Download auditory distortions: origins and functions

Physiol Rev 93: 1563–1619, 2013
doi:10.1152/physrev.00029.2012
AUDITORY DISTORTIONS: ORIGINS AND FUNCTIONS
Paul Avan, Béla Büki, and Christine Petit
Laboratory of Neurosensory Biophysics, University of Auvergne, School of Medicine, Clermont-Ferrand, France;
Institut National de la Santé et de la Recherche Médicale (INSERM), UMR 1107, Clermont-Ferrand, France;
Centre Jean Perrin, Clermont-Ferrand, France; Department of Otolaryngology, County Hospital, Krems an der
Donau, Austria; Laboratory of Genetics and Physiology of Hearing, Department of Neuroscience, Institut
Pasteur, Paris, France; Collège de France, Genetics and Cell Physiology, Paris, France; INSERM UMRS 1120,
Paris, France; and Université Pierre et Marie Curie, Paris, France
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XV.
INTRODUCTION
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HISTORICAL PERSPECTIVE: AUDITORY... 1566
MECHANOELECTRICAL TRANSDUCTION... 1569
NONLINEAR RESPONSE OF THE...
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INTERACTION OF TWO TONES IN THE... 1580
FROM COMPLEX STIMULI TO NO...
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THE PARTICULAR STATUS OF THE...
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MATHEMATICAL MODELS OF...
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AUDITORY MASKING AND THE COCHLEA 1602
CONCLUSION AND PERSPECTIVES:...
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HISTORICAL INSERT: TARTINI’S TONES 1606
APPENDIX I: OTOACOUSTIC...
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APPENDIX II: BACKWARD VERSUS...
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APPENDIX III: COCHLEAR DISTORTION... 1611
APPENDIX IV: CHARACTERIZATION...
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I. INTRODUCTION
A. Scope of the Review
Hearing has evolved to cover a huge range of sound intensities. The mammalian ear is sensitive enough to detect
weak sounds [20 ␮Pa, or 0 dB sound pressure level (SPL)]
when the vibrations of molecules in the transmitting medium barely exceed displacements due to thermal, Brownian motion (19). This impressive sensitivity is achieved by a
subtle sound-amplifying mechanism operated by the sensory cells themselves. This amplifier in the mammalian auditory organ, the cochlea, has been reviewed previously
(228). The acoustic powers processed by the auditory system extend over a thousand billion times above detection
threshold (20 million ␮Pa or 120 dB SPL). Furthermore, in
the presence of noise, the cochlear ability to extract the
characteristics of sound remains robust, thereby allowing
efficient central processing (“cocktail-party effect”; Ref.
39). If amplification were linear, the displacements of
sound-detecting cochlear receptors, of the order of a nanometer around threshold, would reach the potentially damaging micrometer limit around 60 dB SPL, the normal level
of conversational speech. Moreover, the resulting responses
could not fit in the restricted dynamic range of auditory
neurons innervating the sensory cells, as their action potential rates can only vary little, between a few per second, the
spontaneous rate of their activity, and a few hundred per
second at saturation. Actually, amplification gradually diminishes at increasing stimulus intensities, thereby producing the required compression. So, the displacements of liv-
0031-9333/13 Copyright © 2013 the American Physiological Society
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Avan P, Büki B, Petit C. Auditory Distortions: Origins and Functions. Physiol Rev 93:
1563–1619, 2013; doi:10.1152/physrev.00029.2012.—To enhance weak sounds
while compressing the dynamic intensity range, auditory sensory cells amplify soundinduced vibrations in a nonlinear, intensity-dependent manner. In the course of this
process, instantaneous waveform distortion is produced, with two conspicuous kinds of
interwoven consequences, the introduction of new sound frequencies absent from the original
stimuli, which are audible and detectable in the ear canal as otoacoustic emissions, and the
possibility for an interfering sound to suppress the response to a probe tone, thereby enhancing
contrast among frequency components. We review how the diverse manifestations of auditory
nonlinearity originate in the gating principle of their mechanoelectrical transduction channels; how
they depend on the coordinated opening of these ion channels ensured by connecting elements;
and their links to the dynamic behavior of auditory sensory cells. This paper also reviews how the
complex properties of waves traveling through the cochlea shape the manifestations of auditory
nonlinearity. Examination methods based on the detection of distortions open noninvasive windows
on the modes of activity of mechanosensitive structures in auditory sensory cells and on the
distribution of sites of nonlinearity along the cochlear tonotopic axis, helpful for deciphering cochlear molecular physiology in hearing-impaired animal models. Otoacoustic emissions enable fast
tests of peripheral sound processing in patients. The study of auditory distortions also contributes
to the understanding of the perception of complex sounds.
AVAN, BÜKI, AND PETIT
ing auditory detectors are a highly nonlinear version of the
vibrations of the external transmitting medium.
B. Anatomical and Physiological Framework
In mammals, the cochlear duct is divided longitudinally,
from base to apex, by the cochlear partition made of the
basilar membrane (BM) and the overlying auditory sensory
epithelium, the organ of Corti. This partition separates two
of the three internal compartments of the cochlea, scala
tympani (ST) and scala media (FIGURE 1A). Reissner’s membrane separates scala media from the third cochlear compartment, scala vestibuli (SV). Both ST and SV are filled
with perilymph (FIGURE 1A, light yellow area) while scala
media contains endolymph (FIGURE 1A, light-blue area), a
fluid that differs from perilymph and other extracellular
fluids by its high K⫹ concentration and a positive electrical
potential of about ⫹100 mV. Sound pressure waves are
transmitted to cochlear fluids via the outer and middle ears
to SV by an ossicle, the stapes, which, by vibrating in the
oval window of the cochlea, produces a pressure gradient
through the BM.
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It has been suggested in the 1940s (77), then confirmed in
the 1980s that amplification of acoustic stimuli in mammals
stems from a positive feedback mechanism (45) resulting
from bidirectional transduction in OHCs (287), namely,
their ability to respond mechanically to membrane-potential changes produced by MET (30; for a review, see Ref. 8).
Electromotility is due to the presence of a motor protein,
prestin, in their lateral plasma membrane (307). The resulting forces influence the mechanical impedance of the cochlear partition by feeding energy back into its movements.
When applied with the proper timing, this feedback partly
compensates for the loss of energy due to friction with cochlear fluids, thereby achieving amplification in an intensitydependent, nonlinear, compressive manner (228). The
sharpened BM resonance elicited by the positive feedback
process converts the originally passive and broadly tuned
frequency filtering of the cochlear partition into a highly
tuned frequency representation, in which the high and low
frequencies are processed at the base and apex of the cochlea, respectively, forming a tonotopic map along the longitudinal cochlear axis (200). Whereas OHCs are essentially confined to a micromechanical activity, IHCs aligned
along the longitudinal axis in a single row of ⬃3,500 cells
(in humans) and innervated by afferent neurons of the auditory nerve (FIGURE 1A) form the actual sensory cells of the
cochlea.
Hair cells fulfilling comparable functions are found in a
broad spectrum of vertebrate orders, from frogs through
lizards and birds, both in hearing and vestibular organs
(104, 158). Compared with mammals, the ways soundinduced vibrations reach the hair cells and set their stereo-
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Nonlinear systems share a generic definition: the response
to a combination of stimuli is not a simple algebraic sum of
individual responses. However, depending on the design of
a nonlinear system, the mathematical rules describing its
behavior can be very different. For example, compressive
nonlinearity may be achieved by instantaneous clipping of
the stimulus when it exceeds some limit. This strategy has a
drawback. It deforms the stimulus strongly and causes
waveform distortion (WD). An alternative method of compression, the sluggish automatic gain control (AGC) acting
over several stimulus periods, popular in the engineering
and hearing-aid worlds, generates, in contrast, minimal
WD. Finally, critical oscillators poised on the verge of selfoscillation (34, 60) could also combine reasonably fastreacting compressive amplification and limited WD. The
scope of this review is to unravel the origin and design of
auditory nonlinearity. It will lead us to examine how its
manifestations, compression and WD, relate to the mechanoelectrical transduction (MET) process. Its operation inescapably produces WD, and the mechanical coupling of
auditory sensory cells to their surroundings transforms
nonlinear products into cochlear vibrations. However, evolution seems to have managed to reach the best balance
between compressive (nonlinear) amplification, faithful
WD-free processing of sound, and the potential beneficial
effects of WD. Indeed, although nonlinearity results in conspicuous distortions in the frequency spectrum of complex
sounds, with the suppression of existing components and
the creation of novel ones not present in the stimulus, perception puts up with these distortions. As we will discover,
WDs also have a practical implication. Being easy to detect
noninvasively, they may be used to probe the peripheral
auditory function.
The organ of Corti contains the auditory sensory cells, the
inner hair cells (IHCs) and outer hair cells (OHCs). The
transduction of sound is achieved by a unique structural
feature of hair cells, protruding microvilli, namely F-actin
filled processes called stereovilli or commonly, though improperly, stereocilia, which are organized in bundles (FIGURE 1, B AND C). When mechanical vibrations produced by
sound propagate along the BM (see sect. IVA), the resulting
shearing motion between the BM and the tectorial membrane overlying the hair cells displaces stereocilia bundles.
In these bundles, stereocilia organized in three rows are
connected by multiple fibrous links, among which the socalled tip-links extend from the tip of a stereocilium to the
side of the adjacent taller one (FIGURE 1C). When tensed by
vibrational movements, tip-links produce a conformational change of MET channels located at the tip of stereocilia of the small and medium rows (18), thereby modulating their conductance (gating). In consequence,
driven by their positive electrochemical gradient, K⫹
ions, which abound in the endolymph, enter the cells,
together with Ca2⫹ ions. The ion currents lead to cell
depolarization and to interactions between Ca2⫹ ions
and various molecules inside the hair cell (35, 71).
AUDITORY DISTORTIONS
A
B
SV
RM
TM
SM
BM
AN
ST
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C
FIGURE 1. A: schematic cross-section of a mammalian cochlea, showing the three fluid-filled compartments
scala vestibuli (SV), scala media (SM) and scala tympani (ST), the basilar membrane (BM), and Reissner’s
membrane (RM). The organ of Corti contains one row of inner hair cells (IHCs, black arrow) and three rows of
outer hair cells (OHCs, red arrow), overlaid by the tectorial membrane (TM). Afferent auditory neurons forming
synapses at the base of IHCs make the auditory nerve (AN). B: scanning electron micrograph of the organ of
Corti of a mouse. The tectorial membrane (TM) is lifted, revealing the W-shaped stereocilia bundles protruding
from the tops of OHCs organized in three spiraling rows. Beneath the reticular lamina (RL), the cylindrical
bodies of OHCs can be seen behind the thin, winding elongations (phalangeal processes, PP) of supporting cells
(the Deiters cells). The IHCs are not visible; they form a spiraling longitudinal row hidden beneath the tectorial
membrane. On the lower side of the tectorial membrane, the imprints of the tallest OHC stereocilia can be
recognized (one imprint is delineated by arrows). C: scanning electron micrograph of the top of an OHC, with
its stereocilia bundle organized in three rows. The tips of small and intermediate stereocilia are connected to
the sides of their taller neighbors by tip-links (arrows), essential for opening the MET channels during auditory
stimulation. At this early stage (postnatal day 7), the tops of stereocilia are not yet connected by transverse
links (see FIGURE 5). Scale bars: 4 ␮m (B) and 1 ␮m (C). [Courtesy of Aziz El Amraoui (A) and Vincent Michel
(B and C).]
cilia bundles into motion are different, and auditory hair
cells are less specialized than OHCs and IHCs. [Frequency
tuning can stem from the intrinsic resonance of hair bundles
or from electrical properties of basolateral membranes
(158).] Compressive amplification is also ensured by a positive feedback principle even though the somatic, prestindriven electromotility is substituted by “active” hair-bundle
motility (104). The topic of this review pertains to hair
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AVAN, BÜKI, AND PETIT
cell-driven nonlinearities and not to the exact feedback
mechanism affording amplification and frequency tuning.
Examples will be borrowed from nonmammalian auditory
sensory organs when needed.
All manifestations of auditory nonlinearities, compression,
WD, suppression, even spontaneous activity, have been
proposed to emerge from the same processes and coexist or
vanish together. Thus their study should highlight basic
cochlear mechanisms of sound processing including the
analysis of complex sounds. Otoacoustic emissions (OAEs),
sounds emitted by a nonlinear cochlea (121), deserve a par-
II. HISTORICAL PERSPECTIVE: AUDITORY
STIMULUS PROCESSING AND THE
NONLINEARITY PRINCIPLE
The existence of WD in a system supposed to faithfully
convert sound messages into information for the brain is
counterintuitive, e.g., for high-fidelity specialists who obsessively fight any form of distortion. Thus before 1978,
almost all reports of objective measurements of WD, apart
from a few exceptions (193, 289), ascribed them to nonlinearities outside the cochlea, e.g., either in the middle ear
(95) or in the earphones. Actually, the middle ear does not
distort at physiological SPLs (4), and earphone nonlinearities can be controlled (260). However, a few early psychophysical studies, which will be reviewed first, noticed
unique features of WD that identified the cochlea as its
source. With regard to objective measurements, it took two
sets of seemingly unconnected observations to make clear
that an intact cochlea works nonlinearly and is not a passive
resonator (FIGURE 2). The first set of data (228) concerned
the compressive growth of BM motion with the level of the
stimulus at the cochlear places tuned to the frequency of the
probe tone (225). The second set of data described instantaneous WD in normal cochleas, the central topic of this
review.
A. Perceptive Evidence
It has been acknowledged for a long time that when two
tones at frequencies f1 and f2 are mixed, intermodulation
FIGURE 2. Two manifestations of nonlinearity in the cochlea. Left: BM velocity against tone level, at
frequencies equal to and higher than the characteristic frequency, 10 kHz, at the measurement place: the
closer the frequency to 10 kHz, the more compressive (shallower) the BM growth. [From Robles et al. (230).]
Right: spectrum of the sound detected in the ear canal of a mouse in response to a pair of pure tones at
frequencies f1 and f2 (60 dB SPL), displaying distortion products at combination frequencies (solid arrows,
odd-order combinations; dashed arrows, even-order ones).
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The gating principle of a mechanosensitive channel necessarily produces cycle-by-cycle WD (101) and even, clipping
at saturating levels. Because of the feedback process, the
sound wave of the stimulus itself gets contaminated by WD.
Flattening and clipping add harmonics 2f, 3f, etc., to the
sinusoidal wave of a tone at frequency f. If the stimulus
contains a combination of tones, as the processing of auditory stimulation on the BM is spatially distributed, waves at
neighboring and even remote frequencies travel together
and can interact. The response to one tone of a complex
stimulus may result in smaller displacements of the BM than
it would have done if the tone had been presented in isolation. The stimuli undergoing this so-called suppression become less audible. Even though suppression diminishes the
audibility of weak probe tones in the presence of noise, it
has also been suggested that by increasing contrasts, it has
overall positive influences on sound analysis (262). Another
result of WD is that new sinusoids at arithmetic combinations of the original frequencies emerge in the response.
This cochlear nonlinearity is found even at the lowest measurable stimulus levels, which led to call it “essential” (79)
in contrast to the nonlinearities only observed at saturation.
Although WD deforms the frequency spectrum of incoming
sounds, it does not dramatically disturb perception as the
cochlear compressive and filtering activity, applied to WD,
ensures that it remains small in most circumstances.
ticular emphasis. Although usually mixed with the stimulus
that evokes them, they are easily singled out by their characteristic nonlinear properties and long latencies. They have
become unique tools for noninvasive evaluation of peripheral hearing, whether for human audiology or in animal
models with cochlear deficits.
AUDITORY DISTORTIONS
WD results in the production of audible tonal components
at arithmetic combinations (f2-f1, 2f1-f2, 2f2-f1, etc.), absent
in the stimulus, called distortion products (DPs) or combination tones. Their discovery is attributed to the musician
Tartini (267), and the first physically based investigation, to
Helmholtz, although doubts can be raised as to the exact
physiological significance of these early observations (see
sect. XI). The very unusual dependence of DPs on stimulus
level and frequency spacing were first disclosed by psychoacoustical methods (79, 259) ascertaining their cochlear origin.
The importance of Goldstein’s work was not immediately
grasped. Today, its groundbreaking consequences are clear.
Goldstein’s results indicate that there must exist a compressive amplifying device in the cochlea. More precisely, one
has to assume that the amplitude of BM displacement increases by only 0.33 dB/dB increase of the stimulus, to explain why the 2f1-f2 response increases by the observed 1
dB/dB, i.e., 3 times 0.33 dB/dB. Rhode was the first to detect
compression by direct mechanical measurements of BM vibrations (225), but the significance of his results was also
overlooked for years. Another finding of Goldstein pointing
to the cochlear origin of the DP was its strong dependence
on frequency ratio f2/f1. Here was the first explicit statement
that WD, cochlear sensitivity, compression, and frequency
selectivity may be associated (79). Last, the audibility of the
2f1-f2 DP for stimuli down to 20 dB SPL (79) called for
another explanation of this distortion than saturation.
B. Objective Evidence: Otoacoustic
Emissions
Goldstein’s discovery relied on perceptive reports from two
subjects, including himself. Objective evidence of inter-
Pa
mPa
0.3
20
0
-0.3
-20
0
10
20
ms
FIGURE 3. Transient-evoked otoacoustic emission (TEOAE) evoked
by a click and detected by a miniature microphone in the sealed ear
canal of a normal subject. Stimulus ringing (in red, left-hand pressure scale) does not last beyond 3 ms. After this time, the averaging
procedure adds the responses to clicks at opposing polarities and
different amplitudes, chosen in such a way that the overall average
of clicks is zero. Because the linearly growing responses are cancelled, a derived TEOAE remains, demonstrating the existence of a
compressively growing response.
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Goldstein (79) evaluated the most prominent endogenous
DP, i.e., at 2f1-f2, by cancelling it by an adjustable external
tone at exactly 2f1-f2. At a precise combination of level and
phase of the external tone, trained subjects, in whose ear the
experiment was performed, no longer heard the 2f1-f2 component, as it was at the same level and exactly out of phase
with the canceling tone. The concept of “essential nonlinearity” coined by Goldstein stemmed from his key observation that when the amplitude of stimuli decreased, that of
the response at 2f1-f2 decreased in similar proportions (1 dB
per dB decrease of the primary tones at f1 and f2) so that this
nonlinear response was always approximately at the same
dB level relative to the stimuli. In contrast, nonlinearities
due to saturation in a passive system do not behave this
way. It can be shown (see sect. VF) that the amplitude of the
2f1-f2 component depends on the square of the amplitude at
f1 times the amplitude at f2 so that when both increase by 1
dB at the input of a passive, saturating nonlinear system, the
2f1-f2 component should increase by 3 dB.
modulation from the cochlea was provided by the pioneer
studies of David Kemp (120, 121), revealing the existence of
OAEs from the ear. These low-intensity sounds emitted by
healthy ears, either spontaneously or in response to sound
stimuli, could be collected by a microphone sealed in the ear
canal. Three different stimulus paradigms, using transient
sounds (clicks, with a broad frequency spectrum), a single
pure tone or a pair of pure tones, evoke three main categories of sound-evoked OAEs. The first observations were
reported by Kemp (121) as transient-evoked OAEs (TEOAEs) initially called “Kemp echoes” (FIGURE 3). This finding was as groundbreaking a discovery as that of Goldstein
because the existence of OAEs showed that the cochlea
responds to sound by creating vibrations. Kemp echoes
could not be regular linear echoes as they grew nonlinearly,
not in proportion to click intensity. The second category of
OAEs, evoked by a pure-tone stimulus, was found at the
same frequency as the stimulus, stimulus-frequency OAEs
(SFOAEs), which made them technically more difficult to
separate from the stimulus. Last, intermodulation OAEs
showed up in the spectrum of sound in the ear canal, in
response to pairs of pure-tone stimuli with close enough
frequencies f1 and f2, less than half an octave apart (79). In
1977, Kemp (120) had found that the DP heard at 2f1-f2
was also emitted in the ear canal as an OAE, the so-called
DPOAE, yet its weak level in human ears led its discoverer
to set it aside. Henceforth, a distinction will be made between DPs, which encompass intracochlear vibrations at
combination frequencies, their neuronal correlates and
their perception, and DPOAEs, the acoustic correlates of
DPs detected in the spectrum of sound in the ear canal after
backward propagation of DP vibrations through the middle
ear.
AVAN, BÜKI, AND PETIT
Strong doubts were initially raised whether otoacoustic
emissions might be an experimental artifact. The original
experiment was carefully replicated (295, 298), and the
only alternative explanation for the compressive growth of
OAEs with stimulus intensity, middle-ear muscle function,
was excluded. Clear evidence of a close relationship between the presence of OAEs and cochlear sensitivity (7)
ruled out the possibility that emissions were produced elsewhere than in the cochlea.
in the stiffness of the cochlear partition upon tip-link disruption. These changes were large enough to show that
OHC stereocilia can indeed influence the impedance of the
cochlear partition (36). In turn, the cochlear partition can
drive sound pressure in a sealed ear canal, and measurements of the reverse middle-ear transfer function in guinea
pig (157) and gerbil (52) have shown that intracochlear DPs
come out as DPOAEs with a level approximately reflecting
the middle-ear transformer effect, that is, ⫺35 dB regardless
of frequency.
C. Cochlear Origin of Nonlinearities
Because of their large levels above the noise floor in laboratory animals (127, 128), DPOAEs have been extensively
used in pathophysiological experiments conducted between
1980 and 1996, which unequivocally demonstrated their
cochlear origin. In non-mammals and despite the differences in the physiology of their auditory organs, WD and
OAEs were also found (15, 160). Evidence of the physiological links between DPOAEs and OHCs was collected
from a variety of experiments enumerated in TABLE 1, the
principles of which were either to contrast the effects on
DPOAEs of damage to OHCs versus other cells, or to activate known modulators of OHC function. Nonetheless,
when explored more thoroughly, the correlations between
DPOAEs and the status of OHCs at places tuned to stimulus frequencies were less straightforward than initially reported, which required the issue of the places and mechanisms of DP generation to be revisited.
Objections to the idea that OHCs in the cochlea could drive
BM motion strongly enough to produce detectable sound in
the ear canal were lifted by studies revealing large changes
Table 1. Early evidence of the links between OAEs and OHCs
Experiment
OHC Activity
Acoustic overstimulation
Aminoglycoside administration
Medial olivocochlear efferent neural
bundle stimulation
Electrical stimulation of OHCs
Assayed losses of OHCs by
platimun salts
Assayed losses of IHCs by platinum
salts
DP grams in human hearing losses
due to sensory-cell disease
DP grams in human hearing losses
due to neural dysfunction
OAEs
Reference Nos.
Abolished
Abolished
Temporarily affected
Decrease then recovery
Abolished
Slightly inhibited
Abolished
Slightly Decreased
7, 196, 309
127, 241, 252
(Before the mid 1980s, the role of OHCs being
unknown, the effects of noise on OAEs could
be attributed to cochlear, but not OHC
impairment)
28
40, 128, 211
Increased
Abolished
Increased
Abolished
188
269
Unaffected
Unaffected
269
Abolished
Abolished
89, 166
Unaffected
Unaffected
196, 261
OAE, otoacoustic emission; OHC, outer hair cell; IHC, inner hair cell; DP, distortion product.
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It was soon suggested that the links between noninvasively
measured OAEs, regardless of their category, and OHC
functions could be exploited for inferring useful information regarding cochlear frequency tuning. In the frequency
domain, so-called suppression tuning curves of OAEs were
built. These represent for an interfering tone, how the level
required to influence an OAE varies with interfering frequency. They reveal that interference happens only in a
narrow frequency interval with a width similar to that of
cochlear resonators (27). In the time domain, the delay of
OAEs relative to their stimuli betrays the interplay of underlying resonators, such that the sharper their tuning, the
longer they ring, as do OHC responses (200, 228). Nonphysiological, equipment-related causes of distortion show
only a delay corresponding to the time it takes for sound to
travel a few centimeters in the air and through the external
or middle ear (in humans, a fraction of a ms). Conversely,
TEOAEs ring several tens of milliseconds after the click
evoking them has faded away (121) (FIGURE 3). Measurements of the latency of the other two types of OAEs are
more delicate as they are usually produced by continuous
tones with no definite onset. Either special methods utilizing
AUDITORY DISTORTIONS
interrupted stimuli allow DPOAE onset latency to be visualized directly (293), or the frequency dependence of the
phase of DPOAEs reveals their so-called group delays (see
sect. IVF) (130, 185, 186, 221, 231).
D. Essential Links Between Nonlinearity and
Cochlear Activity
The confirmations of a shared origin for cochlear activity
and nonlinearity have changed the status of OAEs since the
1980s. Initially, OAEs were considered as anecdotal byproducts, worth being studied only because they might have
a clinical utility, such as universal screening of congenital
hearing impairment in neonates (22, 108, 290). From the
1990s on, their potential as tools for assessing cochlear
nonlinearities has been understood. OAEs open a unique
III. MECHANOELECTRICAL TRANSDUCTION
AND NONLINEARITY
Mammalian OHCs act within a feedback loop, which processes the auditory stimulation. For modeling purposes,
four functional stages of this feedback loop can be separated, all of them being theoretically able to distort to some
extent. Locally, transverse sound-induced BM vibrations
propagated from the basal cochlea produce a radial shearing motion between the reticular lamina, formed by the
apical cell surface of hair cells and pillar cells supporting
them, and the tectorial membrane, which deflects the stereocilia bundles. The tallest stereocilia embedded in the tectorial membrane are directly deflected in the excitatory direction, away from the modiolus, when the BM moves upward toward scala media (stage 1). The displacement of
OHC bundles away from the modiolus opens MET channels by pulling on the tip-links (see sect. IB; FIGURE 1C). The
endocochlear potential, about ⫹100 mV, combined with
the negative polarization of OHCs, near ⫺40 mV (109),
strongly drives endolymphatic K⫹ ions into the cells
through open MET channels (stage 2). The sound-modulated current through an OHC produces a change in its
membrane potential. As OHCs are endowed with electromotility (see sect. I), forces are generated (stage 3) which, by
acting with the appropriate timing, produce positive feedback and a negative-damping effect that enhances BM vibration (stage 4) (FIGURE 4). In the hair cells of nonmammalian species, as mentioned previously, no somatic electromotility has been reported. The active feedback loop,
instead, involves the stereocilia bundles (104). It has been
proposed that hair-bundle mechanics contributes to the active process also in the cochlea (194, 205), which, in the
four-stage model, would imply active force generation at
stages 2 and 3, that is, not only stage 3.
FIGURE 4. Left: stages of OHCs activity organized in a feedback loop at the origin of amplification (see text
for explanation). Right: at stage 2 (mechano-electrical transduction; MET), the current versus stereocilia
bundle deflection transfer function is a sigmoid-shaped Boltzmann function (plot labeled Po). MET-channel
functioning is also at the origin of the nonlinear stiffness of hair bundles, which reaches a minimum when the
opening probability of MET channels is 0.50 (plot labeled “stiffness”). As a result, the force exerted on a
stereocilia bundle as against its deflection (solid line labeled “force”) departs from a straight line (dotted line
labeled “linear”). [From van Netten (276). Copyright 1997, with permission from Elsevier.]
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Around 2000, several publications independently emphasized that the Hopf-bifurcation formalism provides a parsimonious, generic model of an active ear that unites all its
aforementioned properties. It describes the behavior of critical oscillators, poised (e.g., by negative feedback) for operating near their dynamic instability (34, 60, 105). With
the assumption that the damping term of an oscillator contains some adjustable parameter, a bifurcation occurs when
this parameter sets damping to zero. On one side of the
bifurcation, damping is negative and the (unstable) system
oscillates spontaneously, while on the other side, damping
is positive and the oscillator is stable, yet neither sensitive
nor finely tuned. It is when poised near the bifurcation that
the critical oscillator displays the key features of an active
cochlea, i.e., high sensitivity, compressive amplification,
and level-dependent frequency tuning, combined with the
above-described nonlinearities. Stability is ensured by a distortion-generating damping term increasing with the third
power of oscillation amplitude (see sect. VF). Several specific models of the cochlea present a Hopf bifurcation, even
some of them not explicitly built for this purpose (265).
window on the intimacy of auditory stimulus processing in
a hearing organ.
