Basilar membrane nonlinearity determines auditory nerve rate-intensity functions and cochlear dynamic range

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1 Hearing Research, 45 (1990) Elsevier HAERES Basilar membrane nonlinearity determines auditory nerve rate-intensity functions and cochlear dynamic range Graeme K. Yates, Ian M. Winter * and Donald Robertson Department of Physiology, The University of Western Australia, Nedlands, Australia (Received 25 July 1989; accepted 28 November 1989) In a previous paper (Winter et al., 1990) we demonstrated the existence of a new type of auditory-nerve rate-intensity function, the straight type, as well as a correlation between rate-level type, threshold and spontaneous rate. In this paper we now show that the variation in rate-intensity functions has its origin in the basilar membrane nonlinearity. Comparison of rate-intensity functions at characteristic frequency and at a tail-frequency show that the rate-intensity functions are identical at low firing rates and that the sloping-saturation and straight types deviate from the standard function only at higher firing rates. The frequencies at which the deviations occur, and the change from saturating to sloping-saturation or straight, are closely correlated with the characteristic frequency of the fibre. Using the tail-frequency rate-intensity function as a calibration, it is possible to derive the basilar membrane input-output function at characteristic frequency from the characteristic frequency rate-intensity function. The resulting derived basilar membrane input-output functions are of a simple form and agree well with published direct measurements of basilar membrane motion. They show that the wide dynamic range to which the cochlea responds, about 120 decibels, is compressed by the basilar membrane nonlinearity into a much smaller range of about decibels. General characteristics of the derived basilar membrane input-output curves show features which agree well with psychoacoustic studies of loudness estimation. Auditory nerve; Rate-intensity functions; Basilar membrane nonlinearity; Dynamic range; Loudness perception Introduction Rate-intensity functions for auditory nerve fibres display a variety of forms, but to date no explanation for them has been generally agreed. These functions, generated by plotting mean discharge rate of auditory-nerve fibres against stimulus intensity, are usually sigmoidal and monotonic (Nomoto et al., 1964; Sachs and Abbas, 1974), although nonmonotonic behaviour at high intensities has been reported for the cat (Kiang, 1984). The more common form of rate-intensity (RI) function is symmetrical about some mid-range Correspondence to: Graeme K. Yates, Department of Physiology. The University of Western Australia, Nedlands, 6009, Western Australia. * Present address: M.R.C. Institute of Hearing Research, University Park, University of Nottingham, Nottingham NG7 2RD, U.K. intensity when plotted as action potential (AP) rate on a linear scale against sound pressure level (SPL) on a decibel scale, but a second form of RI function, the sloping-saturation form, has been described by Sachs and Abbas and by others (Nomoto et al., 1964; Palmer and Evans, 1980; Winter et al., 1990). In this paper we show that the whole range of monotonic RI functions may be explained by the nonlinearity presently known to exist in the basilar membrane (BM) input-output (I/O) responses. Such an explanation was originally proposed by Sachs and Abbas when they compared their measured RI functions against predictions from a model based on Rhode's (1971) Mrssbauer observations of the squirrel monkey BM response. Their conclusions, however, were challenged by Palmer and Evans (1980) who claimed no evidence for BM nonlinearities existed in the neural responses. Palmer and Evans based their conclusions on recordings of RI functions from cat audi /90/$ Elsevier Science Publishers B.V. (Biomedical Division)

2 204 tory-nerve fibres, in which they compared the SPL of onset of sloping-saturation across different units having similar characteristic frequencies (CFs). On the basis of Sachs and Abbas's model, Palmer and Evans expected to find an inflection point in the RI curves which occurred at a fixed SPL, and a dynamic range which increased for increasing fibre threshold. Instead, they found that the inflection point correlated more with SPL relative to threshold than with absolute SPL, and they found little correlation of dynamic range with threshold. They did comment, however, that the correlations were significantly better if they included their low-spontaneous fibres and that there appeared to be a 'sub-population of cochlear nerve fibres having different properties from the majority'. This clearly indicated that their low-spontaneous fibres had higher thresholds as well as the wider dynamic ranges typical of sloping-saturation. In a companion paper to this one (Winter et al., 1990) we show the presence, in guinea pig, of a third type of RI function: the straight rate-intensity function. We regarded this as an extension of a continuum of forms ranging from the saturating, through sloping-saturation, to the straight RI form. We also demonstrated correlations between spontaneous rate and threshold and between threshold and RI form. This present paper provides strong evidence that this continuum is the simple consequence of a nonlinear basilar membrane input/output function preceding a variety of synapses of differing thresholds (Sachs and Abbas, 1974). If, as Sachs and Abbas suggested, the nonlinear BM is responsible for the sloping-saturation of neural RI functions then it should be possible to investigate this by comparing RI functions at frequencies on the tail of the frequency-threshold response curve (tail frequencies) and at characteristic frequencies. In making this comparison, we assume that the inner hair cells function as simple passive displacement (or velocity, Patuzzi and Yates, 1987) transducers (see Patuzzi and Robertson, 1988, for review) and that the response of the inner hair cell, synapse and nerve fibre is dependent only on the displacement of the stereocilia and is independent of other frequency-selected effects. This assumption generally is held to be true, at least for first-order effects, with the only known variation being the change from phaselocking to direct current responses as the stimulus frequency is increased above a few khz (Russell and Sellick, 1978). We therefore compared responses from single auditory-nerve fibres over a range of stimulus intensities and at tail and CF frequencies. If the BM is responsible for the transition from saturating to sloping-saturation RI functions then we should expect to find changes with frequency which correspond closely with similar changes in the BM responses and which are closely associated with the frequency selectivity of the neurone. Sachs and Abbas (1974) have made similar comparisons in cat, based on predictions made from their model. Methods The experimental data were taken from the series of experiments which provided data for our companion paper (Winter et al., 1990). Consequently, all experimental methods are defined in that paper. The data were analysed using a variety of techniques ranging from manipulation (scaling, normalisation etc.) with a commercial spreadsheet program (Microsoft Multiplan), graphing with a commercial graphics program (Microsoft Chart) and nonlinear least-squares parameter fitting (curve fitting) using a program written by one of us (G.K.Y.). A short summary Of the details of data collection will accompany each section of the results. Results Comparison of rate-intensity functions at CF and tail frequencies Data were collected using a computer program which applied short (usually 100 ms) tone bursts over a range of stimulus intensities (usually a 96 db range in 3 db steps) at a single frequency. Action potentials were counted during the time each stimulus was on and were added to appropriate bins in computer memory. The frequency was then changed to a second frequency and the same series of stimulus intensities was again presented, but in shuffled order. The order of the two frequencies was then randomly permuted and the

