Cochlear Delays and Traveling Waves: Comments on ‘Experimental Look at Cochlear Mechanics’

Abstract

A. Dancer argued that direct and indirect measures of basilar membrane motion are more consistent with theories of cochlear resonance than with the traveling-wave theory. The present communication reviews empirical evidence that contradicts Dancer’s argument. Such evidence – recordings of mechanical responses of the basilar and Reissner’s membranes to sound – strongly supports the existence of displacement waves that propagate on the basilar membrane from the base of the cochlea toward its apex.

1. Introduction

The traveling-wave theory of cochlear mechanics states that the fundamental cochlear response to acoustic stimuli consists of a displacement wave which propagates along the basilar membrane from base to apex [see reviews in ref. 1, 2]. Apparently first proposed in the late 19th century as an alternative to the resonance theory of Helmholtz [3], the traveling-wave theory received decisive empirical support in the pioneering studies carried out by Békésy [4] in human temporal bones and in the inner ears of dead mammals. Cochlear theory has subsequently been forced to accommodate a variety of findings on basilar membrane motion in live cochleae [reviewed in ref. 5] that almost inescapably imply a role of the organ of Corti in shaping the responses to sound of the basilar membrane [reviewed in ref. 6, 7]. Nevertheless, contemporary mathematical models of cochlear mechanics [reviewed in ref. 8] have retained the traveling-wave concept as a substrate on which to superimpose additional properties which endow the basilar membrane with heightened sensitivity, increased sharpness of tuning and intensity-dependent nonlinearity, and which render its responses physiologically vulnerable.

One important distinction between theories of cochlear resonance and the traveling-wave theory is that in the former all sites of the basilar membrane are stimulated rapidly and simultaneously, irrespective of characteristic frequency, while in the latter stimulation occurs more slowly and sequentially, with regions near the oval window excited first and regions further away disturbed at progressively later times. According to Békésy’s and later measurements in human temporal bones and animal cochleae, and also according to theoretical interpretations of the empirical findings, traveling waves propagate rapidly near the base (at speeds of tens of meters per second) but slow down considerably upon approaching their characteristic places, i.e., the sites where vibrations reach their peak magnitude. In a recent publication, Dancer [9] suggested that direct and indirect measures of basilar membrane motion are more consistent with theories of cochlear resonance than with the traveling-wave theory. The purpose of the present communication is to review empirical evidence that Dancer neglected to consider and which contradicts his proposal and supports the traveling-wave theory.

In the spirit of Dancer’s paper, I shall largely eschew discussion of theoretical issues and will rather highlight ‘experimental measurements performed on actual cochleae’ [9, p. 304]. However, one theoretical underpinning of the following arguments must be emphasized, namely that traveling waves and cochlear resonance do not preclude each other. On the contrary, most theories of ‘passive’ cochlear mechanics describe traveling waves that peak at basilar membrane sites characterized by their resonance frequency (due to damping, the peak response for a given stimulus frequency actually occurs at a location slightly basal to the site of resonance for that frequency) [1, 2, 10–13]. In the words of Lighthill [10]: [the response of the basilar membrane] ‘demands a travelling-wave model which incorporates an only lightly damped resonance’.

2. Definitions of Traveling Wave and Propagation Delays

Dancer [9] used the expression ‘traveling wave’ in the sense that is usual in the context of cochlear mechanics, namely that of a displacement wave that propagates on the basilar membrane from base to apex in response to acoustic stimulation. Since the basilar membrane displacement results from the pressure difference between perilymph in scala vestibuli and scala tympani [11–13], the pressure difference wave itself may also be viewed as a traveling wave that propagates along the cochlea with speed and direction similar to that of the displacement wave. Yet a third traveling wave is the acoustic pressure wave that propagates in the cochlear fluids at the speed of sound. The acoustic pressure wave is much faster than either the pressure difference or displacement waves. The pressure difference and displacement waves also differ from the fast acoustic wave in that they always travel from the cochlear base to its apex [14, 15], irrespective of whether sound enters the cochlea via the stapes, the round window or the otic capsule (by bone conduction). (In the case of otoacoustic emissions that originate inside the cochlear partition, the basilar membrane almost certainly also sustains traveling waves that propagate in a ‘backward’ direction, i.e., toward the base [12, footnote 12, p. 11; 16].) Following Dancer’s usage, in the remainder of this paper ‘traveling wave’ will be used exclusively to refer to the displacement wave that propagates along the basilar membrane.

