Non-linear response to amplitude-modulated waves in the apical turn of the guinea pig cochlea

Hearing Research 174 (2002) 107^123 www.elsevier.com/locate/heares

Non-linear response to amplitude-modulated waves in the apical turn of the guinea pig cochlea S.M. Khanna




Columbia University, Department of Otolaryngology/Head and Neck Surgery, 630 West 168th Street, New York, NY 10032, USA Received 3 June 2002; accepted 19 August 2002

Abstract Mechanical vibrations of the Hensen’s cells were measured in the apical turn of the cochlea in living guinea pigs, in response to amplitude-modulated (AM) sound. The FFT of the input wave consisted of spectral components at the carrier frequency C and two sidebands (C - M) separated from the carrier by the modulation frequency M. The FFT of the velocity response consisted of components at: (i) the modulation frequency M, and harmonics n M; (ii) Carrier frequency C and sidebands (C - n M); (iii) harmonics of the carrier frequency and their side bands (2C - n M); (3C - n M); (4C - n M); T n = 1,2,3,T,10. The carrier and the first pair of side bands were broadly tuned and nearly linear. Other components were sharply tuned and highly non-linear, suggesting a different origin. Evidence is presented that these components are generated in the non-linear stereocilia dynamics. An important function of this non-linearity is to demodulate the AM wave to extract information contained in the modulation. 5 2002 Elsevier Science B.V. All rights reserved. Key words: Guinea pig cochlea; Cochlear mechanics; Apical turn; Amplitude modulation; Non-linear velocity response; Stereocilia non-linearity; Demodulation

1. Introduction In amplitude modulation the amplitude of a higher frequency sine wave (carrier frequency) is modulated with a low frequency wave (modulation frequency). As a consequence the amplitude of the carrier wave changes with time. The degree of increase and decrease in amplitude depends on the depth of modulation. The change is highest at 100% modulation. At 100% modulation the carrier peak amplitude varies between zero and 2. Although the simplest carrier is a sinusoidal wave, it can also be a complex wave; similarly, the modulating wave can also be complex. Amplitude-modulated (AM) waves are common; for example, they bring us speech and music in the AM radio receivers. In this application speech and music is amplitude modulated onto a radio frequency carrier.

* Tel.: +1 (212) 305-3993; Fax: +1 (212) 305-4045. E-mail address: [email protected] (S.M. Khanna).

On receiving these waves at the radio receiver the carrier is demodulated to recover the original speech or music. An AM demodulator is basically a recti¢er in which one polarity of the wave is preferred over the other (Schwartz, 1990). Natural speech is also a modulated wave. The carrier generated by the vocal cords is complex. The motion of the lips, tongue, mouth, etc. modulates the complex carrier both in amplitude and frequency. Clear signs of this modulation can be seen in the spectrograms of all the vowel and consonant sounds. The spectrograms contain frequency bands (formant frequencies) that characterize the sound. The frequency and amplitude of these frequency bands change slowly with time in a way that is characteristic of the sound (Fletcher, 1953). In order to recover the low frequency changes contained in each formant, it is necessary to demodulate the modulated carrier. It is suggested that a basic function of the auditory system is to provide this frequency and amplitude demodulation. In the present paper only the amplitude demodulation is considered. The ear is a sensitive detector of amplitude and fre-

0378-5955 / 02 / $ ^ see front matter 5 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 5 9 5 5 ( 0 2 ) 0 0 6 4 5 - 7

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quency modulation of sound. Small changes in intensity of sound can be detected throughout the frequency and intensity range of hearing (Licklider, 1960). Responses to AM sound have been investigated at all levels of the auditory system: cochlear mechanics (Brundin et al., 1991, 1992); round window microphonics (Verweij and Rodenburg, 1978); auditory nerve (Cooper et al., 1993; Frisina et al., 1996; Greenwood and Joris, 1996; Javel, 1980; Joris and Yin, 1992; Khanna and Teich, 1989; Smith and Brachman, 1980); cochlear nucleus (Frisina et al., 1990a,b; MMller, 1974, 1976; Rhode and Greenberg, 1994; Glattke, 1969); inferior colliculus (Brugge et al., 1993; MMller, 1987; Rees and MMller, 1987; Rees and Palmer, 1989); auditory cortex (Bieser and Muller-Preuss, 1996; Eggermont, 1994; Schulze and Langner, 1999); and scalp (John and Picton, 2000; Lenhardt, 1972). AM must be an important signal because it is represented at every level of the auditory system (Frisina, 2001). Instantaneous velocity amplitude s(t) of an AM wave as a function of time t, may be described by: sðtÞ ¼ A sinð2Z Ct þ P Þf13Bcosð2Z MtÞgT1;

ð1Þ

where A is the amplitude and P the phase angle of the carrier frequency C, and B is the amplitude of the modulating frequency M. The spectrum of the AM wave consists of the carrier frequency C of amplitude A and two sidebands at frequencies (C - M) of amplitude AB/2 (Schwartz, 1990). When A = 1 and B = 1 (100% modulation), the spectrum consists of the carrier C of unit amplitude and two sidebands (C - M) of 0.5 amplitude. The spectral components in the demodulated wave depend on the demodulator characteristics. In an ideal linear demodulator a DC component and the modulation component are present. In a square law demodulator the components are at the modulation frequency (M), its second Harmonic (2M), at twice the carrier frequency (2C), with sidebands (2C - M) and (2C - 2M). The present study is focussed on the understanding of how the non-linearity in the inner ear processes the AM signal. Mechanical vibrations of the sensory cells were measured at the apical turn of the living guinea pig cochlea in responses to AM sound. It was found that the fast Fourier transform (FFT) of the measured velocity^time waveforms contained a very large number of spectral components, including the modulation frequency. The purpose of the present paper is to analyze the responses, and to understand the origin and signi¢cance of the di¡erent sets of spectral components present in the response.

