Biophysics of Hearing 2023 PDF

Biophysics of hearing The text under the slides was written by Peter Nagy 2023 This instructional material was prepared for the biophysics lectures held by the Department of Biophysics and Cell Biology...

Biophysics of hearing The text under the slides was written by Peter Nagy 2023 This instructional material was prepared for the biophysics lectures held by the Department of Biophysics and Cell Biology Faculty of Medicine University of Debrecen Hungary https://biophys.med.unideb.hu 2  In order to understand the biophysics of hearing we are going to introduce you to those physical properties of sound, which are necessary for learning the process of hearing. During the lecture it will be necessary to understand how the ear detects these properties (mainly the strength and frequency of sound and its spectrum). After briefly reviewing the anatomy of the ear we are going to discuss the roles of the outer, middle and inner ear in hearing. At the end of the lecture different scales used for characterizing sensation of loudness will be described. These scales are required and important because sensation of loudness is not linearly proportional to sound intensity. Psychophysical characterization of loudness is required for characterizing hearing loss. Although most of the material of this lecture belongs to the “bio” part biophysics, some physical principles, discussed previously, will be required for understanding hearing. Acoustic impedance, learned in relation to ultrasound imaging, is important in the operation of the middle ear, since it determines the fraction of sound reflected from the boundary of different media in this case as well. Membrane potential changes in auditory receptor cells (hair cells) are described by the Goldman-Hodgkin-Katz equation. 3 (1) In order to understand the process of hearing, the stimulus itself, i.e. sound, must first be characterized. From among the physical properties of sound intensity and frequency, and the relative strength of individual frequency components in the case of compound sounds (spectrum), will be primarily discussed. The extent to which sound stimulates the sound receptors in the inner ear is characterized by the stimulus strength, the intensity of sound (). (2) Vibrations of air particles are first converted to vibrations of certain components of the middle and inner ear, which is converted to an electrical signal as a result of membrane potential changes of the receptor cells (hair cells) in the inner ear. (3) The membrane potential change of receptor cells leads to action potential of the auditory nerve due to neurotransmitter release. The auditory nerve conducts the stimulus to the brain. (4-5) Auditory sensation arises in the auditory cortex (part of the temporal lobe). During this process the presence of the stimulus is sensed. Like any other sensory input, sound is also further processed by the brain taking place in other cortical and subcortical areas. Such processes explain why emotions are elicited when listening to music. During the lecture points 1, 2 and 5 will be mainly discussed. 4  As it can be seen in the insert in the upper left corner sound is the longitudinal vibration of air since air particles are displaced parallel to the propagation of sound. During sound propagation the density of particles changes leading the alterations in the local pressure. Therefore, graphs characterizing sound often display the temporal or spatial variation of pressure. Sound intensity is proportional to the square of sound pressure amplitude. A. First, frequency and spectrum are going to be discussed from among the physical properties of sound. A pure tone is composed of a single sinusoidal component characterized by a single frequency. Frequency tells us how many cycles of vibration are completed in a second. Its unit is hertz (Hz): 1/s. B. Compound sound is composed of components with different frequencies. A compound sound is a musical tone if it consists of the fundamental frequency (the component with the lowest frequency) and the overtones. The frequency of the overtones is an integer multiple of the fundamental frequency. A compound sound (red) can be generated by superimposing different sinusoidal waves (blue, green, black). C-D. In the case of compound sounds it is often useful to break the sound into different frequency components. As it will be seen later, the inner ear also carries out such a frequency analysis. The process of determining the fraction of each frequency component in a periodic vibration is called Fourier analysis. The procedure basically reveals the spectrum of sound. In the case of a pure tone there is only a single frequency component in the spectrum (C). The 5 fundamental frequency and the overtones are present with different intensities in a musical tone (D). Since the spectrum, i.e. the intensity (or pressure amplitude) of different frequency components, unequivocally defines the wave (disregarding differences in phase), the temporal fluctuation of intensity (or pressure) can be reconstructed from the spectrum using the reverse process of Fourier analysis, named Fourier synthesis. 6  If only pure and musical tones are considered, the spectrum defines the relative intensity of the fundamental frequency component (1st harmonic) and the overtones (2nd, 3rd, 4th harmonics), determining timbre, based on which different sound sources, e.g. musical instruments, can be recognized. The formula p(t )  p1 sin t   p2 sin  2t   p3 sin  3t  ... shows the principle of Fourier synthesis, according to which a musical tone can be reconstructed by summing the components with different pressure amplitudes (p1, p2, p3) and angular frequencies (, 2, 3). Angular frequency is frequency multiplied by 2. 