Several definitions of propagation delay are possible [17; 18, pp. 52–58]. In the case of responses to short-duration transient stimuli, such as clicks (fig. 1), the most commonly measured delays are those elapsing between the stimulus and either the beginning of the response (signal-front delay) or its center of gravity (weighted-average group delay). In the frequency domain, the center of gravity of the time-domain response (fig. 1) corresponds to (minus) the average slope of the phase-versus-frequency function, with slopes (group delays) assigned weights proportional to the spectral response magnitudes [19]. For stimulation with low-intensity clicks, the center of gravity of the basilar-membrane response approaches the group delay near the characteristic frequency. The signal-front delay is approximated by the asymptotic slopes of the phase-versus-frequency function at frequencies distant from (either above or below) the characteristic frequency (fig. 2) [17]. Dancer [9] did not specify which of these alternative delays he regards as most appropriate to measure propagation time on the basilar membrane. However, he did explicitly dismiss the steep phase-versus-frequency slope near the characteristic frequency: ‘These phase lags should not be considered as propagation delays’ [p. 310] and highlighted the low-frequency regions of phase curves for basal regions of the chinchilla and guinea pig cochleae [20, 21]. My interpretation of Dancer’s view is summarized in the following two points.

Fig. 1.

Basilar-membrane velocity responses to rarefaction clicks in a relatively normal chinchilla cochlea (left column) and postmortem (right column). The peak negative pressure of the acoustic clicks (top right) is indicated by each trace. Top ordinate scale applies to all basilar-membrane traces but those in bottom row. The bottom traces are compressed by exactly 10 dB relative to the other traces. The dashed lines indicate the time of arrival of the acoustic click at the eardrum. The interval between this time and that of basilar membrane vibration onset (arrows) comprises the basilar-membrane travel time (90 µs) plus a small middle-ear component. The weighted-average group delays are indicated by filled circles. Slightly modified, with permission, from figure 1 of reference 26.

Fig. 2.

Phases of basilar-membrane responses to clicks in a live cochlea (continuous lines) and postmortem (dotted line). Phases, computed by Fourier transformation of the responses illustrated in figure 1, correspond to basilar-membrane displacement toward scala tympani relative to inward stapes displacement. The responses of clicks were digitally deconvolved to reduce the effect of irregularities in the stimulus frequency spectrum (note ‘ringing’ in the click pressure waveform at top right of fig. 1). The thick continuous line represents responses to clicks with a peak SPL of 107 dB; the thin line represents responses to 57-dB clicks. The dashed line has a slope of −125 µs. The down-pointing arrow indicates the characteristic frequency. Modified, with permission, from figure 4 of reference 26.

(1) If a traveling wave exists, its concomitant delays should be reflected in the low-frequency slope of the phase-versus-frequency function. Dancer [9, p. 310] argues that, in fact, ‘… it is not possible to observe the large phase lags and the long delays which … characterize the traveling wave’. Thus, Dancer concludes: ‘… when an impulse is applied to the stapes, it excites at once the whole cochlear partition; all points of the basilar membrane do begin to move at the same time.’

(2) The steep slope of the phase-versus-frequency function in the region of the characteristic frequency largely reflects the delay due to filtering, with longer delays corresponding to sharper filtering. ‘Furthermore, the large phase lags measured in cochleas in good condition (especially at and beyond the characteristic frequency) depend on the presence of outer hair cells and are probably due partly to the action of the cellular active processes’ [9, p. 310]. As a corollary, a normal basilar-membrane site should display a steep phase-versus-frequency slope (long delay) near the characteristic frequency, while a dead cochlea should produce a shallow slope (short delay).

Dancer’s [9] criterion for deciding whether traveling waves exist in the cochlea was based on the presence of a substantial onset delay for responses to clicks (or, in the frequency domain, on the magnitude of the low-frequency slope of the phase-versus-frequency function). In fact, however, traveling-wave theories do not require that the onset delay exceed any particular value. Nevertheless, it will be shown in sections 3 and 4 of this paper that, contrary to point 1, the latencies of basilar-membrane responses to clicks and the mid-frequency slopes of phase-versus-frequency curves for responses to tones are consistent with the existence of a traveling wave. I shall also show (section 5) that, in contradiction with point 2, the steep phase-versus-frequency slope near the characteristic frequency is not a property exclusive to basilar membranes in normal (or ‘active’) cochleae but, rather, can also be demonstrated in traumatized or dead cochleae.