2. Method Experiments were done on the apical turn of cochlea in living guinea pigs. The anesthesia and surgical approach used to access the cochlea has been described earlier (Hao and Khanna, 1996; Khanna and Hao, 1999a). Non-pigmented guinea pigs of Hartley strain, weighing between 220^250 grams, from Charles River Laboratories were used. Approximately one third of the apical turn was exposed, being careful to keep Reissner’s membrane and the stria vascularis intact. The apical turn of the organ of Corti was viewed with an incident light slit confocal microscope (Koester, 1980, 1990; Koester et al., 1989, 1994). Vibrations of the selected cells were measured with a confocal heterodyne interferometer coupled to the microscope (Willemin et al., 1988, 1989; Khanna et al., 1996). Details of the instrument and its application to the cochlear measurements have been described earlier (Khanna and Hao, 1999a,b). The microscope and interferometer share the same objective lens. The objective lens with a custom designed dipping cone for water immersion was optically coupled to the opened apical turn of the cochlea with a thin plastic tube ¢lled with tissue culture medium and forming a £uid-tight seal (Khanna and Hao, 1999a). 2.1. Sound system Signals were generated and the response measured with a 16 bit resolution DSP system using two D/A and two A/D converters (Ariel). The signals were synthesized using a PC. They were applied to the acoustic transducer (Sokolich, 1981) after passing through an anti-aliasing ¢lter (Wavetek 752A). The acoustic transducer was coupled to the ear canal, and the sound pressure was monitored with a Sokolich probe microphone (Sokolich, 1981) positioned approximately 7 mm in front of the tympanic membrane. The probe microphone output was ampli¢ed and passed through an anti-aliasing ¢lter (Wavetek model 716) and applied to the second channel of the A/D converter. Samples of the time waveforms for both channels were stored permanently on an optical disk. 2.2. Time waveform of the acoustic signal The time waveform of the sound pressure generated in a test cavity is shown in Fig. 1 at a carrier frequency of 269 Hz, modulation frequency of 2.44 Hz and 100% modulation. Driving voltage to the sound system in this test was 2.0 V (highest value used in the experiment). Envelope is smooth and its shape is symmetrical.

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Fig. 1. Sound stimulus used in the experiment was an AM wave, modulation frequency was 2.44 Hz. 100% modulation was used. Carrier frequency was from 25 to 2000 Hz. Time waveform of the sound pressure generated in a test cavity is shown at a carrier frequency of 269 Hz. Driving voltage to the sound system in this test was 2.0 V (89 dB SPL), the highest value used in the experiment. Envelope is smooth and its shape is symmetrical.

2.3. Sound pressure in the ear canal The sound pressure produced in the ear canal at the carrier frequency is shown at three signal levels applied to the acoustic system: 2.0, 1.0 and 0.5 V (Fig. 2). The pressure decreased by a factor of 2.5 between 50 Hz and 300 Hz. Above 300 Hz changes in sound pressure were small. The pressure in this frequency region corresponding to 2.0 V peak drive is about 0.5 Pa (88 dB SPL), and to 0.5 V drive is about 0.125 Pa (76 dB SPL).

used in our experiments, the spectrum of the input consists of the carrier frequency C, and sidebands (C - M), second harmonic (2C), and sidebands (2C - M) and (2C - 2M). A large number of additional components present in the cochlear output indicates that the trans-

2.4. Velocity non-linearity does not arise in the experimental hardware The velocity non-linearity is usually seen at high sound pressure levels (SPL s 70 dB). It is important to make sure that the observed velocity non-linearity does not arise in the experimental apparatus. It is clear that the signal generation system or the sound system does not produce the spectral components seen in the FFT of velocity^time waveforms. When terminated in a resistive load and driven at the highest AM signal level

Fig. 2. Sound pressure measured in the guinea pig ear canal at the carrier frequency at three signal levels: 2.0, 1.0 and 0.5 V peak. Above 300 Hz, sound pressure at the highest level is about 0.5 Pa (W88 dB SPL), and at the lowest level about 0.12 Pa (W76 dB SPL). SPL in dB is shown on the right hand ordinate axis.

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formation between the input and output is highly nonlinear. For sinusoidal signals, the non-linearity of the velocity waveforms measured at the reticular lamina goes down dramatically after the sacri¢ce of the animal, although the input acoustic signal level remains the same (Hao and Khanna, 2000). Harmonic distortion of the demodulator was less than 0.1% (360 dB) at a signal frequency of 1000 Hz and a peak deviation of 40 kHz (which re£ects the velocity amplitude). The highest peak deviation in our experiments was about 17 kHz. The distortion level should therefore be lower than 360 dB. For sinusoidal signals, asymmetry of the velocity waveform was higher at OHC1 as compared to a Hensen’s cell, even though the vibration amplitude was lower at the OHC1 (Khanna and Hao, 1999b). The observed asymmetry does not increase with velocity amplitude and, therefore, cannot be produced in the demodulator. The distortion components are not generated in the sound system or the demodulator; they are produced within the inner ear. 2.4.1. Data acquisition system Details of the DSP system have been described in an earlier paper (Hao and Khanna, 2000). In the present study, the sampling rate used for the D/A and the A/D converters was 5 ks/s (sampling period 200 Ws). Samples were captured in 2048 bins, and signal duration was approximately 410 ms. The modulation frequency was 2.44 Hz, 100% modulation was used, the carrier frequency range was 25^2000 Hz, measurements were made at approximately 40 frequencies. Signal level was 0, 36, 312 dB re 2.0 V. Responses were not averaged. The cut-o¡ frequency of the anti-aliasing ¢lters in both channels was set to 5 kHz.

the frequency demodulator will be equivalent to a constant velocity that, if integrated, will produce a false linear displacement with time. The frequency demodulator itself is AC coupled and cannot contain any DC component. The DC component at the output of the A/D converter was therefore subtracted before the velocity was integrated (Teich et al., 1989b). For sinusoidal vibrations, the vibration amplitude X(t) is related to the interferometer output voltage V(t) by the following relationship: X ðtÞ ¼ V ðtÞ=4:356Uf 2.5.1. Polarity of interferometer response When measuring in the apical turn of the cochlea the response was negative when the reticular lamina was moving away from the objective lens, i.e. towards the scala tympani. The results reported in this paper are based on measurements on seven animals in which the DC response was measured. Responses from two of these are reported in the present paper. These were selected because we have the most complete set of data in these two animals. The responses of the other ¢ve animals are similar to the ones reported in this paper. The carrier harmonics and the multiple sidebands were seen in all animals. The Institutional Animal Care and Use Committee (IACUC) of Health Science Division of Columbia University approved the care and use of the animals reported in this study.

