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Physics underlying the physiology of the ear Egbert de Boera) Academic Medical Centre, University of Amsterdam, Amsterdam, The Netherlands (Received 29 July 2015; revised 15 September 2015; accepted 22 September 2015) [http://dx.doi.org/10.1121/1.4932674]

I. INTRODUCTION The author began his university study in physics, mathematics, and astronomy at the University of Amsterdam in 1946, one year after World War II ended. Universities and laboratories in Europe were in a bad state but were rapidly recovering. And so was science in the Netherlands. New paths were sought, for instance, in auditory research and human psychophysics. The author had to work to earn his study. With the state diploma of “radio engineer” he could earn his living and do all the required electronic work for his study. The practical work was done at the Ear, Nose, and Throat Clinic of the Academic Hospital (the Wilhelmina Hospital) of the University of Amsterdam, under the direction of Professor L.B.W. Jongkees. Almost all the needed electronic equipment had to be constructed in the laboratory. The author graduated (in mathematics and physics) in 1953. The Dutch auditory scene at the time was quite exciting, and soon proved to be more and more influential internationally. It was possible to join the moving tide in that field because of a government grant, on the topic of the perception of pitch, which led to a Ph.D. degree in 1956. That topic was closely related to the author’s love of music and his activities in it. More precisely, the title of the academic thesis was “Pitch of Residue Signals,” and that term needs to be explained. What in the Netherlands is called “residue pitch” is generally known as periodicity pitch (or repetition pitch) but in a slightly more refined form. A signal with a periodic (non-sinusoidal) waveform, with a period in the range of approximately 0.2 to 100 ls, can evoke a pitch sensation. Call the repetition frequency f0 . The signal’s spectrum then contains consecutive components with frequencies that are integer multiples of f0 . Further research has revealed that it is not necessary to have all components present to evoke a pitch sensation. Much research on pitch perception has been done with signals comprising a limited number of components. In the author’s work signals with three to five consecutive components have been used. A close study of the perception of these signals shows that it is, with a bit of practice or with special devices or techniques, often possible to hear out the individual components. Signals that cannot be analyzed by the ear in this way were originally called (by Schouten, 1940; Schouten et al., 1962) “residue” signals. Later, this term was used in a more general sense, to denote signals like those mentioned here, but we will continue to speak of “residue signals”—without reference to the question whether individual components can be analyzed by the ear or not. The presence of a component with the repetition frequency f0 , called the fundamental, is not necessary for pitch. Consider such a periodic signal, for instance, one with only the 3rd, 4th, 5th, 6th, and 7th components (harmonics)

present. This signal evokes a clear pitch, which corresponds to a tone with fundamental frequency f0 . Assume f0 to be equal to 200 Hz, the component frequencies then are 600, 800, 1000, 1200, and 1400 Hz, and the pitch corresponds to 200 Hz. Shift all the component frequencies up by a constant amount, for instance, 10 Hz, the component frequencies become 610, 810, 1010, 1210, and 1410 Hz. The frequencies are no longer harmonic, i.e., integer multiples of a frequency near 200 Hz. Nevertheless, the signal has a pitch, a clear pitch, not much less clear than the original one, and that pitch is near 200 Hz. Actually, a bit higher, near 202 Hz. Similarly, when the component frequencies are all displaced down in frequency, the pitch will be lower than 200 Hz. Data from this work are presented in de Boer (1956) and Fig. 1 shows an example, pitch plotted against the frequency of the central component. Hence, non-periodic signals can have a pitch. The signals described above are called pseudo-periodic, and the associated pitch is called pseudoperiodicity pitch. To a first approximation that pitch is proportional to the frequency of the central component (in the above-mentioned case, a central frequency of 1010 Hz leads to a pitch of approximately 202 Hz). Detailed measurements reveal that the actual pitch deviates from that frequency. About the cause of that effect, a lot of discussion has arisen. See for review papers de Boer (1976) and de Cheveigne (2005). The boundary of auditory analysis of components was thoroughly studied by, amongst others, Ritsma (1962). He introduced the concept of “existence region”: a graph depicting the relation between the frequency difference (200 Hz in the example given) and the highest central frequency of the complex to evoke a clear pitch. In summary, inharmonic signals can have a pitch, irrespective of whether individual components can be analyzed or not. An additional puzzling phenomenon has to do with phase effects. The inharmonic signals described above show no beats. However, as soon as the components are no longer precisely equidistant, beats become audible and can be traced to specific phase effects. Again, this is fairly difficult to explain (de Boer, 1961a, 1966). The existence region also plays a part here. At that time, auditory research in the Netherlands was poised to become exciting and highly influential over the coming decades. The same tendency was visible in the surrounding countries. The author obtained a position as audiologist in the Academic Hospital of the University of Amsterdam. In the Netherlands you cannot become an audiologist without a full university education in either physics or engineering (electrical or acoustical). This, perhaps, is one reason for the high standard of audiology in the Netherlands. On an American Fulbright Travel Grant, the author made an extensive trip to the USA in the years 1957 and 1958. Joseph C. R. Licklider, familiarly known as Lick, an excellent psychophysicist (and later well-known in computer technology), was instrumental in the organization of this trip. The gist of his work in pitch perception has been described in Licklider (1979). The author’s work was carried out at MIT, in the Communication Biophysics Lab under Walter Rosenblith. That is where important scientists like Norbert Wiener, Bill Siebert, Jerry Lettvin, Warren McCullough, and others worked and interacted. Nelson Kiang taught the author electrophysiology, and single-fiber work was introduced to me by Jerry Lettvin. The author also taught information theory and nonlinear network theory as an instructor at MIT. Needless to say, this trip enormously contributed to his knowledge and skills.

