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The cochlea plays a crucial role in mammal hearing. The basic function of the cochlea is to map sounds of different frequencies onto corresponding characteristic positions on the basilar membrane (BM). Sounds enter the fluid-filled cochlea and cause deflection of the BM due to pressure differences between the cochlear fluid chambers. These deflections travel along the cochlea, increasing in amplitude, until a frequency-dependent characteristic position and then decay away rapidly. The hair cells can detect these deflections and encode them as neural signals. Modelling the mechanics of the cochlea is of help in interpreting experimental observations and also can provide predictions of the results of experiments that cannot currently be performed due to technical limitations. This paper focuses on reviewing the numerical modelling of the mechanical and electrical processes in the cochlea, which include fluid coupling, micromechanics, the cochlear amplifier, nonlinearity, and electrical coupling.

Models are useful tools to connect our understanding with physical observations. The mammalian cochlea is the organ that converts sound into neural coding and has extraordinary sensitivity and selectivity. It is important to understand the mechanisms of mammalian hearing not only because of the scientific challenges they present but also because such knowledge is helpful in diagnosing and potentially treating the multiple forms of hearing problems from which people suffer. Modelling the mechanics of the cochlea assists in this understanding by allowing assumptions about its functions to be verified, by comparing responses predicted by mathematical models with experimental observations. A cochlear model can be thought of as a tool with which to carry out “numerical experiments,” in which researchers can obtain or predict output response to different stimuli. These predictions can then be used to compare with experimental observations and hence help to refine and validate the model or even to provide a guide on measurements that cannot be performed in experiments due to technical limitations. The type of cochlear modelling undertaken also depends on the purpose of the study and the available data of the cochlea.

This review will focus on numerical modelling of the mechanical and electrical processes that lead to the vibrations of the BM, the cochlear amplifier, and other nonlinear behaviours, in the mammalian cochlea. Some classical cochlear models will be illustrated to give a physical insight into how the cochlea works. This is not to judge which model is the best but to review the progress of cochlear modelling work.

The cochlea can be taken as a frequency analyser residing in the inner ear. The principal role of the cochlea is to transform the hair cell motions induced by the incoming sound wave into electrical signals. These electrical signals then travel as action potentials along the neural auditory pathway to structures in the brainstem for further processing. The whole transformation can be seen as a procedure of a real time spectral decomposition of the acoustic signal in producing a spatial frequency map in the cochlea. Mammalian auditory systems have the capability of detecting and analysing sounds over a wide range of frequency and intensity; for example, humans can hear sounds with frequencies from 20 Hz to 20 kHz and over an intensity range up to 120 decibels. This remarkable performance depends on mechanical and biophysical processes in the cochlea and the peripheral organ of hearing.

The cochlea consists of a coiled labyrinth, like a snail, which is about 10 mm across and has about 2.5 turns in humans, embedded in the temporal base of the skull. It is filled with fluid and divided into three main fluid chambers, as described, for example, by Pickles [

(a) A lateral view of the cochlear structure [

Neither the coiling nor RM is believed to play a major role in the mechanics of the cochlea; the dynamics of which can thus be analysed in terms of two fluid chambers separated by the BM. The motion in the cochlea is driven by the middle ear via a flexible (oval) window at the basal end of the upper fluid chamber, and the pressure at the basal end of the lower fluid chamber is released by another flexible (round) window. It is thus the difference in pressure between the upper and lower fluid chambers that drives the BM. The OC sits on top of the BM and contains two types of hair cells, as shown in Figure