3 ~ o, - '1.o12 o E 300 ~ , o~~~ s~on=2.5/s.018 oo.j.~* =*** 300 ~ 200 (/ aJ spon=l.4/s ~, C.006 c, U _ : ;.008 oq spon=o.4/s '~.008 H " o go Fig. 1. Rate-intensity functions recorded from four auditory-nerve fibres at two frequencies: CF (e) and a frequency on the tail of the frequency-threshold curve (o). The data are represented by points: the lines are functions fitted as described in a later paper. (A) An example of a high-spontaneous unit. The CF and tail RI functions are almost identical. This is an example of a saturating RI function. [20 khz/17 khz] (B) and (C). Two examples of units with medium spontaneous rates. The CF and tail RI functions are very similar up to a certain AP rate and then the CF function flattens to approach saturation more slowly. These are examples of sloping saturation. [19 khz/12 khz and 20 khz/17 khz] (D). An example of a low-spontaneous unit. The CF and tail RI functions match over only a small range, the CF response breaking only a few db above threshold. This is an example of a straight RI function. [20 khz/17 khz] (E-H). As for A-D respectively, but with the tail-frequency RI functions shifted left by 20.7 db, 48.2 db, 29.7 db and 28.7 db respectively. Thus, the absolute db scale on the right-hand panels refers only to the CF functions. These panels demonstrate that the tail-frequency and CF RI functions are identical up to the firing rate at which the break occurs, and that the various forms of RI function simply represent a continuum in the AP rate at which the break occurs. whole procedure repeated. After typically 10 repetitions of each stimulus-frequency combination, the AP counts were normalised to counts per second of stimulus presentation and stored. The two frequencies chosen were the CF frequency (estimated as accurately as possible from a previous frequency-threshold measurement) and a tail frequency. The choice of tail frequency was

4 206 influenced by several factors: it had to be sufficiently low as to be in the tail region of the frequency-threshold curve, it had to be sufficiently high that the RI response could be measured up to high AP rates, and it should, if possible, have been kept the same for all units measured to aid comparison between fibres. It was possible to satisfy this last criterion only for units with similar CFs. Data from the two frequencies were then plotted as rate-intensity functions. Fig. 1 shows a typical result for four separate fibres. The smooth curves are derived by fitting a particular function, similar to but not identical with that used by Sachs et al. (1989), to the data and will be described in more detail in a third paper (Yates, 1990). For the moment they may be regarded simply as a visual aid. We consider only the left-hand column first (A-D). Fig. 1A shows the data for CF and tail frequencies for a unit with a high spontaneous rate. For this unit the shapes of the RI functions for the CF and tail frequencies are clearly similar to one another over the entire intensity range. This response is typical of a saturating RI response. It follows a sigmoidal curve which is quite symmetrical about the midpoint of the curve and saturates quite sharply at about 255/s. The input dynamic range of the unit is about 20 db. Fig. 1B shows a similar plot for a unit with a medium spontaneous rate. Here the two curves are similar for lower firing rates, but when the rate reaches approximately 200/s the CF response suddenly flattens off and approaches saturation more slowly. The tail frequency curve is similar in shape to that in A. This unit is typical of the type classified as sloping-saturation. It has a dynamic range of approximately db, depending upon what one chooses as the arbitrary upper limit of the response. Since the unit is completely saturated at 100, the dynamic range would not have been greater if a wider range of input stimulus amplitudes has been used. Fig. 1C shows another unit of the slopingsaturation type. Here too the responses at tail and CF frequencies are similar until some particular discharge rate (about 100/s) whereupon the CF response again flattens and approaches saturation more slowly. The break-point occurs earlier than in 1B in the sense that it occurs when the AP rate has reached about one third of its maximum possible rate. This unit had a lower spontaneous rate again and a threshold slightly higher than that of the unit in lb. The dynamic range is about db, but apparently would have been greater had we pursued the response to higher stimulus intensities. Fig. 1D shows a representative of the straight RI response group. In this case the RI function at CF appears to match the tail-frequency response only at very low AP rates. The dynamic range is only about 60 db in this case but, as for the unit in 1C, apparently would have been greater had we used higher intensities. The fight-hand column of Fig. I (E-H) shows the same data after the tail-frequency RI curves have been shifted left by various amounts sufficient to bring them into alignment with the CF RI curves near threshold. It is now quite clear that for each panel the CF and tail-frequency RI functions are identical near to threshold and up to the break-point at which the RI functions flatten. In the panels A and E it is clear that shifting the tail-frequency RI function left by 20.7 db brings the two into almost perfect alignment over the whole intensity range. Shifting the tail-frequency RI data of panel B left by 48.2 db (panel F) brings it into close match with the CF data over almost three-quarters of the firing-rate range while the shifts in panels C and D (by 29.7 db and 28.7 db respectively) demonstrate close agreement only for a quarter or less of the AP-rate range. Fig. 1 demonstrates that CF responses are identical with tail-frequency responses up to some fibre-dependent firing rate before the CF response deviates from the tail response. The apparent continuum of RI types, from saturating through sloping-saturation to straight, appears to reflect a continuum of firing rates at which the CF response begins to deviate from the tail response. We conclude from this that the inflection is not an inherent property of the neural process (c.f., Palmer and Evans, 1980), since the inflection point is seen in the responses at CF and not at the tailfrequency. Comparison of rate-intensity functions for frequencies above and below CF Although we have demonstrated a frequency-

5 207.. o.o. 300 I = =t, I ~x3...--r.--~ _* ]= " 12~Z +,4kHz. 16kHz 0 17kHz/..I "o 4.oo l II 18kHz 19kHz 20kHz l 0/ I I / 0 I I I Fig. 2. Rate-intensity functions recorded from one auditory-nerve fibre at several frequencies below, at and above CF. The data are represented by symbols; the smooth lines are calculated from functions fitted by a method described later and are included as an aid to comparison between frequencies. The CF (16 khz) curve is shown thicker than the others. The left panel shows the RI functions as originally recorded. It is clear that for frequencies below CF the RI curves are almost identical while at and above CF they deviate from the tail-frequency curves at progressively lower and lower AP rates. The right panel shows the same data adjusted so that thresholds are approximately equal. The change in shape is now quite clear and the AP rate at which the break occurs is seen to be a monotonic function of stimulus frequency. The inset figure in the right panel shows the frequency-threshold curve with the frequencies corresponding to the above RI functions shown as filled circles kHz **, G4.013 m** **. 13kHz 16kHz o ~ I'ti g (3> 200 o 17kHz = o = 18kHz,, " O m 200 1~ I1.e o O III u II U n- O. II ,4 0 0 N , Fig. 3. As for Fig. 2 but for a unit with a medium spontaneous rate at the same CF. The frequency at which the break is first evident is now at CF rather than above and the change to sloping saturation and straight RI curves occurs at a lower frequency relative to CF.