3. Measurement of Basilar-Membrane Travel Time using Click Stimuli

A simple and direct way to demonstrate the existence of a traveling wave is to measure the latency between the presentation of a punctate acoustic stimulus at the oval window and the onset of a displacement response at the basilar membrane. Békésy [22] performed a series of such measurements at several basilar membrane sites in human temporal bones and showed that latencies increased systematically with distance from the oval window. The fact that Békésy could measure such delays in the ears of cadavers implies that traveling waves are passive properties of the hydrodynamics of the cochlear fluids and the mechanical impedance of the cochlear partition. This implication has been supported by subsequent investigations in the ears of live animals (section 5) [23–26].

Figure 1 shows responses to clicks at a basilar-membrane site, with a characteristic frequency of 9.5–10 kHz, located 3.5 mm from the basal end of the chinchilla cochlea. As seen in the left part of the figure, in relatively normal cochleae the responses vary nonlinearly as a function of click intensity. Responses to low-level clicks have comparatively long latencies and grow at a rate less than linear. As click intensity increases, the responses acquire an initial component that grows linearly with stimulus intensity and thus the peaks of the response envelopes shift to earlier times. Because the initial response component grows faster than later ones, it comes to dominate the overall response at the highest stimulus intensities. If delays peripheral to the cochlea (propagation time between the earphone and the stapes footplate) are subtracted, a 90-µs propagation delay for the cochlear travel time is obtained [26]. Fourier transformation of waveforms such as these yield phase-versus-frequency functions, nearly identical to those obtained by stimulation with tones [26], which consist of three segments: a low-frequency, flat (zero-slope) segment which extends between DC and 1–3 kHz; a mid-frequency segment, with shallow, constant slope between 1–3 kHz and frequencies just below the characteristic frequency, and a high-frequency segment, with steep slope, around and beyond the characteristic frequency (fig. 2). The latency of the time-domain response onset (90 µs) is closely approximated by (minus) the slope (125 µs) of the phase-versus-frequency curve in the mid-frequency region (between 2 and 8 kHz in the case illustrated in fig. 2).

The responses to clicks at the 7-kHz basilar membrane site of the squirrel monkey cochlea [23] have a latency of 300–390 µs after the onset of stapes motion. As for the case of the chinchila cochlea, this latency closely matches (in the mid-frequency region) the slope of the phase-versus-frequency curve for responses to tones. Given that the 7-kHz characteristic frequency probably corresponds to a site of the squirrel monkey cochlea located some 8–9 mm from its basal end [27], a latency of 300–390 µs appears consistent with the 90-µs latency at the 3.5-mm site of the chinchilla basilar membrane.

4. Measurement of Basilar-Membrane Travel Time using Tone Stimuli

An especially convincing demonstration of traveling waves is obtained by measuring basilar membrane responses at two or more sites in the same cochlea, separated by sizable longitudinal distances [28–31]. Figure 3 illustrates such measurements for two places in one squirrel monkey basilar membrane some 1.5 mm apart [28]. It is apparent that responses at the more apical location (with characteristic frequency slightly below 6 kHz) increasingly lag the basal responses (characteristic frequency of 7 kHz) as a function of increasing frequency. This is as expected for a wave that travels between the two sites in the base-to-apex direction. The difference between the slopes of the low-frequency segments (380 and 470 µs) corresponds to a frequency-independent propagation time of 90 µs (or, equivalently, a traveling-wave velocity of 16.6 m/s). Measurements similar to those illustrated in figure 3, carried out in the basal regions of cat [29–31] and guinea pig [31] cochleae, are quite consistent in showing that the phase-versus-frequency slopes at frequencies below the characteristic frequency are progressively steeper with increasing distance from the cochlear base.

Fig. 3.

Transfer functions for two basilar membrane sites, 1.5 mm apart, in a squirrel monkey cochlea, a Magnitude frequency spectra, normalized to the vibration of the malleus. b Phase spectra: basilar membrane displacement toward scala tympani relative to malleus inward displacement. Note that the phase curve is steeper at the more apical site (i.e. the site with lower characteristic frequency; see frequency-distance map in fig. 5). Reproduced, with permission, from figure 8 of reference 28.

Whereas most methods for measuring displacement or vibration are suitable only for the study of a single basilar-membrane location at a time, the ‘fuzziness-detection’ technique [32] permits simultaneous observations of sites along a 3- to 5-mm length. Using this technique in the basal turn of the guinea pig cochlea, Kohllöffel [32] obtained phase-versus-distance plots consisting of a basal segment with a shallow slope and an apical segment with steeper slope (fig. 4). The non-zero spatial slopes indicate that waves propagate on the basilar membrane from base to apex, and that they do so rapidly at the base and more slowly beyond their characteristic places. For frequencies of 5–9 kHz, the basal and apical slopes corresponded to traveling-wave velocities of 15–23 and 5–7 m/s, respectively.