AVAN, BÜKI, AND PETIT
A. Which Stage Is Nonlinear?
Therefore, relevant contributions of OHCs to significant
mechanical nonlinearity and WD should be searched for at
stage 2. Here, two possible mechanisms can be found. First,
the directly measured stiffness of hair bundles is asymmetrical, larger in the excitatory than inhibitory direction (68).
Second, when zooming in on the OHC MET channels, the
plot relating K⫹ ion current versus hair-bundle displacement, the so-called transducer transfer function, is sigmoid
shaped. Experimental plots can be fitted by a first-order
Boltzmann transfer function describing the statistical process by which MET channels in a hair bundle shuffle from
an open to a closed state (288). These two nonlinear contributions, deflection-dependent change in hair-bundle stiffness and MET function, actually are tightly interconnected,
as shown by experiments that used antibiotics to block
MET function and also abolished the asymmetry of hairbundle stiffness (102, 235). Therefore, gating of MET channels and activity-dependent hair-bundle stiffness might be
two sides of the same coin, which suggested the term gating
stiffness (or its inverse, “compliance”) to describe this nonlinearity (see sect. IIIB).
Nowadays, second-order asymmetrical Boltzmann transfer
functions assuming one open and two closed states are more
widely utilized for describing the activity of OHC MET
channels (153). Their point of inflection lies closer to the
lower saturation level than in a first-order function. The
current varies from zero, when all channels are closed, to a
saturated value corresponding to all channels open, and the
operating point (OP) describes the percentage of open channels at rest, thought to be near 50% in OHCs (109, 234).
The slope of the transfer function around the OP is S ⫽
⌬I/⌬␣, where ⌬I is the current through the cell produced by
a deflection ⌬␣ of the hair bundle. The larger S, the larger
the change in membrane potential in response to a given ⌬␣,
thus the larger the strength of feedback onto the BM and the
gain of the OHC-based cochlear amplifier. From the maximum-gain position at the inflection point, if the OP shifts
1570
The exact shape of the transfer function at the OP, where
the hair bundle operates, is of importance because, as will
be shown in section VF, the magnitudes of DPs depend on
the coefficients of its expansion in a Taylor power series.
The lowest-order DP at f2-f1 is called quadratic, and the
next terms at 2f1-f2 and 2f2-f1, cubic, as they depend on the
terms of the series to the powers 2 and 3, respectively. As a
rule, even-order DPs are such that the sum of coefficients of
f1 and f2 in the DP frequency formula is an even integer
(1⫹1, for the f2-f1 DP), whereas for odd-order DPs, this
sum is an odd integer number (2⫹1 for the cubic DPs).
Characteristically, an even-order DP is generated by symmetric components of the transfer function so that the pattern of WDs and their changing content in even-order DPs
betrays the asymmetry of the transfer function which can be
adjusted by changing the OP (21, 29, 70, 253).
In summary, in the widely accepted view that in mammals
the processing of auditory stimuli by OHCs is organized in
a feedback loop, the compressive amplification of stimulus
vibrations stems from the whole feedback process, whereas
the other nonlinear manifestation, WD, is associated with
MET-channel gating. Not surprisingly, cochlear amplification, compression, and WD generally exist and vanish together. For example, in mutant mice with nonfunctioning
MET channels, there is no transduction and no cochlear
activity, and all cochlear nonlinearities are absent (33, 178).
Nevertheless, WD originating from MET-channel gating
can exist in the absence of amplification, in a passive cochlea (9, 183). The most common situation of passive WD
happens in response to low-frequency stimuli of ⬃70 dB
SPL at basal cochlear places which are not tuned to them,
but will vibrate passively (169) (see sect. VIA).
B. Mechanisms of Nonlinearity on a
Subcellular Level
The mechanosensitive channels of auditory sensory cells,
whose gating is directly controlled by hair-bundle deflection
and not by enzymatic reactions as in nonmechanical sensory modalities, can respond to frequencies up to ⬃100
kHz. The stereocilia of a mature OHC hair bundle, arrayed in three rows of increasing height, are coupled
together by tip-links (207), made of cadherin-23 and protocadherin-15 (119) and likely anchored to the MET
channel. The deflection of a hair bundle by the acoustic
stimulus in the excitatory direction increases the tension
in tip-links and related gating springs, putative elastic
elements which pull on MET-channel gates and increase
their open probability (Po).
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At stage 1, the shearing motion between BM and tectorial
membrane contains nonlinear terms due to the complex
trigonometry of their relative displacements but only at
large deflections (⬎10°), beyond the physiological scale of a
few hundred nanometers (deflection ⬍3°). Stage 3 is linear
even though in vitro experiments pinpoint two potential
causes of nonlinearity. The somatic current versus voltage
characteristic of OHCs (97) is indeed complex, due to the
interplay of ionic fluxes through membrane channels of the
cell soma, and the OHC cell-body length relates to membrane potential in a slightly nonlinear manner when its
changes exceed tens of millivolts (8). However, for stimulations above 1 kHz, the somatic current-voltage relationship becomes linear, and in the physiological range of
changes, the voltage-dependent electromechanical activity
of OHCs distorts negligibly (240).
toward a saturating part of the transfer function, its curvature and ability to generate WD change too. The sinusoidal
input tends to get clipped more on one side, as opposed to
symmetrical clipping when the OP coincides with the center
of symmetry of the function.
AUDITORY DISTORTIONS
independent of deflection, which was confirmed experimentally for the first two ones (44, 100). Let K⬁, the sum of
these contributions, be the deflection-independent hairbundle stiffness.
The gating process also introduces a deflection-dependent stiffness term Kg due to the fact that when a MET
channel opens, the associated conformational change
leads to a significant shortening of the associated gating
spring by a few nanometers, which relaxes it partially. As
a consequence of this decrease in stiffness, the gating
springs of the remaining channels bear additional tension, a gating force Fg that increases Po (101). As shown
by the current versus deflection mechanoelectrical transfer function, Po obeys a sigmoid Boltzmann statistical
law (FIGURE 4B)
Po ⫽ 1 ⁄ 共1 ⫹ exp关⫺ Fg共X ⫺ X0兲 ⁄ kBT兴兲
(1)
in which Fg is the gating force, X-X0 is the displacement of
the tip of the hair bundle, kB is the Boltzmann constant, and
T is the absolute temperature. If one considers more than one
open and one closed states for the MET channel, higher-order
Boltzmann statistics replace the first-order one in Equation
1. Thermodynamics then predicts that the stiffness of a hair
bundle carrying N transduction units is (101)
KHB ⫽ K⬁ ⫹ Kg ⫽ K⬁ ⫺ 共N Fg2 ⁄ kBT兲Po共1 ⫺ Po兲
(2)
When all channels are either closed or open, KHB ⫽ K⬁.
When a MET channel is pulled open by a stimulus so that Po
increases, Kg, the nonlinear Po-dependent term in Equation
2, describes a decrease in stiffness, i.e., an increase in compliance, with momentous consequences. Indeed, as the gating force Fg adds up to the stimulus (FIGURE 6), it becomes
easier to deflect the hair bundle and thus to open more MET
channels. The single-channel gating force can be expressed
as a function of the transducer current I(X) and its derivative dI/dX (277)
Fg共X兲 ⫽ kBT ⁄ I共X兲dI共X兲 ⁄ dX
FIGURE 5. Morphology of OHC hair bundles. The low magnification
scanning electron micrograph (top) shows the three rows of OHCs
and their aligned stereocilia. At higher magnification (bottom), the
top connectors are visible, marked by arrows. Scale bars: 2.5 ␮m
(top) and 0.25 ␮m (bottom). [From Verpy et al. (282).]
(3)
Thus both the nonlinear gating stiffness of the hair bundle and the gating force that modulates the stimulus directly relate to the sigmoid Boltzmann transfer function
and to MET-channel operation. As a result, the larger the
size of Kg relative to K⬁ in Equation 2, the more conspicuous the overall hair-bundle nonlinearity. In hair-cell
preparations from the frog saccule, a vestibular organ, Kg
is so large that the overall hair-bundle stiffness becomes
negative within the range of MET channel operation,
which leads to bundle instability and spontaneous oscillations (172). In mammalian hair cells, the linear stiffness
K⬁ dominates so that gating forces elicit much less mechanical nonlinearity (K⬁ and the K⬁/Kg ratio are 10
times larger than in frogs) (277).
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At present, even though attractive candidates have been
proposed for the MET channel (118, 129), the exact molecular architecture of the MET machinery remains unknown.
However, many basic characteristics of its gating have been
inferred from ex vivo measurements of ionic current
through the channel and of stiffness measurements, against
hair-bundle deflection, using calibrated flexible fibers to deflect stereocilia tips (e.g., Refs. 68, 102). To understand the
origin of the deflection dependence of stereocilia bundle
stiffness (68), the first requirement is the identification of its
components. Apart from the tip-link molecular complex
and the stereociliary rootlets around which stereocilia rotate as rigid rods (23, 132), the stereocilia of OHCs are
expected to be stiffened by zipperlike clustered fibrous
links, the horizontal top-connectors that join the upper
parts of adjacent stereocilia within and between rows (81,
280, 282) (FIGURE 5). All contributions are thought to be
AVAN, BÜKI, AND PETIT
F
Fg
C. Horizontal Top Connectors: An
Unexpectedly Large Part
Measurements on mutant mice that lack stereocilin (Strc⫺/⫺
mice), a model for the hereditary recessive form of human
hearing impairment DFNB16 (281), raised a surprising issue regarding the part played by MET channels in the generation of cochlear nonlinearity (282). Stereocilin is a protein associated with horizontal top connectors joining adjacent stereocilia within and between rows, and with the
links that attach the tallest stereocilia to the tectorial membrane (280). In the absence of stereocilin, in Strc⫺/⫺ mice,
horizontal top connectors do not develop, and the tips of
OHC stereocilia are less clearly aligned than in control mice
(FIGURE 7A). Their tip-links persist until postnatal day 15,
P15 (FIGURE 7B), and MET currents are found within normal range, ensuring near-normal auditory thresholds during a few days after the onset of hearing. Yet, DPOAEs and
WDs in the electrical responses of OHCs (FIGURE 7, C, right
panel, and D, dashed line) are abolished in response to stimuli below 90 dB SPL, whereas in wild-type mice they are
evident in the 20- to 90-dB SPL interval (FIGURE 7, C, left
panel, and D, solid line). A strong decrease of masking, in its
suppressive component, is also observed (FIGURE 7E). As
discussed, suppressive masking is viewed as enhancing contrasts among simultaneously present spectral components
(262, 282). Whether decreased suppressive masking is detrimental to the intelligibility of speech in noisy environments remains to be tested in patients affected by DFNB16.
Between P15 and P60, Strc⫺/⫺ mice gradually entirely lose
OHC function, which is attributed to the irreversible loss of
the tip-links. Of note, in a recently published cohort of
affected patients (69), mutations in the stereocilin-encoding
gene, STRC, emerged as a major contributor to pediatric
bilateral sensorineural hearing impairment. Yet it fre-
1572
quently came with only mild increase in hearing thresholds,
which thus indicates that OHC hair bundles without the
stereocilin-composed horizontal top-connectors can still
function in humans.
The intriguing issue raised by the phenotype of Strc⫺/⫺ mice
is that it apparently contradicts the tenet that distortion is
an exclusive property of MET channels in relation to their
ability to open and close normally. By functioning enough
in young Strc⫺/⫺ mice to ensure the detection of low-level
sounds, MET channels must distort. Yet no sign of WD was
detected between 20 and 90 dB SPL. Of note, this divorce
between normal responses at SPLs near the auditory threshold, and the absence of WD and defects in masking (FIGURE
7) may only be apparent, and be compatible with a normal
functioning of MET channels when hair-bundle deflection
is minimum ⫺40 dB SPL for threshold measurements, but
could become abnormal at larger hair-bundle deflections
ⱖ70 dB SPL for WD, DPOAE, and masking measurements.
Several hypotheses may be brought forward to explain
these results. The most radical line of thought would be to
challenge the current view by positing that the recorded
distortion stems, not from MET channel nonlinearity (Kg in
Equation 2 would be too small), but from some passive
nonlinear mechanism of the hair-bundle, so far overlooked,
that involves the horizontal top connectors. This implies
that the molecular assembly containing stereocilin has specific biophysical properties able to confer nonlinear stiffness
to a hair bundle. Notably, some models of hair bundles
predict that nonlinear stiffness can stem from changes in
bundle geometry due to deflection when a cohesive hair
bundle is formed (190). An even simpler view would assume
the horizontal top connectors to be tight when a hair bundle
is deflected in one direction, and relaxed in the other direc-
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FIGURE 6. Principle of gating compliance. Tip-links (green) connect neighboring stereocilia (left). A MET
channel (blue) can be pulled open by hair bundle deflection by a stimulus (force F) in the excitatory direction
(middle). The conformational change due to channel opening (right) shortens the gating spring, a linear elastic
element activating the gate. This shortening reduces tension in the spring, which amounts to a gating force Fg.
The molecular complex anchoring the tip-link to the taller stereocilium cytoskeleton (pink) may adjust its tension
by slipping along the actin filaments, thereby controlling the open probability of the MET channel (MET
adaptation process).
AUDITORY DISTORTIONS
tion, without having any specific nonlinear mechanical property per se. Equation 2 would be replaced by KHB ⫽ K⬁ ⫹ Kd,
with Kd depending on the state of tension or relaxation of
horizontal top connectors, i.e., on the direction of hair-bundle
deflection.
Alternative explanations integrate the role of MET channel
nonlinearity in the generation of WD. The coordinated
opening of MET channels implicating the horizontal top
connectors might combine weakly nonlinear transfer functions of individual MET channels into a markedly nonlinear
mechanoelectrical transfer function of the hair bundle. The
interest of a coordinated motion of bundled stereocilia has
been recently emphasized in the context of transduction
sensitivity rather than nonlinearity, but the same arguments
may be relevant for both topics. One important issue is
whether MET channels work in series or in parallel. In a
mode of action in series of the MET channels, only the
stereocilia in a radial relationship mediated by tip-links are
involved. The opening of one MET channel is expected to
reduce the force exerted on the other MET channels within
the same radial column, which results in negative channel
cooperativity (117). A possible exception was observed in a
model of hair bundle that examined the series engagement
of two transducer channels, with one of them assumed to be
sensitive at small displacements and the second one at larger
ones. In this case, the series mode of activation allowed
complementary activation with improved dynamic range
and intact sensitivity (278). It is noteworthy that hair-cell
activation curves in this two-channel series model showed
greater asymmetry and nonlinearity and much more compression at higher levels, relative to the single engagement
case. Cooperativity between only two channels was sufficient for deeply influencing the nonlinearity of hair cell
transfer functions (278). This strong impact of radial mechanical coupling between MET channels of one column of
a stereocilia bundle suggests that a possible contribution of
the inter-row horizontal top connectors to channel cooperativity and the ensuing nonlinearity might have to be considered in addition to the tip-links invoked (278).
Actually, several investigations of hair-bundle preparations
whereby stereocilia splaying is prohibited suggested that the
transduction channels work mechanically in parallel, rather
than in series. Parallel coupling results in positive cooperativity
in response to a force stimulus, i.e., the more MET channels
open, the larger the force transmitted to the closed ones (116,
136). In hair cells from the bullfrog sacculus, the constraints
on hair-bundle geometry were found to be strong enough for
spontaneous stereocilia movements at opposite edges of the
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FIGURE 7. In the Strc⫺/⫺ mice, which lack stereocilin, the horizontal top connectors are missing. The tops
of OHC hair bundles are less aligned than they should be (scanning-electron micrograph, A), while a close-up
view of stereocilia (B) shows that tip-links are present at postnatal day 14 (arrowheads), which accounts for
preserved MET and auditory sensitivity. C: contrary to wild-type controls at the same age (left), the cochlear
microphonic potential (CM) in Strc⫺/⫺ mice (right) in response to pairs of pure tones contains no WD (arrows
mark intermodulation frequencies). D: Lissajous patterns of CM in response to 80-dB SPL tones display a
sigmoid pattern in Strc⫹/⫹ mice (solid line) and a linear pattern in Strc⫺/⫺ mice (dashed line), despite similar
CM amplitudes suggesting an equivalent function of the cochlear base. E: masking tuning curves plotted at
postnatal day 14 (representing the threshold level of a masker tone, against its frequency, required to
suppress the neural response to a fixed probe tone at 10 kHz) are equally narrow in Strc⫺/⫺ mice and Strc⫹/⫹
mice, yet although probe tones are presented at similar levels (diamonds), masker levels have to be set ⬃20
dB higher in Strc⫺/⫺ mice to exert masking. [From Verpy et al. (282).]
AVAN, BÜKI, AND PETIT
Could the W shape of the OHC hair-bundles have a special
impact on the proposed stereocilin-mediated positive MET
channel cooperativity? The fact that the basal insertion points,
around which neighbor stereocilia pivot, lay at different distances from the bilateral symmetry axis of the hair bundle
generates shearing motion between stereocilia of the same row
when the hair bundle is deflected. Parallel coupling via intrarow horizontal top connectors, possibly reinforced by the
anchoring of the tallest stereocilia in the overlying tectorial
membrane, should allow, when a number of MET channels
open, the tension release of their gating springs to affect the
force acting on other MET channels of the hair bundle. In
the absence of stereocilin in a W-shaped hair bundle, only
the three stereocilia (FIGURE 1C) of the same radial column can
interact via their tip-links, because the lateral constraint between stereocilia of neighboring radial columns has vanished.
[Notably, in hair bundles organized in a straight line, as in
IHCs, whether connected or not, stereocilia of neighboring
radial columns are expected not to slide relative to each other,
or to slide to a far more modest extent.] At hearing threshold,
cooperative opening of channels should be irrelevant so that
one hardly expects the presence of top connectors to make any
difference on hearing sensitivity. But at large enough sound
pressure levels for hair-bundle deflection to engage several
MET channels, in the 20 –90 dB SPL range where WD is explored and was not found in the Strc⫺/⫺ mice, the large difference in nonlinearity between the normal and Strc⫺/⫺ mice
might be related to presence versus absence of positive cooperativity resulting in parallel coupling. At present, horizontal
top connectors can be concluded as essential to the generation
of WD, but which of the proposed mechanisms underlies their
contribution remains to be clarified.
D. Other Consequences of Coordinated
Stereocilia Motion Possibly Contributing
to Hair-Bundle Nonlinearity
The effects of coordinated stereocilia motion have been
mostly studied in terms of interaction between stereocilia
1574
and surrounding fluid. One possibly detrimental consequence of this interaction is damping by friction. The complex hydrodynamics of stereocilia have been studied either
by finite-element modeling or interferometric measurements. Their outcomes concur in showing that in a cohesive
hair bundle such that the fluid trapped between stereocilia is
immobilized, the viscous drag is reduced compared with a
vibration mode allowing relative squeezing among stereocilia shafts (135). An additional advantage of trapping the
viscous fluid between stereocilia might consist of a more
concerted activation of MET channels (135).
Brownian motion in the fluid bathing stereocilia is another
phenomenon that might hamper MET efficiency by introducing noise in MET channel openings. It has been suggested that parallel coupling of transduction elements
should reduce the negative impact of noise by increasing the
synchrony of MET channels (50, 117, 191).
Numerical simulations of hair bundle motion, in the simpler case of the straight rows of IHC hair bundles (258),
also emphasized the importance, for MET-channel dynamics, of controlling the degree of splay among stereocilia. The
authors stressed the role of tectorial-membrane positioning
in this dynamics, and furthermore, predicted the formation
of nanovortices between rows of stereocilia. Of note, nanovortices vanish when stereocilia lose cohesion (Chadwick,
personal communication). Vortices being a known source
of WD, this may provide one more link between hair-bundle cohesion and nonlinearity.
E. OHC Bundles: Their Environment and
Operating Point
If the processes allowing appropriate gating of MET channels form the source of WD and are an essential stage of
nonlinear amplification, this does not preclude that linear
elements in the environment of hair bundles modulate WD
or contribute to transferring it into a detectable form. The
electromotility of OHCs, for example, transfers the nonlinearity of the membrane potential due to MET-channel operation into a mechanical nonlinearity (see sect. IIIA). Despite the absence of DPOAE at low stimulus levels in prestin-null mice mutants lacking somatic motility and
amplification (143), the observation of physiologically vulnerable DPOAEs around 70 dB SPL indicates, however,
that prestin is not necessary to distortion generation when a
high stimulus level overcomes the lack of amplification
(144).
The OP defined by the percentage of open MET channels at
rest controls the gain of the cochlear amplifier and influences the amount and symmetry of WD. What known processes modulate OP position? The percentage of open channels at rest tends to be stabilized by the process of adaptation (101). Conversely, the OP can be shifted by biasing the
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bundle to show high coherence and synchrony (136). In the
same type of hair cell, fast video analysis of hair-bundle motion (117) revealed the sliding displacement of adjacent stereociliary tips, indicative of parallel gating. In these saccular hair
cells, links other than the horizontal top connectors could be
severed without influencing bundle motion, thereby suggesting that sliding motion, at least in these cells, rests on horizontal top connectors. Mathematical models of three-dimensional
motion of a whole hair bundle (190) support the idea that the
asymmetry of the MET current versus hair-bundle deflection
relationship increases with the number of MET channels
working together. Altogether, these studies suggest that strong
lateral coupling among MET channels might impinge the Po
Boltzmann pattern, and thereby may increase distortion
enough to produce the major WD that the Strc⫺/⫺ mice fail to
produce.
AUDITORY DISTORTIONS
BM by infrasound (21, 239); endolymphatic hydrops, i.e.,
swelling of the scala media (11, 268, 270); and modulation
of OHC function by medial olivocochlear efferent neurons
activated by the ipsi- or contralateral presentation of sound
(3, 83, 189). It has even been suggested that the resting
position of the BM and the OP of OHCs are continually
adjusted by a control mechanism that modulates cochlear
sensitivity and tuning (140). Displacement-sensitive experimental setups indeed reveal long-lasting shifts of the BM
from its resting position (31) to which regular, velocitysensitive measuring devices are insensitive. Fluid pressure in
scala media may provide a suitable control mechanism,
with hydrops representing an extreme situation (142).
In summary, overwhelming evidence points to the nonlinear processes that ensure the gating of MET channels in
sensory-cell hair bundles, i.e., their gating compliance and
the cohesive motion of stereocilia, as the source of WD, one
of the manifestations of auditory nonlinearity. In OHCs,
MET processes also are the first stage of a feedback loop
that dynamically controls BM motion and decreases cochlear amplification at increasing input levels. In this way,
MET contributes to the other manifestation of auditory
nonlinearity, the compressive (⫾0.33 dB/dB) growth of responses with stimulus level at frequencies near resonance.
Off resonance, compression no longer exists. This will lead
us, in the next sections, to distinguish between WDs coming
from hair bundles at resonance (nonlinearly growing, “essential” WD) versus off-resonance (more linearly growing,
associated with saturation of MET currents on either side of
the Boltzmann transfer function, i.e., saturation WD), to
IV. NONLINEAR RESPONSE OF THE
COCHLEA TO A SINGLE TONE
This section examines the simplest case of a pure tone at
frequency f. It briefly reviews how acoustic vibrations travel
along the cochlear BM and peak according to the rules of
cochlear tonotopy (200); what nonlinearities the vibrating
OHCs generate; from where and how the resulting OAEs
can propagate backward to the external auditory canal.
A. Forward Travel
The thickness and width of the BM, separating ST from
scala media and on which rests the organ of Corti (FIGURE
1A), gradually vary from base to apex, and so does the local
resonance frequency, which decreases from base to apex.
The Reissner’s membrane, rather uniformly compliant, is
usually thought to play little mechanical part. Incoming
sounds enter SV at the oval window, where the pistonlike
vibration of the stapes produces a compression wave traveling at the speed of sound in water, ⬃1,500 m/s (206);
however, it likely causes little significant motion of sensory
cells. As a general rule, the impedance of a resonant system
driven below its resonance frequency is dominated by its
stiffness. Thus with respect to audible frequencies except
the highest ones, the basal-most cochlea behaves as a stiff
partition. Thus the amplitude of the pressure wave in ST is
small, and there is a pressure difference across the BM.
Energy transfer occurs as the stiff BM responds to this pressure difference by rapid vibrations. As shown by Békésy
(14a), coupled interactions between the BM and cochlear
fluids of ST and SV initiate a transverse, forward-moving
wave traveling along the stiff BM surface at a few hundred
meters per second, which increases in amplitude and gradually slows down to a few meters per second while approaching the site of resonance, called CF (for characteristic
frequency) location (218, 227). This traveling wave excites
auditory sensory cells by efficiently moving their stereocilia
bundle. Toward CF location, the wavelength (ratio of speed
to frequency) decreases and a large phase delay accumulates. Theory indicates that the place of maximum vibration
along the longitudinal cochlear axis is just basal to CF location. Exactly at CF location, at resonance, the stiffness
and mass terms of the cochlear impedance cancel each
other, and the purely resistive impedance absorbs the energy of the tone, by so-called critical-layer absorption (147),
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The tectorial membrane, which embeds the tallest stereocilia of OHCs and incidentally, possibly contributes to the
cohesion of OHC stereocilia bundles when stereocilin is
present, might influence sound-evoked OAEs. Nonetheless,
OAEs do exist in ears of lower vertebrates without tectorial
membrane (17). In Tecta (deltaENT/deltaENT) mice with
mutations in the Tecta gene encoding ␣-tectorin (139), and
a tectorial membrane detached from the cochlear epithelium, DPOAEs are still produced at stimulus levels above 65
dB SPL (151). Their extant level can be explained by an
abnormal mode of OHC stimulation by viscous coupling to
the endolymph bathing the hair bundles, resulting in a
35-dB increase in hearing thresholds. The source of nonlinearity thus requires higher stimulus levels to compensate the
decreased cochlear amplification. It produces DPOAEs that
decrease in amplitude with increasing frequency, whereas
DPOAE amplitude depends little on frequency in control
mice, with an effect of the frequency ratio f2/f1 more irregular than in control mice (151). This supports the view of a
tectorial membrane acting as a mechanical filter shaping the
properties of sound-evoked OAEs, without influencing
their generating mechanism, which depends on OHC hair
bundles with normal MET channels and horizontal top
connectors, as they are in Tecta mutants (280).
examine the spatial distribution of WD sources, and to
review how WDs from different locations propagate and
recombine to form noninvasively recorded OAEs. Although
OAEs conveniently reveal the existence of WD and reflect
many of its important properties, it does not mean that their
generation mechanisms are purely nonlinear, and these
mechanisms will have to be reviewed according to OAE
type.
AVAN, BÜKI, AND PETIT
In sensitive cochleas, the traveling wave is amplified by
OHCs along its way. Their feedback, by compensating
damping, boosts the BM response to soft sounds provided
feedback occurs at an appropriate phase. This happens only
between 1 and 2 mm basal to the CF location (in the chinchilla, at CFs around 10 kHz) (228). The enhancement of
cochlear resonance and sensitivity comes with finer tuning
as the feedback is efficient only at frequencies near f (200).