7 From the stand point of hearing, sound strength is as important as frequency from among the physical properties of sound. Sound strength can be characterized by pressure amplitude or intensity. Intensity is proportional to the square of pressure amplitude. Sound intensity is equal to the energy delivered by the sound to unit surface area in unit time. Its unit is J W. An important property of the human auditory organ is its capability to sense s m2 m2 sound and its strengths with intensities spanning more than ten orders of magnitude. In order to make the numerical expression of such a wide intensity range simpler it is usual to provide the intensity level instead of the intensity itself. Intensity level is proportional to the logarithm of intensity (J1). For the calculation of intensity level a reference intensity (J0) is required. Although it can be arbitrarily defined, it is usually chosen to be the intensity of the weakest perceptible sound with a frequency of 1000 Hz (10-12 W/m2). Accordingly, intensity level (n) can be calculated according to the following formula: J1 n  10 lg J0 The unit of intensity level is decibel (dB). If multiplication with 10 is omitted from the formula, its unit will be bel. 8 Having reviewed the most important physical properties of sound from the stand point of hearing the physical and biological processes taking place in the ear will be discussed. 9  The human ear can be divided into three, functionally and anatomically separate parts (A). The outer ear is an air-filled canal bounded by a cartilaginous and osseous (bony) wall, which conducts sound all the way to the eardrum. The eardrum or tympanic membrane separates the middle ear from the outer ear. The middle ear is an air-filled cavity, in which sound is not conducted by air, but rather by the three, tiny auditory ossicles (or ossicles for short), the hammer (malleus), the anvil (incus) and the stirrup (stapes) (B). The hammer is in connection with the eardrum, whereas the stirrup transmits the vibrations to the inner ear. The middle ear is connected to the pharynx through the auditory tube. This connection fulfills two roles. (1) The auditory tube ensures that the pressure in the middle ear is identical to the outside atmospheric pressure. If these two pressures differ, the movements of the eardrum and the ossicles involved in hearing (to be discussed later) are dampened leading to hearing loss. Everyone may have already had such an experience during take-off or landing of an airplane or in any case when the outside pressure changes. (2) Secondly, the auditory tube drains the mucus produced in the middle ear. If the auditory tube is plugged (e.g. in the case of upper respiratory tract infections), the mucus accumulates in the middle ear leading to middle ear inflammation (otitis media with effusion). The presence of the effusion also interferes with the movement of the ossicles and the eardrum leading to hearing loss. The accumulated mucus can easily be infected by bacteria leading to acute (purulent) otitis media. The inner ear is connected to the middle ear through the oval window and the round window. The base of the stirrup is inserted into the oval window. Vibrations of the stirrup are transmitted to the fluid-filled cavities of the inner ear here. The inner ear contains the cochlea, responsible for hearing, and the semicircular canals detecting head rotations (C). A more detailed anatomical structure of the cochlea will be provided later. 10  Although the outer ear plays a dominantly passive role in conducting sound waves, it also slightly contributes to amplification. Standing waves are generated in the outer ear with their nodes at the eardrum (the displacement of air particles is minimal here, but the pressure amplitude is maximal) and their antinodes at the entrance of the ear canal (the displacement of air particles is maximal here, but the pressure amplitude is minimal). The dashed blue line in the figure shows the pressure amplitude. Accordingly, such standing waves are generated whose wavelength is four times longer than the ear canal. Since the human ear canal is 2.5 cm long, resonance due to standing waves amplifies sound waves with a frequency of approximately 3 kHz (=40.025 m=0.1 m, f=c/=343 m/s / 0.1 m  3 kHz). Interestingly, this frequency falls in the frequency range most relevant for speech (0.5 – 8 kHz). Vibration of air particles is converted to vibrations of the ear drum and ossicles in the middle ear. The middle ear does not simply conduct these vibrations to the oval window (where the base of the stirrup is inserted), but it also amplifies them significantly. This amplification results in the increase of the pressure waves generated by the vibrations. Two mechanisms contribute to this phenomenon. First, the area of the oval window (Aoval3.2 mm2) is approximately 17-times smaller than that of the ear drum (Adrum55 mm2). Consequently, a larger pressure will be generated in a smaller surface area by the same force (F=p·A, F – force, p – pressure, A – surface area). However, since the ossicles form a lever system, torque must also be considered since the lever-like movement of the ossicles also 11 contributes to the amplification taking place in the middle ear. The moment arm of the force exerted by the stirrup is approximately 1.3-times shorter than that of the force exerted by the hammer. Since the torques generated by the two forces must be equal, the force exerted by the stirrup must be larger (M=Fr, M – torque, F – force, r – moment arm). Fdrum and Foval designate the force exerted by the eardrum and the oval window, respectively, on the lever, while r1 and r2 are the moment arms of the respective forces. The equilibrium condition for the torques is displayed in the upper left part of the figure. The solution of this equation tells us that the pressure amplitude at the oval window (pperi) is approximately 22-times larger than at the eardrum (pair). The pressure amplitude at the oval window is designated by pperi because the pressure waves are conducted to a certain part of the inner ear that is filled with a fluid called perilymph (to be discussed later). This pressure amplification is important because vibrations of air particles are converted to waves of the fluid-filled cavities of the inner ear during hearing. Since the acoustic impedance of air and the fluid compartments are substantially different, sound waves would be almost completely reflected from their boundary. Although reflection still takes place from the air-eardrum boundary by inserting the middle ear into the wave conduction pathway, the pressure waves at the base of the stirrup, as a result of the amplification mechanisms of the middle ear, are strong enough so that they set the fluids in the compartments of the inner ear in motion sufficiently to lead to sensation of sound. 