II. AUDIOLOGY AND AUDITORY PHYSIOLOGY a)

Also at: Oregon Hearing Research Center, Oregon Health & Science University, Portland, Oregon 97239, USA. Electronic mail: [email protected]

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After the trip to the US the work in the audiology department of the (ORL) clinic was continued. There was a dispute between psychophysicists

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Europe (the German group around Feldtkeller and Zwicker) and the U.S. (Harvey Fletcher) about the size of the critical bandwidth. It was proposed and proved that this effect is due to the disastrous influence of audible fluctuations (de Boer, 1962, 1966; Bos and de Boer, 1966). A special method was developed to measure the critical bandwidth based on using a noise signal with a variable spectral gap as the masker and a test tone in the gap—a method that is still in use, de Boer and Bos (1962). One thing discovered in this context was that in patients with sensorineural hearing loss, the critical bandwidth is often enlarged (de Boer, 1961b). Much later it was discovered why this is the case, see below. The hearing of hard-of-hearing children equipped with two hearing aids (it took some effort to convince the health authorities to pay for two hearing aids) was studied on the basis of the binaural gain in hearing and understanding speech (de Boer et al., 1967). Later we also started measurements of hearing in infants and newborns with brainstem evoked responses (de Boer et al., 1976, 1977). In the course of time, a little experience was obtained with deaf children who had received a cochlear implant—fairly primitive in those early years, of course. This work can all be put under the heading of (clinical) psychophysics.

III. SINGLE-FIBER WORK, REVERSE CORRELATION Meanwhile research was started on single fibers, recording from fibers in the auditory nerve of deeply anesthetized cats. Based on his electric engineering experience, the author developed the “reverse correlation” technique. That is a method to record from one single auditory nerve fiber stimulated by a (stationary) noise signal and to extract information about the spectral and temporal properties of the transformations that have taken place before the generation of nerve impulses (de Boer, 1968, 1970; de Boer and Kuyper, 1968). The method is related to the technique of cross-correlation. Figure 2 illustrates the technique: segments of the stimulus signal that precede spikes are isolated and used for processing. After many spikes all these signal fragments are averaged, and produce the reverse correlation (or revcor) function, see Fig. 3. Note that this is basically a linear procedure. Fourier analysis exposes the spectral composition of the revcor function. With that analysis method it was possible to extract quantitatively how components of a composite acoustic signal are coded in the impulse train of a single auditory nerve fiber, even if the acoustical stimulus is a wide-band signal. In fact, the method reveals the bandwidth of the signal components that contribute to the firings of one auditory-nerve fiber, a kind of “mapping