3. Results 3.1. Velocity tuning curve

2.5. Method of analyzing the data Velocity^time waveforms v(t) were calculated directly from the interferometer output voltage V(t): 2Z vðtÞ ¼ V ðtÞ; Ls where velocity v is in meters/s ; L = 4 Z/V, V = 0.476 Wm (water) and s = demodulator sensitivity in V/Hz = 1.65U1035 V/Hz. vðtÞ m=s ¼ 1:442U1032 V ðtÞ The velocity spectrum was obtained by taking a FFT of the velocity^time waveform v(t). The vibration amplitude was calculated from the velocity^time waveforms after removing DC o¡sets. The zero of the A/D converter drifts over time, creating a small DC o¡set. A DC voltage at the output of

The velocity of a Hensen’s cell in the ¢rst preparation is shown as a function of frequency in Fig. 3a and the corresponding phase curve in Fig. 3b. Both these measurements were made using pure sinusoidal signals. Characteristic frequency (CF) was 280 Hz. Velocity amplitude at CF was 2.6U1033 m/s/Pa. A smaller peak was located at 550 Hz. The ratio of the velocity amplitude of the two peaks was 2.1. Phase response refers to the sound pressure phase near the tympanic membrane. Below 500 Hz the phase slope is steep (31.51‡/Hz) followed by a shallow slope (30.23‡/Hz) between 500 and 1200 Hz. and a still shallower slope at higher frequencies. 3.2. Quality of the preparations In the ¢rst preparation the characteristic frequency of velocity tuning curve is about 280 Hz, Q10dB of the

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Fig. 3. Normalized velocity amplitude tuning curve of the outer Hensen’s cell. The curve is plotted on a log scale. CF was 280 Hz, velocity amplitude at CF was 2.6U1033 m/s/Pa. Second smaller peak was located at 550 Hz. The ratio of the velocity of the two peaks was 2:1. This ratio suggests that the cochlea was in good physiological condition. The phase response refers to the sound pressure phase near the tympanic membrane. Below 500 Hz the phase slope is steep (31.51‡/Hz), followed by a shallow slope (30.23‡/Hz) between 500 and 1200 Hz and a shallower slope at higher frequencies. Note the phase curve is plotted on a linear frequency scale.

tuning curve is 0.91. Q10dB in previous experiments has ranged from 0.4 (worst) to 1.0 (best). In the present experiment phase slope S1 is 31.51‡/Hz. Phase slope for S1 in previous experiments has ranged from 31.0o /Hz (worst) to 31.6o /Hz (best) (Khanna and Hao, 1999a). Therefore, in the presented results both amplitude and phase measures indicate a good preparation. In the second preparation the characteristic frequency of velocity tuning curve was about 540 Hz, Q10dB of the tuning curve is 0.76. The second preparation was also in quite good condition. 3.3. Velocity^time waveforms of cellular vibration in response to AM sound Velocity^time waveforms measured at a Hensen’s cell at the outer edge of the reticular lamina are shown in Fig. 4 (top). The input voltage applied to the sound system was the same at each carrier frequency (2.0 V peak, W88 dB SPL). Three plots with carrier frequencies of 269, 415 and 464 Hz are shown. Below CF (269

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Hz), the velocity envelope is asymmetrical. Velocity is higher in the negative direction. Above CF (464 Hz) the envelopes are nearly symmetrical. The shape of the tuning curve in Fig. 3 shows the envelope velocity amplitude as a function of frequency. The peak amplitude of the envelope decreases as we approach the notch region of the tuning curve (430 Hz). The amplitude increases again as we go to higher frequencies. At 415 Hz the envelope looks jagged. Displacement waveforms are shown at the same three frequencies in Fig. 4 (bottom). These were obtained by integrating the velocity waveforms that are shown above. The displacement envelopes are irregular (not smooth). The displacement envelope at 269 Hz is asymmetric and displacement is higher in the positive direction. The velocity envelope above from which it is derived was also asymmetric; however, the velocity was higher in the negative direction. At 415 Hz the time waveform is a large displacement at the modulation frequency (note that the vertical scale is di¡erent at 415 Hz). The carrier frequency amplitude is small compared to that of the modulation frequency. At 464 Hz the amplitude of the envelope goes down but its envelope shape remains irregular. 3.4. Velocity^time waveforms of individual carrier cycles Individual cycles of velocity^time waveforms from two di¡erent parts of the AM envelope at 269 Hz are shown in Fig. 5, between 100^110 ms from the start of the envelope (left), and between 200^210 ms (right). Although the carrier in the acoustic stimulus is sinusoidal, the velocity response is distorted. Distortion is less when the amplitude is low (left). Distortion is higher when the amplitude is highest between 200^210 ms (right). Velocity is asymmetric. It is higher during the negative part of the cycle. The positive peak of the wave is broader than the negative peak. Individual cycles of waveforms were examined over a broad frequency range. The distortion was frequency dependent ; it was lower at frequencies above 552 Hz. 3.5. Velocity spectrum The FFT of the velocity^time waveforms shown in Fig. 6 are presented at two carrier frequencies: 269 and 464 Hz. A linear frequency scale is used. At 269 Hz seven carrier and carrier harmonic clusters are seen, each with many sidebands. The carrier harmonics and sideband complexes are present at all carrier frequencies. The total number of carrier harmonics that can be seen is determined by the selection of the upper frequency limit of measurement, which in our case is 2 kHz. Therefore, as the carrier frequency is increased fewer spectral clusters are seen. The eighth spectral clus-