12 After superficially discussing the structure of the inner ear, we must get acquainted with its finer structure, more specifically the anatomy and histology of the cochlea involved in hearing. The cochlea, in accordance with the meaning of its name in Greek (snail shell), is a spiral- shaped cavity with its base attached to the middle ear. The cavity is divided into three compartments by two membranes. The scala vestibuli, the upper compartment in the figure in the red box, begins at the oval window, and through a little hole at the apex of the cochlea (helicotrema) it is connected to the scala tympani, which ends at the round window. The scala tympani is separated from the scala media by the basilar membrane, whereas the scala vestibuli is separated from the scala media by Reissner’s membrane. Although the width of the cavity of the cochlea becomes gradually narrower toward the apex, the width of the basilar membrane changes in the opposite direction: it becomes wider toward the apex. This feature will be important in the frequency-dependent sensing of sound waves. The scala vestibuli and the scala tympani are connected ducts, which are filled with a fluid called perilymph. Pressure waves generated by the base of the stirrup propagate through the scala vestibuli-scala tympani system. Due to the incompressibility of the fluid in this system (perilymph), these pressure waves can only propagate all the way in the perilymph if the other end of the duct (at the round window) is also elastic. The scala media, filled by a fluid called endolymph, is not continuous with the system formed by the other two cavities. The organ of Corti, containing the actual sound receptor cells, is located on the basilar membrane. The fine structure of the organ of Corti will be discussed later. A histological section showing the cochlea can be seen in the lower left corner of the figure. 13 A very important step in the process of hearing is the conversion of mechanical energy (sound) to electric signals. The basilar membrane plays a central role in this process. The first scientifically encouraged model for this phenomenon is attributed to the German physicist and physician Helmholtz. According to his model fibers located in the basilar membrane resonate as a result of the movement of fluid in the surrounding cavities (resonance theory). Since the width of the basilar membrane increases toward the apex, the length of the virtual strings also increases in this direction. As a result, the strings closer to the apex would resonate to lower frequency sound than those at the base. The conditional tense in the previous sentence is warranted, since the model was shown to be wrong. The most important problem with this theory is the fact that the basilar membrane is gelatinous and therefore too tender for strings or fibers to resonate. The model still valid today was developed by György Békésy (1899-1972), according to which propagating surface waves are generated on the basilar membrane as a result of pressure waves of the perilymph. 14  Békésy carried out his investigations on cochlea of human cadavers. He filled the cochlea with a suspension containing silver flakes and recorded the movements of the basilar membrane using strobe photography. He came to the conclusion that the basilar membrane is too flaccid for resonance to take place. The basilar membrane becomes even less stiff toward the apex explaining the decrease in the propagation speed of surface waves from the base toward the apex of the cochlea. If several waves propagate on the basilar membrane, those farther away from the oval window propagate more slowly than the ones closer to it. As a consequence, the faster waves catch up with the slower ones and their superposition results in a large displacement of the basilar membrane at a certain location. A similar phenomenon takes place when surface waves on the ocean approach the shore. In this latter case, gradual decrease in the propagation speed of the waves is caused by the decrease in the depth of water. This phenomenon is illustrated in panel A and in the accompanying video: https://youtu.be/7EEVvk8TBD8. Although the simulation neglects several aspects of the behavior of the basilar membrane, the fact that waves gradually catch up with each other is clearly demonstrated. Békésy spent the last years of his life in Hawaii so the surface waves of the ocean might have inspired his discoveries. However, he had completed most of his observations in Hungary long before he immigrated to the US. It is worth pointing out some of the thought-provoking aspects of the life of Békésy. He was born in Budapest, Hungary, and he defended his PhD 15 degree just after the 1st world war when obtaining a scientific position was extremely difficult. He was offered a position in the experimental department of the Hungarian postal service, where he was given the task of perfecting telephone speakers. Instead of reluctantly doing his job far from his original area of expertise, he was inspired to learn much more about human hearing. He completed most of his famous experiments in the experimental department of the postal service, which led to his Nobel prize. Since he immigrated to Sweden and then to the US, his name is mostly known as Georg von Békésy (instead of György Békésy, his original Hungarian name). 