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The Influence of Cochlear Shape on Low-Frequency Hearing

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The Influence of Cochlear Shape on Low-Frequency Hearing The influence of cochlear shape on low-frequency hearing Daphne Manoussaki*†, Richard S. Chadwick‡§, Darlene R. Ketten‡¶ʈ, Julie Arrudaʈ**, Emilios K. Dimitriadis††, and Jen T. O’Malley** *Department of Mathematics, Vanderbilt University, Nashville, TN 37240; †Department of Sciences, Technical University of Crete, Hania, Greece 73100; ‡Auditory Mechanics Section, National Institute on Deafness and Other Communication Disorders, and ††Laboratory of Bioengineering and Physical Science, National Institute of Biomedical Imaging and Bioengineering, National Institutes of Health, Bethesda, MD 20892; ¶Department of Otology and Laryngology, Harvard Medical School, Boston, MA 02114; ʈWoods Hole Oceanographic Institution, Woods Hole, MA 02543; and **Massachusetts Ear and Eye Infirmary, Boston, MA 02114 Edited by Jon H. Kaas, Vanderbilt University, Nashville, TN, and approved February 13, 2008 (received for review October 22, 2007) The conventional theory about the snail shell shape of the mam- radius of curvature at the apex as a single, simple measure of malian cochlea is that it evolved essentially and perhaps solely to curvature change (and thus, energy redistribution), and show conserve space inside the skull. Recently, a theory proposed that that it is a robust correlate of LF hearing limits for both land and the spiral‘s graded curvature enhances the cochlea’s mechanical aquatic mammals. Contrary to the existing literature that has response to low frequencies. This article provides a multispecies suggested that material properties and geometry local to the LF analysis of cochlear shape to test this theory and demonstrates region control the LF limit (10, 11), we suggest that this measure that the ratio of the radii of curvature from the outermost and of curvature change of the entire cochlea affects the LF limit. innermost turns of the cochlear spiral is a significant cochlear To understand the effect of change of curvature, one must feature that correlates strongly with low-frequency hearing limits. look at how the cochlea functions. Incoming sounds are trans- The ratio, which is a measure of curvature gradient, is a reflection mitted to cochlear fluids by the eardrum (tympanic membrane) of the ability of cochlear curvature to focus acoustic energy at the and three small articulated bones (the ossicles) that connect the outer wall of the cochlear canal as the wave propagates toward the tympanic membrane to a second membrane (oval window) apex of the cochlea. sealing one of the three chambers of the fluid-filled cochlear spiral. The resulting pressure oscillations in the cochlear fluids inner ear ͉ function ͉ mammalian evolution ͉ spiral deflect the basilar membrane (BM), a membranous partition that forms one boundary of the central chamber (scala media) t is often thought that mammalian cochleae are coiled to pack and spans the length of the spiral cochlea. This deflection Ia longer organ into a small space inside the skull and that the propagates along the BM from the oval window toward the spiral cochlear coil increases the efficiency of blood and nerve supply apex as a transverse traveling wave with a time-varying peak through a central shaft (1). Although these spatial advantages of displacement envelope. At high frequencies, the amplitude of a coiled cochlea have been generally accepted, understanding the traveling wave is maximized near the cochlear base where the the effect of shape on hearing itself has been a challenge. BM is stiffest. The traveling wave is effectively terminated Cochlear coiling is absent in reptiles, birds, and monotreme shortly thereafter. At LFs, the wave travels the full length of the mammals, and it appears to have originated in the marsupial and spiral and achieves its maximum amplitude near the spiral’s apex placental mammal lines (2). Coiling allowed the cochlea to where the BM is more compliant (12). The mechanical proper- become longer, increasing the potential octave range, whereas ties of the BM as well as the energy density traveling along the uncoiled cochleae have been associated with relatively limited cochlea control the characteristics of motion of the traveling hearing ranges. Earlier studies suggested that the evolution of wave. coiling enhanced high-frequency hearing (3). This suggestion, However, both the peak amplitude of the traveling wave and however, is not wholly satisfactory for several reasons. Above all, the radial profile of the transverse wave are important. Neuro- increased hearing ranges extended both high-frequency and sensory hair cells seated atop the BM are stimulated by radial low-frequency (LF) hearing abilities in mammals compared with shear forces. Hair cells are therefore responsive to radial vari- birds and reptiles and improved