Harmonic distortion on the basilar membrane in the basal turn of the guinea-pig cochlea

Transcription

1 Keywords: Cochlea, Basilar membrane, Hair cell 7006 Journal of Physiology (1998), 509.1, pp Harmonic distortion on the basilar membrane in the basal turn of the guinea-pig cochlea Nigel P. Cooper Department of Physiology, University of Bristol, Bristol BS8 1TD, UK (Received 6 June 1997; revised 10 January 1998; accepted 11 February 1998) 1. Mechanical responses to pure-tone stimuli were recorded from the basilar membrane in the basal turn of the guinea-pig cochlea using a displacement-sensitive laser interferometer. The harmonic content of the responses was evaluated using Fourier analysis. 2. Harmonic distortion products were observed in many of the basilar membrane responses. Response components locked to twice the frequency of the stimulus (i.e. 2FÑ) were the largest of the distortion products. 3. The second harmonic responses showed a bimodal frequency distribution at low to moderate sound pressure levels: one peak occurred around the preparation s best or most sensitive frequency (i.e. when FÑ 17 khz), and another occurred around one-half of the best frequency (when FÑ 8 5 khz). 4. The absolute levels of most distortion products increased progressively with increasing stimulus strength. When expressed with respect to the levels of the fundamental responses, however, the distortion levels usually decreased with increasing stimulus strength. 5. The levels of the distortion decreased (in both absolute and relative terms) with deterioration in the physiological condition of the cochlea. 6. Maximum second harmonic distortion levels amounted to 3 5 and 28 % of the fundamental responses to tones near and below the best frequency, respectively. 7. The above findings are shown to be consistent with a highly simplified model of cochlear mechanics which incorporates an asymmetric, saturating non-linearity in a positive feedback loop. Healthy mammalian cochleae process sounds in a leveldependent manner: quiet sounds are amplified by active processes in cochlear mechanics, while loud sounds are not (Kim, Molnar & Matthews, 1980; Davis, 1983; for reviews see Patuzzi & Robertson, 1988; de Boer, 1991; Dallos, 1992). The device which is responsible for the mechanical amplification is known as the cochlear amplifier (after Davis, 1983). The precise mechanisms of this amplifier are not yet fully understood, although many of the amplifier s characteristics are well known. At each point along the length of the cochlea, for example, amplification occurs only for certain frequencies of sound: components of a sound with frequencies close to a particular location s best or most sensitive frequency can be amplified substantially, while components with lower frequencies undergo little or no amplification (Rhode, 1971; Sellick, Patuzzi & Johnstone, 1982; Robles, Ruggero & Rich, 1986; Ruggero, Rich, Recio, Narayan & Robles, 1997). The amount of amplification that occurs also depends on the physiological condition of the cochlea: the cochlear amplifier is particularly vulnerable to insults which impair the function of the outer hair cells in the organ of Corti (Patuzzi, Yates & Johnstone, 1989; Ruggero & Rich, 1991; Dallos, 1992). The gain of the cochlear amplifier can be defined as the difference between the sensitivities of the mechanical responses in passive and active cochleae (Davis, 1983; for the purposes of the present report, active can be read as living, and passive as dead). This gain is normally highest at low sound pressure levels (within 20 db of hearing thresholds) and reduces progressively as sound pressure levels increase beyond 20 db above threshold (cf. Nuttall & Dolan, 1996; Ruggero et al. 1997; also see Figs 1 and 4 in the present report). One highly beneficial consequence of this is that the mechanics of the cochlea can compress a very wide range of stimulus intensities into a much smaller range of response amplitudes: in effect, the mechanical compression allows the gap between the thresholds of hearing and those of pain to be expanded by several orders of magnitude (Davis, 1983; Zwicker, 1986; Yates, 1990). The non-linearities associated with the operation of the cochlear amplifier (e.g. its level-dependent gain) have several

2 278 N. P. Cooper J. Physiol consequences in addition to the compression described above. Most notably, the non-linearities must distort the incoming sounds to some extent. One well-known consequence of this is the existence of inter-modulation (or two-tone) distortion products in the cochlea (Goldstein, 1967; Kim et al. 1980; Robles, Ruggero & Rich, 1991). Another likely, or at least possible, consequence is harmonic distortion. This type of distortion has been observed in the receptor potentials of the inner and outer hair cells of the cochlea (Dallos & Cheatham, 1989; Cody & Russell, 1992), as well as in the discharge patterns of single cochlear nerve fibres (see Kiang, Liberman, Sewell & Guinan, 1986). Harmonic distortion can also be perceived (as overtones ) when pure tones are presented in psychophysical experiments (cf. Wegel & Lane, 1924). It is not easy to relate the distortion observed in any of these experiments to the operation of the cochlear amplifier, however, since each stage of the auditory periphery which operates in a non-linear fashion will introduce additional distortion to the signal-processing cascade. Mechano-electrical transduction in the hair cells is likely to obscure any mechanically generated distortion, for example, and synaptic transmission from the hair cells to the afferent nerve fibres is likely to obscure any distortion that is either generated or present at the hair cell level. This is unfortunate, since knowledge of the distortion in the mechanics of the cochlea could provide considerable insight into the workings of the cochlear amplifier (cf. Nobili & Mammano, 1996; Zwislocki, Szymko & Hertig, 1997). Direct observations of harmonic distortion in the mechanics of the cochlea have been made on several occasions in the past (e. g. see LePage, 1987; Khanna, Ulfendahl & Flock, 1989; Gummer, Hemmert, Morioka, Reis, Reuter & Zenner, 1993). Unfortunately, however, most of these observations have been made in physiologically compromised preparations (as evidenced by the lack of compression in the responses to pure tones, for example). Only two series of observations have been reported from truly active (i.e. physiologically