Cochlear amplification is nonlinear, such that at low stimulus levels, the vibration at the CF location is ⬃1,000 times
larger than it would be in the case of damaged OHCs
whereas, above 90 dB SPL, BM vibration is independent of
OHC status (225, 228). The CF-to-stapes vibration ratio is
⬎1,000 at low stimulus levels, as a result of energy concentration due to traveling-wave speed reduction and active
reduction of damping by feedback. Its decrease by more
than two orders of magnitude as the stimulus level increases
(e.g., FIGURE 1D in Ref. 220) means that the growth of the
response at the CF location is strongly compressive (between 0.2 and 0.5 dB/dB increase in SPL), but again only
along the short section on the BM where OHCs efficiently
boost the traveling wave. Where OHCs are far from resonance (passive) or damaged, feedback exerts no amplification and BM vibrations at f grow linearly 1 dB/dB (FIGURE
2). BM displacement thus has two components, near CF
location, a high and sharp peak strongly depending on the
“activity” of the OHC-driven cochlear amplifier, and between stapes and CF location, a broad “passive” tail region,
hardly sensitive to OHC status. Around the peak, amplification ensures a large input to the nonlinearly working
MET channels of active OHCs, accounting for essential
distortion at low SPLs. Along the broad tail, OHC MET
channels still open if stimulated high enough and produce
“passive” nonlinear phenomena including WD. Their existence will be discussed in the next sections. Compared with
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essential distortion from around CF, their growth is expected to be steeper (see sect. IIA), and their time delays
shorter as their generation place is nearer the oval window.
B. Always Forward?
Section IVA may give the false impression that cochlear
waves can only propagate from stapes to CF location. Actually, from a vibrating site on the BM, mechanics prescribe
that its elementary contribution to the wave, a “wavelet,”
starts traveling both ways. The physicist Huygens introduced the principle that every point of a wavefront acts
as the source of wavelets that, by spreading out in all directions, not only that of the incoming wave, combine together
to continue the propagation. Fresnel (70a) later completed
this principle by stating that the amplitude of a wave at any
given measurement point equals the vector sum of the amplitudes of all the wavelets reaching that point. The result of
this summation depends on whether wavelet phases are
similar or widely different. Fast phase variations among
wavelets lead to small contributions even though individual
amplitudes may be large (251). A combination of large
amplitudes and slowly varying phase (phase coherence; Ref.
247) leads to a large outcome. Thus the way that spatially
distributed wavelets can or cannot efficiently combine introduces emergent directionality, critical for accepting the
concept of OAEs. In a linear medium, the wavelets have the
same frequency as the stimulus, but in the nonlinear cochlea, the incoming wave also generates wavelets at frequencies different from its own (see sect. V).
For a regular pure tone coming via the stapes or the bony
wall, from the nondirectional principle of Huygens, forward directionality emerges. Wavelets from basal places,
launched when the incoming stimulus reaches their place,
incur an increasing phase lag while going forward. In this
direction, they meet more apically produced wavelets, in
phase with them because they have been launched with an
increasing phase lag, as the stimulus reached their places
later. The phase lags accumulate in parallel, which favors
wavelet recombination in the forward direction. Wavelets
starting basal-ward, conversely, experience strong cancellation as those from apical places, launched with already a
large phase lag, incur an additional lag while traveling basally and interfere with wavelets produced more basally and
presenting much less phase lags (launched earlier and having less to travel). The case of OAEs generated within the
cochlea is different and will be examined in sections IVF
and V.
C. Vibrations at f Generate Harmonic
Distortion
In the cochlea stimulated by a single tone, WD is revealed by
the production of acoustic harmonic distortion at integer
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so that the amplitude of vibration is less, as the envelope of
BM vibrations at f peak very near CF location. For the sake
of simplicity, no distinction will be made henceforth between CF location and this peak. More apically, the pressure difference between SV and ST vanishes, and BM vibrations decline sharply. If by some means a vibration is generated at frequency f at a location tuned to a frequency
lower than f (it will be shown later on that WD may generate such frequency components), this vibration finds too
slack a BM to propagate in either direction. Frequency is an
important variable controlling the shape of BM vibrations.
Yet, an even more relevant variable is the scaled ratio f/CF
between the stimulus frequency f and the characteristic frequency of the measurement location. From accumulating
observations of BM motion, Zweig (310) was the first to
pinpoint that the amplitude and phase patterns of BM motion are determined solely by f/CF, a property which he
called cochlear scaling symmetry [except in the most apical
part of the cochlea where scaling symmetry apparently
breaks (49, 168)].
AUDITORY DISTORTIONS
produced by spectrally richer sounds, the observed harmonic distortion levels are low. The energy carried by harmonics never exceeds 4% of that of the stimulus. There is
likely an AGC in the cochlea that tends to lower the cochlear gain, when the sound level increases, enough for WD
to be kept to a minimum (315). Harmonic distortions can
be heard only at large stimulus levels (286). Their contributions produced at places tuned to f are filtered out; furthermore, they cannot propagate to their CF locations. As for
“amplified distortions,” they likely suffer from the upward
spread of masking by the louder, lower-frequency tone at f
(see sect. IXA). Second, at variance with the outcome of
two-tone stimulus experiments (see sect. VF) favoring oddorder WD, the 2f (even-order) harmonic is the largest, not
the 3f, 5f . . . harmonics. The balance between odd- and
even-order WD is very sensitive to the position of the OP of
the nonlinear source (see sects. VF and VIII), which may
differ from one experiment to another depending on how
the invasive measurements affect cochlear homeostasis.
D. Waveform Distortions at Off-Resonance
Places
Section IVA suggested the existence of WD even at offresonance basal places. This can be checked easily. An electrode at the round window detects several electric correlates
of the vibrations of basal hair cells, an oscillating change in
electric potential at the stimulus frequency called cochlear
microphonic (CM) potential, and a steady change following
the envelope of the stimulus, called summating potential
(SP). They are thought to relate to the oscillating and steady
components, respectively, of the receptor potentials of hair
cells. The electrode collects spatially averaged responses in
which, at stimulus frequencies below a few kHz, basal haircell contributions to the CM prevail. Being off-resonance,
Two noteworthy facts were reported at basal sites (41).
First, and this will be encountered again in nonlinearities
3
2
velocity (mm/s)
1
0
-1
-2
-3
300
1000
frequency (Hz)
5000
-4
0
2
4
6
8
10
time (ms)
FIGURE 8. Waveform (right) and spectrum (left) of the velocity signal from an apical Hensen cell, a support
cell in lateral position relative to the outer row of OHCs in the mammalian organ of Corti, extracted from
interferometric measurements, in response to a 256-Hz (f) tonal stimulus in a healthy cochlea. Harmonic
components at 2f, 3f, etc. (arrows) are visible at decreasing levels 10 –20 dB below the fundamental
component in relation to distorted waveforms (top right). Harmonics vanished in a few hours after death, while
the Hensen-cell response became almost sinusoidal [From Khanna and Hao (125). Copyright 2000, with
permission from Elsevier.]
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multiples of the stimulus frequency, the fundamental frequency f, i.e., 2f, 3f, etc. (FIGURE 8). Through a tiny opening, a laser beam is focused on the BM (41, 125, 232), or a
displacement-sensitive probe is placed near it (197), and the
stimulus frequency is usually set near the CF of the tested
site, CFsite. Few reports exist on basal harmonic distortion,
possibly because it seems small and labile, absent in Reference 232, ⫺30 dB relative to stimulus level in Reference 41,
and ⫺25 dB in Reference 197. Apical harmonic distortion is
conspicuously larger, ⫺7 dB relative to stimulus level (125),
perhaps in relation to the different geometry of OHCs in the
cochlear apex addressed in section VII. In Reference 197,
the sensitivity of the preparations was not optimal, and the
origin of all detected harmonic pressure components, at 2f
and 3f, was tracked to the place tuned to f (because the
delays of the harmonics and of the fundamental frequency f,
inferred from their phase accumulation against frequency,
were the same). Cooper also attempted to shift the stimulus
frequency to f ⫽ CFsite/2 (41). A response was measured at
CFsite, due to the harmonic 2 ⫻ (CFsite/2). Harmonics,
wherever produced, are vibrations and propagate accordingly. Where had this vibration been generated? It could
hardly be the location tuned to the stimulus frequency, as it
was when CFsite tones were used, because the second harmonic cannot travel along the BM at places tuned to lower
frequencies. The only alternative hypothesis is that it was
produced at basal places and traveled, forward, to the detector. While reaching there, its CF location, it was amplified and became detectable, hence its name “amplified distortion” (41). The existence in this experiment of harmonics generated at their frequency place on the BM and not at
the CF location highlights the importance of considering
basal contributions to WD and not only CF ones.
AVAN, BÜKI, AND PETIT
they are small yet combine in unison, due to the long
wavelength of the traveling wave. In contrast, the larger
responses from hair cells near the CF location undergo
fast phase rotations leading to spatial cancellation. The
CM is largely dominated by contributions from OHCs,
as its amplitude changes little after selective destruction
of IHCs (57). The SP originates from the asymmetry of
the transfer function of the hair-cell transduction process
at the OP, which generates a sustained depolarizing component. The main sources of SP are IHCs, but a sizeable
component from OHCs can be identified all along the
cochlear spiral (57).
E. Emissions Produced by a Single Tone:
Stimulus-Frequency Otoacoustic
Emissions
Reemission of additional sound from within the cochlea is
illustrated by the measurements of SFOAEs emitted at the
same frequency as the stimulus. Although their measurement is straightforward, their properties and relation to
cochlear nonlinearity are complex. It is worthwhile to study
them carefully, however, because they reveal many properties of an active cochlea in a noninvasive manner. It had
been reported by Elliott (61) that auditory threshold plots
against frequency from normal ears display quasi-periodic
ripples of a few decibels. Following up on psychoacoustic
investigations of this auditory threshold microstructure,
Kemp (120) slowly varied the frequency of a low-level pure
tone while recording the SPL in the sealed ear canal. He
detected narrow ripples, superimposed on broad changes in
SPL due to earphone calibration. The ripples signaling periodic changes in the ear’s impedance correlated with auditory threshold microstructure. The most straightforward
interpretation was that of an interference mechanism between the stimulus itself and a pressure wave emitted from
within the ear and likely partially reflected at the middle ear
boundary. Fast phase rotation of the acoustic emission with
frequency would create a regular pattern with periodic
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How can one SFOAE be separated from its stimulus as they
share the same frequency? Their different growths at increasing stimulus level provide a first solution. The relative
size of ripples decreases and eventually vanishes around 70
dB SPL as the SFOAE saturates. The SFOAEs can thus be
extracted by subtraction of the scaled, high-level pattern
from the low-level one. The suppressive action of a tone
superimposed on the stimulus at a slightly lower frequency
affords a convenient alternative (120). Suppression removes the stimulus-related wave traveling inside the cochlea, so the SFOAE may be extracted by subtraction of the
suppressed from the unsuppressed plot (FIGURE 9, A for the
amplitude and B for the phase).
F. Backward Propagation of SFOAEs
Kemp and Chum (123) suggested that the SFOAE came
from the peak of the incoming traveling wave, where nonlinear reflection occurred on some impedance discontinuity.
The current dominant view is that this reflection mechanism
is linear. Its description is necessary, however, as its interplay accounts for many properties of all types of OAEs and
shapes their nonlinear characteristics. How can wavelets of
an SFOAE combine in a regular manner to form a significantly large backward wave (see sect. IVB)? Bragg scattering offered a possibility (264). Strong positive interference
happens in the backward direction when an X-ray wave
falls upon a distributed array of periodic scattering centers,
with a spatial period of half its wavelength. The difficulty is
that there seems to be no regular scattering array of OHCs
in the cochlea (150, 304), unless regularity would arise from
periodic disruptions induced by the curvature of the coiled
cochlea (159), but there are SFOAEs from uncoiled auditory organs.
A breakthrough occurred when a class of models of an
active cochlea was shown (311) to generate a periodic structure of coherent reflecting sites out of spatially irregular
distributions of scattering centers. The theoretical analysis
showed that Bragg-like scattering could occur from a disorderly array satisfying several assumptions. First, if the peak
of the mechanical response to the stimulus is tall, scattering
is localized from the place where the response is largest. At
this place, if the peak of the response is broad enough to
encompass many scattering centers, some will present a
spatial frequency matching the wavelength of the incoming
sound. These will be singled out, by a sort of bandpass
filter-like mechanism extracting a few coherent components
from noise. The requirements of a tall and broad peak may
seem conflicting as, to produce a tall peak, the cochlear
amplifier generates frequency tuning which sharpens the
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The CM waveforms display considerable distortion with
large harmonics in response to a pure tone (193, 289). The
CM WD is attributed to OHCs (38, 139). Its presence
shows that basal OHCs do distort, even though excited by
off-resonance tones. The sinusoidal deflection of stereocilia
produces a saturating, nonsinusoidal current through the
MET channels, which reveals the underlying Boltzmannshaped deflection/current transfer function and the OP of
basal OHCs (FIGURE 7). Mechanically induced position
changes of the organ of Corti, e.g., by gel injections (238) or
application of a biasing force on the bony cochlea (308),
come with shifted OP and modified CM harmonics, but also
an increased SP. This pinpoints the practical interest of
monitoring any of these compound signals from off-resonance places, as they are sensitive to subtle changes in haircell transduction.
crests when emission and stimulus add together in phase,
and troughs when they are out of phase and cancel each
other. This OAE, elicited by a single continuous tone at the
stimulus frequency, was called SFOAE.
AUDITORY DISTORTIONS
peak. Actually, the wavelength is around 0.2 mm at a 16
kHz CF location, in the gerbil (218), and the 1- to 2-mmwide region over which amplification generates a tall peak
(228) covers several wavelengths and many scattering centers.
The backward contributions from scattering centers in the
peak region will add in phase and generate a large backward-emitted SFOAE. Bragg scattering is fundamentally
linear and additive, contrary to what was initially suggested
(123). Nonetheless, cochlear activity requiring the nonlinear action of OHCs is essential for the generation of a tall
and broad peak allowing spatial filtering and, ultimately,
efficient Bragg-like emission of SFOAEs. The nonlinear
characteristics that SFOAEs carry with them are their saturation at increasing stimulus level and their suppressibility.
This view of purely linear coherent reflection has been challenged by recent observations based on time-delay considerations. If an SFOAE results from backward scattering on
irregularities near the peak of the BM response to the stimulus, theory predicts that the time it takes for the SFOAE to
reach the ear canal is the sum of two delays of similar size,
one for the forward journey to the characteristic place on
the BM and one for the backward journey (249). Because an
SFOAE is a steady response to a steady tone, its timing
cannot be directly evaluated. However, a classic way to
derive temporal information from sinusoidal responses is to
study how their phase relative to the stimulus varies with
frequency. Let us consider a system through which the
travel time is t, for example, to some characteristic place
and back. For sound at frequency f, this travel time, or
delay, translates into a phase shift ⌬␸ ⫽ 2␲ft (every period,
i.e., every 1/f, the phase of the sound rotates by one cycle).
At frequency f ⫹ ␦f, the travel time leads to a phase shift
⌬␸= ⫽ 2␲(f ⫹ ␦f)t, assuming that t varies little with f. It
follows that t ⫽ 1/2␲ d␸/df; thus the travel time can be
deduced from how much the phase of the response to a pure
tone varies with frequency (FIGURE 9, C AND D). The delay
measured by this technique is called group delay in signal
processing, as the envelope of a group of waves at different
frequencies (or wave-packet) moving at the group velocity
vg, undergoes a group delay t ⫽ ⌬x/vg when moving by ⌬x.
Average SFOAE delays are ⬃2 ms in cats and guinea pigs at
1 kHz (2 cycles), and even longer in humans, 10 ms at 1 kHz
(10 cycles) (247).
By comparing BM group delays to SFOAE ones in a broad
frequency interval in the chinchilla cochlea, Siegel et al.
(251) found that SFOAEs had too short a delay relative to a
doubling of BM group delays, especially at frequencies ⬍4
kHz. They ranged from 0.7 to 2 ms, well below the esti-
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FIGURE 9. A: amplitude of the residual sound signal corresponding to an SFOAE, plotted against frequency,
in response to a swept tone at 30 dB SPL in a chinchilla ear. Open circles, initial measurement; solid line,
repeated measurement after ⬃1 h. B: phase versus frequency function. C: group delay against frequency,
extracted, at each frequency, from the slope of the phase versus frequency function. Arrows marking the
positions of amplitude notches, ␲ phase reversals and extreme group-delay values, respectively, happen to
coincide [From Siegel et al. (251). Copyright 2005, with permission from Acoustical Society of America.] D and
E: principle of group-delay measurements. The phase rotation of a signal from a local site, when its evoking
stimulus is swept in frequency, accumulates in proportion to the time it takes for the signal to propagate from
its generation place to the measurement point (e.g., the stapes or external ear canal), and therefore, to their
distance. The red and green sine waves start from the same site at different frequencies, thus undergoing
different phase rotations in a given time interval. D: short propagation delay, small phase gradient. E: long
propagation delay, large phase gradient, for the same change in frequency.
AVAN, BÜKI, AND PETIT
G. SFOAEs Without Traveling Wave
The previous explanation refers to the BM and its traveling wave, but the existence of scattering and of SFOAEs
does not require them. For instance, it is possible to
measure SFOAEs in lizard ears, which lack a BM-supported traveling wave (16), resonance being incorporated in hair-cell bundles (161, 204). Lizard SFOAEs,
including their latency, are similar to those measured in
mammalian ears. Obviously, in this case, latency cannot
be attributed to the propagation time of a traveling wave
that slows down while approaching its CF location. Delays arise not only from travel times, but also filter responses. The more finely tuned a filter, the later the beginning of its response and the longer its duration. It has
been shown with the help of a physical model (16) that
the only requirement for long-delay SFOAEs is the presence of a slightly disarrayed battery of active filters. Similar qualitative conclusions have been reached regarding
the presence of long delays in TEOAEs evoked by short
impulse stimuli. Their mathematical principles, simple
and generic, are described in APPENDIX I. They account for
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how MET nonlinearity produces detectable OAEs in all
tetrapod ears, however diverse their anatomy. The basic
requirement is the presence of an active filtering principle, regardless of how it is achieved.
In summary, this section shows how the properties of OAEs
evoked by a single tone can be understood by considering
several key functions of an auditory organ; how waves
propagate and interact with sensory cells; how fast they
travel; what amplification and compression they undergo
and whether a tall and broad peak is produced at CF location; and how finely waves are filtered. By measuring
SFOAE properties, it is possible to probe several aspects of
auditory stimulus processing and perception without disrupting their fragile anatomical support. The next section
elaborates on these methods by examining a spectrally
richer situation.
V. INTERACTION OF TWO TONES IN THE
COCHLEA
When two tones at frequencies f1 and f2, called the primaries (f2 ⬎ f1), are presented simultaneously, the envelopes of
BM motion, that peak at two different places, overlap over
a broad interval. The amplitudes of the peaks at f1 and f2 on
the BM still grow with stimulus level in a compressive manner and harmonics are still produced at 2f1, 2f2, 3f1, etc.
However, as BM and OHC vibrations are nonlinear, two
other important nonlinear manifestations show up from
where the vibrations at f1 and f2 interact, i.e., all over the
basal part of the cochlea and particularly near the place
tuned to f2, the higher frequency. The presence of vibrations
at one frequency tends to suppress the vibrations at the
other frequency, and conversely, intermodulation due to
the combined presence of vibrations at two frequencies generates vibrations at new frequencies not present in the stimulus.
A. Two-Tone Suppression
Two-tone suppression (2TS) occurs when the presentation
of a (suppressor) tone at frequency fs decreases the mechanical response to a probe tone at frequency f, usually set near
the resonance frequency of the measured spot (228). Suppression is one of the mechanisms of masking whereby a
weak probe tone is no longer audible in the presence of a
louder masker sound (see sect. IXA). An objective correlate
is two-tone rate suppression in auditory-nerve fibers (107).
As the firing rate of neurons that code for a probe tone is
reduced by the presentation of a suppressor tone at a different frequency, so is the audibility of the probe tone. Early
single-fiber recordings of auditory neurons revealed two
domains of the frequency versus level map of neuronal activity, on both sides of the excitatory region centered on the
probe frequency, such that a suppressor tone decreased the
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mated round-trip travel between stapes and CF location.
The authors proposed two possible explanations. One is
that the backward travel of the SFOAE is much shorter than
the forward travel of the stimulus along the BM, which
suggests a different propagation mechanism, perhaps by a
compression wave through cochlear fluids (see sect. VH).
Another possibility is that contributions to the SFOAE are
widely distributed, not only near the peak of the traveling
wave, but also basally with shorter delays. These basal
sources, although being smaller in size than the CF-location
ones, would undergo smooth spatial summation in longwavelength cochlear regions, while cancellation in the
short-wavelength region would offset the advantage of
larger levels, finally emphasizing short-latency contributions (251). Since then, the criticism of Siegel has been addressed (248) by qualifying the previous model and incorporating more general considerations no longer based on
“SFOAE delay ⫽ twice the BM delay.” New evidence has
been produced that SFOAEs (in cats) are as sensitive to
low-frequency interference as cochlear responses from the
place tuned to f, which is thus likely their dominant site of
production (145). Other evidence, in the guinea pig, accommodates combined mechanisms of SFOAE generation, by
showing that nonlinear mechanisms (independent of the
presence of coherent reflection sites, but dependent on the
high-level wave at f) can strongly influence the emission at
high stimulus levels, whereas linear coherent reflection
dominates at moderate and low levels (80). Subtle mechanisms such as position-dependent reflectance along the cochlea, multiple intracochlear reflections, or generation of
“intermodulation” SFOAEs (difference tones occurring between harmonics, e.g., 2f ⫺ f or 3f ⫺ 2f; see sect. VB and
Ref. 64) might also have to be taken into account.
AUDITORY DISTORTIONS
activity due to the probe even when too weak to elicit an
activity of its own (236). As IHCs and neurons were not
excited by the suppressor tone, suppression had to be a
micromechanical mechanism, rather than a synaptic or neural one. Direct evidence of 2TS in BM vibrations (226, 233)
confirmed this mechanical interpretation.
In summary, experimental facts concur in suggesting a close
correlation of 2TS with the interaction of OHCs and BM
and with its results, enhanced sensitivity, compressive amplification, and frequency tuning. To give these facts a common theoretical framework requires an analysis of the complex behavior of a nonlinear oscillator driven by a mixture
of input signals at two different frequencies and different
combinations of levels. Of note, the simplest experimental
model eliciting realistic 2TS is a single hair cell from the
bullfrog’s sacculus placed in a two-compartment preparation allowing cell activity and resonance (14), which indicates that 2TS is not specific of the complex architecture of
the mammalian cochlea, but can be ascribed to the active
resonance of a nonlinear oscillator. This suggests the use of
a generic model of a compressive, self-tuned resonator with
mass, stiffness and damping, such as the one provided by
the mathematical formalism of a critical oscillator near a
Hopf bifurcation (105), solvable using reasonable approximations (110). At low stimulus levels, a controlled negative
resistance almost cancelling the viscous drag poises a critical oscillator near self-oscillation and allows cochlear amplification. The resistive term also contains a stabilizing
amplitude-dependent term ⱍzⱍ2z (where z denotes the amplitude) that generates compression near resonance by decreasing amplification at increasing input levels (34, 60).
FIGURE 10. Two-tone suppression in a critical oscillator, predicted by the Hopf-bifurcation formalism. The probe tone is at resonance, and the suppressor frequency is variable. A: when probe and
suppressor tones have the same amplitudes, the response of the
oscillator to the probe tone (red line) decreases as the suppressor
approaches its frequency. In the meantime, the response of the
oscillator to the suppressor (black line) increases when the suppressor gets within the bandwidth of nonlinear amplification, while remaining smaller than what it would be in the absence of the probe
tone (thin line). B: influence of the level of the probe tone on the
responses to the probe (red line) and suppressor (black line), for a
suppressor with fixed amplitude and frequency (near that of the
probe). The thin red line and thin horizontal black line indicate the
responses to the probe and to the suppressor, respectively, when
presented alone. The vertical distance between thin or thick lines
indicates the contrast between probe and suppressor, which is
larger when there are suppressive interactions [From Jülicher et al.
(110). Copyright 2001, with permission from National Academy of
Sciences, USA.]
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The characteristics of 2TS on the BM have been dissected in
the hope that they would shed light on its underlying nonlinear mechanism. The growth of BM response to a probe
tone presented alone is compressive at intermediate intensities, at frequencies near the CF of the measured place.
Extensive experimental studies of 2TS have evaluated how
the BM vibration at the probe frequency changes when a
second suppressor tone is added, at a frequency either
higher or lower than CF (233). Although suppressors presented alone produce responses smaller than the probe, they
induce a clear suppression, monotonically increasing with
suppressor intensity, and decreasing with probe intensity.
In addition to decreasing the sensitivity to the probe at CF,
2TS decreases the compressive nonlinearity of BM growth.
Suppression is also a tuned phenomenon, so that for a probe
at CF, suppressor frequencies close to CF are most efficient
(42, 43). When the probe frequency departs from CF, lowor high-frequency suppressors lose their suppressive influence (233). The phase changes of BM vibrations at CF in the
presence of a suppressor tone are reminiscent of those produced by stimulus-intensity changes (42, 228). Last, 2TS is
labile and requires healthy OHCs (202, 233).
Most characteristics of 2TS can thus be predicted in terms
of a competition between probe and suppressor to drive a
critical oscillator (FIGURE 10) (64, 110). In the presence of a
single tone, the oscillator tuned to it behaves as a sharp filter
with narrow bandpass. Thus, if a second, interfering tone is
added at a frequency far enough from resonance, it is filtered out and bears no influence. When the interfering tone
gets nearer to the resonance of the filter, all the more when
it is louder, the oscillator starts responding to it, which
increases z and the damping, ⱍzⱍ2z. Increased damping decreases the response to the probe tone, which gets suppressed. A critical oscillator becomes linear if it is stimulated at frequencies differing from its resonance frequency,
which explains why the suppressive effect of a low- or highfrequency masker decreases when the probe tone frequencies moves away from CF (233). On FIGURE 10 and in
AVAN, BÜKI, AND PETIT
Whereas, for a single nonlinear oscillator, suppressors below and above the resonance frequency play symmetrical
parts, it is not the case for the cochlea. The pattern of
Békésy’s traveling wave accounts for the so-called upward
spread of masking (see sect. IXA), whereby a low-frequency
suppressor influences higher frequency probes, whenever
the tail of the suppressing envelope presents a significant
amplitude at the place tuned to the probe (see sect. IVA). In
contrast, high-frequency interfering tones, unable to reach
beyond their characteristic place, can suppress only if they
peak near enough to the probe to influence its traveling
wave along the short section where its amplification takes
place (203). The extent of the active mechanism along the
BM can be evaluated in this manner. Patuzzi (203) concluded that activity spans a relatively broad distribution of
OHCs, e.g., 650 ␮m basal to the characteristic place of an
18-kHz probe tone in the guinea pig (203).
It might seem that suppression means “loss of information.” However, by ensuring that when a given filter receives several inputs, the larger one (likely the one at resonance) will win the competition, 2TS also improves spectral
analysis. The spectral components most easily suppressed at
one site of the cochlea are off-resonance ones. Actually, they
will propagate to their own CF locations, where the competition will favor them at the expense of off-frequency
competitors. In this respect, 2TS acts as a contrast enhancer
(14, 262). Nonetheless, very loud low-frequency suppressors will irretrievably suppress all competitors.
B. Distortion-Product Otoacoustic Emissions
Conserved Throughout Evolution
Two-tone suppression can lead to the complete masking of
a soft tone by a loud one, but in the two-tone paradigm
discussed in this section, with similar moderate levels at f1
and f2, 2TS is weak. The second nonlinearity typical of this
two-tone paradigm is an enriched frequency spectrum of
BM motion, with new spectral components at integer com-
1582
binations of frequencies f1 and f2. As already seen in section
II, these audible DPs objectively detected in the ear canal
(120) differ from instrumental nonlinearities by their existence even at low stimulus levels (essential nonlinearity,
Ref. 79) and their spectral content with a predominance of
odd-order DPs (see sect. XI; Ref. 267).
The presence of DPs has been ascertained at many spots
along the mammalian cochlea, and in the first place, on the
BM, with the help of optic interferometric measurements at
basal locations (FIGURE 11) (229, 230) and near the apex,
with slightly different characteristics (43, 86, 125). DPs are
found in the pressure wave in SV throughout the cochlea
(13), and near the BM in ST (54). Last, the CM reflecting
transduction currents through basal OHCs contains DPs
(289).