16 Surface waves of the basilar membrane reach their maximum displacement at frequency- dependent locations. This phenomenon can be seen in part A of the slide in which the basilar membrane is uncoiled. The maximal displacement of the basilar membrane is close to the oval window in the case of high frequency sounds, while low frequency sounds generate the maximal displacement far from the oval window, close to the apex of the cochlea. This correlation is called tonotopy. Part A of the slide also demonstrates that the width of the basilar membrane increases toward the apex. The same principle is demonstrated in part B of the slide. It can be concluded that the place of maximal displacement is frequency dependent, whereas the extent of the maximal displacement depends on the strength of the sound. Coding of sound strength and frequency in the action potential frequency of the auditory nerve is also partially based on this principle (C). As a result of processes to be introduced later, displacement of the basilar membrane leads to generation of action potentials along the auditory nerve. Since individual fibers of the auditory nerve originate from different parts of the basilar membrane, only those fibers will be excited to generate action potentials, which originate from that part of the basilar membrane that produces the highest displacement. Therefore, the excited state (action potential generation) of individual auditory nerve fibers conveys information to the brain about the frequency of the sound. The strength of the sound is partly encoded by the frequency of action potentials, which is approximately proportional to the logarithm of sound intensity (see later the Weber-Fechner law). The principle of how information about the frequency and intensity of sound is transmitted to the brain by the auditory nerve will be further discussed at the end of the lecture. 17 The animation demonstrating the propagation of surface waves on the basilar membrane can be viewed at the following link: https://www.youtube.com/watch?v=dyenMluFaUw 18  Previously the macroscopic anatomy of the inner ear was described. However, the microscopic anatomy of the organ of Corti must be known in order to understand how the propagating surface waves of the basilar membrane, as mechanical stimuli, are converted to electric signals. Part A of the slide reviews the structure previously discussed. In its upper part the view of the cochlea from the outside is presented, while the cross-section of an individual canal of the cochlea is shown in the red box. The organ of Corti is located on the basilar membrane. The schematic structure of the organ of Corti (on the left) and its histologic image (on the right) can be seen in part B of the slide. The most important cellular components of the organ of Corti are the hair cells, which come in two different forms. Inner hair cells are arranged in a single file line on the basilar membrane, while outer hair cells are arranged in three rows farther from the axis of the cochlea. Both types of hair cells form synapses with the auditory nerve. In order to understand the function of hair cells it is important to realize that hair cells sit on the basilar membrane, albeit not directly on its surface, but “in the lap” of supporting cells. Their apical part, more accurately their “hairs” (cellular processes, stereocilia) to which their name is owed, are in connection with another membrane, the tectorial membrane, which can be found in the scala media immersed into the endolymph. The inner hairs cells (in single file, top part of the image) and the outer hair cells arranged in three rows and their stereocilia can be seen in the scanning electron microscopic image of 19 part C of the slide. On average, a human cochlea contains approximately 3500 inner hair cells, which are involved in mechanotransduction, i.e. in the conversion of the mechanical stimulus to an electric signal. Outer hair cells, which number approximately 12-20 thousand in the human cochlea, are responsible for amplification, which will be discussed later. Stereocilia can be found on the surface of both outer and inner hair cells (D). The height of the stereocilia is graded, i.e. it increases from the side of the hair cell closer to the axis of the cochlea towards the side farther away from the axis. Stereocilia are not real cilia, but cellular processes bounded by the plasma membrane. Their typical length (8-10 m) is longer than that of microvilli, and they contain microfilaments as opposed to real cilia, which contain microtubules. 20  After discussing the microscopic structure of the organ of Corti, we are in a position to understand the role of stereocilia of inner hair cells in mechanotransduction. The height of stereocilia on the surface of hair cells is graded (A). In certain animal species a single, real cilium, called kinocilium, containing microtubules can be found beside the longest stereocilium. In mammals, and consequently in humans as well, the kinocilium disappears at around birth after the appearance and ordering of stereocilia according to increasing height. Therefore, only the stereocilia, increasing in height from left to right in image B, are involved in hearing of humans. In the absence of mechanical stimulation, i.e. in a quiet environment, no force is exerted on the hair cells. They are stationary in between the motionless basilar and tectorial membranes (top part of C). A filament, easily discernible in electron microscopic images, connects neighboring stereocilia with each other. These connections, called tip links, are loose in a quiet environment (top part of D). Sound induces the propagation of surface waves on the basilar membrane. As a result, the basilar membrane is displaced relative to the almost immobile tectorial membrane. Therefore, shear force is exerted on the hair cells, which are spanned between the moving basilar membrane and the almost immobile tectorial membrane. The shear force bends the stereocilia leading to stretching of the tip links. 