normal) cochleae (Cooper & Rhode, 1992; Ruggero et al. 1997), and even these are limited in scope. Cooper & Rhode s (1992) observations on the feline basilar membrane were restricted to stimulus frequencies well below the preparation s best frequency, for example, and the distortion at these frequencies only became significant ( > 1 % distortion) at sound pressure levels in excess of db SPL (sound pressure level in decibels, where 0 db SPL = 20 ìpa). This finding is not surprising, as the effects of the cochlear amplifier are known to be limited at low frequencies (see above) and the harmonic distortion levels might well be expected to be low in the absence of significant amplification. The observations made in the chinchilla cochlea by Ruggero et al. (1997) suggest that the amount of harmonic distortion is low even when the cochlear amplifier is hard at work: Ruggero et al. do not give precise values for the distortion evoked by best-frequency tones, but they infer that the distortion is much lower than 20 to 30 db relative to the fundamental responses (the limits of their measurement system under suboptimal recording conditions; cf. Ruggero et al. 1997, p. 2153). The study described in the present report uses a purposebuilt displacement-sensitive laser interferometer to probe the mechanics of the living guinea-pig cochlea. The high fidelity of this equipment permits detailed investigations of harmonic distortion in truly active (i.e. physiologically normal) cochleae. The principal findings are that bestfrequency tones undergo up to 4 % distortion in healthy cochleae, and that this figure drops to less than 1 % in compromised cochleae. The observed distortion patterns are shown to be compatible with a highly simplified model of cochlear mechanics, which incorporates physiologically plausible non-linearities into a positive feedback loop (based on the proposals of Zwicker, 1986). METHODS Experiments Young pigmented guinea-pigs were anaesthetized using combinations of sodium pentobarbitone (25 mg kg, i.p., with additional quarter-doses every 2 3 h), droperidol (Droleptan, 4 mg kg, i.m., supplemented every 2 3 h) and fentanyl citrate (200 ìg kg, i.m., supplemented every min). The animals were maintained in a deeply areflexive state throughout the experimental procedures, and were overdosed with pentobarbitone on completion of the in vivo measurements. Tracheal cannulae were used to maintain a patent airway, and core temperatures were maintained around 37 6 C using a thermostatic heating blanket. The care and use of the animals reported in this study were approved by the Animal Care and Use Committee of the University of Wisconsin Madison, USA. The basilar membrane of the left cochlea was exposed after shaving a small hole into the scala tympani of the basal turn (see Russell & Sellick, 1978). Gold-coated polystyrene microbeads (diameter, 25 ìm; specific gravity, 1 05) were then introduced into the scala tympani and allowed to settle on the basilar membrane. These beads served to reflect the helium neon laser beam that was used to measure basilar membrane vibrations. A small glass coverslip was used to stabilize the air perilymph interface above the beads, but no attempt was made to seal the hole in the cochlea from a hydromechanical point of view. A closed-field acoustic system was sealed into the dissected ear canal to deliver sound stimuli to the eardrum. These stimuli were generated by a 16 bit digital-to-analog converter and transduced by a reverse-driven condenser microphone cartridge. Stimuli were calibrated within 1 mm of the eardrum using a condenser microphone with a 1 mm diameter probe tube. Digital precompensation was used to minimize distortion in the acoustic signal; as a result, all harmonics were at least 50 db below the fundamental component of the stimulus at db SPL. The acoustic distortion could not be measured accurately below 70 db SPL, but was predicted (by extrapolation from higher levels) to be between 75 and 90 db relative to the fundamental (i.e. 15 to 30 db SPL, depending on the stimulus frequency) when the stimulus level was 60 db SPL (cf. Fig. 2A). The physiological condition of the cochlea was monitored using compound action potential (CAP) recordings from a silver wire electrode placed in the round window niche (see Johnstone, Alder,

3 J. Physiol Harmonic distortion in cochlear mechanics 279 Johnstone, Robertson & Yates, 1979). CAP audiograms (2 34 khz in 2 4 khz steps; 10 ms tone bursts, 1 ms raised cosine rise and fall times) were estimated from oscilloscope recordings, and CAP input output functions were derived at selected frequencies (e.g. 10, 16 and 20 khz) using computer averaging techniques. CAP thresholds were defined when peak peak amplitudes were around 5 10 ìv. Full analysis of the mechanical data was restricted to three preparations in which the CAP thresholds were maintained to within 3 db of their initial values (in the 2 20 khz range) throughout the in vivo recording period. Mechanical responses were measured using a displacementsensitive heterodyne laser interferometer (Cooper & Rhode, 1992). The interferometer was coupled to the preparation using an epiillumination microscope with a super-long working distance lens (Nikon SLWD ² 5, NA 0 1). The laser was focused to a spot of 5 ìm diameter in the focal plane of the microscope. This spot had to be aligned perfectly with a reflective microbead in order to make measurements of basilar membrane motion, since the optical sensitivity of the interferometer was insufficient to make measurements from the nearly transparent surface of the basilar membrane itself. The phase difference between (i) the laser light reflected from the selected microbead and (ii) the light reflected from a fixed reference mirror was used to estimate the instantaneous displacement of the object under study (this process is described in full by Cooper & Rhode, 1992). In the present study, the optical phase differences were analysed by two purpose-built single-cycle phase meters which worked in anti-phase. The bandwidth of each phase meter was determined by an eighth-order low-pass Bessel filter with a 3 db cut-off frequency of 50 khz. The outputs of both phase meters were digitized at rates of 250 khz using a 16 bit analog-to-digital converter, and phase unwrapping (the removal of whole-cycle discontinuities in the phase meter outputs) was performed by software. Response amplitudes and phases were corrected