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Fig. 4. Velocity measured at the Hensen’s cell at the outer edge of the reticular lamina (top three frames). Input to the sound system was 2.0 V peak (W88 dB SPL). Response was not averaged. The total duration of the envelope is about 400 ms. Input voltage applied to the sound system was same at each carrier frequency. Three plots are shown with carrier frequencies of 269, 415 and 464 Hz. The velocity envelope is asymmetrical below CF (269 Hz), velocity is higher in the negative direction. Above CF (464 Hz) the envelope is nearly symmetrical. The shape of the tuning curve (Fig. 3) determines the relative response amplitudes at the three frequencies. The peak amplitude of the envelope decreases as we approach the notch region of the tuning curve. The amplitude increases again as we go to higher frequencies. At 415 Hz the envelope looks jagged. Time waveforms of the integrated velocity are shown at the same three frequencies (bottom three frames). The displacement envelope at 269 Hz is asymmetric and displacement is higher in the positive direction. The velocity envelope from which it is derived (top) was also asymmetric; however, the velocity was higher in the negative direction. At 415 Hz the time waveform shows a large displacement at the modulation frequency (note the vertical scale is di¡erent). The range of frequencies over which the modulation component is seen is very narrow.

ter seen around 1.0 kHz is due to the oscillating mirror of the confocal microscope. Stopping the mirror removes this component but leaves the rest of the spectrum una¡ected. Details of the spectral clusters are shown in the expanded view at the top of the Fig. 6. The spectral components around the carrier frequency (269 Hz) and its ¢rst two harmonics (538, 807 Hz) are shown in the top three panels. The frequency scale has been expanded so that the individual sideband components can be clearly seen. The sidebands are separated from the carrier and its harmonics by - nU2.44 Hz (multiples of the modulation frequency), where n = 1,2,3,T,10. In general the amplitude of the sideband component decreases with increasing sideband number.

3.6. Second preparation The FFT of the velocity^time waveforms obtained in a second preparation at three carrier frequencies (268, 390 and 463 Hz) is shown in Fig. 7. The modulation frequency was 9.8 Hz. At 268 Hz, six carrier harmonics (2C, 3C, 4C, 5C, 6C and 7C) and their sideband complexes can be seen. The sidebands are separated from the carrier or carrier harmonics by - nU9.8 Hz, where n = 1,2,3,T,8. At some of the harmonic frequencies up to 16 sidebands can be seen. In this preparation the frequency resolution is 2.4 Hz therefore spectral components between the sidebands can be seen. Within each cluster of the sidebands the noise £oor is raised. At 390

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the modulation component at 2.44 Hz is sharply tuned and the peak is located in the region of the fundamental peak. 3.9. Phase di¡erence between the sidebands and the carrier

Fig. 5. Expanded view from Fig. 4, to show the individual cycles of the velocity wave in two di¡erent parts of the envelope at 269 Hz, between 100^110 ms (left), and 200^210 ms (right). Individual cycles are not sinusoidal, they are distorted. The degree of distortion increases with the amplitude and varies with frequency. Distortion is higher between 200^210 ms.

Hz noise is higher by a factor of 50. The increase in noise £oor suggests that there is ampli¢cation associated with the process that generates the sidebands and this ampli¢cation also raises the noise level. 3.7. Response of the auditory nerve The FFT of a PST histogram from a cat single auditory nerve ¢ber is shown in Fig. 8. The CF of the ¢ber was 1596 Hz, carrier frequency (C = 1596 Hz), modulation frequency (M = 39 Hz), 100% modulation. The sound pressure level was 40 dB SPL. Spectral components can be seen corresponding to the (i) modulation frequency (M) and its harmonics (2M, 3M, 4M), (ii) carrier frequency (C), second harmonic frequency (2C) and third harmonic (3C), (iii) side bands associated with the carrier frequency and second harmonic (C - M), (C - 2M) and (C - 3M). For details of the experiment see Khanna and Teich, 1989 (Note : this ¢gure could not be plotted with higher resolution because the original data ¢le was no longer available.). 3.8. Tuning curves of the carrier and the ¢rst three pairs of sidebands in the second preparation The tuning of the carrier, sidebands and the modulation components in the second preparation are shown in Fig. 9. The carrier is broadly tuned with the CF at about 550 Hz. Tuning of the ¢rst pair of sidebands (C - M) is nearly the same as the carrier, although there are di¡erences between the sidebands (C+M) and (C3M) below 400 Hz. The tuning of the higher sidebands (C - 2M), (C - 3M) is di¡erent from the carrier. They are more sharply tuned. Their tuning peaks are located in the same region as the fundamental peak. The curve for

The phase di¡erences between the carrier and sidebands (C3M), (C32M), (C33M) are shown in Fig. 10. Note the linear frequency scale. The phase di¡erence between (C3M) and C is approximately 180‡. The phase di¡erence between (C32M) and C is frequency dependent. The phase changes between 400 and 600 Hz correspond to the tuning seen in this region. Beyond 1250 Hz the phase slope is about 0.5‡/Hz. The phase di¡erence between (C33M) and C shows a similar pattern but the phase di¡erence is greater. 3.10. Amplitude dependence of the carrier and the sidebands The ratio of the amplitudes determined at two sound levels (88 and 76 dB SPL) for the carrier (C) and the sidebands (C - M and C - 2M) is given in Fig. 11. The ratio of the sound pressure was 4 (12 dB). The ¢gure shows the ratio of the velocity amplitude observed at the higher sound level to that at the lower sound level. The ratio of carrier amplitude (C) was frequency dependent and varied between 3.0 and 5.2, except in the notch region around 420 Hz, where it was 1.2. The ratio of the sideband amplitude (C - M) was also frequency dependent. It ranged between 2.1 (340 Hz) and 5.2 (620 Hz), except in the notch region (40 Hz), where the ratio was 0.8. There were di¡erences between the ratios for (C+M) and (C3M). The ratio for the sidebands (C - 2M) was highly frequency dependent, and ranged between 0.8 and 45. The ratio was di¡erent for (C+2M) and (C32M). The maximum ratio of 45 was at 240 Hz and the second largest ratio of 20 was at 540 Hz. Compared to the ratio for (C) and (C - M), the ratio for (C - 2M) was much larger. The ratio for (C - 3M) sidebands was similarly large (up to 60, not shown). The amplitude dependence is only slightly non-linear at the carrier and the ¢rst pair of side bands. It is highly nonlinear at the higher order of sidebands. 3.11. Amplitude dependence of the modulation component The velocity amplitude of the modulation component determined at two levels of signal input (0.5 V, 2.0 V) at 463 Hz is shown in Fig. 12. At the higher level (solid line) an increased response is seen between 400 and 650 Hz. This increase nearly disappears at the lower level. For an increase in the input by a factor of 4, the veloc-