21  Bending of stereocilia initiates a series of other processes, which can be followed step- by-step in part A of the slide. If stereocilia bend toward the longest stereocilium (“positive stimulation” in part B), tip links become taut. Tip links are connected to an ion channel present in the membrane of stereocilia. Although the channel has not yet been characterized at the molecular level, it belongs to the class of mechanically (or stretch)-gated ion channels, which are opened by the stretching of tip links. Although this cation channel is not only permeable for K+, but since the dominant cation of the intracellular space of hair cells and the endolymph (forming the extracellular space around stereocilia) is K+, the most important consequence of the opening of this mechanosensitive channel is the increased K+ permeability of the apical part of hair cells (where the stereocilia are located). This step leads to the depolarization of hair cells. This is unusual since increased K+ permeability typically hyperpolarizes cells. The fact that increased K+ permeability leads to depolarization is attributable to the ionic composition of the endolymph (C). While the Na+ and K+ concentrations in the perilymph in the scala vestibuli and scale tympani correspond to normal extracellular concentrations, the endolymph, constituting the extracellular space of stereocilia, contains high concentration of K+ and low concentration of Na+. Depolarization of hair cells induces the opening of voltage-gated Ca2+ channels (step 4 in part A) leading to the exocytosis of neurotransmitter (glutamate)-containing vesicles. Hair cells form synapses directly with auditory nerve fibers. Glutamate will, therefore, depolarize auditory nerve fibers, and upon sufficiently large depolarization action potentials will be elicited in the nerve fibers. 22 The figure displays the ionic composition of the perilymph in the scala tympani (ST), scala vestibuli (SV), of the endolymph and the cerebrospinal fluid present in the subarachnoid space. Although the exact numerical values are obviously not to be remembered, the general tendencies once again point out the unusual condition of hair cells, which is the consequence of the fact that they are immersed in the endolymph (at least their part around stereocilia). The ionic compositions of the perilymph (and the cerebrospinal fluid not considered here) are practically identical. Since the K+ content of endolymph is almost as high as that of the intracellular space, the equilibrium (or reversal) potential of K+ (Ueq), calculated according to the Nernst equation, is approximately 0 mV: RT cK,endolymph Ueq  ln F cK,intra where R, T and F are the universal gas constant, absolute temperature and the Faraday constant, respectively. Since cK,endolymphcK,intra (cK,endolymp and cK,intra are the K+ concentrations of the endolymph and in the intracellular space, respectively), the equilibrium potential of K+ indeed approaches zero. Increase in the K+ permeability (as a result of opening of mechanosensitive ion channels) shifts the membrane potential toward the equilibrium potential of K+, which is depolarized compared to the resting membrane potential of -50 – -60 mV of hair cells. 23  We have reached the end of discussion of mechanotransduction putting us in a position to review the major principles with a couple of figures and to point out some further conclusions. The responses elicited by a gradually increasing stimulus can be followed in part A of the slide. The gradually increasing stimulation induces a depolarization increasing proportionally with the stimulus. The extent of depolarization of receptor cells (hair cells) is not linearly proportional to the stimulus. The membrane potential change taking place in receptor cells is called receptor potential. The change in the membrane potential of receptor cells indirectly induces action potentials in auditory nerve fibers. The frequency of action potentials recorded in individual auditory nerve fibers is also proportional, albeit not linearly, to the strength of the stimulating sound. The electric signal conducted by the auditory nerve reaches the central nervous system, where it is converted to sensation. However, these processes are beyond the scope of this material. Square pulses were applied for stimulation in part B of the slide. Individual pulses were of constant strength in time, but consequent pulses got gradually stronger. If the strength of the pulse was under the threshold, no action potential was generated in the auditory nerve. Although such a stimulus induces depolarization of hair cells, this depolarization is not strong enough to induce action potential in the nerve cells. This phenomenon is the consequence of the fact that the extent of depolarization of hair cells is proportional to the amount of neurotransmitter released, which, in turn, depolarizes the post-synaptic membrane (nerve 24 cells) proportionally to its concentration. If the extent of depolarization of auditory nerve fibers is under their threshold, no action potential is generated. As the strength of consequent stimuli increased (B, top black trace), so did the extent of depolarization of hair cells (B, middle blue trace) and the frequency of action potentials propagating along auditory nerve fibers (B, bottom red). Adaptation of receptor cells to a constant stimulus strength can also be observed in this figure. Adaptation is manifested in a gradually decreasing extent of depolarization of hair cells (and a consequent decline in the frequency of action potentials in auditory nerve fibers) in the presence of a constant stimulus strength. 