This electromotility of the OHCs, as it is called, is due to a unique motor protein (Prestin) of the cell membrane that changes its shape when a voltage is applied, much like a piezoelectric actuator. The overall action of each OHC is thus to sense motion within the OC, via its stereocilia, to control the voltage within it, via the gating channels and capacitance, and to generate a response, via electromotility. There are about 12,000 OHCs in the human cochlea and they each act through this mechanism as local feedback controllers of vibration. It is surprising how this large number of locally acting feedback loops can act together to give a large and uniform amplification of the global response of the BM. It is also remarkable how quickly the OHCs can act, since they can respond at up to 20 kHz in humans and 200 kHz in dolphins and bats. This is much faster than muscle fibres, for example, which use a slower, climbing mechanism to achieve contraction. This climbing mechanism is still used within the stereocilia, however, to regulate the tension in the tip links and thus maintain the gating channels at the optimum point in their operating curves [

As previously mentioned, the principal role of the cochlea is to transform the hair cell motion induced by the incoming sound wave into electrical signals. These electrical signals then travel as action potentials along the auditory pathway to structures in the brainstem for further processing. Carterette [

The first finding related to the nonlinearity in the cochlea was back in 1971. Rhode [

(a) The normalised BM amplitude at different sound pressure levels (SPL). All curves converge below 10 kHz, indicating linear response and equal gain, independent of the SPL. Measurements were performed using the Mössbauer technique in the basal turn of the guinea pig cochlea. Maximal response frequency is at about 17 kHz [

It has been discovered that OHCs have a saturation property, which yields nonlinear responses. The relation measured between sound pressure and receptor voltage for OHCs shows a typical S-shape as depicted in Figure

Saturating profile of outer hair cells. (a) The relation between acoustic pressure and outer hair cell receptor potential is S-shaped, saturating at high pressure levels [

One clear difference between cochlear models is the level of detail included in the models. The cochlea is a multiscale arrangement of different cellular and membranous components, whose dimensions vary from 10^{−3} m down to 10^{−8} m, as shown in Figure

Illustrations of the structure of the inner ear at various levels of magnification. The position of the inner ear in the temporal bone is shown in (a). The cross-sectional structure within one turn of the cochlea is shown in (b) with the fluid chambers separated by the basilar membrane and the organ of Corti. The details of the bundle of stereocilia that protrude from the top of the hair cells within the organ of Corti are shown in (c). Finally (d) shows the molecular details of the myosin motors that maintain the tension in the tip links that connect the individual stereocilia within the bundle. The transduction channels (here labelled TRPA1) are now believed to reside at the bottom end of the tip link rather than the top [

The current models of the micromechanics of the OC often use a lumped-parameter representation of the BM, TM, and the structures into which the hair cells are embedded. The other way to study the micromechanics of the cochlea is using numerical methods such as the finite element method which is powerful in modelling complex structures. Determining the optimal complexity of a model is largely dependent on the modelling purpose and available (known) material properties. If the model is too simplistic, it will not embody the important processes of the real system. More details could be included if the needed geometry of the anatomical structure and material properties are available. The analysis time for a system may be inevitably increased with increase of system complexity. Lim and Steele [

Compared to reality, cochlear models may be incredibly simplified, but these crude models can still reflect important components of how the real organ works. The motivations of modelling the cochlea are to represent, within one framework, the results from a large variety of experiments and to explain the functions of the hearing system. In principle, models should also be testable by providing predictions of experiments that have yet to be done. Cochlear models have been formulated and constructed in various forms. These models are concerned with mechanical structures built up with structural elements like plates or beams coupled with fluid [

Models of cochlear mechanics are constructed to replicate basic physiological properties, such as the fundamental and harmonic cochlear responses to a single tone stimulus and then applied to interpret more complex observations and develop valid experimental hypotheses. For example, cochlear modelling was used by Helmholtz (1877) to explore perception of tones and by Gold and Pumphrey [

Most descriptions of the mechanical response of the cochlea involve the forward propagation of a single, “slow,” wave [

If the wavenumber distribution along the cochlea can be calculated from a model, or inferred using an inverse method from measurements [

In the travelling wave theory, the “slow” wave propagates on the BM from base to apex [

In a general way, once we know the wavenumber

From the experimental point of view, studies of the travelling wave were based solely on measurements of BM motion [