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Fig. 6. FFT of the velocity^time waveforms at two frequencies: 269 and 464 Hz. Signal level was 2 V (peak). Linear frequency scale is used. At 269 Hz, seven carrier and carrier harmonic clusters are seen, each with many sidebands. The carrier harmonics and sideband complexes are present at all carrier frequencies measured. The highest number of carrier harmonics that can be seen is determined by the upper frequency limit of measurement, which in our case is 2 kHz. Cluster seen near 1.0 kHz is due to the oscillating mirror of the confocal microscope. In the top three panels, an expanded view of the carrier and the ¢rst two carrier harmonics with their sideband complexes are shown. The ¢rst spectral complex is centered round the carrier frequency (C = 269 Hz), sidebands are separated from the carrier by - M, - 2M, - 3MT, - 11M, where M is the modulating frequency (2.44 Hz). The second spectral complex is located at 2Ucarrier frequency (2C = 538 Hz), and the third one at 3Ucarrier frequency (3C = 807 Hz). Both complexes have an array of side bands. Fewer carrier harmonic complexes are seen at the carrier frequency of 464 Hz due to the 2000 Hz frequency cuto¡.

ity of the modulation component increased (1.4U1034 to 8.7U1033 ) by a factor of 62. The increase in the modulation component amplitude is therefore highly non-linear.

3.12. FFT of the ear canal sound pressure Ear canal sound pressure was measured in parallel with the velocity measurement at the reticular lamina

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Fig. 7. FFT of the velocity^time waveforms obtained in a second preparation at three carrier frequencies (268, 390 and 463 Hz). Modulation frequency was 9.8 Hz. At 268 Hz, six carrier harmonics (2C, 3C, 4C, 5C, 6C and 7C) and their sideband complexes can be seen. At some harmonic frequencies up to 16 sidebands are present. The frequency resolution of the FFT is 2.44 Hz. The noise background is shown by the 2.44 Hz components that appear solid black in the graphs. The magnitude of these components is higher within each cluster as compared to their level between the clusters. The peak amplitude of the noise within a cluster occurs at the carrier or harmonic frequency of the cluster (C, 2C, 3C). At the 390 Hz cluster the peak amplitude of the noise is about 50 times higher compared to the average amplitude on either side of the cluster. This suggests that the ampli¢cation process associated with the generation of the spectral clusters also ampli¢es the noise £oor.

described in Fig. 4. FFT of the sound pressure generated in the ear canal at 2.0 V peak AC input is shown in Fig. 13. The modulation frequency was 2.44 Hz. The carrier frequency, its side bands and six carrier harmon-

ics centered round 2C, 3C, 4C, 5C, 6C and 7C (268, 536, 804, 1072, 1340, 1608 and 1876 Hz) are shown with sideband clusters. Inset shows the details of the cluster centered at 3Ucarrier frequency (804 Hz). 18 sidebands

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Hz), sidebands (C - M), second harmonic (2C), and sidebands (2C - M) and (2C - 2M). All the remaining hundreds of frequency components are absent. This indicates that these components are generated in the inner ear and propagated out to the ear canal.

4. Discussion

Fig. 8. FFT of PST histogram observed in a cat single auditory nerve ¢ber (For details see Khanna and Teich, 1989). CF of the ¢ber was 1596 Hz, carrier frequency (C = 1596 Hz), modulation frequency (M = 39 Hz), 100% modulation. Sound pressure level was 50 dB. Spectral magnitude is shown as a function of frequency. Spectral components can be seen corresponding to the (i) modulation frequency (M) and its harmonics (2M, 3M), (ii) carrier frequency (C), second harmonic (2C) and third harmonic (3C) and (iii) side bands associated with the carrier frequency (C - M), (C - 2M).

can be seen separated from the carrier by - nU2.44 Hz. Where n = 1,2,3,T,9. The level of the second harmonic component (2C) is 43 dB below the level of C, and the level of the third harmonic (3C) is 50 dB below the level of C. When the sound pressure is measured with the sound system terminated in a test cavity and the same driving voltage is applied to the transducer, spectral components are seen only at the carrier frequency C (268

In an AM wave the carrier amplitude changes with time during each modulation cycle. The velocity response at the reticular lamina is non-linear, and therefore the non-linearity changes with time during the modulation cycle. This dynamical change in non-linear motion has been studied earlier using short term Fourier transform (STFT) (Heneghan et al., 1993, 1994; Teich et al., 1994, 1995). In the present study FFT of the time waveforms has been used to estimate the spectral magnitude of di¡erent frequency components present in the response. 4.1. Hensen’s cell velocity response at the reticular lamina shows non-linearity In the present paper responses from the Hensen’s cell are shown because the carrier level is high and the low amplitude sidebands are recorded well above the noise level. Vibratory responses were also observed at the outer hair cells that show similar set of sidebands; however, the data set is limited. Earlier it was shown that for sinusoidal stimulation the harmonics were stronger at the OHC but the harmonic spectrum was similar at the OHC and Hensen’s cells (Khanna and Hao, 1999b)

Fig. 9. Velocity tuning curves of the carrier (C), sidebands (C - M, C - 2M, C - 3M), and the modulation (M) in the second preparation. The carrier is broadly tuned with a peak at 550 Hz. Tuning shape of the ¢rst pair of sidebands (C - M) is nearly the same as the carrier, although there are di¡erences below 400 Hz. Tuning shapes of the higher sidebands (C - 2M), (C - 3M) are di¡erent from the carrier. They are more sharply tuned. The modulation component (M) at 2.44 Hz is even more sharply tuned.