sensitivities compared with even ations in BM motion. We thus suggest that the spiral shape LF specialist fishes (4). Further, the highest-frequency waves are affects strongly these radial motion variations, and by doing so, resolved near the base (entrance) before they propagate far it also affects hair cell stimulation and hearing. enough into the spiral to ‘‘feel’’ the cochlear curvature; it is the lowest-frequency waves that propagate along the cochlea’s coils. Calculation of Sound Energy Focusing at the Apex Earlier work on land mammal ear anatomy (5) found a strong The radial profile of the BM wave is a result of the wave energy correlation between the LF hearing limit of each species and the distribution across the cochlear channel. As wave energy prop- product of basilar membrane length and number of spiral turns, agates along the spiral, energy propagation paths glance off a but did not adduce a mechanistic explanation for this relation- wall with increasing curvature, and thus progressively focus ship. Other data suggested also that longitudinal curvature of the toward the outer wall (8). The energy propagation pathways are cochlear duct generates radial fluid pressure gradients (6) and enhances radial movement of hair cells (1, 7). Recently, a new theory proposed that the cochlea’s graded Author contributions: D.M. and R.S.C. designed research; D.M., R.S.C., D.R.K., J.A., and curvature actually enhances LF hearing (8), similar to a whis- J.T.O. performed research; D.M., R.S.C., and E.K.D. contributed new reagents/analytic tools; pering gallery in which sounds cling to the concave surface of the D.M., R.S.C., D.R.K., J.A., and J.T.O. analyzed data; and D.M., R.S.C., and D.R.K. wrote the lateral wall (9). The cochlear spiral shape redistributes wave paper. energy toward the outer wall, particularly along its innermost, The authors declare no conflict of interest. tightest, apical turn, and thereby enhances sensitivity to lower- This article is a PNAS Direct Submission. frequency sounds. Freely available online through the PNAS open access option. In this article, we test this theory morphometrically. We use §To whom correspondence should be addressed. E-mail: [email protected]. the ratio of the radius of curvature at the base of the spiral to the © 2008 by The National Academy of Sciences of the USA 6162–6166 ͉ PNAS ͉ April 22, 2008 ͉ vol. 105 ͉ no. 16 www.pnas.org͞cgi͞doi͞10.1073͞pnas.0710037105 Downloaded by guest on September 30, 2021 Fig. 1. Focusing of rays by spirals. A uniform distribution of 200 rays (red) entering and filling a spiral duct (walls in blue) at the largest radius is increasingly focused toward the outer wall as the rays progress toward the apex (spiral center). The effect increases with the parameter ␣, which controls the rate of decrease of radius in the exponential spirals described by the formula given in the figure. Insets shown below each spiral, in an expanded scale, indicate the radial gradient of ray or energy density at the apex. These distributions should be compared with a uniform distribution of rays entering at the largest radius to see the energy density focusing effect of the spiral walls. likened to sound propagation in a whispering gallery, only the This result assumes that waves propagate to the LF region. An cochlear spiral, because of this focusing, is more effective. estimate for the energy focusing at a position ␪ along the spiral Regardless of the mechanism for the encoding of the lower can thus be given by the ratio of energy difference at angular frequencies (phase locking or, as for higher frequencies, ampli- position ␪ to the energy difference at the cochlea base: tude-place mapping), changing energy density distribution along ⌬E͑␪͒ R w ͑␪͒2 the radial direction would change the radial profile of the Ϸ base ͩ ϩ bm ͪ ⌬ ͑␪͒ 1 ͑␪͒2 . [4] vibration of the different cochlear structures (7), affecting LF Ebase Rm 6 Rm hearing. In particular, energy density focusing at the outer wall increases the dynamic displacement ␩ ϭ ␩(r, ␪) of the BM The results in Eqs. 3 and 4, to leading order, do not depend on 4 normal to the resting position, and creates an effective radial tilt the wavelength. According to Eq. , energy focusing at an angular position ␪ along the spiral is inversely proportional to the T in the BM motion that is equal to the difference in displace- ratio of the corresponding radii of curvature. The ratio, and thus APPLIED ment ⌬␩ between inner and outer walls divided by BM width energy focusing, becomes greatest at the apex, where the ratio of MATHEMATICS w : bm the radius of curvature at the base to radius of curvature at the ⌬␩ 2 2 apex is greatest, and where low frequencies are resolved: wbm wbm T ϭ ϰ ͩ ϩ ͪ [1] ␭7/2 1 2 , 2 wbm Rm 10 R ⌬E w m apex Ϸ ␳ͩ ϩ bm ͪ ⌬ 1 2 , [5] Ebase 6 Rm where Rm is the radial position of the BM midline and ␭ is the apex wavelength of the associated longitudinal wave (8).
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