FIG. 2. Reverse correlation. Upper trace: series of neural spikes (idealized), recorded from an auditory neuron. Lower traces: three copies of the acoustical stimulus signal. Parts that precede spikes are indicated. All these signal parts are added, and after many spikes produce the reverse correlation (revcor) function. The revcor spectrum is the Fourier transform of the revcor function. of firing patterns in nerve fibers to the sound signals that generate them.” (In the science of vision it would be called the “receptive field.”) This result was new in the sense that it proves that sharp filtering—of complex signals—is already evident at the level of primary auditory nerve fibers, not only for single-frequency stimuli but for complex signals as well. And this in a period in which general thinking was in terms of coarse (Bekesy-like) filtering and subsequent sharpening (in the brain). The reverse-correlation method can be extended to nonlinear systems. For that more general purpose the EQ-NL theorem was developed (de Boer, 1973, 1997a). That theorem tells you exactly under which circumstances you can treat the input-output relation of a nonlinear system as a quasi-linear transformation. Strictly speaking, stimuli used for this work should preferably be wide-band signals with flat spectra. In another trip to the USA, this time to the Bell Telephone Labs in New Jersey, a different method of nonlinear analysis was studied, a method based on power series expansions (originating with Norbert Wiener). This was done in collaboration with Professor Manfred Schroeder. Unfortunately, that study did not provide new solutions, and the author stayed with his own reverse-correlation method and the EQ-NL theorem.

IV. COCHLEAR MECHANICS IN GENERAL, PASSIVE AND ACTIVE

FIG. 1. Pitch of an inharmonic signal. Pitch is measured by comparing the signal with a harmonic one; “pitch” is then the fundamental frequency of that comparison signal. Bx is the frequency f0 in the text, 200 Hz here and fixed. Ax is the central frequency of the complex, and is varied around and between integer multiples of f0 . Ax is the primary variable here and serves as the abscissa. J. Acoust. Soc. Am., Vol. 138, No. 4, October 2015

During the time of the developments described above, the author got firmly interested in cochlear mechanics, the study of how acoustic signals entering the cochlea are physically transformed and coded in the cochlea. Most of the mechanical models used at that time were long-wave models, mainly because those models allowed analytic solutions. Waves in the cochlea are long waves when their wavelengths are of the same order or larger than the cross-section of the cochlear duct. They are called short waves otherwise. In the region of the response peak the cochlear wave is a short-wave phenomenon. A thorough study of the properties of long-wave cochlear models and of models that allowed all kinds of waves was made, several analysis models allowing long as well as short waves were developed and suitable solution methods for them were found (de Boer, 1995a,b,c). In that way the differences between the various kinds of models could thoroughly be described—a topic that is largely neglected nowadays. Again the question of the effective bandwidth arose, this time the bandwidth associated with a mechanical frequency analyzer. It is different for long-wave and other models. It gradually became understood that a model that is “passive” (i.e., does not contain internal sources of energy) is not capable of showing the type of response as had been found in mechanical experiments on the Egbert de Boer

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FIG. 3. Typical result of a revcor operation, comparison of a revcor spectrum and a neural tuning curve, derived from the same auditory-nerve fiber. The revcor function is measured with a continuous wide-band noise signal as acoustical stimulus. The smooth curve is the amplitude of the Fourier spectrum of the revcor function. In the measurement of a tuning curve the intensity of a pure tone is continually increased and decreased around the threshold of detectable spikes from the preparation, meanwhile the frequency is slowly increased. The zigzag curve, a direct recording of the tuning curve made during the experiment, shows the course of the tone intensity (on a dB scale and a linear frequency scale) around its threshold. (In human psychophysics this technique is known as the Bekesy audiogram.) The CF of the fiber (72-08-15) was 2.9 kHz.

cochlea. The passive model has insufficient frequency selectivity. On the basis of Rhode’s high-frequency mechanical results (see Rhode, 1971), Kim et al. (1980) more or less intuitively posed that the cochlea is locally active. That means that the cochlea should contain elements that amplify vibrations. In 1983 the author published a mathematical proof that the cochlea must be “active” in this sense, that is, it must contain frequency-dependent signalamplifying elements. The proof was based on physiological responses—amplitude and phase—as reported in the literature, which were confronted with a stylized model of the cochlea. This work was done for long-wave, shortwave, and general cochlear models (de Boer, 1983a,b, c). Needless to say, that this result considerably fortified the confidence in the intuitive conclusion of Kim (1980). Actual mechanical measurements on this theme came much later. However, the notion of cochlear amplification became wellknown (despite serious criticism), and the term “cochlear amplifier” was launched by Hallowell Davis and is in common use today. Actually, that term is not appropriate because there is no “cochlear amplifier,” not a single physical element that is responsible for amplification that can be isolated. There is a diffuse, distributed process in the cochlea that causes acoustic signals to be amplified in power—for every frequency in an appropriate region of the cochlea. Associated with that type of amplification is sharpening of the frequency selectivity, in other words, shrinking of the bandwidth—here is the bandwidth problem again (de Boer and Viergever, 1984). It is generally believed that the activity effect is due to the outer hair cells (OHCs) acting in that diffuse process.