25  The basilar membrane can be displaced both upward and downward from its equilibrium position (A). While the movements of the basilar membrane are substantial, the displacements of the tectorial membrane are comparatively smaller. It has not been explicitly pointed out so far that the open probability of the tip link-controlled mechanosensitive channels is not zero in the absence of stimulation. The red trace in part B of the figure shows the state of the channel, which can be either open (conductive) or closed (non-conductive). Since the channel has a low, but non-zero probability of opening in the absence of stimulation, the hair cells are sometimes depolarized leading to the generation of action potentials in the auditory nerve. When the basilar membrane is displaced toward the tectorial membrane (“stimulation”), stereocilia bend toward the longest stereocilium, tip links are stretched, and the opening probability of the mechanosensitive cation channels significantly increases. Therefore, a significant amount of K+ enters hair cells causing substantial hair cell depolarization and a consequent rise in the frequency of auditory nerve action potentials. When the basilar membrane is moved away from the tectorial membrane (“inhibition”), stereocilia bend toward the shortest stereocilium and the tip links become completely loose. In this case the opening probability of the mechanosensitive cation channels is zero, and the frequency of action potentials in the auditory nerve practically drops to zero. The non-zero 26 frequency of action potentials in this case is caused by the spontaneous exocytosis of neurotransmitter-containing vesicles. Part C of the slide shows the experimental investigation of phenomena discussed in panels A-B. A hair cell of non-mammalian origin is shown in the figure since it also has a kinocilium. If the stereocilia are repeatedly bent toward and away from the kinocilium, the membrane potential, recorded by patch clamp, perfectly follows these movements (depolarization and hyperpolarization after bending stereocilia in the excitatory and inhibitory direction, respectively). Action potentials, whose frequency is proportional to the depolarization of hair cells, can be recorded on the afferent nerve fibers (afferent=conducting toward the central nervous system from the receptor cell). 27  The figure summarizes the most important steps in the mechanotransduction taking place in hair cells. Surface waves propagating on the basilar membrane lead to bending of stereocilia, which results in stretching of the tip links (if stereocilia bend toward the highest one). Stretching of tip links activates mechanosensitive cation channels and K+ influx depolarizes the hair cells. This membrane potential change will result in the exocytosis of neurotransmitter (glutamate)-containing vesicles through the activation of voltage-gated Ca2+ channels. 28  Coding of the frequency and intensity of sound, i.e. how these properties of sound are delivered to the brain by the auditory nerve, has already been tangentially touched upon. These processes are summarized in this figure. Two fundamental mechanisms exist for coding of sound frequency. (1) tonotopy: the maximal displacement of the basilar membrane is frequency dependent; therefore, different hair cells respond to different sound frequencies. Consequently, different fibers of the auditory nerve conduct action potentials indirectly induced by sounds of different frequencies. (2) If the frequency of sound is low, the frequency of action potentials in individual auditory nerve fibers is equal to the frequency of the stimulus. This is shown for a 0.5 kHz sound in the right part of the figure. In the case of high frequency sounds, the frequency of action potentials does not reach the frequency of the stimulus due to the finite duration of the action potential and the refractory period. In such a case, the summed frequency of action potentials of individual nerve fibers codes the frequency of the stimulus. According to a simplified view, in the presence of a stimulus with a frequency of 6 kHz (hair cell and the connected nerve fibers in red), all the six auditory nerve fibers innervating a single hair cell fire action potential with a frequency of 1 kHz with the action potentials in individual fibers shifted temporally relative to the action potentials in other fibers. If, however, the action potentials of all fibers are considered, their summed frequency is 6 kHz (only three fibers are displayed in the figure due to space restrictions). 29  Sound intensity is also coded in different ways. (1) On the one hand, the frequency of action potentials increases with the strength of sound. (2) If the ear is stimulated with a low intensity sound, the extent of stimulation of hair cells only exceeds the threshold in a small area. Therefore, action potentials will only be generated in auditory nerve fibers innervating this small group of hair cells. If the stimulus strength is large, the level of stimulation exceeds the threshold in a larger group of hair cells, and action potentials will be generated in auditory nerve fibers beginning from this larger area. It can be seen that coding of the physical properties of sound by the auditory nerve is complicated. An obvious source of this complication is the fact that the frequency of action potentials of individual auditory nerve fibers is influenced by both the frequency and the intensity of sound. 30  Understanding the principle of tonotopy made it possible to treat a certain type of deafness that is caused by hair cell loss. According to tonotopy, different auditory nerve fibers conduct information about different frequency components of the sound to the brain. Consequently, if these auditory nerve fibers are artificially stimulated, information about the frequency corresponding to that auditory nerve fiber will be delivered to the brain. The functioning of the cochlear implant, the device used for treatment of deafness due to hair cell damage, is shown in the figure. (A) The device first analyzes the sound according to frequencies, i.e. it determines the intensity of sound components with different frequencies in the compound sound. This frequency analysis is performed by Fourier analysis (see at the beginning of the lecture material). Such a frequency analysis is performed in the human ear by the basilar membrane since its maximal displacement is frequency dependent. (B) Information corresponding to these different frequency components is conducted to the scala tympani in the cochlea by several microelectrodes. (C) The microelectrodes are positioned so that a microelectrode transmitting information about a certain frequency component ends at