6 dependence in the rate-intensity responses of auditory-nerve fibres, it might be that this dependence is a simple function of frequency, such as a global high-frequency effect, and therefore does not provide any direct association with the BM tuning. Fig. 2 and Fig. 3 argue against this. Here we plot RI functions recorded from two fibres at several frequencies above and below CF. Some intermediate frequencies have been omitted for clarity, but all showed the same trends. In the left panel of each figure we plot the RI functions as recorded, while in the fight panel we plot the same data shifted along the intensity axis by an amount sufficient to make the thresholds coincide. Now the frequency-dependence of the RI inflection is clear. In the right-hand panels the tail-frequency data cluster around a single, symmetrical sigmoid, typical of almost all the tail-frequency responses we have measured. For frequencies at or just above CF, however, the RI responses coincide with the tail-frequency responses over their lower firing rates, but then deviate away to approach saturation at a slower rate. Moreover, the AP firing rate at which the inflection occurs is a monotonic function of frequency, the inflection occurring earlier and earlier as the stimulus frequency exceeds CF. Well above CF this trend continues until the inflection point occurs just above spontaneous firing rate and there is no region of coincidence with the tail-frequency curves. Sachs and Abbas (1974) show similar trends with frequency in the cat auditory nerve. This nonlinear behaviour is highly reminiscent of the I/O properties of the BM itself (Sellick et al., 1982; Robles et al., 1986). BM I/O responses are linear below CF and saturate at and above CF. More significantly, the BM I/O responses at CF show an initial linear portion at low intensities which saturates within db of CAP threshold. The BM displacement at which the onset of nonlinearity occurs decreases as the stimulus frequency approaches and passes CF until, well above CF, the initial linear portion is missing altogether. The analogy with the neural RI functions is very close. The data of Fig. 2 are taken from a high-spontaneous unit while that of Fig. 3 are from a medium-spontaneous unit. It is quite clear that in both cases the change from a saturated RI re ~" 0.8- t~ < I:I:: 0.4 T'...., ~', ;r:,i-;,.~.:j.,..,~: ": "..'.."-:..,;.;." " ;"b.,..,:. ".'.'." ". ' I t I t t I t l < O.2 /JII n=2g.,~..,.,.~.~ 0,,_,. ~ p ~ ~, I I I I I I I I Normalised Stimulus Intensity (db) Fig. 4. Rate-intensity functions recorded from auditory-nerve fibres in one animal and normalised to zero spont rate, unity maximum firing rate and for threshold Includes all RI functions recorded at tail frequencies and all CF functions from fibres with spontaneous rates > 5. The top box shows the data as single points measured on the RI curve. The smooth solid curve is the square-law hyperbola to which all RI functions have been fitted. The dashed curve is the equivalent hyperbola: it clearly does not match the data as well as the solid line. The lower box shows the same data, this time connected with lines. This demonstrates that the data are randomly scattered about the normalised curve and that there are few obvious systematic deviations sponse to sloping through to straight is a simple, smooth transition taking place around the CF. The difference is simply that the transition occurs at a slightly lower frequency for the mediumspontaneous fibre of Fig. 3. If the transition of the high-spontaneous fibre had occurred some 2 khz earlier, it would have looked very much like the medium-spontaneous fibre. Fig. 2 and Fig. 3 tie the CF inflection very closely to the frequency-threshold curve, which in

7 209 turn reflects the tuning of the BM response. That is, Figs. 2 and 3 demonstrate clearly that the inflection point occurs at a frequency which is always close to the CF of the fibre. It is not, therefore, a property of the global mechanics of the cochlea, a possibility left open by Fig. 2, but rather depends upon local determinants of the CF. Thus, we may conclude from Figs. 2 and 3 that the inflection in the neural RI responses are closely tied to the properties of the BM responses. Tail frequency and high-spontaneous rate intensity responses To continue further with our analysis we found it convenient to derive a mathematical description of the RI responses. The sigmoidal form suggests a hyperbolic saturating function when plotted on linear axes. That is, if the SPL scale is converted from decibels to Pascals, the RI functions suggest a simple saturating function of the form A 1.x R = A0 + A ~ (1) where R is the AP firing rate in APs/s; A 0 is the spontaneous discharge rate; A a is the difference between the maximum and spontaneous rates; A 2 is the sound pressure (in Pascals) at which the response is half of its maximum value and is therefore a measure of threshold; x is the sound pressure (in Pascals) at which the response is measured. Such a function, however, was found not to be a good description of the observed responses, over-estimating the response near threshold and under-estimating near saturation. It also underestimated the slopes of the RI responses. Experimentation showed that an excellent match to the data could be had by using a similar equation but substituting the square of the sound pressure for the independent variable. That is A 1 X 2 R = A0 + A2 + x2 (2) Where the variables have the same meanings as in eq. 1. This equation was fitted to all complete RI functions measured at tail frequencies, and at CF in fibres with spontaneous rates greater than 5. (The value of 5 was chosen somewhat arbitrarily: above this value all RI functions at CF looked indistinguishable from the tail-frequency functions.) Typically the fit achieved was excellent, with correlation coefficients being in excess of 0.98 for almost all sets of data. Fig. 4 illustrates the quality of the fits achieved. Here we have normalised all such RI functions from the one animal against the fitted parameters A0, A 1 and A 2 such that the reduced RI response had zero spontaneous rate, a maximum discharge rate of 1.0 and a half-maximum firing rate intensity of 0. The top box, Fig. 4A, shows the data as individual points to illustrate the variability about the fitted function (solid curve). The dashed curve is eq. 1 with the same normalised values for A0, A 1 and A2, showing the superior fit achieved by eq. 2. Fig. 4B shows the same data, this time as line-graphs connecting data from the same RI functions. This figure illustrates that the variation seen in Fig. 4A is due to random scatter and not to systematic mismatches between responses and the fitted curve. Since the data in Fig. 4 were not selected in any way other than by excluding RI functions measured at CF for units with spontaneous rates below 5/s, we conclude that Eq. 2 is a good description of almost all RI functions other than sloping-saturation or straight. Close inspection of Fig. 4 shows why we qualify our claim with the word almost. In the stimulus intensity range of approximately -20 to 0 db normalised SPL, we see a single set of data which does not match the curve well. This set is from the tail-frequency response of a fibre with very low spontaneous rate. It seems to deviate quite markedly from the average curve and we note this for future discussion. It should be noted in general however, that since some fibres with spontaneous rates in the range 5-18 spikes/second will show some sloping saturation, it should not be expected that all fibres of medium spontaneous rate will be so well described. Derived basilar membrane responses The previous results indicate that the neural RI

