The Human Ear
2.2 MECHANICAL PROPERTIES OF THE CENTRAL PARTITION
2.2.4 The Half Octave Shift
Hirsh and Bilger (1955) first reported an observation that is widely referred to as the
“half octave shift”. They investigated the effect upon hearing levels at 1 kHz and
70 0 1000
14 00 2000
28 00468 10 12 14 16 18
Ba sal En d A Ap ical E nd
Frequency (Hz)
Temporary Threshold Shift (dB)
B
Figure 2.4 Typical half octave shift due to exposure to a loud 700 Hz tone.
1.4 kHz of six subjects exposed to a 1 kHz tone for intervals of time ranging from 10 seconds to 4 minutes and for a range of sensation levels from 10 dB to 100 dB.
Sensation levels are understood to mean levels relative to the threshold of the individual test subject.
Reporting mean results of their investigation, Hirsh and Bilger (1955) found that for one minute duration in each case and for all sensation levels from 10 to 100 dB the threshold shift at 1 kHz was essentially a constant 6 dB but the threshold shift at 1.4 kHz was an increasing monotonic function of sensation level. At a sensation level of 60 dB the shifts at the two frequencies were essentially the same, but at higher sensation levels, the shift at 1.4 kHz was greater than at 1 kHz. Many subsequent investigations have confirmed these results for other frequencies and for other species as well.
A result typical of such investigations, has been used to construct Figure 2.4. In the figure, temporary threshold shift (TTS) in decibels is shown as a function of frequency in kilohertz distributed on the abscissa generally as along the cochlear duct with high frequencies at the basal end on the left and low frequencies at the apical end on the right.
In the figure, TTS is shown after exposure to an intense 700 Hz tone. It is observed that an 11 dB shift at 700 Hz is associated with a larger shift of about 17 dB at 1 kHz, one-half octave higher in frequency. Significant loss is observed also at even higher frequencies but essentially no loss is observed at frequencies below 700 Hz.
1
2
3 a b (stimulus)
c (half octave above)
Co chlear D uct
Characteristic Frequency (Hz)
Ba sal E nd A B Ap ical En d
M ax im
um velocity M ax im
um displacement
Highfrequency Lo w
frequency Figure 2.5 Half-octave shift model.
Crucial to the understanding of the explanation which will be proposed for the half-octave shift is the observation that the outer hair cells are displacement sensors and the inner hair cells, which provide frequency and amplitude information to the brain, are velocity sensors. While the idea that the outer hair cells are displacement sensors is generally accepted, the quantity that the inner hair cells sense is controversial. Bies (1999) has shown that the inner hair cells are velocity sensors as first proposed by Billone (1972) and that the published papers which have claimed otherwise and have created the controversy have in every case made errors, which negate their conclusions.
For explanation of Ward’s data shown in Figure 2.4, reference will be made to Figure 2.5. To facilitate the explanation proposed here for the half octave shift, points A and B have been inserted in Figures 2.4 and 2.5. In Figure 2.4 they indicate the one- half octave above and the stimulus frequencies, respectively, while in Figure 2.5 they indicate locations on the cochlear duct corresponding to the places, respectively, where the one-half octave above and the stimulus frequencies are sensed. In the two figures, the Apical end and the Basal end have been inserted to remind the reader that low frequencies are sensed at the apical end and high frequencies are sensed at the basal end of the cochlear duct.
Reference is made now to Figure 2.5 on which the ordinate is the characteristic frequency associated with location on the central partition and the abscissa is the
location on the central partition. In the figure, line (1), which remains fixed at all sound pressure levels, represents the locus of characteristic frequency (maximum velocity response) versus location on the duct. The location of line (2), on the other hand, represents the locus of the frequency of maximum displacement response at high sound pressure levels. The location of line (2) depends upon the damping ratio according to Equation (10.15) which, in turn, depends upon the sound pressure level.
Equation (10.15) shows that for the frequency of maximum displacement response to be one-half octave below the frequency of maximum velocity response for the same cochlear segment, the damping ratio must equal 0.5. In the figure, line (2) is shown at high sound pressure levels (> 100 dB) at maximum damping ratio and maximum displacement response. As the sound pressure level decreases below 100 dB, the damping decreases and line (2) shifts toward line (1) until the lines are essentially coincident at very low sound pressure levels.
Consider now Ward’s investigation (Figure 2.4) with reference to Figure 2.5.
Ward’s 700 Hz loud exposure tone is represented by horizontal line (3), corresponding to exposure of the ear to a high sound pressure level for some period of time at the place of maximum displacement response at (a) and at the same time at the place of maximum velocity response at (b). The latter point (b) is independent of damping and independent of the amplitude of the 700 Hz tone, and remains fixed at location B on the cochlear partition. By contrast, the maximum displacement response for the loud 700 Hz tone is at a location on the basilar membrane where 700 Hz is half an octave lower than the characteristic frequency at that location for low level sound. Thus, the maximum displacement response occurs at intersection (a) at location A on the cochlear partition, which corresponds to a normal low-level characteristic frequency of about 1000 Hz, which is one-half of an octave above the stimulus frequency of 700 Hz.
The highest threshold shift, when tested with low level sound, is always observed to be one half octave higher than the shift at the frequency of the exposure tone at (b).
Considering the active role of the outer hair cells, which are displacement sensors, it is evident that point (a) is now coincident with point (c) and that the greater hearing level shift is due to damage of the outer hair cells when they were excited by the loud tone represented by point (a). This damage may prevent the outer hair cells from performing their undamping action, resulting in an apparent threshold shift at the characteristic frequency for low level sound (half an octave higher than the high level sound used for the original exposure).
The lesser damage to the outer hair cells at frequencies higher than the frequency corresponding to one half octave above the exposure tone, may be attributed to the effect of being driven by the exposure tone at a frequency less than the frequency corresponding to the maximum velocity response. Estimation of the expected displacement response in this region on the basal side of point A on the cochlear duct, at the high damping ratio expected of passive response, is in reasonable agreement with this observation.
Here, a simple explanation has been proposed for the well-known phenomenon referred to as “the half octave shift”. The explanation given here was previously reported by Bies (1996).
f(z)'20146e&4 835z&139.8z (2.29)
logef ' loge20146&4.835z (2.30) 2.2.5 Frequency Response
It is accepted that the frequency response of the central partition ranges from the highest audible frequency at the basal end to the lowest audible frequency at the apical end and, by convention, it also is accepted that the lowest frequency audible to humans as a tone is 20 Hz and the highest frequency is 20 kHz. In the following discussion, it will be assumed that the highest frequency is sensed at the basal end at the stapes and the lowest frequency is sensed at the apical end at the helicotrema.
To describe the frequency response along the central partition, it will be convenient to introduce the normalised distance, z, which ranges from 0 at the basal end of the basilar membrane to 1 at the apical end. Based upon work of Greenwood (1990), the following equation is proposed to describe the frequency response of the central partition:
Substitution of z = 0.6 in Equation (2.29) gives the predicted frequency response as 1024 Hz. For z # 0.6, comparison of the relative magnitudes of the two terms on the right hand side of Equation (2.29) shows that the second term is always less than 8%
of the first term and thus may be neglected. In this case Equation (2.29) takes the following form:
Equation (2.30) predicts that, for frequencies higher than about 1 kHz, the relationship between frequency response and basilar membrane position will be log-linear, in agreement with observation.