BM displacement (a) magnitude and (b) phase distribution along the cochlear longitudinal direction, plotted as a function of distance from the apex, in response to a 15 kHz tone over a range of intensities from 15 to 60 dB SPL [

Olson [

The distribution of the real (black lines) and imaginary (grey lines) parts of the wavenumber inferred from measurements of the BM frequency response at seven positions along the length of the cochlea using an inversion procedure [

The method can also be used to reconstruct the BM velocity distribution in combination with the WKB approach, (

The BM velocity distribution reconstructed from the derived wavenumber using the WKB approximation. The reconstructed response (dashed lines), obtained using the WKB approximation, shows good agreement with that from measurement (solid lines) (reproduced with permission from Journal of the Acoustical Society of America, 122, Shera, Laser Amplification with a Twist: Traveling-Wave Propagation and Gain Functions from throughout the Cochlea, 2738–2758, Copyright

The real structure of the cochlea and the components within it are very complicated [

The cochlear walls are immobile and rigid indicating the pressure gradient is zero on the walls [

The effect of “fluid ducts” can be neglected [

The spiral shape of the cochlea is straightened out. This may lose some information in the apical region of the model [

Reissner’s membrane is neglected as it is acoustically transparent [

The two cochlear channels have equal cross-sectional area and shape, so pressures of upper, SV, and lower, ST, fluid chambers are equal with opposite sign [

The boundary condition at the helicotrema is assumed to be pressure release; that is, the pressure difference is equal to zero. This can alternatively be more accurately modelled involving friction terms [

The cochlear fluids have negligible viscosity, so that only the CP dissipates energy [

There is no structural longitudinal coupling along the CP and elements along the CP interact through fluid coupling only [

A simple box model of the cochlea consists of two fluid chambers separated by the BM. The longitudinal coordinate,

In many box models of the cochlea [

It is computationally convenient to divide a continuous system into a number of discrete elements, which may be taken as an accurate representation of the continuous system if there are at least six elements within the shortest wavelength present, which is a condition commonly used in finite element analysis [

The analysis can be generalised to the case in which the radial BM velocity is the sum of a number of such modes [

The single longitudinal variables for the modal pressure difference and the modal BM velocity are spatially sampled as finely as required, dividing the cochlea into

The discrete approximation for a straightened cochlear box model.

The BM, however, is assumed only to extend from element 2 to element

The total pressure vector due to both stapes motion and motion of the BM can be written, using linear superposition, as

An advantage of this discrete formulation is that complicated geometries need to be analysed only once to determine the elements of

The frequency to place mapping that occurs within the cochlea can be described in terms of the propagation of a dispersive travelling wave within it. This wave motion involves interaction between the inertia of the fluid chambers and the stiffness of the basilar membrane. It occurs even for excitation of the cochlea at high sound pressures, for which the active processes within the outer hair cells are saturated and do not contribute significantly to the dynamics. The fundamental wave behaviour can thus be understood in the passive cochlea, in which the feedback loops created by the outer hair cells are ignored. In a simple one-dimensional “box model” for the uncoiled cochlea, as shown in Figure

The distribution of the mechanical resistance, when a constant damping ratio,

Since the wavenumber varies with position and frequency, conventional solutions to the wave equation in (

Simulations of the distribution of the magnitude and phase (plot with respect to the velocity at the stapes,

Idealised representation of the outer, middle, and inner ear, showing the basilar membrane in the inner ear as a series of mass-spring-damper systems distributed down the cochlea coupled together via the fluid shown in blue, together with the distribution of the natural frequencies of these single-degree-of-freedom systems.