8 210 response is a simple function of intensity for the tail-frequency responses, and up to some particular firing rate at CF frequencies. It therefore offers a technique for quantifying the underlying nonlinear drive to the hair-cells. We now make the assumption that the neural responses are a simple passive function of the displacement of the inner hair cell stereocilia. It is clear from intracellular studies of IHCs that this is indeed the case for the receptor potential (Russell and Sellick, 1978; Patuzzi and Sellick, 1983; Dallos, 1985), and there is no evidence to suggest that the synapse can detect the frequency of stimulation. Small, second order effects such as extracellular receptor potentials and extracellular K + (Johnstone et al., 1989) might be present, but these are not likely to have a strong effect on the responses. Thus we may assume that differences between tail-frequency and CF-frequency RI functions reflect differences in the drive to the stereocilia. Accepting this, then, we may use the tailfrequency response as a measure of stereocilia deflection and calculate the input drive as a function of intensity at CF. That is, we make the assumption that the AP rate is a function only of the mean displacement of the stereocilia of the IHCs. Consider now the tail-frequency RI re- sponses of Fig. 1. Since we know that the BM displacement, and presumably the drive to the stereocilia, is a linear function of intensity at frequencies well below CF (Sellick et al., 1982; Robles et al., 1986), we may take the abscissa to represent the BM displacement multiplied by an unknown scale factor. The RI function therefore defines the rate at which APs are generated in the fibre when the BM is displaced with various amplitudes. Thus we may choose a particular AP rate and read off the SPL necessary to produce the BM displacement giving this rate. Looking now to the RI function at CF, the SPL necessary to produce the same discharge rate at CF may be read off and interpreted as the BM displacement at CF. By choosing various AP discharge rates we may then construct a representation of the BM I/O curve (see Fig. 5). Such derived BM I/O curves were constructed for all tail/cf pairs of RI functions measured. The method was simply to draw a smooth curve by eye through both the tail-frequency and CFfrequency responses and then read off pairs of SPL values, one for CF and one for tail-frequency, for various AP rates. Plotting the tail-frequency SPL as ordinate and the CF SPL as abscissa produced the derived BM I/O function CF and Tail RI Functions ' Derived BM I/O Curve APs /sec 250 2OO 150 SPL (Tail) loo, /sec----~. 0 I J! J! I I, i I I l 60 i I i I i ~ O Tail----~ 70 CF Tail (db) CF Fig. 5. Derivation of basilar membrane CF input-output function from auditory-nerve rate-intensity functions measured at two frequencies; CF and a tail frequency. An AP rate is chosen and stimulus intensities at CF and the tail frequency are read from the rate-intensity data (left panel). The two intensities are then plotted, tail-frequency intensity against CF intensity (fight panel). When repeated for a range of AP rates this procedure generates a representation of the basilar membrane CF input-output function. The ordinate is arbitrary, since choice of a different tail frequency would shift the curve vertically. The abscissa is not arbitrary, since the main feature of the derived curve, the intensity at which it breaks from linearity, is defined by the absolute SPL.

9 _j 80 0_ oo (0 IJi.-= 4C I--- AN 20 2'o // I at CF ANF23.012:(20/17) 4- ANF23.018:(19/12) ANF23.006:(20/17) O ANF23.008:(20/17) Fig. 6. Four BM I/O curves derived according to the method of Fig. 5. Each curve is calculated from the RI functions of the four fibres of Fig. 1. The four different RI functions are seen to represent four different regions of a simple BM I/O function consisting of two straight (on log-log co-ordinates) segments. The ordinate is arbitrary and the vertical position of each curve depends upon the choice of tail-frequency and upon the unit threshold. (e) from Fig. 1A, the saturating fibre, forms the initial segment with a slope of 1. (+) from Fig. 1B, the sloping-saturation fibre, is mostly linear, but bends to form a second straight segment of slope approximately 0.2. (*) from Fig. 1C, the sloping-saturation fibre with the earlier breakpoint, consisting of a small segment of slope 1 and a longer segment of slope 0.2. (o) from Fig. 1D, the straight RI fibre, similar to the previous fibre but a little further up the BM curve. Fig. 6 shows a typical example of four derived BM I/O curves obtained from a single CF region (19-20 khz) of one guinea pig. In fact the data are from the four fibres of Fig. 1. All curves take on some aspect of a general form: an initial slope of unity, indicating a linear relationship between SPL and BM amplitude, turning over more-or-less abruptly to assume a second, straight, section with a slope of about The entire range of tail-cf RI pairs reduces to a set of curves lying over some part of this general form. That fibre which we classified as a saturating RI response produces a derived BM I/O curve consisting entirely of the initial, linear region (e). This fibre is that of Fig. 1A for which the RI function at CF is the same as for the tail-frequency. The sloping-saturation RI fibres of Fig. 113, C generate derived BM I/O curves which fall on the transition region (+ and *), with the low-intensity part of their RI functions forming the linear I part of the BM curve and the higher-intensity part of the RI functions forming the flatter part of the BM curve. Finally, the straight RI fibre of Fig. 1D gives rise to a derived BM I/O curve which is mostly on the flat region (o), corresponding with the CF RI curve which deviates from the tailfrequency curves at very low AP rates. This figure encapsulates the explanation of the various RI function types (Sachs and Abbas, 1974): saturating fibres have the highest spontaneous rates and the greatest sensitivity and therefore produce an increase in firing rate over the initial, steep, portion of the BM intensity response. Sloping-saturation units have intermediate sensitivities and operate about the transition region of the BM response, at first in the linear portion and then in the sloping portion. Finally, the straight RI fibres have even higher thresholds and operate mostly on the sloping portion of the BM I/O curve. The enhanced dynamic range of the sloping-saturation and straight fibres is a direct consequence of the nonlinearity of the BM. The vertical alignment of the four curves is of little significance: each curve is positioned according to the choice of tail-frequency and the sensitivity at that frequency. In fact, the vertical position is at most an indication of the overall sensitivity of the fibre, and then only if the tail-frequencies are the same. The alignment along the abscissa is, however, of far greater significance. Given that the curves of Fig. 6 come from a single CF region they should all share a similar break-point intensity, since that is a property of the BM and is presumeably the same for all of them. In fact, the three curves of Fig. 1B-D break within 6 db of one another, at 46, 52 and 52 db respectively, while the data of Fig. 1A imply a break-point at least greater than 40 db. Consequently, the upper three curves of Fig. 6 break within 6 db of one another and the lower curve has not reached a break-point at 45. This variation in break-point intensity seems quite reasonable to us. Sachs et al. (1989) describe a similar variation in the precise location of the breakpoint intensity estimated by curve-fitting to RI functions. They observe variations of between 6 and 10 db over a CF range of 20%. Fig. 7 shows 17 derived BM curves from another animal over a slightly wider range of CFs (16.2

