HANDBOOK OF THE SENSES Audition Volume Chapter 7: Mechano-Acoustical Transformations Sunil Puria1,2 and Charles R. Steele1 1Stanford University, Mechanical Engineering Department, Mechanics and Computation Division, 496 Lomita Mall, Durand Building, Room 206, Stanford, CA 94305 2Department of Otolaryngology-Head and Neck Surgery, 801 Welch Road, Stanford, CA 94304 1 1. KEYWORDS 2. BIOMECHANICS, OTOBIOMECHANICS, COCHLEA, COLLAGEN FIBERS, EXTERNAL EAR, IMPEDANCE, INNER HAIR CELLS, LEVERS, MATHEMATICAL MODELS, MICROCT, MIDDLE EAR, ORGAN OF CORTI, OUTER HAIR CELLS, PRESSURE, TYMPANIC MEMBRANE, VIBRATION 2 I. Outline In this chapter we examine the underlying biophysics of acoustical and mechanical transformations of sound by the sub components of the ear. The sub components include the pinna, the ear canal, the middle ear, the cochlear fluid hydrodynamics and the organ of Corti. Physiological measurements and the deduced general biophysics that can be applied to the input and output transformations by the different sub components of the ear are presented. II. Abstract The mammalian auditory periphery is a complex system, many components of which are biomechanical. This complexity increases sensitivity, dynamic range, frequency range, frequency resolution, and sound localization ability. These must be achieved within the constraints of available biomaterials, biophysics and anatomical space in the organism. In this chapter, the focus is on the basic mechanical principles discovered for the various steps in the process of transforming the input acoustic sound pressure into the correct stimulation of mechano-receptor cells. The interplay between theory and measurements is emphasized. III. Main Body A. Introduction The auditory periphery of mammals is one of the most remarkable examples of a biomechanical system. It is highly evolved with tremendous mechanical complexity. What is the reason for such complexity? Why can’t hair cells tuned to various frequencies just sit on the outside and detect motion due to sound? Clearly, the complexity serves the animal by providing greater functionality. This can be appreciated by looking at simpler auditory systems. One of the simplest hearing organs is that of the fly (Drosophila melanogaster), which has a tiny feather-like arista that produces a twisting force directly exerted by sound. This sound receiver mechanically oscillates to activate the Johnston’s organ auditory receptors with a moderately damped resonance at about 430 Hz (Gopfert and Robert 2001). The level required to elicit a response, due to wing-generated auditory cues involved in courtship, are in the 70 to 100 dB SPL range (Eberl et al. 1997). An example of a simpler anatomy with more complex function than that of the fly is the Müller’s organ of the locust. This invertebrate is capable of discriminating sounds at broadly tuned frequencies of approximately 3.5-5, 8, 12 and 19 kHz corresponding to the four mechanotransduction receptors attached to the tympanic membrane (Michelsen 1966). The best threshold for the receptor cells is about 40 dB SPL. The resonances of the tympanic membrane and attached organs provide the greater number of frequency channels than the fly (Windmill 3 et al. 2005). Amphibians evolved to have a basilar papilla with a few hundred receptor cells in a fluid medium. Other amphibians, birds and mammals have many thousands of hair cells. Other such examples, where structure that is more complex leads to greater hearing capability, are found in some of the other chapters in this volume. The peripheral part of the auditory system comprising of the external ear, middle ear and the inner ear systematically transform and transduce environmental sounds to neural impulses in the auditory nerve. The precise biophysical mechanisms relating the input variables to the output variables of some of the subsystems are still being debated. However, there is general agreement that these transformations lead to improved functionality. Five of the most important functional considerations are described below. 1. Sensitivity. The human ear is most sensitive to a range of sounds from the loudest at about 120 dB SPL to the softest at about –3 dB SPL. At its most sensitive frequency near 4 kHz, the displacement at the tympanic membrane is less than 1/10th the diameter of a hydrogen atom. At this threshold, the amount i -18 of work that is done at the eardrum is 3 x 10 J. In comparison, the amount of work done for the perception of light at the retinaii is 4x10-18 J, which is close to the calculated value at the threshold of hearing. This suggests that at its limits, the two sensory modalities have comparable thresholds. 2. Dynamic range. The dynamic range of psychophysical hearing in humans is about 120 dB SPL corresponding to environmental sounds and vocalized sounds. However, the neurons of the auditory nerve have a dynamic range that is typically less than 60 dB SPL. The organ of Corti mechanics must facilitate this dynamic range mismatch problem. 