Because the common source of DPs is the nonlinear mechanics inherent to the gating of mechanosensitive channels, it is not surprising to observe DPs and DPOAEs in
lower vertebrates, as ubiquitous as SFOAEs (15). Even in
insects, DP equivalents in vibrating structures have been
described (82, 285). The idea to take advantage of this
evolutionary-rooted similarity has led to the development
of simple ex vivo systems in which DPs and attendant characteristics, activity, frequency tuning, and compressive gain
can be studied together in the absence of complex propagation issues specific of the cochlea, such as BM resonance,
propagation of vibrations, interaction of DPs from many
neighboring cells, and recombination from discrete DP sites
of generation. For example, hair cells from the neuroepithelium of bullfrogs, in settings containing two-fluid compartments preserving their natural ionic environment, can be
excited individually either by fluid jets or calibrated glass
micropipettes acting on their stereocilia bundle. In this nonmammalian, nonauditory hair cell, the MET channel gating
compliance is qualitatively similar to that of OHCs, and
amplification activity exists even though it involves only the
hair bundle.
DPs have been observed in the current through hair cells, in
their membrane potentials (106), and in hair-bundle displacements (14). The latter case offers a neat illustration of
how 2TS and DP properties emerge together from the active
or passive properties of hair-bundle motion (14). The nonlinearities in the mechanical response of the frog saccular
hair bundle were explored. FIGURE 12A, which is modified
from Barral and Martin (14), depicts the compressive increase of the hair bundle’s displacement observed for a stimulus at the characteristic frequency of spontaneous oscillations of the hair bundle (its resonance frequency). An interfering tone of about the same frequency produces strong
2TS and decreases the compression (FIGURE 12A). Off-resonance, the mechanical response of the hair bundle to a
single stimulus, is decreased at low stimulus levels but
grows faster with increasing level than near resonance, as
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Reference 110, it can be seen how the general critical-oscillator framework accounts for the outcomes of most experimental situations of 2TS mentioned above. The model also
reveals how suppression acts the other way round as the
probe tone itself, by also influencing the damping term,
competes with the responses to simultaneously presented
tones and tends to suppress them. A general rule that holds
regardless of the exact mathematical model of nonlinearity
is that for suppression to be significant, the suppressor level
at the input of the nonlinearity must get within 10 dB of the
probe stimulus (64), which can be achieved either when the
suppressor frequency moves closer to CF or, at fixed frequency of the suppressor, when its intensity increases. Thus
the fact that a realistic 2TS stems from the parsimonious
formalism of Hopf oscillators does not refute other models
of cochlear nonlinearity (see sect. VIIIA).
AUDITORY DISTORTIONS
compression does not exist (FIGURE 12B). Likewise, 2TS no
longer acts (FIGURE 12B). The narrow range of efficiency of
2TS was as predicted by the model of Jülicher et al. (110)
(FIGURE 10). As regards intermodulation, with stimuli near
resonance, the DPs differed from the off-resonance case by
the shallow slope of their decrease when stimulus levels
decreased [near 1 dB/dB, as in Goldstein’s experiment (79)
vs. ⬎2.7 dB/dB off-resonance], that ensured their persistence at low stimulus levels, and the fact that fewer DP
frequencies stood out, essentially the nf1-(n-1)f2 and nf2-(n1)f1 odd-order series (with n ⫽ 2, 3, 4 . . .) flanking f1 and f2
(FIGURE 12, C VERSUS D) (14). As we will see in sections V,
C–F, and VIIIA, many of these characteristics of single-hair
cell DPs also show up in mammalian DPOAEs.
C. DPOAEs, a (Not so Transparent) Window
on the Inner Ear
Direct access to cochlear micromechanics and OHC operation to assess the enhanced sensitivity, compression, and
filtering requires the opening of the bony cochlea, with a
high risk of damaging the vulnerable active mechanisms.
Even minimal damage would affect the validity of measurements. Furthermore, only a few cochlear spots are accessible near the round window and at the apex. Conversely,
DPOAE measurements are totally harmless and straightforward over a large range of frequencies, and DPOAEs are
easier to extract than SFOAEs as their frequencies differ
from those of the stimuli. Yet, their relation to cochlear
amplification is not straightforward, and this requires the
issues of sites of generation and of propagation mechanisms
[absent from the single hair-cell model of Barral and Martin
(14)] to be teased apart.
A prerequisite is to understand the logics behind the conspicuous changes in DP amplitudes and phases when stimulus characteristics are modified. Consider for example the
2f1-f2 DP, usually the largest one. For a given f2, its amplitude varies considerably when the f2/f1 ratio varies, from
1.60 to near 1.00 (so-called f1-sweep). This dependence
clashes with the intuitive assumption that the dominant
contributions to a DP come from where maximum interference occurs between f1 and f2, near the peak response of the
BM to f2. As the DPs of the f1-sweep come from the same
OHCs, should not their amplitude be constant? Experimentally, a “bandpass filter”-like characteristics is observed,
such that the DPOAE at 2f1-f2 reaches a maximum near
f2/f1 ⫽ 1.20 –1.25 (FIGURE 13) (5, 88). The heard DP is not
constant either, but gets louder at decreasing ratios (79).
A tentative explanation for the “bandpass filter”-like outcome of an f1-sweep is the interplay of a mechanical filter
near the generation site, tuned to a lower frequency. At
optimal ratios, 2f1-f2 ⬇ f2/1.4, so that a suitable filter would
have to be tuned at half an octave below the main filter. The
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FIGURE 11. When pairs of equal-level tones at f1 and f2 are presented near CF (8 kHz, in a chinchilla cochlea)
at frequencies such that 2f1-f2 ⫽ CF and f2/f1 ⫽ 1.05 (left), the spectrum of BM vibrations at CF site displays
a series of equidistant peaks flanking the primary responses, the odd-order DPs at intermodulation frequencies. The largest DPs are the cubic ones at 2f1 ⫺ f2 and 2f2 ⫺ f1, and DPs decrease in amplitude at increasing
orders, forming a tentlike pattern. At decreasing stimulus levels (in dB SPL on the left side of each spectrum),
the DP levels show a slow decrease, all approximately at the same rate in dB/dB. At higher f2/f1 (1.25 on the
right side), only the 2f1-f2 response is visible in addition to small peaks at stimulus frequencies, because the
primaries and DPs other than 2f1-f2 are too far in frequency, and filtered out at the BM place of measurement.
Again, its decrease in amplitude, in dB/dB decrease of stimuli, is compressive [From Robles et al. (230).]
AVAN, BÜKI, AND PETIT
tectorial membrane does present such a characteristic radial
resonance (84, 314), which suggested it as a possible candidate (5, 26). Another, nonexclusive possibility, would be
that at f2/f1 ratios near 1, the 2f1-f2 and other lower band
DPs, by falling near the primary frequencies, undergo 2TS
(113) strongly decreasing their amplitudes. In these frameworks, the noninvasively accessible DPOAEs would uncover the influence of subtle cochlear micromechanics. Yet,
the most conclusive interpretation so far relies on the phase
behavior and directionality of interfering traveling waves
from distributed DP generation sites (65, 244), developed in
the following section.
D. Distributed Sources and Propagation(s?)
of DPOAEs
Potential DP generation sites are distributed over a broad
interval where the envelopes at f1 and f2 overlap and OHCs
respond to both frequencies, particularly, but not only near
f2. The lowest stimulus levels in most publications range
from 40 – 60 dB SPL, and at these levels, the envelope of BM
motion definitely has a broad tail (228). If we consider a
DP-generating region along the BM, stimulation evokes
DP-wavelets, each at a particular location of this region,
from which it travels in two directions, apically to the CF
location and basally. At the stapes or at their CF location,
all wavelets are vectorially added and, depending on their
respective phases, cancellation or enhancement occurs. The
DP level at the stapes determines the DPOAE level, and the
DP level at CF determines the loudness of the heard combination tone.
The propagation of DPs from their generation places follows the same principle as pure tones and SFOAEs (see sect.
IV, B and E). As intermodulation creates new frequencies
and wavelengths, the prediction of phase relationships between the wavelets emitted from different generation sites
at the frequency of each DP is significantly more complex.
Its mathematics (244, 266) are outlined accessibly in APPENDIX II. The particular case of 2f1-f2, the largest DP, is examined there without loss of generality.
Due to scaling symmetry, the vibration patterns of f2, f1,
and 2f1-f2 are just translated versions of each other along a
logarithmic representation of the cochlea. The phase at f2
rotates fast when this primary nears its CF location, while at
the same spot, f1 and 2f1-f2 accumulate less and less phase
with increasing f2/f1 ratio, as the peaks at f1 and 2f1-f2 move
farther apically (see APPENDIX II). A DP wavelet emitted at
some place along the region of generation, starts with a
phase lag due to the phase lag of the stimuli on their way to
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FIGURE 12. A: growth function of the motion of the stereocilia bundle of a hair cell from a frog sacculus in
response to a tone at frequency f1 (f1 ⫽ 13 Hz, near the characteristic frequency of spontaneous oscillations,
11 Hz), in the presence of a suppressor tone at frequency f2 ⫽ 12 Hz, added to the test tone at increasing
amplitudes (symbols: right panel). B: same hair bundle as in A, tested off-frequency, with f1 ⫽ 130 Hz and f2 ⫽
120 Hz. C: frequency spectrum of hair-bundle motion in response to a two-tone stimulation at f1 ⫽ 20 Hz and
f2 ⫽ 22 Hz, near resonance, with rather high stimulus forces (red labels: main intermodulation frequencies).
D: same cell as in C, with off-resonance two-tone stimulation at f1 ⫽ 200 Hz and f2 ⫽ 220 Hz. At lower stimulus
forces (FIG. 3, A AND C, in Ref. 14), the two-tone stimulus of C still elicited visible DPs, whereas that of D did
not. [From Barral and Martin (14). Copyright 2012, with permission from National Academy of Sciences,
USA.]
AUDITORY DISTORTIONS
E. One Nonlinearity, Several Paths
the generation site, then accumulates another phase lag
while traveling from its generation site to the measurement
place, either the stapes or the CF. Here the DP results from
vector addition of wavelets from all generation sites. The
dependence of the DP amplitude on the f2/f1 ratio is explained by the computation, for each ratio, of the difference
in overall phase among wavelets. A large DP stems from
wavelets with the same phase accumulation, phase coherence (244). In the absence of phase coherence, the DP is
small. Inspection of the vibration patterns of f2, f1, and
2f1-f2 leads to predict an f2/f1 ratio-dependent directionality, properly accounting for a large DPOAE amplitude (corresponding to backward propagation of wavelets) at intermediate ratios (near 1.25), and for a loud heard DP cubic
difference tone at small ratios (near 1).
The fact that a theory robustly explains an experimental
fact is encouraging yet constitutes no definite proof. This
theory makes use of the powerful mathematical wavelet
description of Huygens, the main difficulties of which are to
make sure that all possible wave sources are included, and
that all propagation modes are properly described. It cannot be excluded that new sources be uncovered and that the
reverse propagation of DP waves eventually turns out to use
another mode than the BM traveling wave (see sect. VH,
although we will see there that it is unlikely). Then the
theoretical basis of the ratio dependence of DPOAEs and
DPs would have to incorporate the new sources and propagation paths.
As for the second path, the model of coherent reflection on
local BM irregularities (246) implies a suitable frequencydependent phase. One alternative to it has been proposed
recently (217), in which the Reissner’s membrane (FIGURE
1A) is assumed to support a bidirectional propagation
mode, the backward component of which would contribute
to the overall DP at the stapes. Reichenbach’s model postulates that distortion on the BM evokes a traveling wave
along the Reissner’s membrane which, below 1 kHz but not
at high frequencies, can significantly interact with the BM.
It also predicts that DPs, both at 2f1-f2 and 2f2-f1, can travel
along the Reissner’s membrane, forward and backward, by
a short-wavelength mechanism similar to capillary waves at
the surface of a water tank. Interestingly, along the nonresonant Reissner’s membrane, these waves do not incur the
limitation that prohibits propagation across a resonant position. For the BM-propagated 2f2-f1 DP, backward travel
from the place where maximum nonlinear interaction between primary tones occurs is impossible, as it is apical to
the 2f2-f1 CF location. Upper-band DPs experience a shift of
their generation place toward their own CF location (163,
165), where nonlinearity is less. The Reissner mode, by
letting 2f2-f1 DPs travel from the place tuned to f2 along the
whole cochlea, might produce a larger outcome. Experimental validation, by interferometric measurements in rodents (217), was restricted to showing the presence of a
1-kHz 2f1-f2 DP on the Reissner’s membrane, traveling forward beyond its CF location with a fast rotating phase, over
places where the BM could not vibrate at 2f1-f2. Important
unsettled issues are whether large DPs could show up on the
Reissner’s membrane, particularly at 2f2-f1, and not only
travel backward, but also influence the BM-propagated mo-
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FIGURE 13. Average “bandpass filter-like” pattern of DPOAEs in
normal mice (top: CBA/J strain) and slightly hearing-impaired mice
(bottom: CD1) when an f1-sweep is performed at fixed f2 so that the
ratio f2/f1 varies from 1.1 to 1.5. The level of the DPOAE at 2f1-f2
elicited by a 60 dB SPL equal-level pair of (f1, f2) tones displays a
blunt peak for f2/f1 around 1.25, whereas lower and higher ratios
yield smaller DPOAEs. In mice with 20- to 40-dB loss in sensitivity
due to progressive degeneration of OHCs near the places tuned to f1
and f2, the residual DPOAEs have a similar behavior. [From Le Calvez
et al. (138). Copyright 1998, with permission from Elsevier.]
According to the two-mechanism model of Shera (246), the
forward-propagating wavelets at 2f1-f2, when their phases
are coherent enough to let them reach their CF location, are
back-scattered on cochlear irregularities there, by linear coherent reflection (see sect. IVF). The DPs reflected by this
mechanism, primarily arose from the basal nonlinear interference between f1 and f2. At the stapes, the DP from the
coherent reflection site combines with the DP directly emitted backward from the places of nonlinear interaction. The
two mechanisms and propagation modes identified by this
model for the measured DPOAEs share the same initial
nonlinear stage. As we will see, experiments reveal two
distinct phase behaviors of the DPOAEs: one with a uniform phase independent of primary frequencies, and a second one with a phase that rapidly rotates with primary
frequencies. This accords quite well with a two-path model,
either the one developed by Shera (246) or any alternative
one predicting the two phase behaviors. The first path is
straightforward and its existence unchallenged. Part of the
DP directly comes backward from the place of nonlinear
interaction, and its uniform phase relates to scaling symmetry (see sect. VD).
AVAN, BÜKI, AND PETIT
tion at 2f2-f1, even at higher frequencies where the motion
of Reissner’s membrane is supposed to influence BM motion very little.
FIGURE 14. Maps of DPOAE level (A and C) and phase (B and D) against DPOAE frequency (horizontal axis)
and f2/f1 frequency ratio (vertical axis) in the ear of a normally hearing rabbit. The horizontal dashed white line
(corresponding to f2/f1 ⫽ 1.00) splits the plots into an upper and a lower part representing the characteristics
at 2f1-f2 and 2f2-f1, respectively. Different colors are used to represent DPOAEs with amplitudes or phases
falling in different dB or 45°-wide intervals. Horizontal banding in the phase map means that DPOAE phase at
fixed f2/f1 ratio is independent of DPOAE frequency: it is determined by f2/f1 and obeys frequency scaling.
Vertical banding means that DPOAE phase, regardless of the f2/f1 ratio, is determined by some coherently
reflecting microstructure, e.g., at the (fixed) place tuned to the DPOAE. Yet, compared with A and B, C and D
are plotted in the presence of an interfering tonal stimulus near 2f1-f2, meant to jam the coherent-reflection
process at the place tuned to 2f1-f2. Its failure to influence the banding pattern suggests that vertical bands
may be due to coherent reflection on a more basal microstructure. [From Martin et al. (168). Copyright 2009,
with permission from Acoustical Society of America.]
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The extent of generation sites and mixture of propagation
paths are essential issues for the outcome of DPOAE measurements from a locally damaged cochlea to be interpreted
usefully. An unfortunate combination of stimulus parameters, favoring the coexistence of several contributions with
similar levels and different phases in a normal cochlea,
might generate intractable complexity offsetting the advantage of DPOAEs as local probes of active cochlear mechanics (51). Local damage removing only one contribution
might even fail to be noticed on the overall map of DPOAE
levels. This is why intense research has been devoted to
identifying choices of stimulus parameters that emphasize
one mechanism and path at the expense of the other ones. It
is helpful in practice that these choices for the 2f1-f2 DP
extend to all other lower-band DPs (3f1–2f2, 4f1–3f2, etc.).
Phase criteria have been developed for separating the
sources when they coexist (134).
As seen above, the lower-band DPs traveling straight back
from the nonlinear place dominate at intermediate f2/f1 ratios (⬎1.10 for 2f1-f2). Their principle of generation and
propagation essentially depends on how waves at f1 and f2
overlap. When f2 and f1 vary at constant f2/f1 ratio, scaling
symmetry applies, vibration patterns are conserved, and
the phase at any point moving with the traveling wave
envelopes changes little, thus the generated DPs are
“wave-fixed” (134). In two-dimensional plots of phase
against DPOAE frequency at varying f2/f1 ratios (133,
168) (FIGURE 14), the independence of DPOAE phase on
DPOAE frequency at constant f2/f1 ratio translates into a
horizontal banding pattern (FIGURE 14B, top). Conversely,
the DP components traveling forward from the generation
place then backscattered on local irregularities near the DP
CF place (dominant at small f2/f1 ratios) are “place-fixed”
as they strongly depend on the irregular places. As the stimuli are swept in frequency, the excitation patterns move
along the BM and different reflecting sites are encountered,
with different phase rotations as a result. The reflectionmechanism DPs are thus characterized by a vertical banding
AUDITORY DISTORTIONS
pattern (FIGURE 14B, bottom) contrasting with the horizontal pattern of wave-fixed DPs. Within one vertical band, the
measured phase is independent of f2/f1 as, for a chosen DP
frequency, the scattering places remain fixed (since coherent
reflection is supposed to occur at the place tuned to the DP
frequency).
With regard to both upper- and lower-band DPs, it is noteworthy that at high primary levels, the tails of the envelopes
of BM motion at f1 and f2 extend far basally, where the
absence of compression allows a steeper growth than at
resonance. At ⬎70 dB SPL, they are almost as large as they
would be more apically at the CF location. Conceivably, the
interactions in OHCs responding at both frequencies remain nonlinear enough, and phases along the interaction
interval vary smoothly enough, that significant DPs could
come from basal places. Some authors found no evidence of
it (302), while others did (170). The complexity of wave
combinations is such that DPs from basal sites may be easily
overlooked because their removal (by basal lesions or by
application of a third interacting tone well above f2; see
APPENDIX III) hardly influences the overall DP levels. Vector
subtraction has to be used for separating the basal contributions from the standard ones. Computations of vector
differences between 2f1-f2 DPOAEs without versus with an
interfering tone (that turns off the contribution from the
place where it peaks) recently led to the observations that in
A currently active issue is whether this complex behavior of
DPOAEs in a cochlea can be found in other vertebrates
without tonotopically organized BM, electromotile hair
cells, or tectorial membrane (15). In birds in which the
auditory sensory-cell dichotomy is less marked than in
mammals, the dependence of DPOAE phase on primary
ratio is similar to mammals, with clear place- and wavefixed components. In geckos with no BM traveling wave,
more similarities with the DPOAEs of mammals have been
found above than below 1 kHz, as for the latter frequencies,
no evidence exists for the distortion versus reflection classification. In frogs with no flexible BM, no wave-fixed behavior has been found (15), and the question is open whether a
traveling wave could still exist along another flexible structure. Solving these issues would help to establish what minimum architecture leads to a physiological complexity comparable to mammalian DPOAEs, which in the previous sections had been explained mainly in terms of bidirectional
wave propagation from distributed places, with little reference to anatomy.
F. Spectral Characteristics and Growth
From an engineering standpoint, the spectral pattern of distortions coming out of any nonlinear black box provides
specific insights into its distorting mechanism, nonlinear
transfer function, and OP. A caveat for applying this ap-
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A caveat is that, even though the lower-band DPs arising
from coherent-reflection at the place tuned to their frequency are expected to generate a vertical-banding pattern,
evidence has been produced recently that some vertically
banding DP components originate from another place,
basal to f2, using methods more extensively described in
section VIA and APPENDIX III. For example, FIGURE 14D
shows a persistent vertical banding pattern despite the presence of an interference tone, which by being placed near the
DP frequency, suppresses coherent reflection occurring at
the DP CF place (hence the decrease in overall DPOAE level
seen in FIGURE 14C). In other words, although consistent
with coherently reflected DP components, vertical banding
is not a signature of apical coherent reflection (167). This
issue is also raised by upper-band DPs. In their case, only
place-fixed components have been observed, with fast
phase rotation and vertical banding at all frequency ratios
(see FIGURE 14B, bottom). It is a common observation in
mammals that their levels are smaller than their lower-band
cousins of the same order, but not in chicken, lizards, or
frogs (15). A first explanation considers that, as the place
near f2 with maximum nonlinear interaction has a resonance frequency less than DP frequencies, the BM is too
slack for launching these DPs directly toward the stapes.
Upper-band DPs would be more efficiently produced more
basally (163, 169), with a region of place-fixed reflections
occurring somewhere between the DP generation region
and the DP CF place (133).
small rodents, there is no significant contribution from the
CF place of the DPOAE; and that a large DPOAE component sensitive to high-frequency interfering tones, one-third
octave above f2, thus likely generated basally, displays a
vertical banding pattern (167). The proposed explanation
still rests on the idea that vertically banding components
come from a relatively fixed location when primary frequencies are swept, but assumes that what determines this
fixed basal location is some constraint to which the
DPOAEs have to conform. For example, for the 2f2-f1
DPOAE, contributions apical to its CF location would be
absorbed when traveling over this place, and for the 2f1-f2
DPOAE, suppression by high-level primary stimuli (113)
would spare only the contributions from sites well basal to
stimulus CF locations (167). Without contradicting the
existence of coherent reflection, and without the need to
invoke the Reissner’s membrane as an alternative propagation medium (217), such evidence strongly supports a
complementary place-fixed mechanism involving basal
generation sites. At increasing stimulus intensities (70 dB
SPL or more), other combinations of interactions have
been suggested. For example, 2f2-f1 could be produced in
two stages, generation of the harmonic 2f2, which in turn
could interact near its own basal peak with f1, to generate
a quadratic product [2f2] ⫺ f1 (64). However, these
mechanisms can likely be neglected as long as stimulus
levels are kept ⱕ65 dB SPL, as was the case for rodent
ears measured in Reference 167.
AVAN, BÜKI, AND PETIT
proach to the cochlea is the requirement that DPs come
directly from one emission place, which we now know is at
best an approximation. Nonetheless, it reveals several relevant features.
ADP ⬇ A12A2关3 ⁄ 4 a3 ⫹ 5 ⁄ 8 a5共2A12 ⫹ 3A22兲
(4)
⫹ 105 ⁄ 64 a7共A14 ⫹ 4A12A22 ⫹ 2A24兲 ⫹ . . . 兴
A similar treatment of all terms of the Taylor series predicts
the distortion spectrum as a series of equidistant peaks at
intermodulation frequencies.
Indeed, this is observed in physiological responses from
auditory sensory organs (230), the largest DPs being oddorder ones at (n ⫹ 1)f1 ⫺ nf2 (lower band) and (n ⫹ 1)f2 ⫺
nf1 (upper band) (n ⫽ 1, 2, . . .) surrounding the main two
peaks at primary frequencies. The higher the order (defined
by p ⫽ 2n⫹1), the smaller the DP as the terms in the Taylor
series (assuming small amplitudes) get increasingly smaller
with increasing exponent. At increasing primary levels, the
observed tentlike pattern (FIGURE 11) changes little as all
DP peaks grow at approximately the same rate of decibel
per decibel increase of stimuli. In linear physical units, the
amplitude of the pth-order DP grows as the amplitude of
the input to the nonlinear system to the power p (thus at p
dB per dB increase in input). Other features are not generic
but specific of the distorting system, and here, of its physiology, as already seen for the slow average growth of DPs
with SPL, which betrays the compressive amplification (see
sect. IIA; Ref. 79).
Another specific property of physiological DPs is that evenorder components, such as the quadratic DP at f2-f1, are
smaller and more labile in many species (37). In contrast to
odd-order DPs that depend on asymmetric components of
the nonlinear transfer function of the MET channels (a3, a5,
a7, etc., in Equation 4), even-order ones rely on symmetric
components (70). The local symmetry of the MET transfer
function and the exact content in odd- and even-order terms
1588
Experimentally, low-frequency biasing of the cochlear partition has been used as a tool for displacing the OP along the
cochlear transducer function and reconstruct its shape (21,
29). Other experiments have exploited the changes in the
balance between odd- versus even-order DPs to monitor
endolymph volume disturbances produced by direct fluid
injection in scala media (253). The outcome of these manipulations in terms of complex combinations of 2f1-f2 versus
f2-f1 DP changes was similar to those of former experiments
attempting to modulate the cochlear amplifier by decreasing the endocochlear potential using furosemide, a loop
diuretic (180, 187, 253), which suggests that volume-induced OP shifts occurred in such circumstances. Even-order
DPOAEs may also provide a convenient tool for monitoring
the medial olivocochlear efferents, a neuronal pathway that
innervates the OHCs via cholinergic synapses. Physiological activation of these efferents by the presentation of sound
in either ear produces a decrease in the gain of the cochlear
amplifier, likely beneficial in the presence of background
noise (83). By affecting OHC function, efferent activity influences DPOAEs which therefore can serve for assaying the
efferent control (40, 131, 187, 211). The 2f1-f2 DPOAE
responds to contralateral noise by a disappointingly small
decrease in amplitude, but the initially weak f2-f1 DPOAE
component can almost reach the level of its 10-dB-larger
2f1-f2 counterpart when medial efferents are activated by
low noise levels (30 dB SPL) (3). As, in this experiment,
efferent activation also strongly modified the phase of
changes exerted by a biasing infrasound, a common contribution of OP shift in both paradigms was suggested (3). The
changes in even- versus odd-order DPOAEs after continuous low-level sound stimulation, noticed long ago, had already been attributed to efferent activity (25).
G. Nonmonotonic DPOAE Changes With
Stimulus Parameters
Despite the prediction, in the previous section, of a simple
growth of DPs with SPL, real-life data disclose conspicu-
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At small enough input amplitudes, a nonlinear transfer
function can be developed as a Taylor power series for
which the coefficient of the nth power term is the nth derivative of the transfer function evaluated at the OP and divided by n! (For a linear system, only the first derivative
differs from 0 as the transfer function is a straight line.)
Input signals considered throughout this section are of the
form A1sin2␲f1t ⫹ A2sin2␲f2t. In the power series, terms
such as A12A2⫻sin22␲f1t⫻sin2␲f2t show up and, with the
help of some elementary trigonometric formulas, can be
converted into sums of terms corresponding to all combination tones, among which sin2␲(2f1 ⫺ f2)t, the 2f1-f2 DP
(its cubic nature mentioned in sect. IIIA reveals itself in its
coefficient A12A2, proportional to A3 in the commonest
situation with A1 ⫽ A2 ⫽ A). The resulting expression of the
amplitude at 2f1-f2, for example, is far from simple, except
its lowest-order term (62, 66)
of its Taylor expansion, depend on the OP position. It is
thus expected that its displacements influence the whole
series of DPs in a coordinated manner. With the assumption
that the OP moves along a second-order Boltzmann function, it was modeled that when the OP is at the inflection
point where the steepest slope ensures maximum cochlear
amplification (at low sound levels), the even-order f2-f1 DP
is minimum whereas the 2f1-f2 is large (70). This provides
one explanation of the predominance of odd-order components in usual DPOAE spectra (FIGURE 2). However, as
soon as the OP moves away from the inflection point, the
predicted DP pattern dramatically changes with a sharp
increase of the f2-f1 DP, together with the 2f1 and 2f2 (evenorder) harmonics, whereas the 2f1-f2 DP sharply decreases
in amplitude. Larger OP shifts lead to reciprocal changes of
the even- versus odd-order DPs (70).