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4.2. E¡ect of velocity integration The AM velocity wave v1 (t) is given by: v1 ðtÞ ¼ V sing c tf13 b cosðg m tÞg , where gc and gm are the carrier and modulation frequencies, the carrier and modulation peak velocities are V and b, respectively. The velocity asymmetry adds a demodulated component of amplitude k to the wave (for simplicity all other spectral components are left out). v2 ðtÞ ¼ V sing c tf13 b cosðg m tÞg þ kcosðg m tÞ Fig. 10. Phase di¡erence between the sidebands and the carrier. Note linear frequency scale. Phase di¡erence between (C - M) and C is approximately 180‡. Phase di¡erences between (C32M), (C33M) and C are frequency dependent. For (C32M) phase changes between 400 and 600 Hz correspond to the CF region. Beyond 1250 Hz the phase slope is about 0.5‡/Hz.

The envelopes of the velocity waveforms measured at the Hensen’s cell in response to the AM sound are asymmetrical. The velocity of the negative half of the wave is higher. Individual cycles of the time waveform within the AM envelope also show this asymmetry. The degree of asymmetry is frequency and level dependent. It is higher at frequencies below CF. Velocity asymmetry was also observed in previous experiments in which a sine wave stimulus was used. It was shown that the velocity is higher during the negative half of the cycle when the reticular lamina moves towards the scala tympani (Khanna and Hao, 1999b).

Velocity is converted to displacement by integrating the instantaneous velocity. In this conversion, the velocity amplitude at the carrier frequency is divided by the carrier frequency while the modulation amplitude is divided by the modulation frequency. The carrier frequency is much higher than the modulation frequency {(gc /gm ) s 100}. Even though the velocity amplitude of the modulation frequency component may be relatively small, the displacement amplitude of the modulation frequency component can be relatively large, and therefore in the integrated time waveform the modulation component may dominate the response. 4.3. Tuning and amplitude dependence of the carrier and the sidebands. For AM waves, the FFTs of the time waveforms display a complex spectrum that consists of the modu-

Fig. 11. Amplitude dependence of the carrier and sidebands. Ratio of the amplitude of the spectral components C, (C - M) and (C - 2M) measured at 2 V and 0.5 V input level. If the system behaved linearly the ratio of amplitudes should be 4, and independent of frequency. The ratio of carrier amplitude (C) is nearly 4 over most of the frequency region. Below 600 Hz it varies with frequency between 1.2 and 5.2. The ratio of sideband amplitude (C - M) is also nearly 4 over most of the frequency region. Below 700 Hz it varies between 0.8 and 5.2. The ratio of sideband amplitude (C - 2M) is highly frequency dependent. It varies between 0.8 and 45. Maximum ratio of 45 is at 240 Hz, second largest ratio of 20 is at 560 Hz. Ratios are di¡erent for (C+2M) and (C32M) components. The ratio for (C - 3M) sidebands are similarly high (up to 60, not shown).

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Fig. 12. Amplitude dependence of the modulation component. Velocity amplitude of the modulation component at two signal levels: 0.5 V (dotted line) and 2.0 V (solid line). At the carrier frequency of 463 Hz the velocity of the modulation component increased from about 1.4U1034 to 8.7U1033 , a factor of 62. The modulation component increase is therefore highly non-linear.

lation frequency and its harmonics, carrier frequency band multiple harmonics of the carrier frequency each surrounded by side bands that are separated from the carrier frequency by multiples of the modulation frequency. The tuning characteristics and the amplitude dependence of the carrier C and the ¢rst pair of sidebands (C - M) are similar. They are broadly tuned, and their amplitude dependence is nearly linear at high sound pressure levels (W90 dB SPL). The tuning characteristics and amplitude dependence of the higher sidebands (C - 2M and C - 3M) are quite di¡erent from those of the carrier and the ¢rst pair of sidebands. They are more sharply tuned and their level dependence is highly non-linear. Their phase characteristics are also di¡erent. In the apical turn of the cochlea the velocity of the reticular lamina is higher than that of the basilar membrane. After the animal is sacri¢ced the velocity amplitude of the reticular lamina decreases while its tuning becomes broader, at the same time the amplitude of the

Fig. 13. FFT of the sound pressure wave in the ear canal. 2.0 V AC input. Measured at the same time as the velocity response shown in the Fig. 6. Carrier C and six carrier harmonics 2C, 3C, 4C, 5C, 6C and 7C with sidebands clustered around them can be seen. Inset shows the details of the cluster at 3Ucarrier frequency. 17 sidebands can be seen.

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basilar membrane below increasing and its tuning becoming sharper. The magnitudes of the velocity amplitude change and their time course can be explained under the assumption that reticular lamina vibrations are fed back to the basilar membrane to produce negative feedback (Khanna and Hao, 2000). A fundamental property of the negative feedback is to make the ampli¢cation linear even if the ampli¢cation without feedback was non-linear (Bode, 1945). This suggests that the non-linear mechanics located above the reticular lamina couples to it. As a consequence, the non-linear components generated in the stereocilia mechanics are seen at the reticular lamina. 4.4. Non-linear components in the ear canal sound pressure The sound pressure measured in the ear canal contains a series of spectral components: (i) modulation frequency (M) and its harmonics (n M); (ii) carrier frequency and sidebands (C - n M); (iii) harmonics of the carrier frequency and their sidebands {(2C - n M; 3C - n M; 4C - n M; T); n = 1,2,3,4,5,6,7,T}. These additional components originate in the cochlea and propagate back to the ear canal. An extensive array of sidebands has been observed in the ear canal of gerbils in response to two tone stimulation between 2000^6000 Hz and from 50 to 70 dB SPL (Kemp and Brown, 1985). 4.5. Stereocilia non-linearity as the source of non-linear vibrations In previous studies we found that, for sinusoidal acoustic input, the magnitude of harmonic distortion components is highest at OHC. The harmonics reduce dramatically or disappear after sacri¢ce (Khanna and Hao, 1999b; Hao and Khanna, 2000). This suggests that the harmonics are associated with outer hair cells. Non-linearity associated with the hair bundles is well established both at the mechanical and receptor potential levels in isolated OHC (Flock and Strelio¡, 1984; Hudspeth and Corey, 1977). The bundle motion is nonlinear due to the asymmetrical sti¡ness of the bundles (Strelio¡ and Flock, 1984) and the opening and closing of channels (Howard and Hudspeth, 1988). The apical ends of the outer hair cells are attached to the reticular lamina. The roots of the stereocilia bundles are embedded ¢rmly in the cuticular plate of the OHC (Tilney et al., 1980), and the tips of the tall stereocilia are embedded in the tectorial membrane (Lim, 1986). Nonlinear mechanical motion of the stereocilia bundles is therefore mechanically coupled to the cuticular plate and the reticular lamina.