3. Frequency range. Hearing has about 8.5-octave frequency range in human and in some other mammals this range can be as wide as 11.5-octaves (ferrets). To handle this processing mechanically, the sensory receptors should have physical variations on a similar scale. However, the large range is achieved over an extraordinarily small space in comparison to the wavelengths of sound. 4. Frequency resolution. One of the most important functions of the cochlea is the tonotopic organization, which maps different input frequencies to its characteristic place in the cochlea. Like a Fourier frequency analyzer, each characteristic place has narrow frequency resolution, which provides greater sensitivity to narrow-band signals by reducing bandwidth and thus input noise at the individual mechano-receptor hair cells and thus the auditory nerve. 5. Sound localization. The physical characteristics of the pinna and head diffract sound in a spatially dependent manner. The diffraction pattern provides important cues that allow the more central mechanisms to localize, segregate and stream different sources of sound. In this chapter, we follow the chain of acousto-mechanical transformations of sound from the pinna to modulation of tension in inner hair cell tip links which is the final mechanical output variable of the cochlea from our vantage point. The tension in the tip links opens ion channels in the stereocilia, which then starts a chain of biochemical events that leads to the firing of the auditory neurons. We designate the output of a given 4 sub system a proximate variable. The chain of proximate variables leads to the ultimate output variable, the tip-link tension in hair cells. Input variables are sound pressure level, morphometry of anatomical structures, and mechanical properties of those structures. Biomechanical processes combined with the input variables lead to the proximate variables, which are physiologically measurable. B. Theories of sound transmission in the ear Starting with Helmholtz, mathematical models have played a key role in improving our understanding of the underlying biomechanical processes of the auditory periphery. In comparison to using natural languages to describe the observed phenomena, mathematical formulations have advantages and disadvantages. The advantages include a methodology for the possibility to describe compactly a correspondence to reality. The disadvantages are that the description may be incomplete or its validity difficult to test. A mathematical model is also often a statement of a scientific theory that captures the essence of the current state of the empirical observations. The power of a specific model is its ability to evolve as more facts become available and to be able to predict facts not yet observed. Thus the interplay between theory and experiments allows one to test different hypotheses and generate new hypotheses. In this chapter we provide a foundation for physiological measurements in the form of mathematical models. Below we present some common principles, found all through the auditory periphery, to transform the input variables to the ultimate output variable of hair cell tip link tension. Several general concepts are presented. These include how levers are formed, how Newton’s laws apply not only to celestial mechanics as originally formulated but also in otobiomechanics, how sound transmission through different materials is described by transmission line formulations, and how modes of vibrations arise in structures. A combination of these principles is used to understand how the ear improves sensitivity, frequency range, frequency resolution, dynamic range, and sound localization within the constraints of biological materials and anatomic space. 1. Mechanical and acoustic levers One of the simplest transformations of energy is achieved with a simple mechanical lever. There are numerous places in the auditory periphery where levers produce force and velocity transformations through anatomical changes in lengths and areas. These transformations take place in the context of improving sound transmission at the interfaces of different anatomical structures where there is a change in the impedance. An example of a change in impedance is the low impedance of air to the high impedance of the fluid filled cochlea. Examples of the lever action at work include an increase in sound pressure due to a decrease in area of the concha of the pinna to the ear canal, increase in pressure from the decrease in surface area from the tympanic membrane to the stapes footplate, increase in force due to the lever ratio in the ossicular chain, increase in volume velocity from the stapes footplate to the basilar membrane due to a decrease in surface area, and transformation of the basilar membrane displacement to hair cell stereocillia tip link tension due to relative shearing motion between the reticular lamina and the tectorial membrane. 5 2. Newton’s second law of motion A key principle in describing dynamic transformation of forces in mechanical systems to accelerations is the well-celebrated Newton’s second law of motion elegantly written as F = ma , (Eq. 1) which states that a force F acted upon a mass, results in acceleration a.