedaoce instead.

HEquation (37) for 0 implies that for perturbations AA • proportional to frequency (i.e., for perturbations in the damping} the factorization of the scattering integrand into a product of place- and wave-fixed factors is close to exact.

•In lhe nonlinear regime. the analogue of the transfer function is called the describing function.

L•In one-dimensional models ,•f cochlear mechanics

AR•R•a•- R•'eaak'•- 2(-•-• e T•(X- X}, where the chan•e in the impedance of the organ of Corti due to the mass • is given by AZ= ioeM 8(X- Xbeaa}.

•The frequency dependence of liAr seen here in data pooled across subjects is also apparent in single individuals possessing many emissions. (Talmadge et al report 2 snbjects with more than 20 emissions in each ear.)

•SAt frequencies of 1-2 kHz in the rhesus monkey measurements of stimulus-frequency emissions {Lensbury-Martin et aL. 1988} indicate that flAf• 13--compared with roughly 15 in humans--suggesting that species differences may be small among primates.

•6lf h represents the width of a hair cell, then in humans, t%12 rr•llh•720, using I •'7.2 mm (Greenwood, 1961: Greenwood, 199l}} and h • 10 /•m. Thus, near 1300 Hz •/t½• &l•ch • 15•720 - 0.02.

•?Although their transiently evoked emissions are small, rodent ears gener-

2045 J. Acoust. Soc. Am., VoL 98, No. 4, October 1995 G. Zweig and C. Shera: Origin of spectral periodicity 2045

ate acoustic distortion products much larger than those of primates. The magnitude of their evoked emissions thus appears unlikely to reflect any simple "deficit" in the reverse-transmission characteristics of the rodent middle ear.

•SNo symmetry under a cemetery wall. løThe neglected phase factor implies that the transfer function does not

strictly scale, even in a scaling-symmetric model. The phase factor contributes an additional weak frequency dependence to R in the form of a

phase shift, 8Nto/toco. Note, however, that this phase shift has the wrong sign (i.e., it rotates counterclockwise)--and, for reasonable values of N, is far too small in magnitude--to yield the empirical group delay of roughly 15 cycles computed from Eq. (10). For example, at frequency •0 the factor eSNi•'/*'co contributes a time advance of 8N/•o% seconds. At I kHz this amounts to less than a third of a cycle (using N=5 and •%/2*r = 20 kHz). Since the frequency dependence of this phase factor is weak, we have neglected it in the text.

2øThroughout Appendix D, the variable /• represents the ratio w/%(x) described in Note 8. Consequently, the transfer function peaks at the value /•0.97 and the saddle point described below occurs at ,8•0.96 (see Fig. 03).

2•To evaluate the derivative in Eq. (09), note that since K is real, d Re{o4/dK=Re{doddK }. Evaluating the total derivative yields

do• c•a c•ot d•

dK - dK q' c•ff dK' where, by definition, d•/dK vanishes at the saddle point [see Eq. (D5)].

22The location at which the imaginary part of the wavelength vanishes is close to but not identical with the location of the peak of the basilar- membrane transfer function. The function .B/:• 3/2 multiplying the exponential in Eq. (C5) shifts the peak of the transfer function slightly apically from the point defined by the zero crossing of the imaginary part of the wavelength.

23Equation (DI5) has been simplified by assuming that

an inequality well satisfied because of the large value of

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2047 J. Acoust. Soc. Am., Vol. 98, No. 4, October 1995 G. Zweig and C. Sbera: Origin of spectral periodicity 2047

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