10 t 400 2tO 4'0 do CF SPL Fig. 7. Basilar membrane CF input-output curves from one animal, derived by the method of Fig. 5. Although the location of each curve varies somewhat in both the horizontal and vertical directions, the shape of the curves is quite uniform. Each is some part of a smooth curve which has a slope of unity at low stimulus intensities and a slope of at higher intensities. In spite of the huge variety of RI functions observed, the underlying differences between CF and tail-frequencies are quite simple and uniform. khz-20.3 khz). Again and in spite of the wide variation in RI shapes, each curve lies over a part of the presumed BM I/O function and shows similar slopes in the corresponding regions. Fig. 8 shows four more examples of derived BM I/O curves. These data were recorded from four units with two different CFs: 17.5 khz and 14 khz. The resulting derived curves exhibit two 100 I I I I A 1100 distinct break-point intensities, again illustrating a common origin for the break-point at each CF. Also plotted in Fig. 8 are MSssbauer measurements of BM I/O curves recorded from chinchilla (Robles et al., 1986) and from guinea pig (Sellick et al., 1982). The comparison with the neurally-derived results is evident: both types of data show the same initially linear portion changing smoothly to a straight (on a log-log plot) line with slope around 0.2. Conversion between response types If the differences between the various forms of RI functions are simply a consequence of different fibre thresholds then it should be possible to convert one type of response into another by altering the sensitivity of the synapse. Forward masking is known to reduce the sensitivity of a synapse for short periods of time (c.f. Harris and Dallos, 1979) and should enable conversion of saturating to sloping-saturation, or sloping-saturation to straight RI response. This manipulation is very difficult because of the short times for which synaptic sensitivity can be reduced, but it should be possible to produce small threshold shifts and consequently some degree of shift in RI type. Fig. 9 diagrams the stimulus parameters we used to measure forward-masked RI responses. A 100 ms fixed-amplitude tone at CF was followed by a 15 ms probe tone of varying amplitude. The Fixed Masker Variable Probe if) g o 6c II :~ 4- Robles Mo33 I~ kHz 19kHz 40 2'0 20 do IOO OF SPL Fig. 8. Derived basilar membrane input-output functions from the khz CF region in two other animals. Also shown (in grey) are MSssbauer measurements of BM motion in the chinchilla (Robles et al., 1986) and in the guinea pig (18 khz, 19 khz, from Sellick et al., 1982). The dotted lines indicate slopes of 1.0 and 0.2. c Fig. 9. Stimulus parameters used in forward masking an auditory-nerve fibre. The masker tone was always presented for 100 ms at a level of 80. It was followed after a 2 ms delay by a 15 ms tone at various intensities ranging from 0 to 99 db SPL. RI functions were measured by counting APs during the probe-tone period. A 400 ms period preceded the following stimulus cycle.

11 O Q o 3.5 khz ~"~ o 300 t 6 khz o 3.5 khz O) 20C Z: Q_ < C i I I oooo ~.o I I I I00 dbspl Fig. 10. Forward masking changes a fibre from an almost saturating response to a sloping-saturation response. The left panel shows the rate-intensity function for the no-masking condition at CF and a tail frequency. The tail-frequency curve and data have been shifted left by 33 db to facilitate comparison of the two sets of data. The response is close to a saturating type, with a very small sloping-saturation region starting at about 40. The right panel shows the same unit in the masked condition. Both RI functions have altered but it is clear that the CF RI response is now of the sloping-saturation type. The onset of the sloping saturation occurs at about 46, the same as for the unmasked condition. The tail-frequency response has been shifted by the same amount (33 db) as in the left panel. Break-point intensities for CF curves are 44 db (unmasked) and 50 db (masked), as estimated by a least-squares fitting program. frequency of the probe tone was either CF or a convenient tail frequency. RI functions were constructed from the total AP counts within the 15 ms probe tone duration for CF and tail-frequency probes, both with and without the forward-masker tone. Typically the forward masker intensity was set at 80 and the probe tones varied from 0 db to 100. Fig. 10 shows a typical result. Each panel shows the RI curve for the CF and tail-frequency responses, with the tail-frequency curve shifted left to bring it into alignment with the CF curve at threshold. The amount of shift is described in the figure caption. The absolute firing rates are higher than in previous figures because the very short tone durations do not permit significant adaptation to occur, but apparently this does not affect the shape of the RI functions greatly (Smith and Brachman, 1980). The left panel is for the unmasked condition and shows a RI function with a small degree of sloping-saturation. The right panel shows the same two conditions but in the presence of the forward masker. The tail-frequency curve has been shifted by the same amount as in the left panel. The two curves now appear remarkably similar to the data of Fig. 1, which compares RI functions for sloping-saturation fibres. The initial portions of the two curves overly one another but the CF RI curve falls away from the tail-frequency curve at less than half the maximum firing rate. The almost-saturating fibre has been converted to a sloping-saturation response by the forward masker. The forward masker shifted the threshold of the fibre by approximately 15 db (as determined from the least-squares parameter fits) and maintained exactly the same threshold difference (33 db) between masker and tail. In the unmasked condition the nonlinear break in the BM I/O curve clearly occurs at approximately 50 when the tail and CF responses diverge The masked condition has shifted the threshold above this level and we expect the breakpoint to occur close to the foot of the RI function. There does, however, appear to have been a small ( -- 6 db) shift in the break-point, since the RI function is still of the sloping satura-