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4.6. Generation of spectral components by the non-linear motion of the stereocilia bundles A modulated wave applied to the basilar membrane is ampli¢ed to produce the reticular lamina motion that drives the stereocilia bundles of the OHC. The nonlinear vibrations produce components at the modulation frequency (M) and its harmonics (n M), carrier frequency (C) and sidebands (C - n M) and carrier harmonics and sidebands (m C - n M), where n = 1,2,3,T, and m = 1,2,3,T. Because of the mechanical coupling between the stereocilia and the reticular lamina these spectral components couple back to the reticular lamina. The net motion of the reticular lamina is therefore determined by the combination of the motion of the reticular lamina driven by the input signal and the non-linear motion produced by the stereocilia bundles that couples back to reticular lamina. The amplitude of the components coupling back at C and (C - M) is small compared to the motion driven directly by the input that has components at these frequencies. Therefore, the amplitude dependence of the carrier and the ¢rst pair of the sidebands at the reticular lamina remains nearly linear over most of the frequency range. The components at the modulation frequency (n M) and the sidebands (C - 2M), (C - 3M), (2C - 2M) and (2C - 3M) that couple back remain highly non-linear because there are no input components present at these frequencies to alter them. 4.7. Estimate of the rise time from the experimental data The amplitude of the harmonics in the FFT spectrum decreases slowly with increasing harmonic number, suggesting that harmonics were generated in a repetitive process with fast rise/fall time. Examples with sinusoidal input, in which the velocity^time response at OHC1 looks like a negative going pulse train with fast rise/fall time, were shown earlier (Khanna and Hao, 1999b; Hao and Khanna, 2000). Fourier coe⁄cient Cn for the nth harmonic gn for a pulse train of peak amplitude A, repetition rate f and pulse width d, are given by Schwartz (1990) : C n ¼ A d sinðg n d =2Þ=g n d =2; where n = 1,2,T; gn = 2Zn/T; and T = 1/f. The observed harmonic amplitude vs. harmonic number data can be ¢tted to this equation. For the third and higher harmonics at 146 Hz we get a best ¢t for d = 1 ms. At 146 Hz the period T for the carrier is 6.8 ms. In another experiment at 256 Hz, where the period of the carrier is 3.9 ms, the best match occurs at d = 100 Ws. In our present experiment the sampling rate used

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(200 Ws) sets the upper frequency and time limit of resolution. If the rate was set higher it is likely that d may be even smaller. 4.8. Fast rise/fall times are most likely related to channel opening and closing The stereocilia contain gated channels that open and close depending on their de£ection. The kinetics of the channels is fast. Rise times are in the sub-millisecond region (Corey and Hudspeth, 1983). The channel opening and closings are mechanically coupled to the stereocilia bundles and thus re£ected in their movement (Howard and Hudspeth, 1988; Benser et al., 1993). The channel openings and closings are so fast that their instantaneous action cannot be observed with a probing ¢ber tip with a time constant of 250 Ws (Howard and Hudspeth, 1988). The fast channel openings in response to stereocilia defection may be the fast rise time mechanical event that produces the numerous harmonics and sidebands observed. Stereocilia dynamics have two e¡ects. (i) Mechanical : its non-linear e¡ects can be observed in the time waveforms and spectra observed at the reticular lamina; and (ii) electrical : generation of the receptor potential in the outer hair cell by the opening of the gating channels (Howard and Hudspeth, 1988). The receptor potential is low pass ¢ltered by the OHC membrane time constant and thus in the activation of the nerve ¢bers the higher frequency components are removed (Russell and Sellick, 1978). The response of the a¡erent nerve ¢bers therefore should show the same cluster of frequency components at the ¢rst few harmonics but not at the higher harmonics seen in the mechanical response at the reticular lamina. 4.9. Neural response to AM waves Response of single auditory nerve ¢bers to AM tones have been determined in the cat ear using post stimulus time (PST) histograms. The spectrum of the response was obtained by taking the FFT of the PST histogram. The spectral components of the FFT consisted of: (i) modulation frequency (M) and its harmonics (n M); (ii) carrier frequency (C) and sidebands (C - n M); (iii) harmonics of the carrier frequency and their sidebands (2C - n M; 3C - n M;T); n = 1,2,3,4,5,T(Khanna and Teich, 1989). In this paper FFT spectra for the AM wave was shown for a nerve ¢ber with CF of 2299 Hz. Up to four pair of sidebands associated with the carrier frequency were seen at 50 dB SPL. The modulation component and its harmonics were present in the response over a wide amplitude range from 30^90 dB SPL and a wide range of characteristic frequencies (CF) 200 Hz to

