96
D.R. Ketten
ness to width provides a surprisingly close approximation of the static stiffness gradient for a mammalian cochlea (von Bekesy 1960; Ketten 1984).
For most species, therefore, the BMF equation devolves to, not surprisingly, a simple expression that reflects the exponential gradient of most
cochleae: f = A e(ax), where A is a stiffness coefficient derived from the
thickness: width ratio, a is the species size factor dictated by the basilar
membrane interturn radii, and x is the intracochlear position (Ketten 1994;
Ketten et al. 1998 for detailed discussions). This equation, for obvious
reasons, has the same form as Greenwood's analyses; the fundamental difference is that it is cochleocentric rather than homocentric and, therefore,
does not presume a generalist format and constant gradient.
On the other hand, this equation does presume a regular spiral and membrane substructure. While the equation is sensitive to membrane gradients,
at this stage it does not accomodate multiple gradients. For specialized
species like CF bats and, possibly, Type I odontocetes with dichotomous
membrane profiles, more than one expression is required. Even more
important, as the curves for the kangaroo and mole rat in Figure 2.8 demonstrate, a t/w ratio-based equation addresses one aspect (stiffness) of a fundamental mechanism (membrane resonance) and can differentiate between
generalist and specialist ears for which a stiffness irregularities internal to
the basilar membrane are the principal variable, but it is blind to auxilliary
structural effects. Mass-loading is just one alternative side to laminar buttressing coin. Certainly, there are more sophisticated and computationally
complex models (see de Boer 1996 for review) that attempt to address
these issues, but few are based in the anatomy and even fewer are aimed
at understanding species-specific variations. For a comprehensive morphometric model, a third step is now required-and like most interesting mathematical issues raised in book chapters, the solution is left, of course, to
the student.
There has been comparatively little work done on inner ear correlates
of low-frequency hearing, but at least one interesting correlate with canal
configurations has been reported. Dallos (1970) found radically different
magnitude and phase responses in two high-frequency species (cat and
chinchilla) and two low-frequency species (guinea pig and kangaroo rat)
that have similar middle ear transfer functions. The differences were consistent with differences in the acoustic input impedances of the cochlea,
helicotrema dimensions, and cochlear spiral turns. Low-frequency sensitivity was inversely related to both helicotrema area and cochlear turns. The
guinea pig and kangaroo rat had areas approximately one-tenth those of
the cat and chinchilla. They also had scala vestibuli that decreased rapidly
in area towards the apex and cochleae with greater than 4 turns. Cat and
chinchilla by contrast had slower rates of decrease in scala vestibuli and
cochleae with less than 3 turns. The rate of change in sensitivity functions
at low frequencies were twice as large (-12dB) in animals with large
helicotrema (cat and chinchilla) as in the animals with small helicotrema
(-6dB). Dallos suggested that these features are consistent with and, in the
D.R. Ketten
ness to width provides a surprisingly close approximation of the static stiffness gradient for a mammalian cochlea (von Bekesy 1960; Ketten 1984).
For most species, therefore, the BMF equation devolves to, not surprisingly, a simple expression that reflects the exponential gradient of most
cochleae: f = A e(ax), where A is a stiffness coefficient derived from the
thickness: width ratio, a is the species size factor dictated by the basilar
membrane interturn radii, and x is the intracochlear position (Ketten 1994;
Ketten et al. 1998 for detailed discussions). This equation, for obvious
reasons, has the same form as Greenwood's analyses; the fundamental difference is that it is cochleocentric rather than homocentric and, therefore,
does not presume a generalist format and constant gradient.
On the other hand, this equation does presume a regular spiral and membrane substructure. While the equation is sensitive to membrane gradients,
at this stage it does not accomodate multiple gradients. For specialized
species like CF bats and, possibly, Type I odontocetes with dichotomous
membrane profiles, more than one expression is required. Even more
important, as the curves for the kangaroo and mole rat in Figure 2.8 demonstrate, a t/w ratio-based equation addresses one aspect (stiffness) of a fundamental mechanism (membrane resonance) and can differentiate between
generalist and specialist ears for which a stiffness irregularities internal to
the basilar membrane are the principal variable, but it is blind to auxilliary
structural effects. Mass-loading is just one alternative side to laminar buttressing coin. Certainly, there are more sophisticated and computationally
complex models (see de Boer 1996 for review) that attempt to address
these issues, but few are based in the anatomy and even fewer are aimed
at understanding species-specific variations. For a comprehensive morphometric model, a third step is now required-and like most interesting mathematical issues raised in book chapters, the solution is left, of course, to
the student.
There has been comparatively little work done on inner ear correlates
of low-frequency hearing, but at least one interesting correlate with canal
configurations has been reported. Dallos (1970) found radically different
magnitude and phase responses in two high-frequency species (cat and
chinchilla) and two low-frequency species (guinea pig and kangaroo rat)
that have similar middle ear transfer functions. The differences were consistent with differences in the acoustic input impedances of the cochlea,
helicotrema dimensions, and cochlear spiral turns. Low-frequency sensitivity was inversely related to both helicotrema area and cochlear turns. The
guinea pig and kangaroo rat had areas approximately one-tenth those of
the cat and chinchilla. They also had scala vestibuli that decreased rapidly
in area towards the apex and cochleae with greater than 4 turns. Cat and
chinchilla by contrast had slower rates of decrease in scala vestibuli and
cochleae with less than 3 turns. The rate of change in sensitivity functions
at low frequencies were twice as large (-12dB) in animals with large
helicotrema (cat and chinchilla) as in the animals with small helicotrema
(-6dB). Dallos suggested that these features are consistent with and, in the