V. A CHANGE OF PERSPECTIVE: MECHANICAL MEASUREMENTS Mandatory retirement from the university at the age of 65 years afforded the opportunity to start intensive and frequent collaboration with Professor Alfred L. Nuttall (first at Ann Arbor, MI, later on in Portland, OR). Actually, this productive and successful cooperation has been continued over more than 21 years now. In Dr. Nuttall’s lab mechanical experiments on the cochlea were conducted, in which movements of the basilar membrane (BM) were measured with a laser interferometer, these experiments were conducted in deeply anesthetized guinea pigs. The author developed special analytical and digital analysis techniques for this work whereby, as earlier in the reverse-correlation work, wide-band noise signals were often used as standard acoustical stimuli. The cross-correlation technique was applied to the stimulus signal and the acquired BM signal, and to deal with nonlinearity the aforementioned EQ-NL theorem was used where needed. In most of that work noise signals were used as stimuli. In addition, the stimulus signal was divided into several frequency bands, both in experiments and in analysis. With this combined technique it was possible to get a 2556

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good overview of the analysis properties of the segment of BM under study—a region operating at high frequencies (mostly above 15 kHz). Figure 4 shows a typical result of the mechanical measurements. The legend describes the procedure and the layout of the results. A feature to be noted is the systematic variation of the bandwidth with stimulus intensity. In this way this figure gives a clear presentation of the effects of nonlinearity on cochlear amplification and selectivity; more on nonlinearity will follow. Note that at this point nothing has been said about the specific nature of nonlinearity. Many subjects were studied with this method, amongst others: tone versus noise responses, impulse responses, otoacoustical emissions, wave propagation of partial waves, and, in particular, nonlinear effects for several types of stimulus. It was also the time that cochlear models were further refined. A cochlear model essentially consists of a description of the structure with a fitting description of the hydrodynamics of the fluid contained in it, characterization of the BM by its mechanical impedance, and a description of the ways the system communicates with the outer world (through the round and oval window and the vibration of the bone); all of this in terms of mathematics. When the functioning of the model is close to that of the actual cochlea, the computed responses can be trusted, and we say, that we “understand the cochlea.” An understatement, of course. Quite a few different mathematical techniques have been developed or tried out as solutions. Then, several special problems were attacked, such as: forward and reverse traveling waves (primary signals, internally reflected waves, as well as distortion products created from multi-tone stimuli), effects of electrical stimulation of the cochlea, and responses to complex multi-component signals such as used in pitch research (see above). For a couple of references outside of the central field, see Grosh et al. (2004) and Summers et al. (2003).

VI. THE BM MECHANICAL IMPEDANCE, FORWARD AND INVERSE SOLUTIONS, COCHLEAR NONLINEARITY In all this modeling by the author, a central role is played by the BM impedance. Measuring or estimating this impedance led to a few surprises and this gave a new view on “activity” and the subject of cochlear amplification. In this analysis, all variables, which originally are functions of frequency, are transformed into functions of location (x). This was done from the earliest publications (e.g., de Boer, 1997c; de Boer and Nuttall, 1999, 2002b) on. Which reference frequency should we choose in this transformation? The choice is arbitrary. We have generally chosen the best frequency for the lowest level of stimulation at the location of the observations. We consider the cochlea to consist of one channel which is divided by the cochlear partition. The main mechanical element of the partition is the BM. The Egbert de Boer

difference in pressure that arises across the partition gives rise to movement of the partition, and the complex quotient of pressure difference across the membrane and velocity is defined as the BM mechanical impedance. In acoustics, various types of acoustical impedance are used; what we have here is the specific acoustical impedance which we will refer to simply as “acoustical impedance.” How do we now find the pressure difference across the BM? If pBM ðxÞ is the pressure on one side (let us call it the upper side) of the BM, it is given by pBM ðxÞ ¼ ixq