for the frequency response of the recording system post hoc. The noise floor of the recording system was<5pm( Hz). With very few exceptions, the total harmonic distortion that could be attributed to the recording system was < 0 1 %. Basilar membrane tuning andïor input output functions were determined using 30 ms long tone bursts with 1 ms raised cosine rise and fall times. Stimulus repetition periods were 100 ms. Mechanical responses were digitized and averaged over sixteen to sixty-four presentations of a single stimulus. The averaged responses were then analysed by Fourier decomposition. Response components at integer multiples of the stimulus frequency, FÑ (i.e. 2FÑ, 3FÑ, etc.), Figure 1. Mechanical tuning characteristics of a site on the basilar membrane (BM) 3 mm from the base of the cochlea in two guinea-pigs A and C, normalized response amplitudes as a function of stimulus frequency for pure-tone stimuli ranging from 10 (or 30) to 100 (90) db SPL in 10 db steps (as labelled). Post-mortem data are indicated by &; these were obtained around 30 min post mortem at a single sound pressure level (80 db SPL). Data shown elsewhere indicate that the sensitivity of the post-mortem responses was independent of the SPL over most of the range tested in vivo (cf. 01 in Fig. 4). The sensitivity difference between the in vivo and post-mortem responses in A and C gives one measure of the gain of the cochlear amplifier. The maximum gain at each preparation s best frequency is 53 db, as shown by the double arrows. B and D, response growth rates derived from panels A and C, respectively. Compressive non-linearities (growth rates of less than 1 db db ) are evident at frequencies above 12 khz in guinea-pig GP204 and 14 khz in guinea-pig GP192. Increasing SPLs are indicated by increasing thickness of the line, with the two extremes of the range being labelled. The preparation s best frequencies (17 khz in GP204, 19 khz in GP192) are indicated by dotted lines.

4 280 N. P. Cooper J. Physiol were compared against components at nearby (but inharmonic) frequencies to determine their statistical significance. Data were considered reliable (P < 0 05) when their amplitudes exceeded the vibratory noise floor by more than two standard deviations (the noise floors were evaluated at eight points between ± 30 and ± 300 Hz from the frequency of each harmonic, thus controlling for any frequency andïor time dependencies in the background vibration level). Control responses were measured from the ossicles of the middle ear when the basilar membrane measurements had been completed. The ossicular measurements were usually made less than 100 ìm from the tip of the long process of the incus, but some measurements were also made from the footplate of the stapes. RESULTS Data from single locations on the basilar membranes of two guinea-pig cochleae illustrate the main findings of this study (Figs 1 4). Less extensive results from a third preparation were entirely consistent with the illustrated data. CAP threshold losses at the times when the first mechanical data were collected from each preparation were minimal (see Methods). In agreement with previous findings for cochleae in good physiological condition, the responses of the basilar membrane to tonal stimuli were sharply tuned at low SPLs and more broadly tuned at high SPLs (Fig. 1A and C). Compressive non-linearity (fundamental growth rates of less than 1 db db ) was evident only for tones around the best or most sensitive frequency of the preparation (Fig. 1B and D). The best frequencies for the preparations illustrated in Fig. 1A and C were 17 and 19 khz, respectively. The gain of the cochlear amplifier for low-level tones at each preparation s best frequency was 53 db (see double arrows in Fig. 1A and C). Figure 2 illustrates the harmonic content of the responses of the basilar membrane to two series of moderately loud tone bursts (60 db SPL in Fig. 2A, 80 db SPL in Fig. 2B). The components locked to twice the fundamental frequency of the stimulus (2FÑ; 12 ) are the largest of the response harmonics in either case. In fact, the higher harmonics of the responses (3FÑ, 4FÑ, etc.) rarely exceeded the noise floor of the recordings by statistically significant amounts (see Methods; statistically significant data are indicated with numeric symbols in Figs 2 5). The 2FÑ responses in Fig. 2A exhibit two clear peaks; one for stimuli around 8 5 khz (i.e. one-half of the preparation s best frequency) and another for those around 17 khz (the best frequency, cf. Fig. 1A). The maximum 2FÑ amplitude in the lower frequency peak (0 086 nm at 8 5 khz) is only 3 5 times lower than the FÑ amplitudeatthesamefrequency (0 304 nm). However, it is questionable whether this amount of distortion should be referred to as 28% of the FÑ response (i.e. 2FÑ = 11 db relative to FÑ, as indicated in Fig. 2A), since the 2FÑ responses to the 7 10 khz stimuli occur at frequencies which are subject to the effects of the cochlear amplifier, whereas the FÑ responses to the same stimuli do not (cf. Fig. 1A). An alternative way to assess the level of the distortion in the low-frequency 2FÑ peakisto estimate the levels of the stimuli in the khz range (whose frequencies are twice as high as those in the lowfrequency 2FÑ peak) which evoke fundamental responses with magnitudes similar to the 2FÑ response components in the low-frequency peak (e.g nm at 2 ² 8 5 khz). By analogy with other studies of the auditory periphery (e. g. Goldstein, 1967; Robles et al. 1991), these stimulus levels Figure 2. Harmonic distortion in basilar membrane responses to two series of moderately loud tone bursts Amplitudes of the fundamental (FÑ; 1, thick line), second (2FÑ; 12, medium thickness line) and third harmonic (3FÑ; 13, thin line) responses are shown for 60 db SPL stimuli in GP204 (A) and for 80 db SPL stimuli in GP192 (B). Symbols are plotted only for data lying more than two standard deviations above the frequency-dependent noise floor. The vertical reference lines (dotted) indicate the best frequencies of the two preparations (17 and 19 khz, respectively). The curves formed by the small & symbols overlying the 12 symbols in the 7 10 khz frequency range show the fundamental response amplitudes evoked by 10 db SPL tones (A) and 40 db SPL tones (B) of between 14 and 20 khz (the 10 and 40 db SPL data are plotted at one-half of the stimulus frequencies for comparison with the 2FÑ data; see text for reasoning).