Fig. 4.

Spatial distribution of postmortem responses to a 5-kHz tone on the basilar membranes of two guinea pig cochlea. The upper panels indicate the relative magnitudes of vibration (expressed as stimulus attenuations, in decibels, required for a constant response) as a function of distance from the oval window. The lower panels show the variation of response phases (expressed in radians) as a function of distance. Reproduced, with permission, from figure 3 of reference 32.

Phase-versus-frequency functions are available for a number of basal basilar-membrane locations in the guinea pig [31, 33] and the cat [29–31], as well as for the basilar-membrane sites in the squirrel monkey [34, 35] and the chinchilla [20, 26] discussed in section 3. The mid-frequency segments of the phase-versus-frequency curves have slopes corresponding to delays ranging from less than 20 µs to greater than 400 µs. These delays appropriately reflect cochlear location; for example: extreme basal locations of the guinea pig cochlea [31] yield the shortest delays (about 20 µs) while measurements at a site in the squirrel monkey cochlea some 8–9 mm from the oval window produce a 334- to 414-µs delay [33, 35].

Because apical sites of the basilar membrane are rather inaccessible, no in vivo recordings at such locations are available for any mammalian species. As a substitute, in vivo measurements of the vibration of Reissner’s membrane have been obtained at several locations in the squirrel monkey cochlea [35]. For sites near the middle of the cochlea, the delays (0.75–1.25 ms, derived from the mid-frequency slopes of phase-versus-frequency functions) are substantially larger than delays for basal locations (fig. 5). At the cochlear apex, delays are even larger (> 2.5 ms).

Fig. 5.

Travel times in the squirrel monkey cochlea as a function of location. The data points represent the negative of the mid-frequency slope of phase-versus-frequency functions for the basilar membrane (□) [34] or Reissner’s membrane (▲) [35], or the onset latency for basilar-membrane responses to clicks (○) [23], in each case measured relative to motion of the middle-ear ossicles. The abscissa is the logarithm of characteristic frequency. Also shown is a scale indicating cochlear locations according to the frequency-distance map proposed by Greenwood [27].

In his review, Dancer [9] argued that discrepancies between travel times calculated using Zwislocki’s [36] hydrodynamical model and Békésy’s [22] mechanical data for human temporal bones invalidate the traveling-wave theory. Similarly, Dancer [9] contrasted Zwislocki’s [37] calculations for travel time in guinea pig cochleae with delays derived from cochlear microphonics [9, fig. 1; 38] and concluded that the measured delays are inconsistent with the existence of traveling waves. It seems to the present writer that it is illogical to reject the existence of traveling waves merely on the basis that travel times measured in (or derived from) some selected experiments are shorter than those predicted by a particular mathematical model (such as Zwislocki’s). As shown in this and the preceding section, all available direct measurements of the vibration of the cochlear partition, including those cited in Dancer’s [9] review, actually reveal that displacement waves propagate on the basilar membrane at finite speeds, albeit very rapidly at the base of the cochlea. Lastly, it is worth noting that the use of the phases of cochlear microphonics to derive cochlear delays [as favored by Dancer, 9] is fraught with uncertainty because of the significant distance between the recording electrodes and the widely distributed signal sources (the outer hair cells).