12 khz o 6.25 khz o 4kHz 30C 300- o 0.,< o J FM ~ C Fig. 11. As for Fig. 10. These data are from another fibre in the same animal. The tail-frequency data have been shifted left by 30 db. Break-point intensities for CF curves are 49 db (unmasked) and 57 db (masked), as estimated by a least-squares fitting program. tion type, although the break-point is much closer to threshold than in the unmasked condition. Given the statistics of the data we cannot be sure this effect is real, but we know of no evidence to deny the possibility that some small shift in BM sensitivity at CF might occur in the short period following presentation of a loud sound. Nonetheless, it is clear that when the threshold is shifted up 15 db, the RI function shifts closer to the break point and assumes the shape characteristic of sloping-saturation. Fig. 11 show a second example of the conversion from almost saturating to sloping-saturation. We conclude from these experiments that the type of RI function exhibited by a fibre is a function of the threshold of the synapse. Similar data were collected from another 4 animals and the same pattern was observed in each. Discussion We have presented several lines of evidence to support the original suggestion by Sachs and Abbas that saturating and sloping-saturation RI functions are a result of an interaction between different fibre thresholds and the BM nonlinearity. The evidence is more than circumstantial: it is completely consistent both qualitatively and quantitatively with the known BM nonlinearity in the same species as we have worked with. To summarise the evidence: (a) For high-spontaneous fibres the RI functions at both CF and tail frequencies are identical in form and magnitude. (b) For medium-spontaneous fibres the RI functions at CF and tail frequencies are identical up to some AP discharge rate and then the CF response deviates away, rising towards saturation at a slower rate. We have found in the guinea pig (Winter et al., 1990) a third form of response which consists mostly of the flatter region of the RI curves and which we call the straight response. Our comparison of CF and tail frequency RI curves shows that these fibres are an extreme form of the sloping saturation type. It is interesting that Sachs et al. (1989) mention '... if [the unit threshold] is large enough that [the input to the saturating nonlinearity] is in the nonlinear range throughout the dynamic range of the fibre, then no break will be observed, but the rate function will have a reduced slope compared to the function of Eq. 2 (or equivalently to the flat saturation function) throughout its dynamic range'. The resulting RI shape should correspond with our straight fibres but their model graphs (Sachs and Abbas 1974, Fig. 9) do not look much like them, indicating perhaps that their parameter values are not appropriate to the guinea pig. (c) The discharge rate at which the CF RI curve deviates from the tail-frequency curve is, for a

13 215 given fibre, closely connected with its frequencythreshold curve. Below CF the RI functions are all similar while at or above CF, depending upon the threshold of the fibre, the RI functions show progressively more sloping saturation until they reach the straight form. This result is similar to that of Sachs and Abbas (1974). (d) The assumption that the response of the fibre to stimulation of its stereocilia is independent of frequency (apart, of course, from the obvious BM frequency-response) permits the use of the tailfrequency RI response as a calibration to derive BM I/O functions from the CF RI curve. These BM I/O curves match published BM data quantitatively. (e) Altering the sensitivity of the synapse can change the CF RI response from saturating to sloping-saturation, or, presumeably, from sloping to straight. The shape of the neural RI functions Rate-intensity functions at tail frequencies were almost always of identical form and varied only in the spontaneous rate, maximum firing rate and threshold. An occasional exception was found (such as the obvious misfit in Fig. 4), but these were rare. Even in such cases, however, it usually turned out that an inappropriate choice of tailfrequency had been made and that the tailfrequency was too close to CF. In very high threshold fibres this can result in a small amount of nonlinearity being evident in the tail RI responses. A possible explanation for the remainder could lie in the nonlinearity of the inner hair cell receptor potential (Patuzzi and Sellick, 1983) which shows a slower-than-linear growth at high intensities. These fibres were rare, however, the overwhelming majority being very well described by Eq. 2. The square-law form of Eq. 2 is not without precedent. The Schroeder-Hall model of adaptation (Schroeder and Hall, 1974) suggested a simple hyperbolic saturation would describe the relationship between the electrical stimulus to the synapse and the AP rate. Sachs and Abbas, however, found it necessary to use an exponent of 1.77 on the stimulus intensity when modelling cat neural RI functions and Yates (1987) found it necessary to use an exponent of 2.0 on the stimulus intensity when comparing modulation responses of auditory-nerve fibres with predictions from the Schroeder-Hall model. There are at least two possible explanations for this. Patuzzi and Sellick (1983) found a square-law relationship between BM displacement and inner hair-cell receptor potentials, but only at very low intensities, typically below neural threshold. Over most of the input-output range of the IHC they found a linear or saturating relationship between intensity and receptor potential, and so this is unlikely to be the cause. The second possible origin of the square-law relationship with intensity could occur in the synapse. If the post-synaptic receptor required 2 molecules of transmitter to open a conductance channel then we would expect a square-law relationship between transmitter concentration and post-synaptic generator potentials. This could lead to a similar relationship between IHC potential and AP rate. The RI functions at CF are a consequence of the neural threshold relative to the BM break-point intensity. Above the break-point, the AP rate becomes relatively insensitive to stimulus intensity, the effective gain being reduced to a factor of about 0.2 db/db. Our range of fibres did not include units with thresholds actually above the BM break-point intensity, but given the relative difficulty of recording from low-spontaneous units with thresholds up to the breakpoint (see Winter et al., 1990), it is conceivable that such units might exist and we have missed them. If so, their thresholds at CF might be excessively high. For example, if the threshold of unit ANF of Fig. 1D and Fig. 6 was about 20 db higher (as measured at the tail frequency) then its threshold to sound stimuli at CF would have been around db higher, since it would not respond until the BM response was saturating. Such units might then be classified as very high threshold units (Liberman 1978), but they would exhibit very flat RI functions at CF. Liberman did report that RI functions for the low-spontaneous fibres appeared to saturate at lower rates than those for highspontaneous fibres and this might be a consequence of the flatter RI functions for low-spontaneous fibres, but insufficient data has been published to permit a detailed comparison.