AUDITORY DISTORTIONS
ously nonmonotonic DPOAE growth functions with deep
minima (notches) that separate two portions with different
slopes (66, 152, 173, 180, 292). Such notches are consistently observed in small mammals for primary levels around
60 –70 dB SPL, at many combinations of primary frequencies (179, 292), but they are less frequent in human ears
(291). A single measurement at the notch could lead a naive
experimenter to conclude that absent DPOAEs pinpoint
damaged OHCs, whereas a slight arbitrary shift of stimulus
level would have produced different results. The theoretical
background of these notches, therefore, has to be explained;
otherwise, one would have to deny the practical validity of
DP measurements.
FIGURE 15. Dependence of the 2f1-f2 DPOAE level (A) and phase (B)
on primary levels in the ear of a guinea pig before (thick line) and at
varying times (as indicated in the inset) after injection of furosemide, a
loop diuretic which, by decreasing the endocochlear potential, affects
cochlear amplification. A deep notch splits the initial growth function
into a shallower low-level segment and a steeper high-level segment (A),
with a half-cycle difference in phase between the two segments (B).
Furosemide essentially affects the low-level segment, as if two different
sources of DPOAEs were combined, an active one at low levels, vulnerable to furosemide, and a passive one at higher levels and with a
steeper growth, not physiologically vulnerable (180). Their combination
would generate a notch when their levels are similar because they
happen to be out of phase. An alternative explanation assumes a single
nonlinear transfer function with changing gain and operating point. This
function would generate both segments and even predict the observed
shift (⌬L) in notch position with decreasing amplifier gain. [From Lukashkin et al. (152). Copyright 2002, with permission from Acoustical
Society of America.]
Nevertheless, an alternative generic explanation of growthfunction notches has taken shape, implying a single compressive nonlinearity and changing level (66, 153). Several
mathematical models of nonlinear transfer functions have
been tested for their influent parameters, in search of signatures of a single-site mechanism producing a notched pattern. The origin of nulls has been interpreted in terms of the
cancellation of terms in the Taylor series describing the
nonlinear transfer function, due to the mixing of an oddorder nonlinear term with the next highest odd-order nonlinear term (66). In this framework, nulling turned out to be
a commonplace result of the nonlinear nature of a transfer
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A first generic explanation of a notch has been brought
forward (179), drawing on the characteristic 180° phase
change observed whenever a nulling happens (152, 180,
182, 292) (FIGURE 15). This pattern suggests interference
between two DPs of opposite phases coming from two different sites or mechanisms. When they happen to have equal
amplitudes, cancellation happens. On one side of the notch,
one mechanism dominates while on the other side, the other
mechanism takes over, and their physiological meanings
may be totally different. The observation of different slopes
on both sides of a notch has suggested that the DPs from
active compressive mechanisms, the ones discussed so far,
dominate at lower levels while at high levels, passive DP
sources with a steep 3 dB/dB growth would take over. Their
lack of vulnerability to cochlear damage whereas at the
same time, low-level, slow growing DPs have vanished,
might point to their being valueless. Actually, there is
mounting evidence that the nonlinear sources of slow- and
fast-growing DPs likely are the same MET channels of the
same OHCs. When impaired, these OHCs no longer exert
compression so that the DPs that they produce are only the
fast-growing ones (9, 183). The reported lack of vulnerability depends on what means were used to “silence” OHCs.
Loop diuretics or transient hypoxia, for example, by decreasing the endocochlear potential without affecting OHC
structures, kill low-level DPs without affecting high-level
ones (181, 292). Conversely, factors destroying OHCs and
their MET channels [aminoglycosides (28, 292); genetic
defects (33, 96); excessive noise (9)] readily remove any
trace of the “passive” DP component, right where OHCs
have been damaged, while in the same cochlea, places where
OHCs survive in an inactive state still produce DPs at high
stimulus levels (9). In hair-bundle preparations from the
bullfrog’s sacculus (14), the DPs produced when a hair bundle is passive, either when stimulated far from its intrinsic
resonance or at high level, keep providing specific information on the function of MET channels, even though the
hallmarks of activity have been lost. In the more complex
situation of the cochlea in vivo, the possibility that significant DP contributions come from basal OHC-stimulated
off-resonance is now strongly substantiated (170).
AVAN, BÜKI, AND PETIT
Table 2. Possible (not mutually exclusive) explanations for the presence of notches in DPOAE (2f1-f2) growth functions and of
microstructure in DPOAE versus frequency plots
Explanation
Reference Nos.
Two emission principles, one active and vulnerable, one passive and robust
Two emission sites, one near f2 or basal to it (distortion), one at the place tuned to 2f1-f2 (coherent reflection)
and/or two propagation modes with different phase rotations
One generation site, two propagation modes at different speeds along BM and Reissner’s membrane
One generation site, one propagation mechanism, notch due to a change in saturation on the transfer
function at increasing level
One generation site, one propagation mechanism, shift in OP
Retarded action of an AGC on each primary, generating two spectral series, and interference between them.
Single hair bundle in its passive (off-resonance) regime, with a gating compliance strong enough to elicit
negative stiffness
180, 292
133, 170, 246
217
152
70
271
14
function. Several properties of this function (even symmetry, degree of asymmetry, severity of clipping) could equally
easily produce nulling, while a mere shift of the OP could
profoundly alter the ability to produce it (TABLE 2).
Small changes in the frequencies of stimuli, not only their
levels, can also lead to large changes in DPOAE levels. A
so-called fine structure shows up in the DP grams (DP vs.
frequency plots) of human ears, consisting of up to 20-dBdeep troughs (92) (FIGURE 16) that modulate the overall
level-versus-frequency pattern in a quasi-periodic manner.
Notches are more densely packed in human ears than in
other species, about 1/10th of an octave apart (92). It appears that their density is inversely proportional to DP time
delay, of the order of 10 ms around 1 kHz in humans. The
fine structure persists in an interval of primary levels over
20 dB wide (FIGURE 16, A VERSUS B). Its acknowledged
explanation is an already familiar one (see sect. VE), building on the existence of the two discrete DP contributions
from the nonlinear and coherent-reflection sites (111, 133,
263). As their relative phases rotate with frequency, an
interference pattern of regularly spaced nulls is created in
the DP gram. The lack of fine-structure in small mammals
comes with the lack of reflection component from the 2f1-f2
place, inferred in Reference 167 from the failure of an interfering third tone (see APPENDIX III) to exert any change in
the DP pattern when set near 2f1-f2, where it was meant to
suppress the reflection mechanism.
Human ears contrast with rodent ears in that no notch in
growth function is observed around 60 –70 dB SPL primary
levels; nonetheless, another type of notch in the growth
function has been reported in close relation to the commonly observed deep fine structure (92). These notches correspond to the fact that the fine structure tends to shift in
frequency when the primary level changes so that the DP
grams at different primary levels are not parallel. At fixed
frequency, this generates irregular growth functions with
dips around 50 dB SPL, with little difference between the
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slopes below and above a dip (92). For human DPOAEs,
growth notches and fine structure thus rest on a common
background, the distortion versus coherent reflection dualism. In rodents, the explanation of growth notches is more
contentious. To decide whether the notch in a DPOAE
growth function is due to the interference between two
contributions from different sites, not the coherent-reflection one but other possibilities have been invoked (170), or
FIGURE 16. The high-resolution plot of DPOAE amplitude against
frequency at fixed stimulus level and frequency ratio commonly displays pseudo-periodic notches, 5–10 dB deep (example in one human ear; A: high-level stimuli, f1 ⫽ 70 dB SPL and f2 ⫽ 60 dB SPL;
B: low levels, f1 ⫽ 55 dB SPL and f2 ⫽ 40 dB SPL). Salicylate intake,
which decreases cochlear amplification by affecting the electromotility of OHCs, affects the depth of notches and shifts their frequency
(open circles: initial measurement; dashed and dotted black lines:
measurements at various times during salicylate treatment; closed
circles: post-control showing the stability of DPOAE fine structure).
[From Rao and Long (213). Copyright 2011, with permission from
Acoustical Society of America.]
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DPOAE, distortion product otoacoustic emission; BM, basilar membrane; AGC, automatic gain control.
AUDITORY DISTORTIONS
H. DPOAEs: Their Timing and Backward
Propagation
Evaluation of cochlear tuning is an important challenge in
audiology, as decreased tuning may account for impaired detection of sounds in noisy environments. Direct measurements
of tuning on the BM or in single-neuron responses are highly
invasive, thus prone to damage the system. Behavioral measurements of tuning (psychophysical tuning curves or notched-noise
masking experiments; Ref. 184) are time-consuming even in
experienced subjects. Indirect objective measurements are
suggested by the rule that the more narrowly tuned a filter,
the longer the delay of its response. The cochlea can be
considered as a filterbank, which led Shera (248) to analyze
how cochlear tuning, cochlear filter delay, and OAE delay
covary. Thus the topic of this section, i.e., how to measure
the timing of OAEs, might provide reliable information
about the other two.
A thorny issue is that filter delay is only one part of OAEdelay beside the time traveling waves need to reach the
nonlinear place(s) and the backward propagation time(s) of
OAEs from the generation place(s) to the recording site in
the ear canal (FIGURE 17). This is why so much energy had
to be devoted to the first task of identifying the sites where
OAEs are produced and their propagation pathways to the
detection system. Strong evidence is now available concerning the existence of distributed sites of DPOAE production,
all along the sites where primary envelopes overlap and not
only where they peak. One direct backward pathway from
the nonlinear generation sites and one forward then backward pathway ascribed to coherent reflection have been
described, both influenced by the f2/f1 ratio (see sect. VE).
Noninvasive measurements of OAEs only provide access to
their round-trip travel time, and this indirectly, except when
viewed in the time domain using special averaging techniques (293). The phase plot of DPOAEs evoked by a fixed
tone and a swept-frequency tone is approximately linear
and from its negative slope, the DPOAE group delay in the
ear canal can be inferred (see sect. IVF). The filter delay is
the DPOAE group delay minus the travel times of the stimulus to the generation place and of OAEs along their propagation path(s). The forward travel time is given by the
onset delay of BM response (231). The mainstream hypothesis is that DPs reach the stapes via retrograde BM traveling
waves by the same mechanism and at the same speed as the
forward traveling wave, in which case, DPOAE delay ⫽
filter delay ⫹ 2 onset delays. In a thorough analysis of
available data, Ruggero (231) listed all possible alternatives
to this view, among which that whereby OAEs propagate
via very fast acoustic compression waves in the cochlear
fluids, implying a different prediction: DPOAE delay ⫽ filter delay ⫹ onset delay. For DPs to move the stapes and
produce DPOAEs, the important parameter is the pressure
difference between stapes footplate and round window, not
BM motion as was the case for sensory cells, and a compression wave might achieve pressure transmission efficiently (296). The stapes footplate of mammals is far from
the basal BM so that to reach the oval window, DPs have to
travel through the bulk of SV fluid at least for the last stage
of their journey (219).
A first observation is that the formula giving the filter delay
combines several components, and cumulative errors can
lead to largely erroneous estimations. The most recent experimental attempt at improving the accuracy of OAE delay
measurements (176) used a modified emission technique
designed for emphasizing the contribution of the return
journey to the emission delay. Partial suppression of the
SFOAE at frequency f2 was produced by a complex of
closely spaced low-frequency tones around frequency f1,
inspired from Reference 192. The resulting modulation of
the f2 SFOAE is nothing but intermodulation, which thus
translates into DPOAEs consisting of a series of spectral
components, at f2 plus the frequency separation between
the components of the f1 complex. Frequency f1 itself bears
no influence in this series thus can be shifted at will, and for
instance can be set very low. Then, the forward journey of
an f1 tone to the place tuned to f2, where OAEs emerge, is
shortened to a minimum, and the OAE phase behavior is
dominated by the reverse travel of the OAE complex. The
authors found a reverse travel time of ⬃400 ␮s or 2.5 cycles
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to the dynamic behavior of a single nonlinear source, a
suitable experimental test would be to perform well-defined
local damage, e.g., to OHCs in the basal cochlea, and examine whether it affects the notches produced by primary
frequencies with more apical CF locations. Alternatively,
DPs produced by the same stimuli could be measured at
different places to check whether a notch is stable at all
locations, in which case it would be an inherent part of a
single nonlinearity. The two-site model predicts that the
notch exists because of a particular 180° phase difference
which would not hold if the measurement site moved closer
to one site and further from the other one. The latter test
was performed in a few guinea pigs as a complement of the
protocol of Reference 13, the 2f1-f2 signal being measured
in the ear canal as a DPOAE and in the perilymph of SV1 as
a pressure wave, and it revealed that the notch in the DP
growth function was present at both measurement sites (P.
Avan, P. Magnan, and A. Dancer, unpublished data). In
contrast, in the cat, the stable dip visible in the DPOAE fine
structure at one f2/f1 ratio vanished when the intracochlear
DP was noninvasively assayed using a third tone as a calibration probe (245), in support of a two-mechanism model.
This discrepancy, yet to be explored, suggests that indeed,
notches in DPOAE plots may arise from different mechanisms, among those theoretically available (TABLE 2), thus
bear different significances.
AVAN, BÜKI, AND PETIT
f2 f1
τback-
compression
2f1-f2
Stapes
τbasal
τforward + filter
τstapes-microphone
τbackward-BM
Microphone
site of
DP-frequency
FIGURE 17. Main contributions to the delays of cochlear responses from different locations and mechanisms, when the cochlea is excited by a pair of pure tones at frequencies f1 and f2 (vibration patterns in blue
and red, respectively; delays, likely similar at f1 and f2, in violet). The 2f1-f2 DPOAE propagates both forward and
backward via different putative mechanisms associated with different speeds and delays (vibration patterns
and delays in green). Upward-pointing arrows mark measurement sites between which propagation delays can
be evaluated. The basal site from which the propagation delay to the stapes is ␶basal, actually corresponds to
multiple locations, likely broadly distributed between the stapes and the generation site tuned to f2. Straight
horizontal lines indicate propagation via a fast compression wave, and undulating ones, via a slow transverse
traveling wave.
(176), seemingly shorter than the forward travel time of a
BM wave at the same frequency, estimated at 900 ␮s (223).
As indirect measurements are inconclusive even when confounding factors are kept to a minimum, invasive experiments are required to definitely establish by which mechanism DPOAEs reach the stapes to exit the cochlea: slow
backward traveling waves along the BM, fast compression
waves through SV fluid, or alternative paths such as that
recently suggested along the Reissner’s membrane (217).
The common goal of the scarcely published experiments
was to assess the timing of DPs along their way, from comparisons of phase data collected at different spots along the
cochlear partition. Three different experimental protocols
were used. In the first one (13), the authors inserted a sensitive piezoresistive hydrophone in the SVs and STs of the
first and second turns of the guinea pig cochlea. A second
group (219) performed interferometric measurements of
gerbil BM motion at different places extending for ⬃1 mm
in the longitudinal direction (93) and of stapes motion (94).
Last, spatial variations in intracochlear pressure were measured with the help of a fiber-optic pressure detector at
different places close to the BM in ST in the basal turn of
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gerbil cochleas (53). All investigators measured DPs in response to intermediate level primaries (40 –70 dB SPL) in
cochleas with preserved sensitivity, with the thresholds of
neural responses after the DP measurements being within a
few decibels of their initial values.
What sparked the controversy were the consistent findings
by Ren and co-workers (94), with continually improving
experimental setups, of DPs reaching the stapes earlier than
any other BM measurement spot in the cochlea (FIGURE 18A).
Measurements at different longitudinal places of the BM,
thought to be well basal to the place where the DP was
allegedly generated (the f2 CF place), should have revealed a
backward traveling wave with an increasing phase lag toward the stapes. Instead of that, systematically increasing
phase lags were found at increasing distances from the stapes, suggesting that on the BM, DP waves traveled in the
forward direction and not the reverse one. This led Ren to
propose that the DPs reach the stapes and round window
almost instantaneously via a compression wave through the
fluid, and that from there, the asymmetric motion of cochlear windows due to their difference in impedance generates a forward traveling wave along the BM.
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generation
AUDITORY DISTORTIONS
In the other experiments using a fiber-optic pressure sensor
(53), the technique, although requiring stimulus levels ⱖ70
dB SPL, allowed physiologically vulnerable DPs to be
mapped near the BM at one cochlear location approximately tuned to 20 kHz. The complexity of DP patterns
confirmed the combination of basally produced DPs traveling forward and apically produced DPs going backward.
The fact that the DP level fell off when the tip of the pressure
sensor was moved away from the BM by a few hundred
microns (FIGURE 18B) revealed the existence of a slow DP
traveling wave along the BM, without precluding a fast DP
pressure wave near the noise floor of the system, i.e., 55– 60
dB SPL. Previous data with the same setup (54) had shown
evidence of a fast, spatially invariant compression wave
dominating at ⬎100 ␮m from the BM, whereas the slow
backward propagating wave took over at closer distances.
Most of Ren’s findings were thus confirmed including, in a
number of configurations, DPOAEs leading in phase their
corresponding BM DPs. The interpretation clashed with
Ren’s in terms of which DP wave was most relevant, as the
larger slow wave near the BM was emphasized. The phase
lead of DPOAEs was explained, not in terms of the stapes
vibrating at the DP frequency before the BM, but by assum-
ing the combination of a forward traveling DP produced
basal to the measurement point and propagating toward its
CF location, and of a flat-phased DPOAE obeying scaling
symmetry, i.e., the wave-fixed component of Kemp.
The fiber-optic work, though showing DP waves traveling
on the BM in the reverse direction, could not assess which,
these or the fast compression waves away from the BM,
dominate in the external ear canal. Another experiment
(13) addressed this issue, by measuring the properties of the
pressure wave far from the BM, near the bony wall of the
cochlea, both in SV and ST in turns 1 and 2 of the guinea pig
cochlea (sites SV1 and SV2 tuned to 9 and 3 kHz, respectively). The single pressure detector could be moved back
and forth without perturbing cochlear sensitivity and earcanal DPOAEs, and its sensitivity allowed stimuli at 50 dB
and 60 dB SPL to be used for generating DPs at 2f1-f2 well
above the noise of the detector. Phases were compared between the two measurement sites for stimulus frequencies
between 18 and 1.5 kHz (FIGURE 18C). The DPs, larger in
SV than ST by ⬎10 dB, were comparable in magnitude
(55– 60 dB SPL in SV) in the first and second turns, within a
few decibels on average (FIGURE 18D). Their phase differ-
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FIGURE 18. A: delay between stapes and BM responses to a DP at one basal cochlear site computed from
phase measurements, suggesting that the BM response lags, as if the stapes were reached almost instantaneously by the DP via a compression wave, before the DP started traveling forward as a transverse wave.
[From He et al. (94). Copyright 2010, with permission from Elsevier.] B: DP pressure against DP frequency in
the ST of one preparation, at two distances from the BM (thick line, 37 ␮m; thin line, 117 ␮m). The DP
amplitude drops off by ⬎5 dB with increasing distance from the BM, suggesting a dominant traveling-wave
mode close to the BM, as a compression mode should fill up the ST. [From Dong and Olson (53). Copyright
2008, with permission from Acoustical Society of America.] C: phase difference between pressures at the DP
frequency in SV of the second and first turn of guinea pig cochleas (SV2 vs. SV1), against DP frequency. For
f2 ⫽ 5 kHz, the average difference is 0 (dashed vertical line) as the DPOAE site of generation falls midway
between the two measurement points. A propagation speed of 40 m/s from apex to base is inferred, in line
with published interferometric BM measurements at similar frequencies, i.e., between 20 and 100 m/s
(227). D: the DP level in SV displays little spatial variations and matches the DPOAE level in the ear canal when
taking into account the reverse middle-ear transfer function. [From Avan et al. (13). Copyright 1998, with
permission from John Wiley & Sons.]
AVAN, BÜKI, AND PETIT
ence between SV2 and SV1 was about ⫹90° for f2 ⫽ 3 kHz,
i.e., with the main DP generation site near SV2, thus the DP
pressure wave took ⬃160 ␮s to travel the 6 mm from its
generation site toward SV1, at an average speed of ⬃36 m/s.
For comparison, the pressure difference between SV and
ST at stimulus frequencies traveled from SV1 to SV2 at
⬃40 m/s. As for levels, as the reverse middle-ear transfer
function applies a flat 35-dB loss (52, 157), the 55- to
60-dB SPL of the DP wave filling up the SV, correctly
matched the DPOAE level in the ear canal, i.e., 22 dB
SPL. At f2 ⫽ 9 kHz with the dominant DP generation site
near SV1, the DP wave at SV2 lagged by 90°, as expected
from a DP wave traveling apically from its generation
place.
Going back to data modeling, the most thought-provoking
finding of Ren is undisputed, the negative slope of phase
data. As the initial explanation, i.e., that the DPs reached
the oval window ahead of the BM responses, has lost
ground, this negative slope calls for an alternative explanation. A recently proposed one (250) rests on the outcome of
the so-called Allen-Fahey paradigm (6), invoked by Ren in
support of the compression-wave theory (222). This consists in keeping constant, not only the DP CF site, but also
the DP level at this site by monitoring the responses of
auditory neurons tuned to the DP frequency. The ensuing
measurements of DPOAEs under different combinations of
stimulus frequencies and ratio f2/f1 reveal a strong influence
of the latter parameter. At f2/f1 ⬇ 1, wavelets from distributed sources recombine much more strongly toward the DP
CF site than toward the stapes. This can be explained only
if wavelet phases undergo the large rotations that come with
1594
At present, evidence that the dominant mode of OAE backward propagation is by slow transverse waves along the BM
has become overwhelming, supported as it is both by theoretical models and experiments, even though hints of a fast
compression mode and perhaps, of a second slow transverse
mode along the Reissner’s membrane have been reported.
However, this propagation controversy had the positive
consequences of fostering innovative experimental setups
allowing fine measurements of longitudinal BM responses
and intracochlear sound pressures, and of emphasizing the
inescapable complexity of DPs. For a model to realistically
predict even the simplest outcomes (f2/f1 ratio, Allen-Fahey
paradigm, suppression patterns), it must acknowledge that
DP sites of production span a broad cochlear interval, much
broader than the place tuned to f2, and that DPs travel and
recombine along intricate paths. Thus the naive idea that a
DP gram provides an “undistorted” map of cochlear function no longer safely supports audiological applications. On
the other hand, DP grams provide much more useful insights into the intimacy of cochlear micromechanics, for
experimenters willing to spend some time designing proper
paradigms and deciphering their outcomes.
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This experimental setup reveals a clearly shorter backward
travel time than the one derived from OAE studies, including recent ones (160 vs. 400 ␮s; Ref. 176). It conclusively
indicates that the measured pressure throughout SV is the
relevant signal at the origin of DPOAEs beyond the stapes
footplate, thus addressing a question left unanswered in
Reference 53. Nonetheless, it contradicts the compression
wave theory in several respects. In the first place, 40 m/s is
40 times slower than the speed of sound in water, and
actually comparable to the average speed of transverse traveling waves at the cochlear base (227). Second, a compression wave should spread through the whole cochlea and
produce, almost instantly, similar pressures in SV and ST,
whereas the finding of a ⬎10-dB difference suggests that the
BM, acting as a high-impedance device, was coupled to the
SV pressure wave, as would a membrane serving as support
for a backward traveling wave. The relationships between
the space-filling pressure wave in SV measured in Reference
13 near the bony wall and the pressure measured in Reference 53 near the BM on its ST side, at two places where the
wavelengths were clearly very different, remain to be clarified, i.e., how do sound waves propagate between these two
places, and how fast?
slow wave retro-propagation (see sect. VD). A hybrid
model might solve the conundrum by assuming DP source
coupling into both slow and fast waves, with the DPOAE
arising predominantly by fast-wave coupling, and the DP at
its CF location by slow-wave coupling. The simplest class of
hybrid models fails to improve the accuracy of predictions
of the Allen-Fahey experiment. More complex models, that
remain to be designed, are criticized on the grounds of suspiciously ad hoc requirements such as the need to radiate
similar amounts (250). Nevertheless, both fast and slow
waves have been detected and their interferences produced
notches (54), which suggests that they do radiate similar
amounts. Another recent theoretical approach (254) examines several conventional models of sound propagation in
the cochlea, which consistently predict that slow reversetraveling transverse waves can produce negative phase
slopes if two conditions are met: a broad enough distribution of DP sites of generation and a reasonably large stapes
reflectivity. Even more recently, application of a two-dimensional nonlinear hydrodynamic cochlea model (283) to
the analysis of Ren’s experiments suggested that the DP
generation sites were well basal to the measurement sites,
thus reconciling the negative phase slopes with the possibility of slow-wave backward propagation. The distribution
of generation sites basal to the f2 CF location was estimated
to 0.5 mm for f2 ⫽ 12 kHz, which emphasizes, once again,
two important caveats. What relates DPOAE levels to local
cochlear status is not a simple proportionality relationship,
as contributions undergoing significant phase rotations
within the generation sites interfere in a complex manner;
and DPOAE versus lesion maps may not coincide as DPs
basal to local lesions still contribute (170).
AUDITORY DISTORTIONS
VI. FROM COMPLEX STIMULI TO NO
STIMULUS
A. Three or More Tones
The suppression paradigm has been refined to explore all
possible sources of DPs and all possible nonlinear interactions in the cochlea, and the maps now include the effect of
suppression on DPOAE phases (64, 169). The addition of a
third tone f3 to a two-tone stimulus (f1, f2) at the input of a
nonlinear system also creates a huge increase in complexity
by multiplying the possible arrangements of combination
tones. Their full analysis is outlined in APPENDIX III. It has
revealed the importance of basal generation of DPs, more
than one octave above f2 even at moderately loud stimulus
levels (103, 168, 169) (FIGURE 19). These results stress the
need to take into account the broad distribution of nonlinear sources to correctly explain initially unexpected properties of DPs (see sect. VH for the paradoxical phase lead of
DPs at the stapes, for example).
Novel combinatorial possibilities that lead to the production of additional tones have been discovered or inferred
from three-tone studies, extending the two-mechanism taxonomy of nonlinear versus coherent-reflection contributions (246). The need for faster paradigms, e.g., for collecting phase-gradient data to infer from them cochlear travel
times and tuning information, has also inspired attempts at
replacing f1, in the (f1, f2) pair of stimuli, by a group of
closely spaced tones with carefully chosen spacings (176,
272). A whole new set of DPs has been disclosed in this
manner, thus opening a new field of investigations (see APPENDIX III).
B. Impulses
FIGURE 19. Interference response areas in a rabbit ear, obtained
as color maps of the changes in amplitude (top) and phase (bottom)
of a 2f1-f2 DPOAE (primary tones near 3 kHz, vertical arrows, at 60
dB SPL), in the presence of a third interfering tone swept in frequency (horizontal axis) and level (vertical axis). In addition to a first
suppression lobe in the (f1, f2) region, a second prominent lobe
peaks about an octave higher. The phase change in the high-frequency suppression lobe differs from that in the primary region (dark
red vs. blue-green), suggesting the presence of two different DPOAE
sources with different phases, each in turn being removed by the
interference tone at different places. [From Martin et al. (169).
Copyright 1999, with permission from Elsevier.]