ðL

! Gðx; nÞ vBM ðnÞdn þ SðxÞ vst :

(1)

0

Here, Gðx; nÞ is the Green’s function, vBM ðxÞ is the BM velocity, SðxÞ is the stapes propagator for the point x on the BM, and vst is the virtual stapes velocity associated with the movements of the BM. The functions Gðx; nÞ and SðxÞ are determined by solving a three-dimensional fluid dynamics problem

FIG. 4. (A) Response functions—amplitudes—of the BM derived from measurements performed in the basal turn of the guinea pig cochlea with wide-band noise signals as stimuli. The stimulus levels are varied, the numbers actually indicate the sound pressure level per octave of bandwidth. Data are referenced with respect to the stimulus amplitudes. That they nearly coincide at low frequencies shows the near linearity of the data in that range. The curves measured at lower levels are sharper and rise to higher levels. Conversely, frequency selectivity decreases with increasing stimulus level, a manifestation of cochlear nonlinearity. (B) Phase curves corresponding to two of the curves of pane (A). J. Acoust. Soc. Am., Vol. 138, No. 4, October 2015

(following Mammano and Nobili, 1993). Equation (1) is central to the solution of cochlear mechanics problems and can be applied in different forms. The pressure pBM ðxÞ and velocity vBM ðxÞ are directly related by the BM impedance. If that impedance is known, vBM ðxÞ can be solved from Eq. (1) given the stapes velocity vst . Let us define this as the forward solution to cochlear mechanics. It is, however, also important to derive the BM mechanical impedance from actually measured responses; this is defined as the inverse solution. The same equation is used for the inverse solution. The technique of the inverse solution has been developed in various forms. In short, vst and vBM ðxÞ are known (in fact, derived from measurements) and pBM ðxÞ is computed from Eq. (1). Of course, the two functions Gðx; nÞ and SðxÞ must be known, computed, or estimated. From pBM ðxÞ and vBM ðxÞ follows the impedance (this is a most simplified description), and this constitutes the inverse solution. Remember that in this analysis all variables are functions of location x, and most of these have to be derived from measurements in the frequency domain. For more details, see de Boer and Nuttall (1999, 2002a,b). Figure 5 shows an example, the BM response measured at the lowest level in the upper panel and the BM impedance corresponding to that response in the lower panel. In the impedance panel the solid curve shows the real part and the dashed curve the imaginary part. The ordinate scale is deliberately made nonlinear, values close to zero are depicted linearly, and larger values are nonlinearly compressed, positive and negative values are treated with the same nonlinear scaling. The course of the imaginary part is

FIG. 5. Response and BM impedance, one experiment. Upper panel: response amplitude (in dB) and phase (in units of p). Lower panel: components of the BM impedance, real and imaginary parts. In the impedance panel the data are shown on a nonlinear scale that smoothly compresses the larger values (in both directions). The central region of the ordinate scale, between 1 and þ1, is linear. The dip in the real part of the BM impedance indicates the region where power goes to the BM. Egbert de Boer

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not surprising (except in one respect, see further on). It is negative throughout, denoting stiffness; note that stiffness of the BM is an important feature, known since the days of von Bekesy. We might have expected, however, this component to go through zero near the point of the response peak, signifying a true resonance, but that is not so. Surprisingly it remains negative (over a large region), and this is the exception just mentioned. In fact, we can consider it as a manifestation of short-wave behavior. The real part of the BM impedance shows a completely different course. On the extreme right, it is fairly regular, and the same appears to be true on the extreme left, a behavior that is to be expected. In the region of the response peak, however, this real part shows a pronounced dip which reaches into the region of negative values. This means that in this region of the abscissa the BM produces power which is given to the fluid wave. In other words, power amplification takes place in this region. In fact, the behavior of the real part of the BM impedance gives the clearest evidence that the cochlea is capable of amplification. For locations slightly more basal, we can interpret the situation in that the BM here produces power but not enough to compensate the innate losses in the BM and its associated structures. The entire area of the dip signifies the area of “excess” power generated. In summary, we observe in the BM impedance a detailed manifestation of cochlear amplification. In addition, this result indicates that cochlear amplification is frequency-specific. The shape of the negativegoing dip gives rise to a few important comments. We realize that cochlear amplification is causing a power increment of several tens of dB, and that occurs by way of positive feedback. Why, then, does the system not immediately go into oscillation? The answer is that the power created in the region of the negative dip is dissipated in the neighboring regions to the left and right. Details of this process have not yet been studied. Figure 6 shows a collection of impedance curves for various values of the stimulus intensity, from 20 to 100 dB sound pressure level. For the imaginary part, only two curves are shown because the variations are relatively small. In the real part, however, we observe large and characteristic variations. With increasing stimulus intensity, the size of the negative-going dip diminishes. In network terms, the impedance is seen to consist of two components, the “passive” part and the “active” part, whereby the latter strongly depends on stimulus level. We do not have to search long for a possible explanation of this intensity effect. Hair cells are nonlinear; with increasing stimulus signal amplitude the hair-cell response is less than proportional to the stimulus. Therefore, the size of the negative-going dip in the impedance can only become smaller for higher levels. Consequently, amplification diminishes.