5 J. Physiol Harmonic distortion in cochlear mechanics 281 define the effective levels of the distortion. The procedure for estimating the effective levels of the 2FÑ distortion for one particular frequency of stimulation (8 khz) is illustrated below, using the data of Fig. 3. Figure 3A shows amplitude growth functions for the various harmonics of a response to an 8 khz stimulus. The response components locked to twice the frequency of this stimulus ( 12 in Fig. 3A) occur at 16 khz and reach an amplitude of 0 05 nm at 60 db SPL (dashed line labelled a on graph). Corresponding growth functions for the responses to a 16 khz stimulus are shown in Fig. 3B (two data sets are actually shown in this panel to illustrate the repeatability of the measurements). In this case it is the fundamental components of the responses ( 1 and 01 ) which occur at 16 khz, and these reach an amplitude of 0 05 nm at 10 db SPL (dashed line labelled b on graph). The effectivelevelofthe2fñ responses to 60 db SPL tones at 8 khz is hence 10 db SPL, or 50 db relative to the stimulus level (dashed line labelled b a in Fig. 3C). Returning now to Fig. 2A, the effective levels of the 2FÑ distortion for all of the stimuli in the 7 10 khz range were estimated to lie around 10 db SPL. This is illustrated by the close correspondence between the 2FÑ data ( 12 ) and the curveformedbythesmall&symbols in Fig. 2A. This latter curve plots the amplitudes of the fundamental responses to 10 db SPL tones in the khz range at one-half of the actual stimulus frequency (e.g. the FÑ response evoked by a 16 khz, 10 db SPL tone is plotted at 8 khz for direct comparison with the 2FÑ response evoked by an 8 khz, 60 db SPL tone). The effective level of all of the distortion in the low-frequency 2FÑ peak of Fig. 2A can hence be considered to be 50 db lower than the 60 db SPL stimuli. The 2FÑ responses to the higher level tones shown in Fig. 2B are more evenly distributed across frequency than the ones shown in Fig. 2A, but they still peak slightly around 8 5 khz. In this case, the low-frequency 2FÑ peak does not occur at exactly one-half of the preparation s best frequency (19kHz, cf. Fig. 1C). However, the 2FÑ responses to the Figure 3. Effective levels of harmonic distortion on the basilar membrane A and B, amplitudes of the fundamental (FÑ; 1 and 01, thick lines), second (2FÑ; 12 and 02, medium thickness lines) and third harmonic (3FÑ; 13 and 03, thin lines) responses to 8 and 16 khz tones, respectively. Two sets of data are shown in B to illustrate the repeatability of the measurements. The data shownbytheopensymbolsinbwere collected 25 min before those shown by the filled symbols. Symbols are only plotted for data lying more than two standard deviations above the frequency-dependent noise floor. The sloping reference lines (dotted) indicate growth rates of 1 db db. C, the effective levels of the second harmonic distortion in the responses to the 8 khz tones. These levels were computed by subtracting the stimulus SPLs (e.g. a in panel A) from the SPLs of the 16 khz tones (e.g. b in panel B) that were required to evoke FÑ responses with the same magnitudes as the 2FÑ responses to the 8 khz tones. The dotted line in C shows the amount of 2FÑ distortion that could be attributed to the stimulus generation system (see Methods).

6 282 N. P. Cooper J. Physiol khz stimuli still occur at frequencies where the cochlear amplifier has a marked effect, while the FÑ responses to the same stimuli do not (cf. Fig. 1C). The response amplitudes in the low-frequency 2FÑ peak should hence be compared with the fundamental responses evoked by stimuli at frequencies twice as high as those shown in Fig. 2B (see above discussion of Figs 2A and 3). The effective levels of most of the 2FÑ responses in this case were around 40 db SPL (see curve formed by small & symbols in Fig. 2B), or around 40 db relative to the 80 db SPL stimuli. The method of estimating the effective levels of the distortion described above could not be applied to the responses evoked by tones much higher in frequency than one-half of the preparation s best frequency, as the steep high-frequency cut-off slope of each preparation s tuning characteristics (Fig. 1) meant that responses to very highfrequency stimuli (FÑ > 22 khz) were too small to measure below 100 db SPL (the maximum level used in these experiments). Hence, the only way to express the distortion in a potentially meaningful manner for high-frequency stimuli (e. g. when FÑ > 11 khz) was to relate the amplitudes of the harmonics directly to those of the fundamental responses (the validity of this method is considered further in the Discussion). For the largest of the responses shown in Fig. 2A and B, the absolute amplitudes of the FÑ responses were 3 nm (at 16 and 15 5 khz, respectively), while those of the 2FÑ responses were 0 07 and 0 1 nm. The second harmonic response levels for these high-frequency tones can hence be considered to be 33 and 29 db below the fundamental responses (see vertical arrows at 16 and 15 5 khz in Fig. 2A and B, respectively). Physiological vulnerability Figure 4 shows amplitude growth functions for the various harmonics observed in response to tones near the best frequencies of two preparations. Two sets of growth functions are shown for each preparation: one of these was recorded in vivo, when the cochlear amplifier was intact (curves 1, 12 and 13 ), and the other was recorded post mortem, when the amplifier was defunct (curves 01, 02 and ). The maximum sensitivity of the fundamental in vivo 03 data is 53 db greater than that of the post-mortem data in Fig. 4A, with the corresponding value in Fig. 4C being 42 db (see arrows comparing horizontal positions of lowamplitude FÑ responses in Fig. 4A and C). These values are directly comparable to the cochlear amplifier gains illustrated by the vertical arrows in Fig. 1, since the slopes of the relevant growth functions are all around 1 db db at low levels. The reason that the in vivo gain (or sensitivity enhancement) in Fig. 4C differs from that depicted in Fig. 1C is that the stimulus frequency in Fig. 4C is 1 khz Figure 4. The effects of SPL and physiological condition on basilar membrane distortion A and C, amplitudes of the fundamental (FÑ; 1 and 01, thick lines), second (2FÑ; 12 and 02, medium thickness lines) and third harmonic (3FÑ; 13 and 03, thin lines) responses to tones near the best frequencies of two preparations, in vivo (open symbols) and post mortem (filled symbols). Symbols are only plotted for data lying more than two standard deviations above the frequency-dependent noise floor. The sloping reference lines (dotted) indicate growth rates of 1 db db, and the horizontal lines depict the gains that can be associated with the cochlear amplifier in each preparation (53 db at 17 khz in GP204, 42 db at 18 khz in GP192). B and D, the same data with the amplitudes of the harmonic distortion expressed relative to the amplitudes of the FÑ responses.