14 216 The shape of the BM 1/0 function The derived BM I/O curves of Figs. 6-8 are the same shape as measured directly on the BM using the M~Sssbauer technique (Sellick et al., 1982; Robles et al., 1986). The initial linear part of the response saturates suddenly at about 20 db above the best neural thresholds and then continues at a slope of about 0.2. Robles et al. report a typical slope of 0.3, but their most complete data (curve Mo33) shows a slope of about 0.2 and, given the likelihood of damage during their measurements, the flatter slope is more likely to be accurate. What physical mechanisms could produce such a shape? Patuzzi et al. (1989) offered good evidence that the BM nonlinearity was a consequence of saturation of the active feedback process, via saturation of the outer hair cell mechano-electrical transducer (Zwicker, 1986). They suggested that nonlinearity of the OHC receptor current reduces the drive to the active process and thereby reduces the forward loop gain. The reduction in the gain of the BM response is much greater than the reduction in OHC sensitivity since positive feedback loops, of which the BM mechanics appears to be an example, are hypersensitive to the closed-loop gain. The same authors also produced a model of the BM I/O response which assumed a saturation of the OHC current with intensity resulting in an I/O function which had two linear regions separated by an intermediate region with a slope of about 0.5. This may provide a clue as to the mechanism by which the BM I/O function is compressed to accomodate an input dynamic range in excess of 100 db into an output range of only db. Consideration of this is left to a third paper (Yates, 1990). Palmer and Evans (1980) expected to find the break-point intensities to be closely confined at any one CF region. This is certainly to be expected of a global determinant of RI shape, but some variation is possible, because of the variation of thresholds with time during an experiment, of the sensitivity of the breakpoint to the exact choice of frequency relative to CF, and of the fluctuations in threshold which can occur over very small variations in CF. This latter factor, known as the threshold microstructure, can be responsible for more than 10 db variation in threshold over a frequency range of 100 Hz (Kemp, 1979; Wilson, 1980; Long, 1984). It is presumed to be due to variation in the mechanical sensitivity of the cochlea and therefore might be expected to affect the shape of the BM curves on this scale. Overall, it might be reasonable to expect variations of as much as 10 db: we find a variation of about 6 db with the extreme range being 10 db in any one CF region. Given these factors it is difficult to establish why our results are at variance with those of Palmer and Evans. Their data appear to have few fibres with very obvious sloping-saturation and it may be that statistical variations in the data caused them to misclassify some responses. Also, Sachs et al. (1989) point out that it is difficult to judge visually the correct break-point, presumeably because of statistical fluctuations, and in this regard it is our experience that digital smoothing of the type employed by Sachs and coworkers and by Palmer and Evans serves only to increase this confusion. Loudness perception and intensity discrimination Assuming that similar BM dynamics occur in humans, the compression of input dynamic range should be detectable in psychoacoustic estimates of intensity difference limens and loudness estimates. Many estimates of intensity difference limens (IDLs) show a steep change of IDL with intensity near threshold (Saunders et al., 1987; Viemeister and Bacon, 1988) and a flatter region starting about db above threshold. It is tempting to identify the change in IDL slope with the change in BM I/O slope which occurs at very similar intensities in the guinea pig and chinchilla. The latter species in particular has been shown to exhibit a similar break in the IDL at around (Saunders et al., 1987). Given that the changes in nerve activity occur against a background of spontaneous discharge, detectability of a given percentage increase in intensity might grow monotonically with stimulation level of the IHCs and hence give rise to the IDL curves observed. Similarly, the perception of loudness in humans might be directly related to auditory nerve activity and hence may be closely related to BM displacement (see Delgutte, 1987 and Ryugo and Rouiller, 1988 for discussions). Recent estimates of loudness perception (Hellman and Zwislocki, 1968;

15 217._1 13_ O0 rn -o Fig. 12. Three derived guinea pig BM I/O curves plotted for comparison with a human loudness-estimate curve from Viemeister and Bacon (1988). The figures is parentheses are the stimulus frequencies used in deriving the BM I/O curves. The higher frequency is the CF. Viemeister and Bacon, 1988) show that, typically, loudness grows linearly with intensity for low stimulus intensities and then changes to a flatter power-law slope at about db above threshold. The observed slope of the power-law region is somewhat variable, with later results tending to be somewhat flatter than the earlier: Hellman and Zwislocki report a value of 0.54 while Viemeister and Bacon recently found slopes of 0.182, and (Viemeister and Bacon report their slopes referred to power input while we referred our derived BM I/O curves to displacement input. Hence, we have converted their slopes to pressure input, the same reference used by Hellman and Zwislocki. This requires that the slopes reported by Viemeister and Bacon be doubled.) Their loudness estimates cover approximately one order-of-magnitude for a stimulus range of up to 100, a figure very close to our 1.5 orders-of-magnitude for the BM displacement. Fig. 12 plots three more derived BM I/O curves along with loudness-estimates from one of Viemeister's subjects. The resemblance is striking, as the loudness-estimate curve shows all the main features of the BM I/O curves including the slope of 0.2 over most of the hearing range. It also shows a return to linear slope at the high-intensity range, a result predicted by the feedback-saturation model (Zwicker, 1986; Patuzzi et al., 1989), I observed by Johnstone et al., 1986, but not yet observed in the derived BM I/O curves. Of the three sets of data published by Viemeister, two are very similar (Fig. 12 shows one of them) and the third is quantitatively different but shows the same general form. It appears likely then, that loudness perception may be based on a simple coding of BM displacement amplitude at CF. This hypothesis has been advanced previously (see Viemeister 1988) based on psychoacoustic evidence and modelling studies, but has found little physiological support. There remains a small problem in that the full range of loudness seems, in the guinea pig at least, to be coded by two or more sets of fibres, each with a different dynamic range. Simple summation of the responses of a set of fibres would not lead to the psychoacoustically-observed loudness functions, but a relatively simple gating scheme under which the more-sensitive set of fibres was gradually switched out as stimulus intensity was increased and other, less-sensitive, fibres began to respond, could well do so. Conclusions This paper has demonstrated the relationship between the auditory-nerve rate-level functions measured at CF and at frequencies below CF, and has identified the processes which produce the various types of CF RI functions. The three forms currently recognised, saturating, sloping-saturation and straight, are all produced by the same mechanisms operating in slightly different ranges of the stimulus amplitude. The three groups together cover the entire range of stimulus amplitudes between absolute threshold and probably to in excess of 100. Thus, coding of stimulus amplitude requires no other strategy than simple AP rate. A method of plotting the RI functions at CF and a tail frequency produces a curve which we interpret as being a representation of the basilar membrane input-output curve at CF. These derived curves match published BM I/O data in considerable quantitative detail and their interpretation permits much greater confidence in quantitative definition of the BM behaviour with increase in stimulus strength. Comparing the