And here we have what underlies the remarkable increase in the critical bandwidth that had been observed in cases of hearing loss (see above). It is due to diminution of cochlear amplification that is associated with the appearance of hearing loss. This finding agrees, of course, with what had been deduced earlier on the basis of secondary cues (masking data, otoacoustic emissions) about the cochlea in those clinical cases. It is possible from a stylized version of the nonlinear input-output function of hair cells to predict the size of the negative-going impedance dip for all stimulus levels. From the predicted impedance modification we can compute, “resynthesize,” the response of the model for all stimulus levels. We have used this method to confirm the validity of the impedance approximation. It is clear how valuable this procedure is. In summary, nonlinearity of hair-cell responses explains, via its influence on cochlear amplification, how the response varies as a function of stimulus level. It is important to note that this process can be imitated in a model and followed quantitatively. More aspects of the additivity of impedance components can be found in review papers de Boer (1997b) and de Boer and Nuttall (1999). A close relation exists, of course, between nonlinearity, stability, and spontaneous activity. In this connection, we report that Dr. Nuttall’s group has discovered at least one example of a spontaneous mechanical cochlear oscillation (Nuttall et al., 2004). This evidence could be linked to the theory of coherent reflection (Zweig and Shera, 1995, de Boer and Nuttall, 2006).

VII. MODERN TIMES The modeling story, as it has been unfolded above, is currently undergoing a pronounced revision. In recent times it has become possible to measure more details of movements of structures inside the organ of Corti (OoC). This is done with the technique of optical coherence tomography (OCT) (Chen et al., 2007; Choudhury et al., 2006; Tomlins and Wang, 2005; de Boer et al., 2014b). Movements of structures within the OoC, yes, even within the fluid channel between the reticular lamina (RL) and BM, can now be detected and measured. The data obtained from this type of work—although far from complete—lead to remarkable and unexpected consequences. In the region of maximal response it has generally been found that the oscillations of the RL are larger than those of the BM. In that region, the maximum difference is on the order of 6 dB. Furthermore, the response at the BM has a phase lag with respect to the RL. Both of these features are illustrated by the four panels of Fig. 7(A) for the amplitude (level differences are expressed in dB) and Fig. 7(B) for the phase differences (in units of

FIG. 6. Response and BM impedance, effect of stimulus level v. Experiment: 7611. Left panel: dashed curves, original response amplitudes; solid curves, BM impedance ZBM(x, v), real part, recovered by inverse solution. Right panel: dashed curves, response phase. The slope of the phase curve is smaller at higher levels of stimulation. Solid curves, imaginary part of impedance. Stimulus levels: 50, 60, 70, 80, and 90 dB for live animal, 100 dB for dead animal. At higher levels of stimulation, the response peak shrinks and the negative dip in the real part of the BM impedance decreases in size. In fact, the transfer of power to the BM diminishes. This is the principal manifestation of cochlear nonlinearity.

