94
D.R. Ketten
average, similar reductions occurred in both optic and vestibular systems in
whales.
6. Gedanken Experiments
6.1 Functional Predictions from Anatomy
Greenwood's equations (1961, 1990) are the most commonly used methods
for estimating the frequency distribution map (range and location of frequencies along the basilar membrane) in different species. They are based
on the distribution of critical bands in the human and on von Bekesy's
(1960) elasticity-position-frequency measurements for six mammals and
one bird. Greenwood's equation for resonant frequency at point (x) of the
basilar membrane is: F = A (wax - k). The empirical values for the related
constants for humans are A = 165.4, k = 0.88, a = 0.06. For all species,
ax = 2.1 for 100% length. Using these values, it is possible to estimate the
distribution of frequency along the cochlea. To estimate basilar membranefrequency (BMF) maps for other mammalian species, A is calculated as:
Aanimal = (A human )(human length/animallength)2
Greenwood's formulae have one free parameter (length) and one
assumption: all membranes are isomorphic with the human. Therefore, the
subject membrane is represented in the calculation as a proportion of
average human length. As discussed earlier, length is an indirect representation for stiffness in generalist ears; Greenwood's calculated curves have
the same form as von Bekesy's membrane-elasticity curves. Fay's extrapolation (1992) of Greenwood's work shows that the BMF distribution equation can be used to derive estimates of critical bands (CB), critical masking
ratio (CRB), and frequency discrimination thresholds (FDT) that are
comparable to psychophysical values for species with generalized ears.
They have recently been shown, with limitations, to be applicable also at an
individual level (Ketten et al. 1998).
However, none of these estimators are robust for specialized ears, particularly not for aquatic echolocators. Some specialized ears are in a sense
cryptomorphic in that their key features are difficult to extract from their
predominately generalist structure. Type II odontocetes fall into this category. Based on conventional measures, Type II odontocetes have few structural deviations from a general terrestrial mammal ear. Nonetheless, these
are specialized ears that violate Greenwood's primary assumption: stiffness
and mass do not covary with length with the same function as land mammal
ears. For example, standard land mammal length-derived hearing models
(e.g., Greenwood 1961,1990) predict an upper limit of hearing of approximately 15kHz for the bottlenosed dolphins, T. truncatus, based on basilar
membrane length of 39 mm (Table 2.1). T. truncatus actually has a functional
D.R. Ketten
average, similar reductions occurred in both optic and vestibular systems in
whales.
6. Gedanken Experiments
6.1 Functional Predictions from Anatomy
Greenwood's equations (1961, 1990) are the most commonly used methods
for estimating the frequency distribution map (range and location of frequencies along the basilar membrane) in different species. They are based
on the distribution of critical bands in the human and on von Bekesy's
(1960) elasticity-position-frequency measurements for six mammals and
one bird. Greenwood's equation for resonant frequency at point (x) of the
basilar membrane is: F = A (wax - k). The empirical values for the related
constants for humans are A = 165.4, k = 0.88, a = 0.06. For all species,
ax = 2.1 for 100% length. Using these values, it is possible to estimate the
distribution of frequency along the cochlea. To estimate basilar membranefrequency (BMF) maps for other mammalian species, A is calculated as:
Aanimal = (A human )(human length/animallength)2
Greenwood's formulae have one free parameter (length) and one
assumption: all membranes are isomorphic with the human. Therefore, the
subject membrane is represented in the calculation as a proportion of
average human length. As discussed earlier, length is an indirect representation for stiffness in generalist ears; Greenwood's calculated curves have
the same form as von Bekesy's membrane-elasticity curves. Fay's extrapolation (1992) of Greenwood's work shows that the BMF distribution equation can be used to derive estimates of critical bands (CB), critical masking
ratio (CRB), and frequency discrimination thresholds (FDT) that are
comparable to psychophysical values for species with generalized ears.
They have recently been shown, with limitations, to be applicable also at an
individual level (Ketten et al. 1998).
However, none of these estimators are robust for specialized ears, particularly not for aquatic echolocators. Some specialized ears are in a sense
cryptomorphic in that their key features are difficult to extract from their
predominately generalist structure. Type II odontocetes fall into this category. Based on conventional measures, Type II odontocetes have few structural deviations from a general terrestrial mammal ear. Nonetheless, these
are specialized ears that violate Greenwood's primary assumption: stiffness
and mass do not covary with length with the same function as land mammal
ears. For example, standard land mammal length-derived hearing models
(e.g., Greenwood 1961,1990) predict an upper limit of hearing of approximately 15kHz for the bottlenosed dolphins, T. truncatus, based on basilar
membrane length of 39 mm (Table 2.1). T. truncatus actually has a functional