7 J. Physiol Harmonic distortion in cochlear mechanics 283 lower than the preparation s best frequency. The main point to note in Fig. 4isthattheamountsofharmonicdistortion are far greater in vivo than post mortem. This is true in both absolute and relative terms (Fig. 4A and C, andband D, respectively). It is important to recognize that the in vivo distortion levels are still remarkably low, however. The highest absolute 2FÑ levels equatetolessthan1%ofthefñ responses, for example (see the 100 and 95 db SPL data in GP204 and GP192, respectively; 1 % = 40 db on the ordinates of Fig. 4B and D), and even the most distorted responses contain less than 4 % distortion (see the 40 and 60 db SPL data in Fig. 4B and D; 4 % = 28 db on the ordinates of Fig. 4B and D). It is also important to note that the in vivo distortion levels usually decreased, relative to the FÑ responses, with increasing stimulus levels. Typical rates of decrease were between 0 25 and 0 5 db db, with the 2FÑ levels typically varying from 30 db to 40 db relative to the FÑ responses over a db range of stimulus levels (as shown in Fig. 4B and D). Clear exceptions to this finding were only observed when relatively lowfrequency tones were played at low-to-moderate stimulus levels (one example is shown in Fig. 3A; in this case the distortion grew slightly, relative to the FÑ responses, for stimuli between 55 and 70 db SPL). DISCUSSION The cochlear amplifier as a source of harmonic distortion The harmonic distortion observed in this report has two characteristics which are particularly revealing: it is highly frequency dependent, particularly at low to moderate SPLs, and its magnitude decreases (relative to the FÑ responses) with increasing stimulus strength andïor deteriorating physiological condition. These characteristics imply that the distortion originates within the cochlea itself: an extracochlear source (e.g. in the sound delivery system, or in the vibrations of the ossicles of the middle ear) would most probably impart little frequency dependence, increasing distortion with increasing stimulus strength, and little or no dependence on physiological condition (see Goldstein, 1967, for related discussion on the issue of inter-modulation distortion in the cochlea, and Rosowski, 1994, for a review of the response characteristics of the middle ear). The dependence of the observed distortion on stimulus level and on physiological condition is very similar to that of the cochlear amplifier; both the gain of this amplifier and the amount of distortion (relative to the FÑ responses) are highest at low stimulus levels in vivo. Most researchers consider the cochlear amplifier to involve some form of feedback from the outer hair cells in the organ of Corti to the basilar membrane (see Kim et al. 1980; Zwicker, 1986; Yates, 1990; de Boer, 1991; Nobili & Mammano, 1996). Like many other types of hair cell (e.g. see Hudspeth & Corey, 1977), outer hair cells are known to operate non-linearly in their forward (i.e. mechanoelectrical) transduction mode (Kros, R usch & Richardson, 1992). The outer hair cells may also operate non-linearly in reverse (i.e. electro-mechanical) transduction (Santos-Sacchi, 1989; Evans, Hallworth & Dallos, 1991), although several lines of evidence suggest that reverse transduction is linear within the cells normal operating environment (e.g. Santos- Sacchi, 1993; Ren & Nuttall, 1995). For the purposes of this discussion, however, the exact details of the reverse transduction process are somewhat irrelevant; the most important points are that the outer hair cells are known to operate in a non-linear fashion in vivo (Dallos, 1986; Cody & Russell, 1987), and that they play an important role in cochlear amplification. The outer hair cells hence present themselves as an obvious source of the distortion observed in this report. Indeed, previous reports have suggested that outer hair cell non-linearities can account for most of the non-linear mechanical behaviour of the cochlea (e. g. see Zwicker, 1986; Patuzzi, Yates & Johnstone, 1989; Nobili & Mammano, 1996). Data interpretation: a highly simplified model Without further consideration, the idea that the cochlear amplifier could be responsible for the harmonic distortion observed in this report might appear to be inconsistent with some of the observations made in Figs 1 and 2. In particular, the clear peak in the 2FÑ distortion for stimuli around 8 5 khz (Fig. 2) might appear to clash with the absence of amplification for stimuli below khz (cf. Fig. 1). When the amplification is considered as just one part of an integrated hydromechanical filtering system, however, the observed responses can be understood quite easily. In order to illustrate this, one highly simplified model of cochlear mechanics is presented in Fig. 5A. This model can account for almost all of the salient features in the observed data using nothing more than two filters and a static non-linearity (it should be stressed that this model is not intended to replicate the physics of the cochlea in detail its purpose is simply to illustrate the types of signal processing that might be involved). The model illustrated in Fig. 5A is a highly simplified version of the feedback saturation model proposed by Zwicker (1986). The main simplification in this model involves the removal of Zwicker s original transmission line, such that all of the model s passive filtering characteristics are represented in a single processing block (Filter 1 in Fig. 5A). For the purposes of the present discussion, the characteristics of this filter have been adjusted to match the post-mortem tuning observed in one of the experiments from this report (GP204, cf. Fig. 1A). (Notethatonlythe amplitude transfer characteristics of Filter 1 are of relevance in the simple model, so the phase characteristics are ignored. Also note that Filter 1 includes the effects of middle ear transmission, whereas the data of Fig. 1donot.)Theoutput of Filter 1 is fed to a summing junction in the model, with the output of this summing junction representing the vibration of the particular location studied on the basilar membrane. A non-linear function f(u) is used in a positive feedback loop to represent the function of outer hair cells in