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p radians). The data are shown for four different stimulation levels. In most of the frequency range, the response of the BM is smaller than that of the RL, therefore, the amplitude level difference data shown in the figure lie mostly below the zero line. Assuming that the effective widths of BM and RL are equal. We conclude that during the oscillations caused by sounds, the volume of the channel (between RL and BM) at the longitudinal region of interest does not remain constant. The first problem raised by this result is, where does that excess volume of fluid go? And where can we find the net effect of those movements? The second point is, what is the reason for this difference? The latter point receives an easy but perhaps incomplete answer: we attribute it to the fluid mass inside the channel of Corti (CoC). The phase difference between RL and BM can then simply be explained by inertia (of the fluid). The third point is how to account for the more complex fluid flow inside the CoC. At present these problems are only partly solved. Let us, for a moment, consider the situation from another perspective. Pressure is exerted on the OoC on both sides, “above,” on the RL, and “below,” on the BM. The fluid inside will move under the influence of the (algebraical) difference of the two pressures. By taking this into account, the

organ structure between the RL and the BM can be associated with an impedance. In the publication(s) of de Boer et al. (2014a,b) that impedance was tentatively derived from measurements and shown to be relatively constant and dominated by a mass. This result strengthens our intuition expressed above: it is mainly a (passive) mass of fluid that is being moved between the RL and the BM, and this tallies with the phase difference between RL and BM. Taking into account this phase effect, it becomes possible to understand why the RL shows a higher best frequency (BF) than the BM. Let us stress that we observe little to no evidence of power transfer or amplification inside that channel. Note the general correspondence between the four panels in each part of Fig. 7 and think of the large differences in oscillation amplitudes (up to 40 dB) that are involved in the original data. In other words, cochlear amplification does not involve what is going on in minute fluid movements inside the CoC. For the moment this is as expected because the fluid by itself is passive; in other respects it is somewhat surprising. It is not to be excluded that, later, more accurate and more complete measurements give slightly different results or lead to a different and more detailed interpretation. In our work, the velocities of BM and RL have been measured in the high-frequency region. In other laboratories, regions tuned to lower frequencies have been and are being explored. In that work it was not necessary to open the cochlea by breaking away the bone. Results cannot yet be compared to those described here. It is clear now that the problem of finding out how the inner ear works is far from solved. This is, in particular, true of the mechanism of cochlear amplification. We know quite much, but essential details are still missing. Just to mention one point, the mechanism of cochlear amplification is frequency-selective but we do not know how that is achieved. Further, how does power amplification work? Where does it gain its energy? Via motility of OHCs or via active mechanical movements of stereocilia—as is often presumed to occur in non-mammals? One more (but certainly not the last) question, how is the inherent low-pass filtering of intracellular voltage of OHCs overcome? To present and future scientists, the cochlea will remain a focus of attention, and it can be expected that its study will remain a source of wonder and satisfaction at every stage of the solution of the cochlear-mechanics problem. The author, at least, has enjoyed and is still enjoying his work immensely, and he is happily surprised by every new bit of information extracted from this remarkable organ.

ACKNOWLEDGMENTS Many of the author’s former collaborators have vanished (or died). One exception is Dr. Luc-Johan Kanis. From his earlier years at the Universita la Sapienza, Rome (Italy) he brought a degree of maturity that is rare among doctoral students. This has been beneficial for his and our research. Talents for music and physics or mathematics often go together. Luc-Johan is no exception. Vera Prijs and Paul Kuyper are colleagues in both respects. Vera has repeatedly participated in the execution of some of the author’s musical compositions, whereas Paul had extended musical discussions with the author while we were playing the music of Darius Milhaud on two pianos. The later part of the author’s career has been tightly connected with Professor Alfred L. Nuttall and his group in the U.S. Without Dr. Nuttall’s collaboration and his initiatives, most of what has been described above would have been impossible. Dr. Nuttall is warmly thanked for all he has done, including frequent and stimulating discussions, and for the great hospitality and friendship he and his wife Bonnie bestowed on the author. It is stressed that this gratitude is extended to Dr. Nuttall’s entire lab; all the members of his “crew.”

FIG. 7. Responses of the RL and the BM, shown at four stimulation levels. (A) Amplitude, shown are the differences between the levels of RL and BM amplitude responses, expressed in dB. Positive means RL amplitude is larger than BM amplitude. The dotted lines indicate a decrease of 6 dB per octave. (B) Phase. Differences between RL and BM phases are shown, in units of p. Negative means BM is lagging. J. Acoust. Soc. Am., Vol. 138, No. 4, October 2015

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Egbert de Boer