The briefer a sound impulse (a click), the broader its spectral range. Thus a click spreads over most of the BM. An
advantage is that the entire cochlea can be probed for its
ability to emit OAEs simultaneously by collecting the TEOAEs generated by this sudden excitation (121). The tradeoff is that, while the response of a linear system to a click is
a mere superposition of the responses to each spectral component of the click, the click-evoked response of the nonlinear cochlea bears complex relationships to the responses
to tones. These already require a careful analysis of complex
issues regarding the processing and propagation of soundinduced vibrations in the cochlea (see sects. IV and V), and
TEOAEs add two degrees of complexity in relation to nonlinearity and traveling-wave dispersion along the BM, i.e.,
the fact that the lower frequency OAE components being
processed more apically will increasingly lag the higherfrequency ones and spread out in time. The wide popularity
that TEOAEs have gained as an audiological tool, for giving
an idea of OHC function over a broad range of frequencies
in a single recording, justifies a brief overview of their prop-
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Suppression induced by cochlear nonlinearity provides a
simple means for probing DP generation sites. As suppression arises from compressive amplification, which is tuned,
a suppressor tone is most efficient near the place tuned to its
frequency. Thus when a signal from an unknown site undergoes strong suppression, this site is likely to be located
near the characteristic place of the suppressor. The addition
to two DPOAE-generating primaries of a third tone swept
in frequency and level allows suppression tuning curves to
be built, depicting the suppressor level required to induce a
given amount of DPOAE level change, against suppressor
frequency (FIGURE 19). These curves display a sharp tip
around f2, indicating that DPOAE suppressibility is tuned.
It confirms the sensitivity of the 2f1-f2 DPOAE to cochlear
status around f2, the place of maximum interaction between
primaries (27, 163, 165). The suppression patterns of
DPOAEs also display a fine structure, notably confirming
the existence of the secondary DP mechanism (the coherentreflection one) around the place tuned to 2f1-f2 (72).
AVAN, BÜKI, AND PETIT
erties in relation to cochlear nonlinearity and other types of
OAEs.
Another typical feature of TEOAEs, as conspicuous in the
time domain as in the spectral one, is an idiosyncratic distribution of beats among higher and lower frequency oscillations, and of rather irregular peaks and troughs in the
spectrum (210). Such irregularities suggest that TEOAEs
arise from place-fixed perturbations along the BM, as their
exact distribution, differing from cochlea to cochlea, would
strongly influence how backward-scattered wavelets would
recombine to form the final TEOAE. Indeed, simple hardware or computational models of banks of filters (300, 313)
excited by impulse stimuli produce outputs closely resembling natural TEOAEs once some degree of disarray has
been introduced in their frequency-to-place map. Such
models also called into question the idea that narrow-band
frequency components extracted from a TEOAE could be
literally interpreted in terms of round-trip travel time of the
corresponding frequency component along the BM (which
does not mean that it does not relate to it). First, some
models of the cochlea as a travel-less filterbank accurately
predict the observed spectral structure and temporal delays
of TEOAEs (300), and second, the temporal properties of
oscillations generated by filter disarray reflect the way oscillations from neighboring filters, by interfering with one
another, produce waxing and waning envelopes, the peak
delays of which depend on the frequency differences among
filters and not on their central frequencies (313).
One anticipates that TEOAEs may relate to SFOAEs, in
response to every single frequency component of the click,
1596
The dominant view is that TEOAEs arise mainly from (linear) coherent reflection (112, 246). If TEOAEs are principally made of a mixture of SFOAEs, a close correspondence
is expected between frequencies such that OHCs tuned to
them are healthy (efficient emitters giving rise to coherent
reflection) and frequency bands represented in the TEOAE,
and indeed often reported (210). The structure of TEOAE
spectral peaks, highly stable in a given ear, supports the idea
that place-fixed irregularities serve as primers for coherent
reflection. Furthermore, TEOAE and SFOAE growth functions at fixed frequency, and TEOAE and SFOAE spectra at
fixed stimulus intensity tend to match one another in a given
ear (112). The rapid phase variation of TEOAE phase with
frequency and that already reported with SFOAEs closely
resemble each other and differ from the nearly frequencyindependent phase of DPOAEs (244).
Yet, a few discrepancies suggest a more complex picture for
the mechanism of TEOAEs. For example, the correlation
between the map of healthy OHCs and the TEOAE spectrum is only approximate. After basal OHCs had been selectively exposed to acute damage, TEOAE responses were
found to be decreased in lower-frequency intervals, although OHCs tuned to these frequencies had been unaffected (10) (FIGURE 20). Cogent evidence of a nonlinear
mechanism other than coherent reflection has been brought
about by the finding in TEOAE responses of frequency components not present in the stimulus and, thus, produced by
a distortion kind of mechanism (305). Such a mechanism
predicts a not-so-straightforward mapping of healthy versus damaged cochlear intervals, as indeed observed (10).
Insight into the coherent reflection versus distortion contributions (place- versus wave-fixed), was provided by reports
that in guinea pigs and humans, with an open-canal recording technique allowing the onset of TEOAEs to be reliably
analyzed (301, 303), TEOAE generation appeared to shift
from a wave-fixed to a place-fixed mechanism over the time
course of the response, with the phase slope shifting with
poststimulus onset time from a shallow to a steep pattern.
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Recordings of TEOAEs in human ears display a complex
pattern of oscillations (FIGURE 3) generally lasting 10 –20
ms after click onset, with the lowest frequency components
tending to fade out the latest. The onset of TEOAEs is
generally hidden by the fact that the recording equipment
removes the first millisecond so as to ensure that the processed response cannot be contaminated by stimulus ringing in the external ear canal. This procedure may remove a
sizeable part of the short-lived, high-frequency TEOAE
components (⬎6 kHz, typically). The spectrum of TEOAEs
thus extends from 6 kHz down to ⬃0.5– 0.7 kHz, below
which middle-ear transfer is thought to be unfavorable.
Characteristically, the growth of TEOAEs is compressively
nonlinear, much less than 1 dB per dB increase in click level,
which inspires the design of the so-called nonlinear-artifact
rejection protocol. In the original equipment designed by
Kemp (122), one block of regularly spaced stimuli is made
of three clicks at level L and positive polarity, followed by
one click at level L⫹10 dB (i.e., three times larger) and with
negative polarity. In the averaging process, linearly growing
contributions, among which the echo of the stimulus on
passive structures, get cancelled and only the nonlinearly
growing part of the TEOAE remains.
or to DPOAEs, responses to any pair of components of the
click (fi, fj) such that nfi ⫺ mfj ⫽ f, with n and m integers, or
to both, which raises the issue of which relationship is closest. As SFOAEs mainly stem from a coherent-reflection
mechanism (part IV) whereas DPOAEs result from a mixture of contributions from a nonlinear, wave-fixed mechanism and from (linear) coherent reflection (part V), which
mechanism accounts best for TEOAE properties must be
examined. The mechanism issue is also important for interpreting the pattern of changes in TEOAEs produced by
OHC impairment over some frequency interval along the
BM. It has been seen with DPOAEs that the nonlinear
mechanism of generation can spread over a broad basal
interval where OHC damage may impinge on OAE levels at
lower frequencies.
AUDITORY DISTORTIONS
(167), which suggests a weak or nonexistent coherent-reflection mechanism. In addition to accounting for large rodent DPOAEs only made of the nonlinear wave-fixed component, combined with the conclusion of Reference 301, it
explains why the TEOAEs of rodents are small and shortlasting, as only their early, wave-fixed component exists.
The longer-latency TEOAEs relying on coherent reflection
are strong only in primates. The short latency of TEOAEs in
small rodents finds its counterpart in the large-frequency
spacing of the periodic crests found in SFOAE spectra, as it
is inversely proportional to the emission latency.
This framework helps understand the differences between
TEOAEs in humans or non-human primates, e.g., in Reference 164, and small laboratory rodents (TABLE 3). In the
latter, TEOAEs are smaller in amplitude and much shorter
in duration (e.g., Refs. 10, 12, 295). Conversely, DPOAEs
usually have much larger levels in small laboratory animals
(30 – 40 dB SPL are common) than in humans (⬃10 dB
SPL). These observations, summarized in TABLE 3, fit the
two-mechanism model of Shera (246) if one assumes that
the coherent-reflection mechanism is larger in primates than
nonprimate mammals. Indeed, in four species of small rodents, the presence of an interacting tone at the frequency of
the DP, meant to suppress the coherent-reflection component, actually does not affect any of the DP properties
Taking advantage of the sharp onset of the TEOAE-evoking stimulus, the timing of suppression of active mechanisms has been demonstrated directly, which was impossible with other types of OAEs. A click suppressor presented at varying times before the click stimulus (114)
produces a TEOAE change depending on postsuppressor
delay (90, 279). This change discloses some typical features of an AGC, a system able to adjust its gain according to the input level integrated over the immediate past
and such that the larger the integrated level, the smaller
the updated setting of the gain. Experiments revealed the
presence of a cochlear AGC acting with a time constant
of about two periods, with a controlling slope of ⬃0.5
dB/dB (90, 279). Its general significance will be examined
more closely in section VIIIB.
C. Spontaneous Vibrations
Even without stimulation, the ear can spontaneously emit
continuous tones, SOAEs (210). They were objectively detected and identified for the first time by Kemp (120). It was
initially argued that SOAEs might be mere filtered noise;
however, the statistical properties of the temporal distribution of SOAE amplitudes strongly suggest the existence of a
Table 3. OAE mechanisms in humans versus rodent ears, in TEOAEs versus DPOAEs
Human Ears
Distortion mechanism
Coherent reflection mechanism
DPOAE amplitude
DPOAE growth notches
DPOAE fine structure
TEOAE duration
TEOAE amplitude
TEOAE structure
Present
Present, yet not strong: ⫺3 to ⫺15 dB relative
to the distortion component (1); present in
only 3 of 10 ears (167)
Small (120, 167):ceiling 15 dB SPL
Few, around 50 dB SPL (92)
Conspicuous (91)
Long (121)
Large (121)
Two parts, the early one attributed to
distortion, the later one to coherent
reflection (303)
Rodent Ears
Present
Absent or weak (167)
Large (127, 167):ceilings 35–40 dB SPL
Many, around 60–70 dB SPL (291)
Weak or absent (167)
Short (10)
Small (if integrated over the same 20-ms
window as for human ears) (10)
Only the early (distortion-related) part (10)
TEOAE, transient evoked otoacoustic emission; DPAOE, distortion product otoacoustic emission; SPL, sound
pressure level. Reference numbers are given in parentheses.
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FIGURE 20. TEOAE amplitude spectrum in the ear of a guinea pig
before versus after exposure to a loud high-frequency sound (top
part of the graph; dotted line, before exposure; solid line, after
exposure) against audiometric changes induced by the exposure
[changes in compound action potential (CAP) thresholds between 1
and 30 kHz: inverted triangles]. While CAP threshold changes did
not extend below 4 kHz, a decrease in TEOAE (solid line, bottom part
of the graph) occurred even in the 1.5- to 3-kHz interval. [From Avan
et al. (10). Copyright 1995, with permission from Acoustical Society
of America.]
AVAN, BÜKI, AND PETIT
Despite his conceptual error, Gold had made an important
point. In the engineering domain, families of limit-cycle
oscillators (e.g., the van der Pol oscillator) poised on the
verge of auto-oscillation can, in the absence of input other
than the inherent noise, generate outputs presenting many
highly nonlinear characteristics of SOAEs (104), even
though isolated oscillators cannot emulate all SOAE properties (274). A pure tone in the ear canal can enhance or
suppress an SOAE at a neighboring frequency, and entrain
SOAE frequencies (148). Repeated click stimuli synchronize the existing SOAEs that show up among the averaged
TEOAEs as stable, inordinately long-lasting oscillations.
Changes in cochlear status, whether general [aspirin- or
temperature-induced (148, 149)] or targeted to a specific
frequency interval (exposure to high-level sound) affect the
SOAEs by decreasing their level and shifting their frequency,
before recovery brings them back to their initial state. Minor
changes in the boundary conditions at the stapes, produced by
a change in ear-canal, middle-ear, or intracochlear pressure
(47), similarly affect SOAEs (FIGURE 21). When the cochlear
partition is biased by the application of a loud infrasound,
SOAEs exhibit dynamic behaviors consistent with the shifting of the OP of OHCs (20). Privileged frequency spacings
between adjacent SOAEs have been reported (24), similar
to the average frequency spacings between the crests in
SFOAE and TEOAE spectra (313). All these properties illustrate typical characteristics of cochlear micromechanics,
i.e., subtle impedance adaptation for optimal wave propagation; strong nonlinearities adjusting the amplification
such that near threshold, when the cochlea amplifies optimally, cochlear oscillators are close to instability; as a result
of the former two properties, a tendency for coupled oscillating elements to lock to a common frequency linear coupling (297, 299), in sharp contrast, always pulls resonance
frequencies apart.
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FIGURE 21. Frequency spectrum of the sound collected by a
microphone in the sealed ear canal of a human subject, initially
displaying three peaks (SOAEs) in the 1,150- to 1,350-Hz frequency
interval (bottom). Sound recordings were then repeated every 10
min while the subject was moved from an upright to a ⫺20° headdown posture, before being taken back to upright (horizontal arrows
on the left mark the times when his body was tilted). Body tilt
produces an increase in intracranial pressure that stiffens the
boundary between inner and middle ear. This change strongly influenced the SOAE spectrum, as the initial three SOAEs vanished while
two novel spectral peaks appeared in turn (vertical arrows, a then b).
When the subject went back to his initial posture, the three initial
peaks immediately reappeared, yet with amplitudes differing from
the initial recordings, the central peak being smaller while its neighbors were larger and narrower. [From de Kleine et al. (47). Copyright 2000, with permission from Acoustical Society of America.]
Two theories of SOAE generation have been proposed. The
first one is inspired by the behavior of van der Pol oscillators, and supported by experiments demonstrating that isolated hair bundles can spontaneously oscillate. This striking
phenomenon has been described in sensory cells from bullfrog saccules surviving in controlled ionic environments
(171). Oscillations relate to the existence of an unstable
interval of negative stiffness, calcium-dependent and due to
the influence of the gating compliance. Every time a deflected bundle moves toward its resting position, it enters
the unstable region and flips back and forth without stopping. If deflection-independent stiffness terms exceed the
negative stiffness associated with gating compliance, which
is likely the case in mammalian cochleas (277), spontaneous
bundle oscillations may no longer occur. An alternative
mechanism for SOAE generation is that of cochlear standing-wave resonances. In this view, SOAEs result from mul-
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genuine intracochlear source of energy (19). Many species
from all four tetrapod classes have SOAEs so that the presence of a basilar or tectorial membrane is not critical. These
SOAEs are usually made of one or a few narrow spectral
peaks, 0.15– 40 Hz wide in humans (19, 274), with highly
stable frequencies and levels. Apart from rare exceptions
(76), a normal cochlear sensitivity is required for SOAEs to
be present in humans. However, a sizeable percentage of
normally hearing individuals have no SOAE, from 35% in
human neonates to ⬃70% in adults (210). As expected,
SOAEs have correlates on the BM (195). In his pioneer
work, Gold (77) predicted that spontaneous oscillations
would occur in his local regenerative oscillator model, close
to the region of instability. However, he wrongly assumed
that SOAEs would be objective correlates of tinnitus. His
efforts at recording them in tinnitus-complaining subjects
failed because tinnitus is a phantom neural percept in relation to peripheral deafferentation due to sensory-cell failure
that precludes OAEs.
AUDITORY DISTORTIONS
tiple internal reflections of forward- and backward-traveling waves between the stapes boundary and a place with
disarrayed reflection sites giving rise to coherent reflection.
In its passive version, the energy comes either from sounds
from the environment or physiological noise, while in the
active, prevailing view, the cochlea acts as a biological “laser cavity” maintaining stable standing waves by coherent
wave amplification (243).
VII. THE PARTICULAR STATUS OF THE
APICAL COCHLEA
The apical DPs, not only harmonics but also intermodulation products at f2-f1 and 2f1-f2, can be heard, have correlates in the responses of neurons in the auditory midbrain
(2), and usefully underpin psychophysical tasks (99, 174,
209). Below some corner frequency, measured at 1.3 kHz in
gerbils (2), DPOAEs are smaller than would be expected
from the large intracochlear DPs. This might be due to a
decrease in middle-ear gain, as happens in human temporal
bones (212). Yet, large interspecies differences in the frequency dependence of the middle-ear gain have been reported, perhaps in relation to differences in ear-canal volumes (157, 212), making it difficult to find a general explanation for the issue of low-frequency DPOAE levels.
The specificity of low frequencies affects not only the tuning
mechanism, but also wave propagation. A break of scaling
symmetry has been pinned down at the cochlear apex (49).
Although suppression experiments show that human
DPOAEs below 1.5 kHz still come directly backward from
the nonlinear site of generation, and not from secondary
coherent reflection, they lose their characteristic phase invariance at constant f2/f1, the one underpinning the horizontal-banding pattern of FIGURE 14B. A possible explanation is suggested by the recent model of cochlear mechanics
positing two sound propagation modes, the classical one
along the BM and another one along Reissner’s membrane
(217). Although the latter wave may be of uncertain relevance at high frequencies, the model derives from impedance and wavelength considerations that, at low frequencies, it can interact with BM motion and disturb BM resonance and DP propagation.
VIII. MATHEMATICAL MODELS OF
NONLINEARITY
The ultimate sources of peripheral auditory nonlinearities
are clearly identified as the MET channels of hair cells.
Their nonlinear behavior can thus be incorporated in models of cochlear mechanics that attempt to predict realistic
characteristics. Many of these models correctly describe the
nonlinear compressive amplification and frequency tuning.
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Experimental data from the cochlear apex are scarce,
mostly from interferometry (43, 85, 125). Little amplification can be observed even in a sensitive cochlea, and the
growth function is said to be “linear,” a misnomer only
referring to the absence of compression. Broad and symmetrical tuning curves are observed, in stark contrast with the
high-frequency pattern of a sharp tip with a steep cutoff on
the high-frequency side and a U-shaped low-frequency tail.
Nonetheless, WD is definitely present and physiologically
vulnerable, and large harmonic components have been recorded from the motion of the apical organ of Corti in
healthy guinea pig cochleas (125). A revealing pattern of
changes was observed after death, at different spots in the
organ of Corti where the probing laser beam was focused.
Whereas the motion of reticular lamina connecting the tops
of OHC bodies decreased, as intuition would have suggested, BM motion increased almost a hundredfold, with a
sharp increase in tuning. It suggests that at variance with the
rest of the cochlea, the apex is subject to a different feedback mechanism with a different purpose. Active negative
feedback (instead of the positive basal one) exerted by
OHCs would tend to decrease BM motion. Fundamental
principles of negative feedback are to reduce noise, to stabilize the amplification, and to make the growth of BM
motion “linear” even if it was compressive without feedback (124, 125). Feedback manipulations did not affect the
size of harmonics, which the authors saw as an indication
that their hypothesized feedback loop coupling fluid velocity acting on the BM to the Hensen support cells on which
they focused their interferometer, did not include the WDgenerating OHC stereocilia bundles (125).
Another model exploring the peculiarities of apical responses has been proposed (216), hypothesizing a ratchet
mechanism that strives to explain how frequency selectivity
may be achieved in low-frequency auditory-nerve fibers,
despite the fact that the resonance frequency of the apical
BM is likely higher than the frequencies processed apically.
Indeed, predictions from realistic values of mass and stiffness indicate a discrepancy between apical BM resonance
and the actual limit of low-frequency hearing. The ratchet
model also examines why the steep high-frequency cutoff
that characterizes wave propagation along the basal BM is
not visible apically. The ratchet mechanism (216) is such
that every second half-cycle, OHC electromotility uncouples hair-bundle forces from the BM. The resulting unidirectional coupling is unfavorable to amplification of BM
motion, hence its observed lack of significant compressive
nonlinearity. The main perceptive interest would be that
contrary to BM motion, hair-bundle motion would be
tuned, with open and symmetrical tuning curves, as observed in auditory neurons, extending the hearing sensitivity to frequencies well below the low-frequency limit of BM
resonance. This ratchet model may well be anatomically
grounded. The orientation of OHCs in the apical cochlear
turn strikingly differs from basal ones in that, instead of
standing perpendicular to the BM, their axis is tilted by
⬃67° relative to the reticular lamina joining the apexes of
hair cells (125), which may favor periodical uncoupling of
hair bundles from the tectorial membrane.
AVAN, BÜKI, AND PETIT
A. Generic or Specific?
The last decade has witnessed thriving literature using the
Hopf-bifurcation formalism to describe the motion of auditory hair cells or of their stereocilia bundles as critical
oscillators poised on the verge of self-oscillation (34, 60,
105). The advantage of this model lies in its ability to describe many features of auditory stimulus processing by hair
cells, compressive amplification and nonlinearity with limited risk of instability, just by introducing a control parameter in the behavior of a mass-stiffness resonator with
damping. This control term allows viscous damping to be
offset at very low oscillation amplitudes. Damping contains
an additional nonlinear term varying as the third power of
displacement. Its sharp increase at higher amplitudes reduces the gain of the oscillator and avoids instability. The
dynamics of a critical oscillator described by the complex
variable Z, in the presence of a force f(t), is described by the
differential equation
ⱍⱍ
dZ ⁄ dt ⫽ ⫺ 共r ⫹ i␻0兲Z ⫹ B Z
2
Z ⫹ ei␪ ⁄ ⌳ f共t兲
(5)
where ␻0 is the resonance radian frequency, the real parameter r controls the proximity to the bifurcation (which happens when r ⫽ 0). The multiplicative factor acting on the
external force expresses the existence of friction (⌳) and of
a phase shift ␪ at which f acts on Z. The amplitude-dependent |Z|2. Z cubic damping term predicts compressive amplification at a rate of 0.3 dB/dB, consistent with experimental data. By slowing down the decrease of DPs with decreasing stimulus level, this rate also underpins the prediction of
essential nonlinearity with the characteristics that Gold-
1600
stein described long ago. Two-tone suppression, DPOAEs,
and many of their characteristics can also be correctly predicted, at least within the approximation of small displacements as a Taylor expansion. Refinements such as the incorporation of a traveling wave for exciting the resonators
have been successfully implemented (56, 110).
Other systems of time-dependent differential equations can
be built from attempts to model a realistic auditory organ
with its BM and OHCs. For example, they can include MET
transfer functions as sole source of nonlinearity, longitudinal coupling between OHCs for introducing some degree of
delayed feed-forward, and a combination of hair-bundle
dynamics and OHC-body electromotility reconciling the
two alternative models of an amplifying mechanism (265).
A Hopf bifurcation may even emerge from such systems,
without the need for individual oscillators to be critically
poised near a bifurcation (265). Modeling the active nonlinear cochlea is still a very open field in which many details
remain to be sorted out. To choose the most suitable set of
models, the ability to simulate realistic nonlinearities may
be a good criterion. It is thus important to establish which
characteristics are generic and which are specific to a particular model. For example, does the success of Hopf-bifurcation views of auditory stimulus processing in hair cells,
for predicting realistic cochlear nonlinearities and not only
the compressive growth, definitely vindicate the underlying
model?
Actually, theories published in the 1930 –50s in electrical
engineering (55, 64) have shown that several properties of a
(time independent) nonlinear system mapped by an input/
output sigmoid transfer function automatically derive from
a minimum set of hypotheses. Any transfer function can be
modeled as a combination of odd and even-order terms [an
odd function is such that f(x) ⫽ ⫺f(⫺x), while f(x) ⫽ f(⫺x)
for an even one] that underlie odd and even DPs, respectively. Experimentally derived voltage/displacement haircell transfer functions share with any arbitrary nonlinear
function displaying large second and third derivatives,
many nonspecific properties of their combination tones. For
example, the condition allowing two-tone interference to
generate maximum DPs is always that the amplitudes of
both tones be equal at the input of the nonlinear system,
that is, at the interference place (137). The growth of a DP
with the amplitude of one primary tone when the amplitude
of the second primary is kept constant is also stereotyped.
With A1 (amplitude at f1) held constant, the amplitude of
the 2f1-f2 DP increases linearly when A2 (at f2) increases
from a small value. With A2 constant, the amplitude of the
2f1-f2 DP increases in proportion to A12, reaches a maximum, then decreases. As for suppression, it always requires
that the suppressor drives the nonlinear function into its
saturation region, and that if the suppressed signal is, itself,
into saturation, the suppressor must be no less that 10 dB
below it. Not surprisingly, Equation 5 predicts these fea-
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Auditory WDs are more difficult to accurately predict, as
they present distinctive features such as a rate of growth
that depends on the passive or active modality at the generation site, and the preeminence of odd-order distortions.
These idiosyncracies were ignored at the time of Helmholtz
when he tried to propose the first mathematical model of
auditory nonlinearity (95), which led him to dismiss some
of Tartini’s observations that contradicted the outcome of
his model (see sect. XI: Historical Insert). In the field of
cochlear mechanics, modeling efforts endeavor to address a
few questions, e.g., what is the most parsimonious model of
active resonators in which nonlinear compressive amplification, frequency tuning and distortions are coupled and
present realistic features? Which parameters of this model
are critical for accurate predictions? If the cochlea gives
sluggish responses to brief excitations, is it due to mere filter
delays that come with any battery of highly tuned oscillators, or does it betray the presence of stimulus envelopeextracting devices from which an AGC can be built, with
the advantage that its compression is almost WD-free? Last,
if a model predicts a correct outcome, does it vindicate the
architecture of the model or can its correct outcome stem
from generic considerations that any other model of a similar class would also satisfy?
AUDITORY DISTORTIONS
tures equally well. Another intriguing, yet generic property
is the occurrence of sharp nulls in the output of the system
(TABLE 2). Although sometimes interpreted as proof of interference between sources that would happen to be equal
in amplitude and out of phase, they can actually mark transitions between saturations from one to both sides of a
nonlinear transfer function (64, 154).
B. Instantaneous or Sluggish?
Another intriguing question has been little explored, namely,
whether cochlear nonlinearities are static or dynamic (59,
155, 315). Above, this review has largely rested on descriptions of the nonlinearity associated with hair bundle motion
as instantaneous, with immediate distortion of the current
through the bundle. A dynamic nonlinearity would introduce some degree of sluggishness, due to its dependence on
the past history of the system. The most typical example is
provided by an AGC, a design that engineers use to obtain
compressive amplification, similar to that in the cochlea,
while minimizing waveform distortion. It has already been
mentioned that the moderate levels of distortion in the ear
are striking, in view of the strong nonlinearity of the transfer function of MET channels. In an AGC, the gain is adjusted as a function of the average level of the output from
the system. This is extracted with the help of some low-pass
filter, providing an integrated view of the stimulus over a
finite interval of recent history. Different integration times
define a whole continuum of AGC systems, and even a
Hopf-bifurcation system implicitly assumes sluggish gain
control (271). However, neither sluggishness nor the moderate distortion level suffices to imply the existence of an
AGC. Regarding the time issue, any tuned system stores
energy and responds to a stimulus with a delay that increases with the degree of tuning. As for the pattern of
distortions, their frequencies, particularly those of harmonics and even-order intermodulation, are far enough from
the CF of their generation site to be filtered out more than
the odd-order intermodulation components, again in pro-
Several experimental observations strongly support the
presence of a noninstantaneous control of cochlear gain.
For example, in the auditory nerve, the period histogram of
neuronal discharges in response to a pure tone, up to a few
kilohertz, remains a half-wave sinusoid over a broad range
of levels (the sinusoid is rectified because there cannot be
any action potential when hair bundles move in the inhibitory direction). Yet the growth of the synchronized (i.e.,
phase-locked) rate of action potentials with stimulus level is
highly compressive (74). With a system displaying an instantaneous nonlinearity, even though its tuning made it
sluggish, it would be paradoxical that such a compression
result in a sinusoidal, thus undistorted period histogram,
but not with an AGC system.
Initially, the proposed AGC was a neural mechanism, either
a reservoir of neurotransmitter or a possible slow modulation of cochlear gain by an efferent control of OHCs via the
medial olivocochlear neural pathway (155). Later, indirect
evidence of a cochlear AGC has been produced (315). Electric responses to tone bursts from supporting cells, thought
to be quite similar to OHC responses yet easier to record
directly, displayed several signatures of an AGC in addition
to small WD despite high compression, i.e., asymmetry of
the on- and off-transients; a frequency increase with time
during the off transient, within a modulated envelope. In an
AGC, the gain is turned off during signal onset so that the
on-response builds up faster, while the increase in gain after
signal termination generates a complex shape in the offtransient.