8 284 N. P. Cooper J. Physiol the organ of Corti. As described by Zwicker (1986), these cells are thought to augment the pressure acting on the basilar membrane in a non-linear fashion. For simplicity, the non-linear function of the model was constrained to the form of a first-order Boltzmann function: f(u) =áï(1+âe uïã ) áï(1+â). (The transduction characteristics of real hair cells are better described by second-order Boltzmann kinetics (cf. Kros et al. 1992), but these kinetics were considered too complex to be included in a simple model.) A second filter (Filter 2) is included to tune the feedback loop to the model s best frequency (17 khz, by analogy with that observed in vivo in GP204). The correlate of Filter 2 in Zwicker s (1986) model is a resonant mass spring damper circuit with a fixed Q 3dB, or quality factor, of 10. (It should be noted that the resonant circuits in Zwicker s original model were integral to the basic transmission line (i.e. Filter 1), such thattheeffectsoffilters1and2werenotentirelyseparable; as will be seen later, separating Filter 1 from Filter 2 introduces several shortcomings to the present model, but it also makes the model easier to understand.) The Q3dB of Filter2inthemodelofFig. 5A was set to 2 (i.e. the filter was slightly under-damped) to approximate the leveldependent tuning characteristics shown in Fig. 1A. Note that the phase transfer characteristics of Filter 2, unlike those of Filter 1, do affect the performance of the model in Fig. 5; the phase-lag of Filter 2 is zero, leading to perfect feedback, only at the model s best frequency. With the characteristics of the two model filters being fixed to match the data of Fig. 1A, the only real variables in the model of Fig. 5 are the three parameters of the Boltzmann function (á, â and ã). These three parameters were adjusted Figure 5. Model results A, highly simplified model comprising two filters (1 and 2), a static non-linearity and a positive feedback loop. The two filters are modelled on the fundamental tuning characteristics observed in one particular experiment (GP204; see Discussion for details), and the output of the summing junction represents the vibration of a single location on the basilar membrane. Amplitude and phase characteristics of Filter 2 are shown by thick and thin lines, respectively, with arrows indicating the appropriate ordinates. The phase characteristics of Filter 1 have no significance, and are hence omitted from the model. The non-linearity comprises an offset Boltzmann function: f(u) =áï(1+âe uïã ) áï(1+â).vertical reference lines in the two filter panels indicate the model s best frequency (17 khz). B, harmonic tuning characteristics at 60 db SPL. C, harmonic growth functions for best-frequency tones. Continuous lines in B and C show model predictions with active feedback (á = 15 0 Pa, â = 1 7, ã = 3 5 nm). The dotted line in C shows passive model predictions (á=0pa;themodel behaves linearly in this case, so harmonic distortion is precluded). For comparison, data from Figs 2A and 4A are replicated using numeric symbols in B and C, respectively.

9 J. Physiol Harmonic distortion in cochlear mechanics 285 by trial and error (using an interactive computer program) to approximate the harmonic distortion data in Figs 2 4. á was always adjusted to match the gain of the cochlear amplifier for low-level tones at the model s best frequency (note: for most realistic parameter combinations, the maximum closed loop gain of the model is determined by theslopeoftheboltzmannfunctionwhentheinputlevelis zero; denoting this slope as ö, the maximum closed loop gain is 1Ï(1 ö); to obtain a maximum closed loop gain of 53 db (as in Figs 1A and 4A), á was adjusted to make ö = Pa nm ); â provided a horizontal offset or asymmetry factor which was primarily responsible for introducing even-order distortion products (2FÑ, 4FÑ, etc.) into the model s output; and ã provided a horizontal gain or saturation factor which was responsible for controlling the odd-order distortion products (3FÑ, 5FÑ, etc.) as well as setting the threshold for compression in the fundamental responses. It is important to recognize the significance of the feedback loop in the model of Fig. 5. As described above, this loop was adjusted to make the closed-loop gain for low-level tones at the model s best frequency match the gain apparent in the data of Figs 1A and 4A. When the amount of feedback in the loop was reduced (relative to the input signal), either through the actions of Filter 2, or through the saturation of f(u) at high input levels, the closed-loop gain fell dramatically (cf. Yates, 1990). For the parameters given in Fig. 5, a feedback attenuation of just 6 db (i.e. a halving of the maximum amount of feedback) decreased the overall amplification by 47 db (i.e. from 53 to 6 db). Distorted amplification, amplified distortion, or both? The model described above can account for many of the features observed in the real cochlea, including both the bimodal distribution of the 2FÑ responses across frequency, and the general dominance of the 2FÑ responses over the other harmonics (cf. Fig. 5B and C). The general dominance of the 2FÑ responses is a straightforward consequence of the asymmetry that was introduced through the term â in f(u) (see above, and also the forthcoming section Odd- vs. evenorder distortion ). The bimodal distribution of the 2FÑ responses in Fig. 5B, on the other hand, has more complex origins. In order to illustrate this, it is instructive to consider the behaviour of the model in response to various frequencies of stimulation. This behaviour should allow us to distinguish quite clearly between distorted amplification, amplified distortion and the more general concept of nonlinear amplification in the cochlea. Naturally, stimuli at frequencies near to the model s best frequency will pass through both of the model s filters with little attenuation. These stimuli can hence drive the model s analogue of the outer hair cells (i.e. the function f(u)) well into its non-linear range (where the function is both asymmetric and saturating) even at low to moderate stimulus levels. As a result, considerable amounts of harmonic distortion can be generated at the output of the non-linearity. Not all of this distortion actually feeds through to the output of the model, however, due to the characteristics of Filter 2. In the case of the responses to near-best-frequency tones, all of the higher harmonics are bound to fall well above the centre frequency of Filter 2, and will hence be attenuated to some extent. The fact that Filter 2 attenuates the harmonics under these conditions not only