about the location where this frequency component is sensed by the cochlea. These microelectrodes stimulate auditory nerve fibers in their vicinity, and these nerve fibers will deliver the information about the different frequency components to the brain. Hearing experience provided by cochlear implants is not perfect, since the frequency of action potentials in the auditory nerve is determined by several factors (as discussed previously). Many persons having cochlear implants describe the sound they hear as robot- like. 31  The function of outer hair cells is not related to mechanotransduction, but to amplification of motions of the basilar membrane. Outer hair cells respond to the passive displacement of the basilar membrane identically to how inner hair cells do. As a result, outer hair cells are also depolarized, but this depolarization initiates other processes in them. This difference is related to the protein prestin that is expressed in outer hair cells, as demonstrated by the immunofluorescence image (A). The conformation of prestin is dependent on the membrane potential; therefore, when the cell is depolarized, prestin changes its shape (B). This shape change is relayed to the cell membrane resulting in the shortening of outer hair cells. If the membrane potential of outer hair cells is externally modified by patch clamp, the cell will contract (C). Contraction of outer hair cells is relayed to the basilar membrane. Since contraction of outer hair cells is extremely rapid, the basilar membrane movements they induce will be in phase with the displacement of the basilar membrane induced by the sound. Therefore, vibrations of the basilar membrane induced by outer hair cells will augment the sound-induced vibrations of the basilar membrane. Outer hair cell-induced amplification of displacement of the basilar membrane is called cochlear amplifier. The cochlear amplifier is a typical example for positive feedback, since the input (basilar membrane movement) is augmented by the response. An animation similar to the one available in the lecture material can be viewed at the following link: https://www.youtube.com/watch?v=rcj9QqQW6Lg 32  Since Békésy investigated cochleas from cadavers, he did not see the cochlear amplifier in action, which very significantly augments the displacements of the basilar membrane (D). The cochlear amplifier not only increases the displacements of the basilar membrane, but it also makes the maximal displacement more localized. Without the action of outer hair cells, hearing threshold increases by 50 dB. Since the intensity level of speech is 50 dB above the threshold, a person loses the ability to understand speech without the cochlear amplifier. 33 The auditory nerve conducts the electric signal through several intercellular junctions (synapses) to the auditory cortex found in the temporal lobe. Here, the electric impulses are converted to sensation. Several other brain areas take part in further processing of the sensory input. Listing to pleasurable music leads to dopamine release in ancient, subcortical, mesolimbic reward centers similar to certain drugs. These latter processes are involved in eliciting emotions while listening to music. 34 It has already been pointed out several times that the intensity of the stimulus () and the strength of the sensation () are not linearly related to each other. The psychophysical laws describing the relationship between the strengths of the stimulus and the sensation were first studied scientifically by Ernst Weber and his disciple, Gustav Fechner, in the 19th century. The word “psychophysical” unequivocally refers to the fact that such laws describe the relationship between an objectively measurable, physical quantity (e.g. sound intensity) and another, subjective quantity, currently not objectively measurable, that can only be determined by self-declaration of the investigated individuals. The Weber-Fechner law formulated based on the findings of the two German scientists in valid for all sensory organs, but it will be introduced here for hearing. According to the relationship sensation of loudness () is proportional to the logarithm of the stimulus intensity ():    const lg (1) 0 where 0 is the intensity of the just noticeable sound under optimal conditions, i.e. complete quietness. Although the investigations of Weber and Fechner were pioneering, the equation above has been shown to be valid only in a narrow intensity range. In the 20th century Stevens also carried out investigations to reveal the relationship between the strength of the stimulus and the sensation. According to his results sensation is a power function of the intensity of the stimulus: 35  n      const   (2)  0  where n is a constant characteristic for a certain sensation modality (n=0.3 for hearing). The Stevens law predicts the relationship between the strength of the stimulus and the sensation appropriately for a wide intensity range. Supplementary material: The limited validity of the Weber-Fechner law formulated in equation 1 is attributable to the way Weber and Fechner carried out their investigations. They did not directly measure the sensation () elicited by a certain intensity of sound (), but they determined the minimal variation of the background that was audible (just noticeable difference). If the change of the background noise is just noticeable, then this change leads to a tiny alteration, denoted by d, in the sensation. According to Weber and Fechner this change in the intensity of the stimulus (d), required for this minimal alteration of sensation, is proportional to the background intensity (). The mathematical formulation of the aforementioned principle is the following equation: d d   const (3)  The Weber-Fechner law is the solution of this so-called differential equation. The reason for the limited validity of the Weber-Fechner law is the fact that although Weber and Fechner did not directly measure the relationship between  and , but the law named after them expresses such a relationship. Assumptions, not detailed here, that are required for deriving the Weber-Fechner law (equation 1) from equation 3 are not fully justified. On the other hand, since Stevens directly measured the relationship between  and , his law is applicable to a wider intensity range. It is, however, worth noting that the just noticeable difference (d) and the associated change in the stimulus strength (d) were both relatively accurately, quasi- objectively measurable quantities in the measurements of Weber and Fechner. However, in Stevens’ measurements sensation was determined from the self-declared perception of the investigated individuals, which is a source of uncertainty. In conclusion, although the Stevens law is more widely accepted nowadays, it also has limitations and it is subject to criticism. 