Direct evidence of the size of the AGC-like delay has
emerged from several paradigms. The first one (see sect.
VIB) consisted in measuring the timing of TEOAE changes
when suppressed by a click emitted a few milliseconds before the regular TEOAE-evoking stimulus (90, 115). A time
constant of two periods was inferred. Another protocol
(273) studied the timing of 2TS in auditory neurons tuned
to a broad interval of frequencies, by evaluating the group
delays of their responses. The delay of suppression was
larger than the delay of excitation by several hundred microseconds, which suggested an AGC with, again, an integration time of about two periods at CF location. Last,
responses of the BM of guinea pig cochleas to white noise
were analyzed by cross-correlating the input noise to the
BM velocity response (214), which yields a so-called “firstorder Wiener kernel” (see sect. VIIIC, APPENDIX V). In a
linear system, this is a standard procedure providing its
impulse response. Comparison of the BM responses to noise
and clicks, therefore, evaluated by how much the cochlea
departed from linearity. It was found to depart very little so
that cochlear filtering could be said to be “quasi-linear.”
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However, modeling issues are far from straightforward because in addition to local properties of nonlinear oscillators,
several other aspects of auditory processing in the cochlea
shape the outcome of nonlinearities. Wave mixing from
distributed cochlear sources of DPs is one compounding
factor. A second one is that the input to the nonlinear transfer function producing cochlear nonlinearities is not the
sound in the external ear canal, but what reaches the DPgenerating system after propagation, amplification, and
compression have occurred. Likewise, a measured DP is
what eventually reaches the measurement spot, and after
leaving its site of generation, the DP has undergone several
modulating interactions, such as filtering at the generation
place (tuned to another frequency), absorption at its own
CF location, coherent reflection, reflection at the impedance
discontinuity at the cochlea/middle ear boundary.
portion to the narrowness of tuning. To probe the existence
of an AGC experimentally, more distinctive features must
be sought.
AVAN, BÜKI, AND PETIT
This is as surprising as the fact that cochlear responses to
tones exhibit so little harmonic distortion (43, 224, 230),
despite the high measured compression rate. If cochlear
processing worked as a bandpass linear filter combined
with an instantaneous nonlinearity, the wave shapes of responses to noise could not be accurately predicted by a
linear method. Thus it is not only convenient but also necessary to invoke an AGC in the cochlea.
C. From Particular Stimuli to a General
Characterization
Most published investigations of cochlear function have
used only a very restricted set of stimuli, i.e., either pure
tones, pairs of pure tones for the specific topic of DPOAEs,
or brief impulses. What is straightforward for a linear system, that is, to predict the response of the system to a spectrally complex stimulus from the responses to each spectral
line of this stimulus, becomes only approximately true in
the case of a nonlinear cochlea. One has to consider that the
prominent nonlinearities of the cochlea are small enough
for being mere perturbations in an almost linear cochlea.
When using a pair of tones, for example, 2TS is conveniently evaluated by measuring how the amplitude of the
response to the probe tone changes in the presence of a
second, suppressing tone. Likewise, it is possible to study
the interaction of two simultaneous tones at similar levels
by detecting the combination tones that their interference
generates, and perturbation theory is valid as combination
tone levels are generally found at more than 30 dB below
primary levels, i.e., they are more than 30 times smaller.
The Wiener-Volterra kernel formalism provides a powerfully general method in strongly nonlinear systems. Briefly
presented in APPENDIX IV, it describes the responses of a system as a sum of functionals unfolding the nonlinearities at
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IX. AUDITORY MASKING AND THE
COCHLEA
Auditory masking is one typical nonlinear phenomenon. It
is revealed by the psychophysical task of detecting a probe
tone, first alone and at a level slightly above detection
threshold, enough to be easily detected, and then, at the
same level but mixed with a second sound with varying
frequency and level, called masker. The masker characteristics are then adjusted in order that in the masking configuration, the probe tone is no longer heard, or the auditory
potential evoked by the probe is no longer detected. Differences between masking and nonmasking configurations
probe the nonlinearity of the system. Mixtures of sounds at
different frequencies and levels are frequently encountered
in everyday life, either when the vibrations from different
sources are mixed in the transmitting medium before reaching the ears, or when a single source produces a combination of components at different frequencies (e.g., a musical
instrument or a voice). Nonlinear interactions among frequency components of the mixture allow some components
to mask others, according to well-acknowledged empirical
rules (58, 286). The detection of speech alone or in the
presence of background noise can become difficult if crucial
components get masked. The study of masking should help
quantify the perceptive consequences of cochlear nonlinear-
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The existence of an AGC drastically reduces harmonic
distortion, but interestingly, without necessarily precluding intermodulation. Assume an AGC with a response
time extending over several periods, and an amplitudemodulated signal. Its envelope, extracted by the AGC
filter, adjusts the gain. A sine wave is transformed, after a
few periods, into a sine wave with smaller amplitude. The
amplitude-modulated signal, conversely, gets distorted as
its envelope is flattened by the decrease in gain triggered by
the growth of this envelope. This generates DPs, and it can
even be shown that these DPs, even when generated at a single
location, present a rather realistic microstructure suggestive of
interference (this time among terms with different mathematical origins; Ref. 271). Overall, a variety of experimental clues
point to the existence of a functional AGC in the cochlea, even
though an open issue is the actual identity of the low-pass filter
mechanism extracting the average level of the output from the
system, to provide an integrated view of the stimulus over
about two periods.
increasing orders, extracted from cross correlations between the output of the system, and its input stimulus at
varying time delays between input and output. Gaussian
white noise is the privileged input, as its mathematical interest is to provide Wiener kernels that are independent of
each other and can be numerically computed. Its ecological
interest is that, better than tones or clicks, it allows both
simultaneous and prolonged stimulation of the whole cochlea. First-order Wiener kernels from a linear system
would give its impulse response, but in a nonlinear one, it
contains the impulse response plus corrective terms. Second-order Wiener kernels give a measure of how much the
presence of a first impulse stimulus influences the response
to a second one (162). They have sometimes been successfully applied to characterizing the temporal coding of auditory nerve fibers even when the carrier frequency of the
stimulus is too high for phase-locking of the action potentials to occur. In this case, it is from the envelope modulations also produced by the stimulus that, indirectly but efficiently, temporal coding in these neurons can be inferred.
The extended use of Wiener kernels for studying cochlear
mechanics, seldom attempted (156, 275), usually requires
their outcome to be compared with that of a model, in
search of an optimal match ensured by adjusting the model
parameters. Thus the advantages of these general methods,
unconstrained by the limitations of low-level approximations to explore the nonlinearities of a system, are somewhat offset by the need to start with a few hypotheses
regarding the block diagram of the system under study.
AUDITORY DISTORTIONS
ity. However, whether measured by perceptive tests in subjects or by neuronal recordings, masking actually results
from a blend of three mechanisms, only one of which, the
suppressive masking, is of cochlear origin while the other
two mechanisms are neuronal ones. This section examines
which are the distinctive features of the suppressive masking, from which information on cochlear nonlinearity can
be derived.
A. Basic Characteristics of Auditory
Masking
The line-busy explanation is a sketchy one, positing that, as
neurons have a limited rate of action potentials due to their
refractory period of a few milliseconds, if the masker drives
them at the maximum rate, the occurrence of the probe
tone, which when presented alone increases the discharge
rate, can no longer influence it as it is already saturated.
This explanation does not hold when the masker is weak
enough not to saturate the neuron, and yet masking still
occurs. The term swamping is more conservative, suggesting that masking results from the fact that the probe tone
increases the discharge rate, but not enough for being detected because the probe-induced activity is swamped by the
masker. Swamping happens for two reasons. The first one is
that the rate-level functions have a compressive slope so
that the increment in discharge rate is smaller when the tone
is mixed up with the masker than when alone. The second
reason is the increase in variance of the discharge rate of a
nerve fiber when its mean rate increases, which makes it
more difficult to detect the slight increase in rate due to the
probe tone (48). The swamping masking mechanism requires that the masker itself excite the neurons responding
to the probe tone (excitatory masking). It is not cochlear,
and not even specific of hearing. Yet, whenever masking
experiments investigate the nonlinear cochlear mechanisms, their results will most often be contaminated by this
nonlinear behavior of neurons.
B. Line-Busy Masking
C. Adaptation
The earliest systematic investigations of the mechanisms of
masking relied upon recordings of auditory nerve fibers (48,
107, 256), and contributed to the identification of the three
masking mechanisms. The earliest to be invoked (67) is
purely neuronal, the line-busy mechanism. Let us consider a
probe tone presented alone, above the threshold of a recorded auditory nerve fiber. This induces an increase in
discharge rate of this fiber. If, when the same probe tone is
added to a continuous masker, no change happens to the
discharge pattern produced by the masker alone, a simple
interpretation is that the neuron is busy responding to the
masker so that the probe tone can no longer modify its
activity; thus it can no longer make itself detectable. [A
caveat is that even though the neuronal activity in response
to the probe tone is no longer detectable in the group of
auditory fibers tuned to the probe, this does not mean that
the probe can no longer be detected at all. It may still acti-
The second mechanism by which masking happens in auditory neurons is called adaptation, again an excitatory, specifically neuronal mechanism happening when a masker has
been presented long enough for adapting the neurons responding to it, thus decreasing the response of these neurons when the probe tone is presented (87, 257). Adaptation still goes on for a few milliseconds after the masker has
been turned off. This phenomenon is called nonsimultaneous masking, here, forward masking.
D. Suppressive Masking
The third mechanism, suppressive masking, is one inherent
property of nonlinear cochlear mechanics, whereby the
masker suppresses the mechanical response to the probe
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For a tone, serving as a probe, presented with a narrowband masker (it is not recommended to use a pure tone as
masker as the amplitude modulation of the mixture of
probe and masker might provide unwanted perceptive cues)
the threshold of masking increases roughly in proportion to
masker level for probe frequencies around the masker frequency region. In other words, starting from a probe tone
just detected in the presence of a masker, when the level of
this masker increases by 10 dB, the probe tone has to be set
10 dB higher to be heard again. In contrast, thresholds
increase more rapidly for probe frequencies well above the
masker frequency region (58, 286). This so-called “upward
spread of masking” expresses the detrimental effect of highlevel, lower-frequency maskers on the detection of higher
frequency probe tones. As the probe must increase by over
1 dB/dB increase of masker level, to remain audible, psychophysical references often call this masking “nonlinear,”
as opposed to a “linear,” 1 dB/dB race for audibility between probe and masker levels. With reference to the meaning of linear and nonlinear throughout this review, these
terms are misnomers, and as such, when necessary, will be
mentioned with quotes, as even the “linear” masking is an
essentially nonlinear effect.
vate other neurons enough to modify their activity. This
detection is called off-frequency listening as it is mediated by neurons not tuned to the probe. The perceptive
properties of the detected signal may of course differ
from what they should be if detected through the neurons
tuned to it, in terms of pitch for example, but since the
psychoacoustic tasks are usually based on mere binary
detection, the fact that tested subjects exploit off-frequency listening to achieve a task may be difficult to
guess even by the subjects themselves.]
AVAN, BÜKI, AND PETIT
Suppression is strong enough that, for a probe tone at the
CF of a neuron, the level required to produce a given average discharge rate may shift upward by 40 –50 dB in the
presence of a suppressor (48, 73, 75). Despite this large
effect and more generally, the impact of nonlinearity on the
cochlear processing of sound, traditional psychophysical
models rest on an excitatory picture of masking explaining
the influence of a masker by the way its pattern of excitation
spreads along the cochlea to the place of the probe tone (67,
286), and swamps neuronal activity in response to the
probe. Accordingly, many models used for interpreting
masking, loudness, and frequency and intensity discrimination do not take nonlinear interactions into account. Nowadays, the accuracy of these linear masking models is increasingly called into question (48). Recent reports of
strong changes in masking, in the particular case of Strc⫺/⫺
mutant mice with no suppressive interaction despite normal
cochlear sensitivity and frequency tuning, raise the same
issue (282) (see sect. IIID). In an extensive study at the level
of the auditory nerve, Delgutte (48) therefore sought to
differentiate between excitatory and suppressive mechanisms over ranges of masker levels and of frequency spacings between probe and masker. Comparisons were made
of masking thresholds obtained in simultaneous and forward masking. Suppression may contribute to the former,
but in no way to the latter one. Thus the differences in
masker levels required to mask a fixed signal at the CF of the
recorded fiber, could be attributed to suppression. For
masker frequencies well below the probe frequency, the
amount of suppressive masking was found to be large, increasing with masker level more rapidly than did excitatory
masking. The results of Delgutte, obtained with probes ⬍40
dB SPL, thus indicate that the upward spread of masking is
largely due to the growth of suppression rather than to that
of excitation. This view, supported by other physiological
studies, e.g., Reference 199, suggests that psychoacoustic
models should be revisited to incorporate suppression. Accordingly, physiologically motivated models that describe
cochlear compression with the help of nonlinear, velocitydependent feedback characteristics similar to those dis-
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cussed in section VIIIA, now achieve realistic predictions of
OAE properties, of 2TS and of the outcome of some psychophysical tasks (63).
The psychoacoustic finding by Oxenham and Plack (198) of
a strong “nonlinear” (i.e., ⬎1 dB/dB) upward spread of
masking in nonsimultaneous masking apparently contradict the physiological results of Delgutte (48), who found
no suppression in the nonsimultaneous masking condition,
no strong upward spread of masking, and a merely “linear”
(1 dB/dB) growth of masking. A likely cause of this discrepancy is that Oxenham’s probes fell between 40 and 90 dB
SPL. It is in this range of levels that the compressive behavior of the BM is most intense, whereas at ⬍40 dB SPL,
compression decreases and probably vanishes below 20 dB
SPL (228). In the region tuned to the probe tone, BM vibrations in response to maskers between 40 and 90 dB SPL and
near the probe frequency (“on-frequency”) grow slowly as
they undergo strong compression, whereas the responses to
off-frequency maskers, masker frequency well below that of
the signal, grow at a rate of 1 dB/dB increase of the masker
level. This differential growth of BM responses to the probe
versus the masker suffices to explain a large growth of
masking for probes with a frequency above the masker
frequency even in the nonsimultaneous, i.e., nonsuppressive
masking conditions observed by Oxenham and Plack. In
addition, and in agreement with physiological studies, the
same authors did also find a major role for suppression in
determining thresholds at high masker levels, for an offfrequency masker (198). Playing with simultaneous versus
nonsimultaneous-masking conditions and with on- versus
off-frequency maskers turns out to be suitable for separating the various causes of masking and study their distinctive
properties.
When more than two tones are presented simultaneously to
the cochlea, a more complex situation than 2TS pertaining
to the intelligibility of speech sounds, unmasking phenomena may occur and reinforce auditory-contrast sharpening.
A two-tone stimulus can be less effective to mask a probe
tone than one of its components (98, 242). In a nonsimultaneous masking experiment, the masker, made up of a
mixture of two sounds, is presented first, then a gap, then
the probe tone. This probe tone can be better heard if mutual suppression happens between the two simultaneously
present components of the forward masker, as this suppression decreases the efficiency of the overall masker. Conversely, simultaneous presentation of all sounds would lead
to no clear result because the signal, being present simultaneously, would be suppressed by the masker, and not only
the weak component of the masker by the stronger one.
This explains an apparent paradox. It is widely accepted
in the psychoacoustical literature that effects of suppression are not revealed in simultaneous masking (98),
which might sound weird since suppression requires simultaneous presence of suppressor and probe. The par-
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tone along the basilar membrane, by jamming the active
mechanism by which the probe tone is processed. The simplest protocol revealing this masking mechanism is 2TS,
described in section VA. In contrast to line-busy and adaptation mechanisms, suppressive masking is not caused by an
excitatory neuronal mechanism. Even before direct measurements on the BM and hair cells revealed the mechanical
nature of the suppressive nonlinearity (201, 233), experiments using microelectrode recordings of neuronal activity
showed that a masker can suppress the response of a neuron
to a probe tone despite not exciting this neuron when presented alone (236). More complex situations than the 2TS
paradigm have been studied, i.e., noise (126) and speech
sounds (237), for which evidence of suppression in the discharge rate of auditory neurons has also been obtained.
AUDITORY DISTORTIONS
adox disappears when one distinguishes the suppression
exerted by the masker on the probe (“masker-to-probe
suppression”) from the suppression among frequency
components of a complex masker (“within-masker suppression”) (48).
X. CONCLUSIONS AND PERSPECTIVES:
NONLINEARITIES AT THE CORE OF
AUDITORY MECHANISMS
While the basic functional principles of auditory nonlinearity are increasingly well understood from mammals to
lower vertebrates, the molecular assembly surrounding its
core, the MET channel, remains incompletely established.
The solution to this protracted molecular hunt would not
radically change any physical principle of the gating process
leading to the generation of WD, anchored as they are in
thermodynamic laws. However, it would make it easier to
dissect the interaction of MET-machinery subparts, e.g., the
pore, gating spring, tip-link, and the mechanisms that reset
the resting position of the MET channel. Our knowledge of
how interstereociliary links work also remains imperfect.
An improved understanding of hair-bundle architecture
might help to explain why DPs become undetectable in the
absence of top connectors even though MET channels still
operate, and more generally, how the shape of the MET
transfer function and the characteristics of cochlear nonlinearities can depend on a coordinated motion of the stereocilia bundle.
The most readily detected WDs are the DPOAEs, yet their
propagation from stereocilia bundles to the stapes is a complex topic so that, in practice, it may be useful to choose
eliciting stimuli such that one generation site or one propagation mode dominates the DPOAE pattern. The controversy regarding backward propagation of sound by fast
compression waves versus slow traveling waves has led to
considerable refinements of measuring devices and theoret-
The perceptive benefits of WD remain an open issue even
though the coherence of all cochlear nonlinear properties
grants it, at least, the status of a sensitive probe. Several
mechanisms tend to maintain WD at low levels, i.e., compression by controlling their growth; filtering (14) as WDs
are produced at places not tuned to them; and an AGC
mode of functioning. One likely advantage of WD is that
the resulting suppressive interactions may improve the analysis of complex sounds by enhancing local contrasts and
favoring resonant components (262). Suppressive interactions within the spectral components of complex maskers
might decrease their masking effect on a probe tone (98,
242). Last, DPs may usefully contribute to the processing of
complex sounds as the activity of auditory centers contains
a robust representation of the f2-f1 and 2f1-f2 components
(2). Pitch extraction is likely assisted by neuronal responses
to these DPs, as suggested by masking experiments that
decrease pitch salience and neuronal representations of DPs
in parallel (209, 255). [The missing-fundamental perceptive
phenomenon is different (see sect. XI, Historical Insert), as
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While the physiological and psychophysical relevance of
compressive amplification in the nonlinear cochlea is well
acknowledged, WD, the other manifestation of auditory
nonlinearity, for the time being, seems to draw its major
interest as a tool for probing cochlear mechanisms. A coherent picture has gradually emerged from three decades of
investigations of the active mechanics in auditory organs
and of OAEs, relating amplification and frequency tuning
to all manifestations of nonlinearity. An amplitude-dependent positive damping term ensures mechanical stability
and produces compression. Nonlinearity itself is a mandatory consequence of MET channel functioning, and WD
and its consequences afford a privileged insight not only on
the processes that ensure an appropriate gating of MET
channels, but also on the integrated mechanism of feedback
which, within auditory sensory cells, actively processes auditory stimuli.
ical tools. The part played by slow traveling waves seems
dominant; however, intermediate structures and paths ensuring DP propagation between hair bundles and the stapes
remain a matter of active research. DPs present a strikingly
complex microstructure that challenges simplistic interpretations, e.g., of a change in DP amplitude. Some DP features, deep notches for example, are reminiscent of interference patterns among different sources, yet some possibly
are mere products of the complexity of the nonlinear transfer function of a single source, with its saturations, asymmetries, and mobile OP. No less convincingly, other DP
features have been shown to stem from genuine interference
among sources or paths. Is it possible that still other paths
and mechanisms exist that have been missed? Propagation
is only one aspect of the broader issue of DPs as noninvasive
windows on cochlear frequency tuning. Their use heavily
draws on the possibility to extract reliable timing data
which, with steady tones as stimuli, remains a nagging issue.
In particular, despite evidence that the cochlea operates as a
(minimally distorting) AGC exploiting a noninstantaneous
nonlinearity, surprisingly little hard data can be found regarding the timing and hard-wiring of what controls the
noninstantaneous extraction of stimulus level to adjust the
gain. Several reports point to an interaction time of a few
periods (90, 273), while thought-provoking, extreme models incorporate the whole history of cochlear activity over
hours (142), by assuming that the scala media may act as a
pressure accumulator, with pressure and the resulting shift
in resting position (140) as a control parameter of cochlear
operation. In contrast, Hopf bifurcation-based models predict noninstantaneous distortion, but its mechanisms are
not explicit, and at present, a tangible way to control critical oscillators, e.g., by Ca2⫹ ions entering the MET channels, is suggested only in models of nonmammalian ears.
AVAN, BÜKI, AND PETIT
it relies on central processes that persist when DPs have
been cancelled.]
XI. HISTORICAL INSERT: TARTINI’S TONES
The composer Giuseppe Tartini (1692–1770) reported the
perception of an additional tone (“terzo suono”), when two
tones at frequencies f1 and f2 played in a loud and sustained
manner by two violins or oboes were listened to by an
experimenter, himself, placed midway, near enough for the
sounds to be loud and with similar levels, thus 5– 6 steps
away. He published a detailed account of his experiments in
Trattato di Musica secondo la vera scienza dell’armonia
printed in 1754 in Padua, Italy. This captivating book
(available online as a free e-book) classically leads his name
to be used in the alternative designation as “Tartini tones”
of intermodulation WD components at arithmetic combinations (2f1-f2, 3f1–2f2, f2-f1, etc). A few years before, the
organist Sorge had published similar findings in his book
Vorgemach der musikalischen Komposition (Lobenstein,
1745–1747).
However, doubts arise whether Tartini and, especially, Tartini’s readers were actually interested in combination tones.
From Tartini’s own words, “one has discovered a new harmonic phenomenon . . . if two sounds of just intonation be
sounded clearly and loudly together, there will result a third
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Pitch, the key element invoked in all these reports, is the
quality that distinguishes two different notes played on the
same instrument with equal force, and that allows notes to
be organized on a scale from low to high. While the pitch of
a pure tone and its frequency are isomorphic, the pitch of a
mixture of tones relates to physical cues in a less straightforward manner. The so-called “periodicity pitch” (146)
derives from the detection in central auditory nuclei of periodic patterns of discharges in afferent neurons at the period 1/f of the fundamental component, when several harmonics of the same fundamental f (mf, nf, etc., with m and
n integers) coexist. The pitch of a sound complex (mf, nf)
matches that of f presented alone, as the main periodicity of
the discharge patterns is the same regardless of the presence
of energy at f. When it is missing, the so-called “missing
fundamental” (306) shows up as a phantom perception
from the neural auditory system. This phenomenon is universally experienced by telephone users. Most telephone
lines cut low frequencies ⬍300 Hz so that all male voices
are deprived of their fundamental, but the perception of
their pitch is not in the least affected. The identification of
the fundamental pitch can even persist, although in a
weaker form, if each harmonic of a pair is presented in a
dichotic manner, showing that the reconstruction of pitch
from the two ears is made in the central auditory system.
Both the missing fundamental and the Tartini tones are
spectacular perceptive effects involving additive mechanisms in relation to the occurrence of a series of regularly
spaced spectral lines. Yet they are fundamentally different
in that a Tartini’s “terzo suono” is physically produced in
the cochlea irrespective of any particular relationship between f1 and f2, while a missing fundamental can be perceived, centrally, only if the primaries are consonant.
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Regardless of its influence on perception, WD still deserves
the efforts required to disentangle its intricate origin and
propagation mechanisms, as its detection moved from the
laboratory to everyday use in audiological setups decades
ago. Several examples deserve to be mentioned owing to
their everyday usefulness. With sound-evoked OAEs, universal screening of sensorineural hearing impairment has
been implemented in neonates, which most often includes
OHC dysfunction (22, 122). Being intracochlear sources of
sound, OAEs act as a virtual calibrated earphone on the
internal side of the stapes, probing the influence of acoustic
impedance mismatches along sound-propagation pathways. Impedance changes due to increased pressure in labyrinthine fluids creating endolymphatic hydrops, or even to
causes outside the field of otology such as increased cerebrospinal fluid pressure, can be monitored and applied to
clinical and physiological investigations (32). Longitudinal
studies in large human cohorts have suggested that soundevoked OAE decline with noise exposure might detect latent damage ahead of audiometric changes, thus serve as a
forewarning of hearing loss (141). In mutant laboratory
mice, sound-evoked OAEs rapidly sieve the auditory function and help unveil the associated impaired subcellular
structures. Last, central modulations of cochlear function
via the olivocochlear medial efferent nerve bundle, possibly
affording protection against noise overstimulation and attention-driven improvement of signal-to-noise ratio, can
also be assayed by OAEs (83).
sound lower in pitch than the other two, and which will be
the fundamental sound of the harmonic series of which the
first two sounds form an integral part . . .” The suspicion
increases at the reason invoked by Tartini in support of his
claim to have discovered combination tones well before
anybody else, in another of his books published in 1767,
De’ principii dell’armonia musicale contenuta nel diatonico
genere. There he tells that many people could testify that
when aged 22, at Ancona, he had discovered the third
sound, later (1728) used in Padua to teach pupils how to
obtain pure intonation on the violin. When playing double
stops (by pressing two strings simultaneously), the pupils’
ability to hear the “terzo suono” served as a guide to correct
intonation, since its occurrence indicated that the intonation was pure. As a music teacher, Leopold Mozart had in
mind a similar scope when he wrote that Tartini tones are a
device that “a violinist can use in playing double-stopping,
and which will help him to play with good tone, strongly,
and in tune.” Tartini also reported that any two consecutive
sounds of the series of harmonics produced by a vibrating
body invariably produce the same “terzo suono,” and often
used the term “basso fondamentale” throughout his works.
AUDITORY DISTORTIONS
A century later, Hermann von Helmholtz was the first to
propose a mathematical theory of Tartini’s tones, based on
a Taylor expansion of the mechanical response of a nonlinear system to two primary tones, limited to the quadratic
term assumed to generate the largest contribution (it does,
unless the nonlinear system displays a peculiar kind of symmetry). Regardless of any harmonic or inharmonic relationship between f1 and f2, components at f2-f1 and f2⫹f1
emerge (Die Lehre von den Tonempfindungen, APPENDIX
XVI). Helmholtz thus ascribed Tartini’s observations to the
f2-f1 quadratic combination tone. Intriguingly, however, he
noticed discrepancies between his work and Tartini’s, subtle yet most revealing. According to Helmholtz, Tartini had
been an unreliable observer . . . In his chapter IV, Helmholtz wrote that “Tartini notated all secondary pitches an
octave too high.” In hindsight, now that the properties of
cochlear distortion are understood, this rebuke provides a
forceful vindication of Tartini’s outstanding skills as an
experimenter, and of his indisputable discovery of Tartini’s
tones as a characteristic cochlear phenomenon: not only did
Tartini notate his pitches correctly, but this remark shows
that Helmholtz could not have detected Tartini’s tones in
his studies!
Among the ratios tested by Tartini were 4:3:(2), 5:4:(2),
6:5:(2), 8:5:(2) and 5:3:(2), the first two numbers of each
series referring to the primary sounds played by each violin,
deliberately chosen as integer multiples of the same fundamental frequency, and in parentheses, the terzo suono identified by Tartini (177). With the difference tone at f2-f1 in
mind, Helmholtz predicted that the difference tones reported by Tartini should have been 4:3:(1), 5:4:(1), 6:5:(1),
8:5:(3) and 5:3:(2), thus in many cases, secondary pitches
different from Tartini’s. Helmholtz thought, wrongly, as we
now know, that higher-order combination tones did not
exist as such, but were first-order combination tones of
particular harmonics of the primaries (208). For reasons
explained in section V, it is now acknowledged that physiological odd-order DPs are much louder than even-order
ones, the 2f1-f2 tone reaching ⫺15 dB below the primary
tones while the f2-f1 is not audible at stimulus levels below
50 dB SL (79). It is therefore likely that Tartini detected the
(n⫹1)f1-nf2 combination tones, with the right pitch and not
an octave too high, and to give another example, 8:5:(2 ⫽
2 ⫻ 5 - 8) and not at all the 8:5:(3 ⫽ 8 ⫺ 5) of Helmholtz.