reduces the amount of distortion which can reach the output of the model, it also prevents the distortion from being amplified substantially by the feedback loop. In fact, the only component of the model s responses that can be amplified substantially by the feedback loop is the fundamental component. Of course, if this component is amplified, it will in turn generate more distortion at the output of the model s non-linearity, but the distortion itself will still not be amplified (substantially) by the feedback loop. Under conditions of stimulation with near-bestfrequency tones then, most of the distortion observed at the output of the model can be considered to represent distorted amplification, as opposed to amplified distortion (the fundamental responses may be amplified, and this in turn may generate more distortion, but the distortion itself is not amplified directly). The harmonic tuning characteristics for near-best-frequency tones will hence reflect the tuning characteristics of the fundamental responses, giving a clear peak near the model s best frequency for low- and moderatelevel stimuli (as shown in Fig. 5B), and a more low-pass characteristic for higher level tones (not illustrated). Thesituationisdifferentfortoneswhichfallwellbelowthe model s best frequency. These tones may be attenuated slightly more than near-best-frequency tones by Filter 1, but the amount of attenuation is small in comparison to the dynamic range of the auditory system. The low-frequency tones can hence drive the model s analogue of the outer hair cells in much the same way that best-frequency tones do, particularly when the low frequency tones are presented at slightly higher input levels. The low-frequency tones can hence generate just as much distortion at the output of the model s non-linearity as the (lower-level) higher-frequency tones do. In the case of the low-frequency tones, however, the amount of distortion that feeds through to the model s output (via Filter 2) is not always lower than that which is generated by the non-linearity. In the special case of tones which fall near subharmonics of the model s best frequency, one particular component of the responses (e. g. the second harmonic, if the stimulus frequency is exactly one-half of the model s best frequency) will pass through Filter 2 without being attenuated. This particular component will hence be amplified substantially by the feedback loop (at least when the stimulus levels are sufficiently low), even though the fundamental responses (and the other harmonics) are not amplified. Most of the distortion seen under such conditions can hence be considered to be amplified distortion, as opposed to distorted amplification. The distortion is still generated by the outer hair cell non-linearity, but it is also amplified (directly) by the feedback loop. The tuning

10 286 N. P. Cooper J. Physiol characteristics associated with the harmonics of the lowfrequency tones can hence differ considerably from those associated with the fundamental responses (the fundamental tuning characteristics vary monotonically with frequency below khz, for example, while the second harmonic tuning has a clear peak at around 8 5 khz). The distinctions that can be made between amplified distortion and distorted amplification in the above model may well have counterparts in the real cochlea. If this is the case, then the two metrics used to quantify the levels of distortion in this paper (one expressing the distortion in terms of an effective level, the other expressing it with respect to the FÑ responses) may both be useful. The effective level metric should be particularly meaningful when the distortion components are amplified more than the fundamental component of the responses, as is the case in the amplified distortion described above. In effect, the effective level metric attempts to estimate the amount of distortion that is actually produced by the non-linearities in the system, without being confounded by the effects of amplification. There are two rather obvious problems with the effective level metric, however: firstly, effective levels can only be derived accurately when the stimulus frequencies are much lower than the preparation s best frequency. The reason for this is that the responses of the basilar membrane to stimuli which are well above the best frequency are too small to measure accurately with present-day techniques. Of course, one could use this observation to suggest that the effective levels of the distortion produced in response to high-frequency tones are extremely high (> 40 db greater than the stimulus levels in many instances, cf. Figs 2A and 4). However, such an argument makes little sense from a physical point of view: the hydromechanical filtering that takes place in the real cochlea prevents high-frequency energy from propagating beyond a certain point along the length of the cochlea (cf. Lighthill, 1981; de Boer, 1991), but it does not prevent high frequencies from being generated beyond that point (e. g. in the form of distortion products). In the case of the harmonic distortion generated by tones near to a preparation s best frequency then, the most appropriate metric to quantify the levels of the distortion must be the relative metric. The second problem with the effective level metric is that it involves comparisons between two responses which are evoked under very different conditions: in the case of the distortion whose effective level is to be computed, the various harmonics are always observed in the presence of a large response component at the fundamental frequency of the tone. In the case of the fundamental responses evoked for comparison with the distortionproducts,however,thereisnostimulation(and hence no large response component) at the frequency of the original stimulus. This leads to the possibility that various interactions (similar to those occurring in two-tone suppression, for example; cf. Cooper, 1996) could occur between the fundamental response and the various harmonics when the lower frequency tone is presented, but not when the higher frequency tone is used. This suggestion is supported by data obtained at high SPLs from the model of Fig. 5 (not illustrated in the figure), and also by some of the observations made in the real cochlea (note the slightly non-monotonic growth of the distortion in Fig. 3A, and the correspondingly rapid decrease in the effective levels of the 2FÑ responses