The lumped-parameter model of the cochlea is a simplification of the OC. In this kind of model, the properties of the spatially distributed OC are represented by a topology consisting of discrete entities (masses, dampers and springs) that approximate the dynamic behaviour of the OC under certain assumptions. From a mathematical point of view, the dynamic behaviour of the OC can be described by a finite number of ordinary differential equations with a finite number of parameters. Mechanically, every component in the lumped-parameter model is taken as a rigid body and the connection between each rigid body takes place via springs and dampers. The model can be divided into a finite number of segments in the longitudinal direction with each individual segment having a unique characteristic resonant frequency, decreasing from 20 kHz, at the base, in the human, to about 200 Hz at the apex over the 35 mm BM length, as shown in Figure

Various lumped-parameter models of the OC have been developed by researchers. The simplest one only contains one-degree-of-freedom, in which the TM is assumed only to move with the same motion as the BM. Allen [

An advantage of such lumped-parameter models, however, is that the conditions for stability, which is not guaranteed in an active model and can otherwise lead to misleading results, can be formulated using a state space representation [

Although the finite element cochlear model is an elemental representation of the real continuous cochlea, the flexibility of the finite elements allows the possibility of considering more detailed and complicated cochlear structure than in the elemental model above. In many areas, the finite element analysis is a key and indispensable technology in the modelling and simulation procedures. However, a good understanding of physical, mathematical, and computational modelling plays an important role in utilizing these advantages of the finite element method.

A finite element version of the cochlear box model can be obtained by dividing its length into

The dynamic response of the fluid can also be represented in finite element form [

For a single frequency excitation, proportional to

Finite element techniques have also been applied to problems associated with cochlear micromechanics, including the motion of the hair cell stereociliary bundle [

Kolston and Ashmore [

(a) An oblique view of a small section of the cochlear partition in the 3D FE modeling technique [

Givelberg et al. [

Cai and Chadwick [

Geometry and mesh of cross-sections at apical (a) and basal (b) regions of the cochlea.

Andoh and Wada used a finite element method to predict the characteristics of two types of cochlear pressure waves, fast and slow waves [

(a) 2D FE Model of the OC [

Kim et al. [

Finite element models have also been used to investigate the effects of several longitudinal coupling mechanisms on the coupled BM response [

Our understanding of the cochlea is largely based, either explicitly or implicitly, on the assumption that only a single type of wave propagates along its length. The properties of this “slow wave” can be calculated from a simple model of the passive cochlea that includes a locally reacting BM and 1D fluid coupling. In general, however, there are many other mechanisms, apart from 1D fluid coupling, that give rise to longitudinal coupling in the cochlea, particularly, the higher-order modes associated with 3D fluid coupling [

The discussion of multiple wave types in the cochlea is not new. Steele and Taber [

These authors, and Steele and Taber [

Elliott et al. [

It is only when additional forms of longitudinal coupling are included in the model, such as provided by multiple fluid chambers [

As described in Section

When the box model of the cochlea with a rigid BM, Figure

(a) The box model of the cochlea, (b) the pressure distributions in the upper and lower chambers as

The 1D fluid coupling assumed above is only valid when the height of the fluid chamber is small compared with the wavelength [

Distribution of the total pressure difference, due to both far and near-field components in the fluid coupling matrix, along the length of the cochlea due to excitation of a single element on the BM at ^{−1} at a frequency of 1 kHz.

Generally, a cochlear box model is a three-dimensional representation of the cochlea, since the fluid inside has the ability to move in all directions. Following Steele and Taber [

The bony structures outside the cochlear fluids can be represented by hard boundary conditions on the sides and the top of the cochlear chamber above the BM, so that following relations must hold

The BM velocity is now assumed to have a given distribution across its width, and in the longitudinal direction it has a sinusoidal variation with wavenumber

The distribution of the transverse motion across the width of the BM is complicated and level-dependant in the real cochlea [

The normalised BM velocity distribution,

The pressure field can be described by a summation of modes of the form

In order for each term in the model expansion to satisfy the equation for mass conservation, (