Tartini’s combinations 5:4:(2) and 6:5:(2) correspond to
higher odd-order combinations than 2f1-f2 [that would
have been 5:4:(3) and 6:5:(4), respectively], the 3f1–2f2 and
4f1–3f2 DPOAEs that can be quite large and fully audible.
Moreover, Tartini also understood that the terzo suono was
best heard when the primary tones were of similar level so
that the musicians playing each primary could not hear it as
clearly as himself standing midway. This is now attributed
to the need for the two primaries to have envelopes of
similar size on the BM to maximize their nonlinear interaction and the resulting DPs.
Tartini thus deserves full credit for the discovery of physiological combination tones. His power of analysis led him
to disclose several of their unusual properties, now known
to be signatures of cochlear nonlinearity, i.e., odd-order
predominance and envelope-overlap requirement. But this
raises a new shocking question: could an outstanding scientist like Helmholtz have so deeply misunderstood the issue
of Tartini tones? The (obvious) answer is that he did not.
Helmholtz actually never claimed to have specifically studied physiological combination tones; on the contrary, he
explains thoroughly, in chapters VII and XI of his book,
that many of his sound sources distorted sound, especially
when the primary sounds producing combination tones
were played on the same instrument or by mechanically
coupled sound sources. They also produced higher harmonics. Helmholtz sometimes used purer tones from tuning
forks, but mostly ⬍ 200 Hz (177). Helmholtz also mentioned his misgivings regarding his, and others’ ability to
identify the pitches of combination tones, all the more when
buried in a complex mixture of spectral components, as
seen above, hence his lack of confidence in Tartini’s reports.
In many cases, his measurements were done with the help of
one of his resonators, a cumbersome, yet remarkable ancestor of digital spectral analyzers. This could be done only for
sounds generated, outside the ear, by the aforementioned
nonlinear sources.
Helmholtz’s only mistake was to have assumed that the
characteristics of these sources, and of their carefully studied exogenous combination tones, also applied to the case
of combination tones from the ear. This would have been
justified if the distorting element had been a passive structure, such as the eardrum or ossicles as Helmholtz proposed. Actually, being produced by the MET channels of
OHCs, the endogenous combination tones bear the signatures of a slow rate of increase with stimulus level and of
odd-order symmetry. All the quantitative characteristics reported by Helmholtz, at variance with these signatures,
point to exogenous combination tones, i.e., that they are
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The insistence of Tartini and especially his followers to
focus on pitch may suggest that if they fell on combination
tones, as they were actually interested in harmonic series
and for Tartini, in the use of double-stops in the pieces he
composed, their interest was largely caught by the missingfundamental phenomenon. An incidental paragraph on
page 17 of Tartini’s Trattato, however, shows that Tartini,
as the keen observer he was known to be, did hear genuine
combination tones: “perche il terzo suono si ha non solo
dagl’ intervalli composti da quantita razionale, ma si ha
ancora dagl’ intervalli composti da quantita irrazionale”
(“because the third sound stems not only from intervals
composed of rational quantities, but also from intervals
composed of irrational quantities,” in which case no missing fundamental can exist).
AVAN, BÜKI, AND PETIT
produced by even-order, quadratic distortion (his appendix
XVI); that their growth function is steeper than that of
primary tones (in the second page of his chapter VII; we
know that it should be 2 dB/dB in the absence of compression); that the dominant spectral lines in the case of quadratic distortion are at f2-f1 and f2⫹f1, the summation tone,
that Helmholtz most likely heard, not as an endogenous DP
as this one is inaudible (208), but as the product of a distorting music instrument. The reasons why the properties of
cochlear DPs are drastically different from instrumental
ones stem from the dynamic properties of the hair bundle,
discussed above, and that have only been recently unveiled.
The emission of SFOAEs with long delays by the auditory
organs of all tested tetrapods, i.e., mammals, birds, lizards,
and frogs, regardless of the existence of a BM-supported
traveling wave, raises the issue of the minimum system able
to produce responses with long delays. Delays can be seen
as the sum of traveling and filtering contributions. The current view is that the only requirement for producing realistic
SFOAES is the presence of a slightly disarrayed battery of
tuned resonators (16). This paper starts from a detailed
realistic model of a nonmammalian inner ear, to thoroughly
derive an analytic expression of the SFOAE delay. Other
authors have reached similar qualitative conclusions regarding the presence of long delays in another category of
OAEs evoked by short impulse stimuli [clicks; these transiently evoked emissions are thought to stem, at least partially, from a backscattering mechanism similar to that of
SFOAEs (see sect. IVE)].
The core of their argument rests on minimal assumptions.
Assume a bank of overlapping bandpass filters connected in
parallel, regularly spaced, with gradually increasing center
frequencies. Let the bank be excited by an impulse. Every
frequency component of the impulse preferentially goes
through the filter centered on its frequency. The outputs of
this filter and of all other filters recombine to generate an
output signal. As a whole, the battery passes all frequencies
equally; thus it responds to the impulse by a similar impulse.
However, the output of each individual filter is a slowly
damped oscillation at its center frequency. As the oscillations from neighboring filters gradually shift in frequency,
when combined they oscillate out of phase with one another
and cancel each other, except at the very beginning when
they oscillate in unison, hence the impulse as overall output.
If the filters get slightly disarrayed, then the cancellation
between neighboring components becomes imperfect so
that remnants of individual responses show up, with time
delays corresponding to several periods of the center frequencies of the disarrayed area. Zwicker (312), in his hardware model of the cochlea simulated as a battery of electrically tuned filters, observed this phenomenon and aptly re-
1608
In a workshop held in 1989 in honor of Thomas Gold and
his pioneer idea of an active cochlea, Gold (78) brought
forward the analogy of the sound of a concert piano when
clapping one’s hands nearby. Every string significantly vibrates for several seconds and ringing is heard, but it would
not if the accuracy of frequency mapping was improved by
putting 20 times the number of strings. Conversely, ringing
would get louder if strings were mistuned. Wit et al. (300)
quantitatively modeled the behavior of a disarrayed bank of
gammatone filters and showed that not only do its “emissions” resemble TEOAEs in the time domain, but they also
present, in the spectral domain, a realistic quasi-periodic
pattern of peaks and troughs with the periodicity emerging,
not from the structure of the disarray (there was none as it
was random) but from the tuning properties of individual
filters, just as shown for SFOAEs (16). This happened with
no intervention whatsoever of traveling wave or specific
cochlear micromechanics, just a battery of tall and broad
resonators with neighboring resonance frequencies and
some degree of disarray.
XIII. APPENDIX II: BACKWARD VERSUS
FORWARD PROPAGATION OF
DISTORTION PRODUCTS
The principles allowing DPs to travel backward from their
generation place to give rise to DPOAEs should remind the
reader of those discussed for SFOAEs, even though here
more mechanisms and more nonlinearities come into play.
Assume that potential DP sites of generation are distributed
along the places where the envelopes at f1 and f2 overlap
and OHCs respond to both frequencies. Consider the case
of the 2f1-f2 DP. Likely, its dominant contributions come
from where the displacement is largest at primary frequencies, that is, near the peak response of the BM to f2. This
place covers a broad interval. Let B and A be its limits on the
basal and apical sides (FIGURE 22). Since the generation
segment is long and not punctual, during stimulation DPwavelets are evoked along its entire length, every one of
them at a particular location X between A and B (FIGURE
22). From X they start to travel in two directions, apically to
their CF location D, and basally to the stapes S. At the
stapes or at their CF location, all of them are vectorially
added and, depending on their respective phases, cancellation or enhancement occurs. The DP level at the stapes
determines the DPOAE level, and the DP level at CF
location determines the loudness of the heard combination tone. As seen in section IVA, the phase rotation of a
tone increases during propagation, by one cycle every
time the wave travels by one wavelength, and as the
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XII. APPENDIX I: OTOACOUSTIC
EMISSIONS WITHOUT TRAVELING
WAVE
marked that although the disarrayed system seemed to emit
transient-evoked signals with delays that could suggest the
propagation of traveling waves, these delays, in this case,
merely corresponded to the time it took for the oscillations
of the disarrayed filters to build up.
AUDITORY DISTORTIONS
2Φ1– Φ2 + ΦDP
(≈ –Φ2)
2Φ1– Φ2 – ΦDP
(≈ –Φ2)
S
B
X
A
D
SXS
SXD
B
A
X
B
X
A
X
A
X
2Φ1– Φ2 – ΦDP
S
B
X
A
D
SXS
SXD
B
X
B
A
Coherent
2Φ1– Φ2 + ΦDP
(≈ 2Φ2)
2Φ1– Φ2 – ΦDP
(≈ 0)
S
SXS
B
X
A
D
SXD
B
X
B
A
Coherent
FIGURE 22. Sketch of the interactions between waveforms at f1 and f2 along the BM (envelopes, solid lines;
instantaneous vibrations, dotted lines), that lead to the generation of DPs and DPOAEs. Three different
frequency ratios f2/f1 are represented from top to bottom, with f2 fixed. S, stapes, from where DPOAEs are
collected; D, place tuned to the 2f1-f2 DP frequency; locus X, within the BA segment, hypothetical site of
nonlinear interactions between waves at f1 and f2, along which wavelets at 2f1-f2 are launched forward (toward
D) and backward (toward S). Boxplots on the right and left sides depict, for each f2/f1 ratio, the phase
accumulation of 2f1-f2 wavelets launched from X along the paths SXD (on the right) and SXS (on the left). Phase
coherence is ensured when phase accumulation no longer depends on X between B and A, and allows a large
DP to be measured where all wavelets recombine.
wavelength gets shorter when the incoming wave gets
closer to its CF place, the phase accumulation speeds up.
Scaling symmetry tells that the vibration patterns of f2,
f1, and 2f1-f2 are just translated versions of each other
along a logarithmic representation of the cochlea. At the
apex, where the scaling symmetry is broken, this representation is probably not valid and its predictions fail. It
is important for what follows that the phase at f2 rotates
fast when this primary nears its CF location, while at the
same spot, f1 and 2f1-f2 accumulate less and less phase
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2Φ1– Φ2 + ΦDP
A
AVAN, BÜKI, AND PETIT
with increasing f2/f1 ratio, as the peaks at f1 and 2f1-f2
move farther apically.
The DP amplitude dependence on the f2/f1 ratio can be
explained according to the following line of thought [FIGURE 22; from the sketches of the phase behavior of primary tones and DP, diagrams have been built displaying
the phase accumulation of wavelets as a function of their
generation place X, when they travel either forward
(right-hand side) or backward (left-hand side); 3 different f2/f1 frequency ratios are illustrated from top to bottom].
A. Forward Propagating Wavelets
At small f2/f1 ratios (close to 1.0), the phase accumulations
␸2, ␸1, and ␸DP tend to equalize, despite the necessary phase
rotation along the A-B interval. For more basal Xs, wavelets
experience this phase rotation during their journey forward; in the case of more apical Xs, their stimuli accumulated the same phase change instead. That wavelets experience similar phase accumulations regardless of where they
were generated between B and A (bottom right diagram on
FIGURE 22) is favorable to their adding up coherently to
produce a high-amplitude vibration on the BM at CF location (point D) so that the DP is clearly audible. At large f2/f1
ratios however, the forward phase accumulation of DP
wavelets, 2␸1 ⫺ ␸2 ⫺ ␸DP, is dominated by ⫺␸2. It can be
seen on FIGURE 22 (top sketch) that between B and A, as the
places tuned to f1 and 2f1-f2 are much more apical that that
tuned to f2 at large f2/f1 ratios, ␸1 and ␸DP remain small as
most of the phase accumulations happen apically to A. Conversely, ␸2 rotates fast from B to A, and the term ⫺␸2
dominates the phase difference between DP wavelets
launched from different Xs. The diagram depicting how
2␸1 ⫺ ␸2 ⫺ ␸DP varies against X for wavelets traveling
1610
B. Backward Propagating Wavelets
Backward propagating wavelets are responsible for the
DPOAE in the ear canal. On FIGURE 22, the paths along
which phase rotations are to be compared are SXS (diagrams on the left side of FIGURE 22, from top to bottom
according to the f2/f1 ratio), with the same patterns of vibrations at f2, f1, and 2f1-f2 as for the forward-propagation
case supporting the evaluation of phase accumulation. Its
cumulated value at the stapes S is now 2␸1 ⫺ ␸2 ⫹ ␸DP, with
2␸1 ⫺ ␸2, as before, representing stimulus phase accumulations along SX, their forward path to the interference
places X, and ␸DP representing the phase accumulation of
the DP wavelet at 2f1-f2 from X to S. The sign flip of ␸DP,
compared with the previous case, expresses backward
propagation of the DP.
Once again, the more basal X, the smaller the initial
phase lag with which the DP wavelet is launched. However, this time, the more basal X, the less the DP wavelet
has to travel to the stapes [actually, the quantitative evaluation of ␸DP depends on what mechanism is accepted
for the reverse propagation of DPs. Whether it is a slow
reverse traveling wave along the BM according to the
same mechanism as forward propagation, or a fast compression wave in scala vestibuli (see sect. VH) is a topic of
intense controversy. The slow traveling-wave mechanism
predicts the same ␸DP forward and backward while the
fast compression-wave mechanism predicts a smaller ␸DP
backward.]
At large f2/f1 ratios, the phase accumulation of DP wavelets
at S, 2␸1 ⫺ ␸2 ⫹ ␸DP, is the same as at D, dominated by
⫺␸2, as the resonance at 2f1-f2 occurs well beyond A in the
apical direction so that ␸DP is negligible. This term ⫺␸2 is
again unfavorable to the generation of a large overall
DPOAE. For f2/f1 ⬇ 1, ␸1 ⬇ ␸2 ⬇ ␸DP, as for the forwardpropagation case, but because of the sign flip of ␸DP in
2␸1 ⫺ ␸2 ⫹ ␸DP, the phase difference among backwardpropagating wavelets varies approximately like ⫹2␸2. This
is too fast a phase rotation to build a large DPOAE. At
decreasing f2/f1 ratio, the phase lag (2␸1 ⫺ ␸2 ⫹ ␸DP) varies
monotonically, and this, from ⫺␸2 at high ratios to ⫹2␸2 at
low ratios. The change in sign implies that between these
extremes an intermediate f2/f1 ratio can be found where the
result is 0. Near this ratio, the phases of all wavelets remain
coherent (FIGURE 22, left side, intermediate diagram) and
the resulting DPOAE is maximum. According to the models
and the results of human measurements, this happens
around f2/f1 ⫽ 1.20 –1.25.
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In the case of wavelets traveling in the forward (apical)
direction, to the place tuned to the DP, the total phase
accumulation can be shown to be 2␸1 ⫺ ␸2 ⫺ ␸DP (244,
266). The journey splits into two segments, SX and XD.
Along SX, primary tones f2 and f1 accumulate phase rotations ␸2 and ␸1 while reaching places of nonlinear interaction X, somewhere between A and B near the f2 peak. A DP
wavelet starts from X with a 2␸1 ⫺ ␸2 initial phase, and
travels the next segment of the journey, XD, at a new frequency 2f1-f2. The ␸DP term expresses the phase accumulation of the DP wavelet along its forward path XD. The
nearer X is to B, the smaller the initial DP-wavelet lag,
because both f2 and f1 accumulated less phase, on their way
to their resonance place. Yet the nearer X to B, the longer
XD, so the larger the phase lag that the wavelet accumulates
on its own. Because of the sign “⫺” in (2␸1 ⫺ ␸2) ⫺ (␸DP),
wherever X lays along the BA segment, the stimulus- and
wavelet-associated phase lags at the CF location, D, act in
opposite directions.
forward to D (FIGURE 22, top right) shows a spreading
spectrum of phases, unfavorable to the generation of a DP
at its CF location.
AUDITORY DISTORTIONS
XIV. APPENDIX III: COCHLEAR
DISTORTION IN THE CASE OF THREE
OR MORE TONAL STIMULI
The presence of at least three tones in a nonlinear cochlea,
f1, f2 (the primary tones at the origin of a DPOAE), and f3 (a
suppressor tone probing the two-tone interaction) creates a
large complexity as the number of possible combinations,
notably if sound levels are high enough, increases considerably compared with the two-tone situation. The problem
faced by experimenters is to avoid technical issues such that
different combinations lead to the same spectral line, and
more generally to disentangle the possible combinations to
identify the sources of observed combination tones.
The mechanism of basal generation of two-tone DPOAEs,
revealed by a third tone, calls for an explanation. A first
possibility is a harmonic mechanism (64), for example, a
sound at 2f1 could be produced around the peak response to
f1, and induce a response at its characteristic 2f1 place.
Although small because having to travel in the wrong direction over a too loose BM, in certain circumstances, the 2f1
As for the simpler case of OAEs produced by a single tonal
stimulus (SFOAEs), a similar application of a catalyst principle may account for the outcome of suppression experiments when a suppressor tone at fs is added to the SFOAEevoking probe stimulus at frequency fp (251). Nonlinear
combinations of fs and fp can produce, for example, 3fp ⫺
fs, 2fp ⫺ fs, fs ⫹ 2fp, fs ⫹ fp, etc., distortion components,
from which [(3fp ⫺ fs) ⫺ (2fp ⫺ fs)] or [(fs ⫺ 2fp) ⫹ (3fp ⫺
fs)] combinations produce fp, perhaps more efficiently than
the fp produced by coherent reflection, the strength of which
has been called into question in small rodents (167). The
production of an SFOAE at fp by catalyst processes would
occur at more basal sites than by coherent reflection, thus
with shorter group latency (see the timing and backwardpropagation controversy in sect. IVF).
Recent protocols designed combinations of even more than
three tones. DP phase gradient paradigms (for example, an
f1-sweep at fixed f2) afford a noninvasive means to evaluate
cochlear travel times. With the goal of simultaneously acquiring data for avoiding interrecording variability and improving the computation of phase gradients, the lower of
the two tones of the traditional f1, f2 stimulus can be replaced by a closely spaced group of tones (gi, gj, etc.) (175).
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A first issue is order aliasing, happening when two different
combinations of f1, f2, and f3 lead to the same combination
tone, for example, if there are integers L, M, N and frequencies f3 such that Lf1 ⫹ Mf2 ⫹ Nf3 ⫽ 2f1-f2. The resulting
spectral line at 2f1-f2 is a complex vector addition of two
contributions, one of which comes from places and mechanisms that were not acting in the absence of f3. The choice of
f3 can be made so as to avoid obvious aliasing problems,
and data can be collected using phase rotations of the suppressing tone. For example, if the f3 suppressor is presented
four times with a phase set at 0, ␲/2, ␲, and 3␲/2 in succession, most combination tones containing f3 are cancelled
without the directly produced 2f1-f2 being affected. To
characterize the influence of an interfering tone at f3, interference response areas are built (FIGURE 19) representing
how this f3 tone influences the amplitude and phase of a
given DPOAE, against interfering level (the term interference is more appropriate than suppression, as the DPOAE
amplitude can sometimes increase in the presence of f3). A
striking feature of these areas is the presence, at moderately
loud primary levels, of sharp high-frequency interference
lobes when f3 is set at frequencies that can be more than one
octave above f2 (FIGURE 19) (66, 163, 169). A simple explanation is that there is a secondary site of DP generation in
the high-frequency region in addition to those already described; that the DP generated in this region is comparable
in level to the other sources from the f2 and 2f1-f2 sites; that
enhancement or suppression is observed depending on the
phase difference among the mixed DP contributions so that
when the f3 tone removes the basal contribution, the overall
level can increase if this contribution is out of phase with the
other ones. Therefore, if the plotting procedure takes into
account only level changes and not phase shifts, the interference effect can well be missed (66, 302).
harmonic might remain large enough to interact nonlinearly with the lower frequency f2 stimulus, thereby generating a 2f1-f2 difference tone, from a quadratic mechanism
akin to that producing f2-f1 at the place tuned to f2. Direct
measurements at the cochlear apex confirm the existence of
a very large mechanical component at 2f1 (85), and the
high-frequency cutoff of the BM may be less sharp at higher
levels than it is usually. Another mechanism termed catalyst
by Fahey et al. (64) assumes the existence of richer nonlinear interactions in the presence of a third tone, that could
generate 2f1-f2 and 2f2-f1 responses via intermediate stages
involving f1, f3 and f2, f3 combinations. Contrary to the
harmonic mechanism for which the third tone serves only
for unveiling an existing source, the catalyst mechanism
requires the f3 component. As an example of catalytic pathway, the combination tones f3 ⫺ f1 and f3 ⫺ 2f2 could
combine at the f3 ⫺ f1 place and generate a difference tone
at 2f2 ⫺ f1. Likewise, mixes (f3 ⫺ f2) ⫺ (f3 ⫺ f1) ⫹ f1 or (f3 ⫺
f2 ⫹ f1) ⫺ f3 ⫹ f1 could generate a 2f1-f2 component adding up to the cubic 2f1-f2 produced near f2. Last,
cogent evidence has been produced (170), with the help
of accurately mapped lesions of OHCs, of basal interactions between the tails of f1 and f2 able to produce sizeable
components. Surprisingly, phase characteristics previously
attributed to a coherent reflection mechanism that was
thought to occur at the 2f1-f2 CF place were observed (vertical banding), in humans and rabbits, which suggests that
the two-mechanism taxonomy may have to be completed.
Whether the possible contribution of a traveling mode
along the Reissner’s membrane would better account for
this vertical banding pattern remains to be investigated
(217).
AVAN, BÜKI, AND PETIT
The resulting DPs are made of several groups of simultaneous frequency components from which group delays can be
derived from a single recording. Among the DPs, a novel
category emerged, made of sidebands around the single
tone primary f2. This group has no two-tone equivalent, as
it consists of a set of sidebands around the single stimulus
component f2 at frequencies f2 ⫹ gi ⫺ gj, with i ⫽ j. With
stimulus frequencies chosen such that all possible difference
and sum frequencies were unique (a so-called “zwuis” complex) (272), each i ⫽ j combination resulted in a unique
DPOAE frequency, unambiguously affected to a single
pair (gi, gj).
XV. APPENDIX IV: CHARACTERIZATION OF
NONLINEARITIES BY WIENER
KERNELS
In terms of the mathematics for describing a biological system such as the cochlea, it is customary to test it with the
help of a restricted set of stimuli, for example, pure tones,
then extrapolate the obtained results to predict the response
of the system to any stimulation, however complex, for
example speech. This is straightforward for a linear system,
as it can be equally well described by a transfer function in
the spectral domain (for example, the bell-shaped transfer
function of a bandpass filter), or in the time domain, by its
impulse response (the damped oscillation obtained when
the bandpass filter is excited by a short click). A third
method is to use Gaussian white noise as a stimulus; the
first-order cross-correlation h1(␶) between this stimulus and
the system’s response is called first-order Wiener kernel. In
a linear system, h1(␶) is identical to the impulse response of
the system. In a nonlinear system, the impulse response of
the system also contains other components. The use of
Gaussian white noise as input to the auditory system, with
spikes in auditory nerve fibers as the output, was first introduced by Egbert de Boer (46) who computed the revcor (for
“reverse correlation”) function as the average value of the
stimulus x(t) at time ␶ before the occurrence of a spike:
normalization to the stimulus power spectral density provides the cross-correlation h1(␶).
1612
In general, second-order kernels give a measure of the
“cross-talk” between the responses to two impulses, i.e., of
how much the presence of the first impulse influences the
response to the second one (162). Another prominent interest of the second-order Wiener kernel, regarding auditory
nerve fibers, is that, thanks to even-order cochlear nonlinearities encoded in the low-frequency response envelopes, it still
provides information on temporal coding in auditory-nerve
fibers innervating the entire length of the cochlea, whereas the
revcor function vanishes as neuronal phase-locking decreases
(215, 275). Studies of the nonlinearities themselves have been
less rewarding so far because their translation in physiological terms often requires models that can be probed against
experimental data (215). Two types of conclusions can then
be reached. In Reference 275, the authors, who described
the hearing organ and auditory neurons of frogs with the
help of a sandwich model incorporating filters and nonlinearities in cascade, deduced the orders of the filters from a
careful inspection of the second-order Wiener kernel. Yet,
they also had to acknowledge the failure of their model to
account for the two-tone interactions revealed by Wiener
analysis. In Reference 156, Wiener analysis applied to
noise-evoked OAEs demonstrated the consistence of the
first-order kernel and of click-evoked OAEs. The compressive level dependence of cochlear responses was revealed in
the nonlinear dependence of the first-order Wiener kernel
on stimulus level, but the fact that no emission component
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Another interest of this sideband complex of DPs is that it
persists at wide frequency separation (several octaves) between f2 and gi, whereas with a (f1, f2) pair, the ratio f2/f1
can hardly exceed 1.5–1.6. Moreover, the DPOAE frequencies depend only on the frequency differences gi⫺gj and are
unaffected by a collective frequency shift of the entire tone
complex g. It is therefore possible to efficiently use a quite
low frequency for the average g, such that the forward
travel time of the g stimuli to the place of nonlinear interaction at the f2 CF place is very brief. Thus the time properties of DPs should be associated principally with the reverse journey of DPs, lately a hot matter of controversy
(231) (see sect. VH).
In a linear system, from any of these equivalent descriptions, it is possible to correctly predict any response, for
example, the response to a sum of sinusoids is the sum of the
responses to each sinusoid presented alone. Nonlinear systems are more difficult to analyze as their response to a sum
of sinusoids is not the sum of the responses to each sinusoid
presented alone. The use of perturbation theory is one convenient solution for weakly nonlinear systems. For more
strongly nonlinear systems, a first possibility is to use a
model and derive from its analysis a set of differential equations describing the response of the system. The parameters
of the model can then be adjusted by comparison between
model predictions and experimental data. The topology of a
strongly nonlinear system, however, is seldom well known,
and a more general and systematic method relies on Volterra and Wiener formalisms (284, 294; reviewed in Ref.
59), describing the responses of a system as a sum of functionals including correction terms unfolding the nonlinearities at increasing orders. With Gaussian white noise as input
to the system, its Wiener kernels are independent of each
other and can be computed from input-output cross correlations. So, the outcome of the revcor method of de Boer is
proportional to the first-order Wiener kernel. It provides an
idea of the impulse response of the system connected to the
auditory nerve fiber under investigation, from which by
Fourier transform, the bandpass filter characteristics (i.e.,
frequency tuning) can be derived. For nonlinear systems,
the first-order Wiener kernel is only one component of the
impulse response to which higher kernels also contribute.
AUDITORY DISTORTIONS
was found in the higher order kernels indicated that at each
particular noise level the cochlea behaved nearly linearly, as
would do a system with an AGC once set up after a time
delay.
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basic characteristics. Hear Res 44: 151–160, 1990.
We are particularly indebted to Jean-Pierre Hardelin at Institut Pasteur for his many apt suggestions for improving
the manuscript and for his meticulous editing work. The
recommendations of two anonymous reviewers were of
great help for improving the consistency of arguments from
section to section and better emphasizing the harmony of all
manifestations of distortion reviewed in this text.
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GRANTS
The authors’ research was supported by grants from ERC
2011-AdG nr 294570, Fondation Louis-Jeantet, NovalisTaitbout, Réunica-Prévoyance, ANR MNP P007354 “Presbycusis,” ANR 08-ETEC-001-01 “Audiapic,” Groupe Entendre, Fondation de l’Avenir pour la Recherche Médicale
(ET1-617).
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DISCLOSURES
No conflicts of interest, financial or otherwise, are declared
by the authors.
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