36 Since sensation of loudness is not linearly related to sound intensity, several non-linear scales are used for expressing sensation of loudness. Since, according to the Weber-Fechner law, sensation of loudness depends on the logarithm of the intensity of sound, the decibel scale seemed an obvious choice for expressing sensation of loudness: J Hdecibel  10 lg J0 where Hdecibel is the sensation of loudness on the decibel scale elicited by sound intensity J. J0 is the threshold of hearing at the given frequency. It must be pointed out that the decibel scale is not specific to sensation or hearing. It is often used for expressing quantities spanning several orders of magnitude. Accordingly, intensity level is the intensity of sound expressed on the decibel scale. 37  In order to apply the decibel scale the threshold of hearing must be known at each frequency, e.g. J0,5000 is the threshold intensity of a sound with 5000 Hz expressed in W/m2. This disadvantage is eliminated by the phon scale. The approach is based on the observation that people can relatively easily identify when two sounds of different frequencies sound to be equally loud. If the loudness of a sound of arbitrary frequency (5000 Hz in the worked-out example) is to be expressed on the phon scale, then the intensity of the sound with a frequency of 1000 Hz (J1000) is to be found that induces the same sensation of loudness as the investigated sound with a frequency of 5000 Hz (the corresponding intensities are connected by the red arrow). Given the threshold of hearing at 1000 Hz (J0,1000) the loudness of the sound with a frequency of 5000 Hz can be expressed according to the following equation: J Hphon  10 lg 1000 J0,1000 The threshold of hearing of a pure tone with a frequency of 1000 Hz is 10-12 W/m2. According to the Stevens law a there is a power law relationship between the sensation of loudness and the sound intensity. The sone scale is based on the aforementioned relationship: 0.3 1 J    1000 Hsone  16  J0 ,1000 Its principle of calculation is identical to that of phon values. In the case of a sound with arbitrary frequency, the intensity of the sound with 1000 Hz that induces the same sensation 38 of loudness as the investigated sound is to be found (J1000), and this intensity is compared to the threshold of hearing at 1000 Hz (J0,1000). 39 The phon and sone scales are easily interconvertible to each other: 0.3 Hphon 1 Hsone  10 10 16 In order for the calculated sone values to be rounder, the following approximation is often used: 0.1 H phon  4 H sone  2 Comparison of the sone scale, based on the Stevens law, and the phon scale, based on the Weber-Fechner law, has been carried out. The subjective sensation of loudness of 1000 Hz sounds of different strengths was determined. These real, objective loudness values are shown by the  symbols in the figure. The word measured should be placed in quotation marks since the values were derived from subjective experiences of the investigated individuals. Loudness expressed on the sone scale, which can be read from the left vertical axis, appropriately fits the measured values in the whole range expect at low intensities. Loudness expressed on the phon scale, which can be read from the right vertical axis, turns out to be only a rough approximation of the real loudness values. 40 Human hearing is usually investigated using the equal loudness curves shown in the figure. Before taking a closer look at the curves themselves, let us get conversant with the different modes of expressing sound strengths (three different vertical axes). Sound strength can be characterized by the pressure amplitude, the sound intensity and the expression of the latter on a logarithmic scale called intensity level (n). Since both the intensity level and the sensation of loudness expressed on the phon scale according to the Weber-Fechner law (Hphon) are proportional to the logarithm of the sound intensity, it follows that they are linearly related to each other: H phon  n1000 Hz It is important to understand that although both quantities on the left- are right-hand side of the equal sign are expressed on the decibel scale, the left side shows the subjective sensation of loudness, whereas the right side gives the physical intensity. Therefore, the equal sign only implies numerical equality. 41 Equal loudness curves are shown by the blue and black lines in the figure. They were first established by Harvey Fletcher and Wilden Munson, two US scientists, and therefore, they are also called Fletcher-Munson curves in their honor. The equal loudness curves connect intensities of sounds with different frequencies eliciting the same sensation of loudness. The curves clearly show that sounds of identical intensity, but different frequencies induce different sensations of loudness. By looking at the curves it can be concluded that the human ear is the most sensitive in the 3-5 kHz region. It can be easily read from the curves which sound intensity of arbitrary frequency sounds to be equally loud to another sound, e.g. the sound with an intensity of 10-9 W/m2 and a frequency of 100 Hz is on the 10 phon curve similar to the sound with an intensity of 10-11 W/m2 and a frequency of 1000 Hz. Therefore, both of them sound to be equally loud. 42 43