above 80 db SPL in Fig. 3C, for example). It should be noted here that the model of Fig. 5 does not replicate all of the features observed in the real cochlea: the amount of distortion produced by tones just above the best frequency is severely underestimated by the model, for example, and the slopes of the harmonic growth curves in Fig. 5C are much higher in the model than in the real data. The discrepancies between the model and the real data are most likely to reflect the severe simplifications that were made in the model (as stated earlier, the model was not intended to replicate the physics of the cochlea in detail). A more realistic implementation of the model would have to combine the functions of Filters 1 and 2 into a single transmission line (e.g. as in Zwicker, 1986), or even into a distributed 3-dimensional structure (e. g. as in Nobili & Mammano, 1996). The additional degrees of freedom associated with these types of model should permit abovebest-frequency distortion products to be trapped at their site of generation (mainly through the effects of the mass of the cochlear partition), while allowing below-best-frequency harmonics to propagate freely towards the apex of the cochlea. Preliminary results from one state-of-the-art transmission line model (Geisler & Sang, 1995) indicate that this is indeed the case (C. D. Geisler, personal communication). Odd- vs. even-order distortion One of the most surprising features of the data in this report is that the 2FÑ componentsarethestrongestofthe harmonic distortion products on the basilar membrane. This is surprising for two reasons. In most non-linear systems, even-order distortion (such as that at 2FÑ) is accompanied by large DC shifts. But DC shifts in the basal turns of the cochlea are known to be extremely small, both on the basilar membrane (LePage, 1987; Cooper & Rhode, 1992) and in the receptor potentials of the outer hair cells (Cody & Russell, 1987). In the model of Fig. 5,thelevelsofthe2FÑ responses depend strongly on the asymmetry of the embedded non-linearity (i.e. on â, which controls the horizontal offset of the Boltzmann function), but DC shifts are precluded by the bandpass filter in the feedback loop. In the real cochlea, there is no obvious analogue of a bandpass filter (like Filter 2), but the overall function of the feedback loop appears similar (i.e. the closed-loop gain appears large for response components which fall close to the best frequency, but DC shifts are precluded). It is not clear exactly how this situation arises in the real cochlea, particularly when one considers that the DC shifts appear to be minimized (despite the presence of strong even-order

11 J. Physiol Harmonic distortion in cochlear mechanics 287 distortion) even in the receptor potentials of the hair cells themselves (Cody & Russell, 1992). The fact that the 2FÑ components are the dominant harmonics in the mechanics of the cochlea is also surprising by comparison with findings on inter-modulation (or twotone) distortion in the cochlea. The most prominent distortion products in most studies of inter-modulation distortion are odd-order products (e. g. the cubic distortion products 2FÔ Fµ,3FÔ 2Fµ, etc.,where FÔ and Fµ are the fundamental frequencies of the two stimulating tones). Even-order distortion products (e. g. the quadratic terms Fµ FÔandFµ + FÔ) are undoubtedly present throughout the system (note the oto-acoustic emissions and neural recordings by Kim et al. 1980; the mechanical recordings of Cooper & Rhode, 1997; and the hair cell recordings of Cheatham & Dallos, 1997), but their magnitudes rarely exceed those of the odd-order terms. This should not be taken as evidence that inter-modulation distortion and harmonic distortion are distinct processes within the cochlea, however; once again, preliminary evidence from various mathematical models (e. g. Geisler & Sang, 1995) suggests that both types of distortion can be produced in appropriate quantities by single hair cell non-linearities (C. D. Geisler, personal communication). Comparison with previous work As pointed out in the Introduction, very few studies of cochlear mechanics have considered harmonic distortion in the past. This may be due, in part, to technical limitations. The distortion observed in the present report is limited even when the cochlear amplifier is working hard, and it would fall well below the detection limits of many previous measurement techniques (e.g. the M ossbauer technique, and some forms of optical interferometry cf. Rhode, 1971; Sellick et al. 1982; Robles et al. 1986; Ruggero et al. 1997). The fact that the distortion levels decrease with deterioration in the physiological condition of the cochlea (cf. Fig. 4) might also have contributed to the findings of previous reports. LePage (1987) reports harmonic distortion levels to be45 db below the fundamental responses to best-frequency stimuli in the basal turn of the guinea-pig cochlea, for example. This level is similar to that found post mortem in the present report, and seems quite consistent with the linearity of the fundamental responses (i.e. the lack of compression at any frequency) in LePage s study. Cooper & Rhode s (1992) observations in the hook region of the cat cochlea are also consistent with the present report: they show significant levels of distortion (harmonics > 40 db relative to the fundamental response) only for stimuli above 90 db SPL. Cooper & Rhode s observations were made in reasonably healthy cochlea (i.e. significant amounts of compression were observed at the preparation s best frequencies), but they were limited to low stimulus frequencies by the bandwidth of their recording system. As in the present report, the second harmonic was the largest of the harmonics in the cat cochlea. In conclusion, the present report presents the first detailed measurements of harmonic distortion in the mechanics of physiologically normal mammalian cochleae. The fact that the observed distortion levels are low (< 4 % total harmonic distortion), even when the cochlear amplifier is working hard, is considered to reflect the presence of frequency selectivity in the feedback loop of the cochlear amplifier. The fact that the second harmonic is the largest distortion component in the basilar membrane response to a pure tone suggests that considerable asymmetry exists in the nonlinearities of cochlear mechanics. Cheatham, M. 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