5. Effect of Cochlear Dysfunction or Death on Basilar Membrane Response Phases

Comparison of the left and right parts of figure 1 shows that the delay to response onset is not affected by death. This is consistent with the fact that basilar membrane responses to tones with frequency well below the characteristic frequency are linear and unaffected by trauma or death [28, 34, 39, 40]. On the other hand, it is also clear that death causes a relatively large shift of the center of gravity of the responses to a constant, short delay. That this shift is not due to a commensurate decrease of the group delay (phase slope) near the characteristic frequency is obvious from perusal of the phase spectra for the same data (fig. 2). In general, irrespective of stimulus intensity, the phases of normal responses are very similar to those of postmortem responses. The changes in the centers of gravity both in live and dead cochleae (fig. 1) arise largely from the loss of sharp frequency tuning, as a result of which the low-frequency spectral regions, with shallow group delays, acquire greater weights (relative to the steep-slope high-frequency region) in determining the weighted average (see section 2). However, a small component of the shift of the center of gravity arises from a reduction of the group delay around the characteristic frequency: the postmortem group delay (470 µs) is noticeably smaller (by 220 and 50 µs, respectively) than the group delays of responses to 57- or 107-dB clicks in live, normal cochleae. This shorter group delay is due to a combination of phase lags at frequencies immediately below the characteristic frequency and phase leads at frequencies just above the characteristic frequency. (NB: In healthy cochleae, response phases near the characteristic frequency depend on stimulus intensity [26, 28, 31, 39, 41]: as intensity is raised, responses just below (above) the characteristic frequency lag (lead) responses to lower-intensity stimuli. Such nonlinear lag/lead transition is the phase counterpart of the nonlinear intensity dependence of gain that characterizes normal basilar membrane vibrations [5, 24–26, 28, 31, 34, 39, 40, 42, 43]. Upon death or cochlear trauma both types of nonlinearities disappear so that the features of postmortem basilar membrane responses resemble and exaggerate the features of responses to very intense stimuli in live cochleae, e.g., see bottom panels of fig. 1.) One might view these postmortem changes in group delay as being consistent with Dancer’s [9] argument that ‘the large phase lags measured in cochleas in good condition … are … due … to the action of the cellular processes (of outer hair cells)’. In fact, figure 2 contradicts Dancer’s argument in that, at the characteristic frequency (9.5–10 kHz), there is essentially no change of absolute phase due to death.

A stark demonstration that cochlear dysfunction, far from promoting phase leads at the characteristic frequency (as suggested by Dancer [9]), can actually have the opposite effect is presented in figure 6. Figure 6 shows the effects of an intravenous injection of furosemide, an ototoxic drug, upon the phases of click-evoked vibrations at a basal site of the chinchilla basilar membrane [42]. Furosemide, which interferes with hair cell function by reducing the endocochlear potential, caused drastic changes in both the magnitudes and phases of responses to sound which were confined to frequencies near the characteristic frequency. There were no changes at frequencies below 5 kHz, i.e., at frequencies at which responses were linear to start with. In particular, the slope of the phase-versus-frequency in the mid-frequency region (which corresponds to the delay to response onset) remained unaltered. On the other hand, there was a large reduction in vibration magnitude at the characteristic frequency (not shown) which was accompanied by phase lags as large as 180 degrees. However, since the drug-induced phase lags were relatively uniform over a wide frequency region, the slope of the phase-versus-frequency curve was hardly changed near the characteristic frequency. Thus, the overall effect consisted of a shift of the curve toward lower frequencies, in accord with the drug-induced decrease of (apparent) characteristic frequency.

Fig. 6.

Effect of systemic injection of furosemide upon the phases of chinchilla basilar membrane responses to clicks. Phases were computed via Fourier transformation of the time-averaged responses to 75-dB (peak SPL) clicks. The down-pointing arrow indicates the characteristic frequency (i.e., the frequency of the maximal response at low stimulus levels). The circles indicate the frequencies at which responses to 75-dB clicks were largest before and after furosemide injection. Negative phase values (ordinate, expressed in periods) indicate lags relative to the start of data collection, which preceded the arrival of the acoustic click at the eardrum by 250 µs (fig. 1). Slightly modified, with permission, from figure 6 of reference 42.

With but one exception, all published data on the effects of cochlear dysfunction on basilar membrane response resemble the results illustrated in figures 2 and 6. As a rule, cochlear damage induced by ototoxic drugs [42], acoustic overstimulation [31, 43] or death [26, 34] either elicits little phase change (as in figure 2) or, more commonly (as in fig. 6), causes phase lags at the characteristic frequency [31, 34, 42, 43]. The only clear exception, which Dancer [9, fig. 5a] happened to highlight, is one guinea pig cochlea in which acoustic trauma apparently caused a large phase lead at the characteristic frequency [44].

6. Summary and Conclusion

The findings reviewed in sections 3 and 4 contradict Dancer’s view (point 1) in that they indicate that the onset of basilar-membrane responses to clicks and, equivalently, the mid-frequency slope of the phase-versus-frequency function for responses to tones, are not zero but rather have values which grow systematically with distance from the oval window.

The effects of death or cochlear damage on the phases of basilar membrane vibration (section 5) contradict Dancer’s view (point 2) that ‘… the large phase lags measured in cochleae in good condition … depend on … the outer hair cells …’: in fact, cochlear dysfunction either leaves unaffected the absolute response phase at the characteristic frequency or actually induces lags.

In conclusion: contrary to Dancer’s proposal, there exists robust evidence that the basilar membranes of both live and dead cochleae do, indeed, support displacement waves that propagate from base to apex, i.e., traveling waves.