16 218 BM I/O curves with published data for perception of loudness and for loudness discrimination suggests that these effects may both reflect directly the displacement of the BM. It is now clear that the cochlea accomodates the enormous input range of more than 100 db by compression into a much smaller range, of the order of db (see Fig. 12). At CF the increased sensitivity to low-intensity sounds is accomplished by a nonlinear process of amplifying the weaker sounds more than the louder so that the actual range of stimuli handled by the inner and outer hair cells is approximately the same as for the tail frequencies. This appears to be unique among all the senses: other senses either respond directly to a transformation of their input either with a linearly-related output or by some form of range-change mediated by an adaptation process. We have only circumstantial evidence on which to infer a similar behaviour in humans, but the evidence is strong. Our interpretation explains at least the qualitative details of loudness estimation and of intensity difference discrimination, and the quantitative agreement with recent data is striking. If the case is confirmed, however, then we have plausible explanations for the near-miss to Weber's law of intensity discrimination and for Stevens' power-law of intensity perception. In any case, it would appear that almost our entire range of auditory signal processing, and particularly of speech processing, takes place in the compressive power-law range. It is, moreover, somewhat remarkable that the slope of that power-law, 0.2, is the same for guinea pig and chinchilla, and appears to be the same for humans. Acknowledgements This work was supported by the National Health and Medical Research Council of Australia and The University of Western Australia. We thank Otto Gleich for the data of Figs. 2 and 3, and Brian Johnstone and Robert Patuzzi for critical comments on the manuscript. References Dallos, P. (1985) Response characteristics of mammalian cochlear hair cells. J. Neurosci. 5, Delgutte, B. (1987) Peripheral auditory processing of speech information: Implications from a physiological study of intensity discrimination. In: M.E.H. Schouten (Ed.), The Psychophysics of Speech Perception, M. Nijhoff, Dordrecht, The Netherlands, pp Harris, D.M. and Dallos, P. (1979) Forward masking of auditory nerve fiber responses. J. Neurophysiol. 42-4, Hellman, R.P. and Zwislocki, J.J. (1968) Monaural loudness function at 1000 cps and interaural summation. J. Acoust. Soc. Am. 35, Johnstone, B.M., Patuzzi, R., Syka, J. and Sykov& E. (1989) Stimulus-related potassium changes in the organ of Corti of guinea-pig. J. Physiol. 408, Johnstone, B.M., Patuzzi, R. and Yates, G.K. (1986) Basilar membrane measurements and the travelling wave. Hear. Res. 22, Kemp, D.T. (1979) The evoked cochlear mechanical response and the auditory microstructure - evidence for a new element in cochlear mechanics. Scand. Audiol. Suppl. 9, Kiang, N.Y.-S. (1984) Peripheral processing of auditory information. In: I. Darian-Smith (Ed.), Handbook of Physiology. Section 1, Vol. 3, Sensory Processes. American Physiological Society, Washington, DC. Liberman, M.C. (1978) Auditory nerve response from cats raised in a low-noise chamber. J. Acoust. Soc. Am. 63, Long, G.R. (1984) The microstructure of quiet and masked thresholds. Hear. Res. 15, Nomoto, M., Suga, N. and Katsuki, Y. (1964) Discharge pattern and inhibition of primary auditory nerve fibers in the monkey. J. Neurophysiol. 27, Palmer, A.R. and Evans, E.F. (1980) Cochlear fibre rate-intensity functions: no evidence for basitar membrane nonlinearities. Hear. Res. 2, Patuzzi, R. and Sellick, P.M. (1983) A comparison between basilar membrane and inner hair cell receptor potential input-output functions in the guinea pig cochlea. J. Acoust. Soc. Am. 74, Patuzzi, R.B. and Robertson, D. (1988) Tuning in the mammalian cochlea. Physiol. Rev. 68, Patuzzi, R.B. and Yates, G.K. (1987) The low frequency response of inner hair cells in the guinea pig cochlea:-implications for fluid coupling and resonance of the stereocilia. Hear. Res. 30, Patuzzi, R.B., Yates, G.K. and Johnstone, B.M. (1989) Changes in cochlear microphonic and neural sensitivity produced by acoustic trauma. Hear. Res. 39, Rhode, W.S. (1971) Observations of the vibration of the basilar membrane using the Mossbauer technique. J. Acoust. Soc. Am. 49, Robles, L., Ruggero, M.A. and Rich, N.C. (1986) Basilar membrane mechanics at the base of the chinchilla cochlea. I. Input-output functions, tuning curves, and phase responses. J. Acoust. Soc. Am. 80, Russell, I.J. and Sellick, P.M. (1978) Intracellular studies of hair cells in the mammalian cochlea. J. Physiol. 284,

17 219 Ryugo, D.K. and Rouiller, E.M. (1988) Central projections of intracellnlarly labelled auditory nerve fibres in cats: Morphometric correlations with physiological properties. J., Comp. Neurol. 271, Sachs, M.B. and Abbas, P.J. (1974) Rate versus level functions for auditory-nerve fibres in cats: tone-burst stimuli. J. Acoust. Soc. Am. 56, Sachs, M.B., Winslow, R.L. and Sokolowski, B.H.A. (1989) A computational model for rate-level functions from cat auditory-nerve fibers. Hear. Res. 41, Saunders, S.S., Bhagyalakshmi, G. and Salvi, R.J. (1987) Auditory intensity discrimination in the chinchilla. J. Acoust. Soc. Am. 82, Schroeder, M.R. and Hall, J.L. (1974) Model for mechanical to neural transduction in the auditory receptor. J. Acoust. Soc. Am. 55, Sellick, P.M., Patuzzi, R. and Johnstone, B.M. (1982) Measurement of basilar membrane motion in the guinea pig using the Mossbauer technique. J. Acoust. Soc. Am. 72, Smith, R.L. and Brachman, M.L. (1980) Operating range and maximum response of single auditory nerve fibres. Brain Res. 184, Viemeister, N.F. (1988). Intensity coding and the dynamic range problem. Hear. Res. 34, Viemeister, N.F. and Bacon, S.P. (1988) Intensity discrimination, increment detection, and magnitude estimation for 1 khz tones. J. Acoust. Soc. Am. 84, Wilson, J.P. (1980) Evidence for a cochlear origin for acoustic re-emissions, threshold fine structure and tonal tinnitus. Hear. Res. 2, Winter, I.M., Robertson, D.R. and Yates G.K. (1990) Diversity of characteristic frequency rate-intensity functions in guinea pig auditory nerve fibres. Hear. Res. 45, Yates, G.K. (1987) Dynamic effects in the input/output relationship of auditory nerve. Hear Res. 27, Yates, G.K. (1990) Basilar membrane nonlinearity and its influence on auditory nerve rate-intensity functions. Hear. Res. (in press). Zwicker, E. (1986) A hardware cochlear nonlinear preprocessing model with active feedback. J. Acoust. Soc. Am. 80,