15 000 Hz. A second example from this series of experiments is included in the present paper. Even though the animal model is di¡erent, the same set of frequency components are seen in the cat auditory nerve response as at the guinea pig reticular lamina. This suggests that the non-linear transduction mechanisms are similar in di¡erent animals. 4.10. Comparison of neural and mechanical responses The lowest sound pressure level at which the carrier, carrier harmonics and their sidebands have been observed in the mechanical response was at a SPL of 78 dB, while in the neural response these were observed at 30 dB SPL (lowest level at which measurements were made). The di¡erence can be attributed to the fact that the two events are located in di¡erent paths of the transduction process. The neural response is produced by the transmitted component of the stereocilia mechanical response in the forward direction that includes the generation of receptor potential and the nerve impulses. Both receptor potential and neural responses are known to show non-linearity at low sound pressure levels (Russell and Sellick, 1978; Dallos, 1986; Evans, 1972). The non-linear response seen at the reticular lamina, however, is produced by the stereocilia mechanical nonlinearity coupling back in the reverse direction. The reticular lamina vibration is essentially linear and the coupled non-linear component is relatively small at low sound pressure levels; therefore, the vibration looks essentially linear. With increasing sound pressure the reticular lamina vibration increases linearly while the coupled non-linear component increases non-linearly. Therefore, at high sound pressure levels the amplitude of the non-linear component becomes quite high and the reticular lamina vibration shows non-linearity. 4.11. Relationship with receptor potential Another reason why the reticular lamina non-linearity is seen only at high sound pressure levels is suggested by observations of the receptor potentials in the apical turn of the guinea pig cochlea (Dallos, 1986). The negative part of the receptor potential curve falls steeply with increasing negative pressure and saturates at about 36 mV at a sound pressure level of about 60 dB. The positive part of the receptor potential curve rises steeply, reaching about 19 mV at about the same SPL. Between 60^80 dB SPL the slope of the input output curve becomes increasingly shallower (Dallos, 1986, Fig. 9, T4, OHC). At an outer hair cell with sinusoidal stimulation the start of mechanical nonlinearity can be seen near a SPL of 70 dB (Khanna and Hao, 1999a,b). With an AM wave a rich spectrum with

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many harmonics and sidebands is seen at about 78 dB SPL. The saturation of the receptor potential and the sharp increase in mechanical non-linearity occurs at about the same SPL, and suggests a relationship between the two. Increase in the amplitude of the nonlinear components at the reticular lamina could be due to a sharp increase in amplitude of these components at the stereocilia bundles at high sound pressure levels, or due to an increase in coupling between the stereocilia and the reticular lamina. If the input/output curve of the mechanical vibration of the stereocilia bundles followed the receptor potential curve, their vibration amplitude would saturate at high sound pressure levels and the amplitude of the non-linear components would not increase rapidly. Their mechanical impedance, however, would increase sharply, and therefore an increase in coupling between the stereocilia and the reticular lamina due to the increased sti¡ness of the bundles may be the more likely reason for the non-linearity seen at the reticular lamina at high sound pressure levels. 4.12. Non-linear mechanical bundle impedance? Sti¡ness changes in the bundle mechanics at small static de£ections are well established (Howard and Hudspeth, 1988; Russell et al., 1992). Changes in the stereocilia bundle mechanical characteristics at larger de£ections show asymmetry between the inhibitory and excitatory directions (Flock and Strelio¡, 1984; Strelio¡ and Flock, 1984; Van Netten and Kros, 2000). Most of these studies have been made in isolated hair cells or preparations, and with static de£ections. Dynamic bundle impedance changes may be di¡erent in the intact preparation at the frequencies the hair cells normally operate within the cochlea. 4.13. Signi¢cance of the modulation component The information in an AM wave is carried by the modulation. In order to extract this information the wave must be demodulated. The input does not contain any component at the modulation frequency and its presence in the mechanical response at the reticular lamina and in the neural response of the auditory nerve indicates that a demodulation process has taken place. In the neural response the modulation component was observed over a carrier frequency range of (0.2^14.5 kHz) and sound pressure range (30^90 dB SPL) that was tested. Therefore, the demodulation is taking place over a wide frequency and intensity range. In order to demodulate an AM wave one needs to pass the wave through an asymmetric transfer function that would remove one half of the wave. The asymmetrical characteristics of the hair cell receptor potential (Flock, 1965; Hudspeth and Corey, 1977; Dallos, 1986) show that

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such a non-linear function is associated with the stereocilia bundle mechanics. In an AM radio receiver diodes are utilized for amplitude demodulation. Their function is based on low forward and high reverse resistance. In a demodulator utilizing asymmetrical resistance the thermal noise generated by the resistance limits its sensitivity. This can be circumvented, however, by the use of asymmetrical reactance for demodulation. Pure reactance cannot produce thermal noise and therefore much more sensitive detection could be achieved. Moreover, under suitable conditions substantial ampli¢cation could also be obtained. The asymmetry in stereocilia sti¡ness and observed asymmetry in velocity amplitude suggest that parametric demodulation may be taking place in the hearing organ. A parametric system could provide low noise demodulation (Kouril and Vrba, 1988). A parametric system generates a special set of spectral components as its signature. It was for this reason that we had earlier analyzed several models of the asymmetric reactance (Keilson et al., 1989). A bilinear sti¡ness model with sine wave stimulus showed asymmetry in the time waveform and generation of a large number of harmonics of the signal frequency (Teich et al., 1989a) similar to those observed experimentally in the cochlea.

5. Summary 1. The spectrum of the AM sound wave applied to the ear consists of three components: (i) carrier, (ii) lower sideband and (iii) upper sideband. 2. The spectrum of the mechanical response of the sensory cells at the reticular lamina to this AM stimulation consists of the three components described above and hundreds of additional frequency components. These additional components consist of (i) modulation frequency and its harmonics, (ii) harmonics of the carrier frequency and (iii) sidebands around the carrier frequency and each of the harmonic frequencies. These additional spectral components indicate the presence of a highly non-linear process. 3. The tuning curves and the amplitude dependence of the additional components are quite di¡erent from those of the carrier and the ¢rst pair of sidebands. This suggests that the non-linear components are generated at a location above the reticular lamina. 4. The results suggest that the non-linearity is located in the stereocilia bundles of the outer hair cells and is mechanically coupled to the reticular lamina. 5. The same set of spectral components has been observed in the response of single nerve ¢bers of the cat auditory nerve. The presence of a set of frequency components identical to those observed at the retic-

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ular lamina suggests that the components are generated in a mechanical non-linearity and transmitted through the peripheral auditory system to the auditory nerve. 6. The presence of the modulation component in the mechanical response and in the neural response suggests that a basic function of this non-linearity is to demodulate the AM components of the incoming sound.

Acknowledgements This work was supported by the Emil Capita Charitable Foundation. The author thanks Leslie Hao for preparation of the animals, and Elizabeth Olson and Mats Ulfendahl for their valuable comments on the manuscript.

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