The coefficients

Substituting (

Multiplying each side of (

The modal pressure can be written by analogy with the modal velocity in (

In the wavenumber domain, the pressure difference can be represented by [

Based on the 3D expression of the fluid coupling in the cochlear, 1D and 2D expressions can be obtained by some simplifications. For example, the fluid component can be simplified to a one-dimensional function of only longitudinal position. In two-dimensional models, the height of the fluid is taken into account and in the three-dimensional models the width of the fluid and the width of the cochlear partition are additionally included. For the two-dimensional model, the pressure associated with the first term in (

Using the long-wavelength approximation with the one-dimensional model, in which the wavelength is large compared to

For low values of

The normalised fluid equivalent height

The Green’s function was widely used for calculating the fluid coupling, for example, by Allen [

Only forward travelling waves are assumed,

The normalization constants

The modal amplitude in (

The difference between this formulation and that in the wavenumber domain is that the driving source is initially assumed to be concentrated at a point, rather than the infinite sinusoidal distribution along the cochlea assumed in the wavenumber analysis, and that instead of the wavenumber being a specified value, it is now a variable that changes with the modal order. In the case assumed here, where the fluid is assumed to be incompressible, the modal wavenumber becomes

The pressure in the chamber due to the velocity distribution corresponding to excitation of a single element of the BM with a predefined modal shape

The modal pressure difference due to the far field component is thus due to the plane acoustic wave, corresponding to both

The modal pressure due to the near-field of this vibrating element of the BM can thus be written as

In fact, a reasonable approximation to the averaged near-field pressure due to a single BM element can be obtained using only two terms of the infinite series in (

Continuous distribution of the modal pressure along the cochlea due to the fluid coupling near-field component (dashed line) and the average pressure over each discrete element of the BM (solid line), when excited by a single element at ^{−1} at a frequency of 1 kHz. Also shown (dot-dashed line) is the approximation to this discrete distribution obtained from the sum of two exponentially decaying terms of an acoustic analysis of the fluid coupling, (

The finite element method is a powerful technique that has the advantage of modelling complex structures. In the finite element model, the fluid coupling (of the box model or of a complex geometry such as a coiled model) of the cochlea can be written as

The rectangular box geometry needs to be divided into finite longitudinal sections to fulfil the requirement that there are at least 6 elements within the shortest wavelength, which is a common rule in finite element analysis [

Modal pressure difference on the BM calculated using the FE model for excitation of a single longitudinal segment of the BM at ^{−1} at a frequency of 1 kHz with 8 × 1 elements (dotted lines), 8 × 2 elements (dashed lines), 8 × 4 elements (dot-dashed lines), 8 × 8 elements (solid lines), and analytic solution (red lines) [

An advantage of the finite element method is that since the fluid is modelled using acoustic elements, the compressibility of the fluid, as well as its inertial properties, is taken into account. The widely used theoretical models [

It is interesting to compare the predicted frequency of this quarter wavelength resonance with the upper frequency of hearing in several species [

If the cochlear fluid is assumed to be compressible, then the classical slow cochlear wave can be given by [

Despite the very significant change in the pressure distributions in the fluid coupling calculations due to fluid compressibility, this hardly appears to have any effect on the coupled cochlea response at all. This surprising result can perhaps be understood by returning to the way in which the coupled model is formulated [

Due to the fact that the cochlear components of interests are housed in bone, as shown in Figure

A micrograph of a plastic cross-section of the guinea pig cochlea. The section is cut at a near midmodiolar plane [

Another important geometric factor in cochlear mechanics is coiling. It is believed that the coiled structure is an adaptation to the problem of fitting a long basilar membrane, to provide good low frequency hearing, into the relatively small heads of early mammals [

von Békésy [

An analytic model of the fluid coupling in the coiled cochlea was developed by Steele and Zais [

Ni et al. [

The representation of the cochlea can be discussed in terms of its micromechanics and its macromechanics. The term “micromechanics” refers to the dynamic behaviour of a radial slice of the cochlea at the microscopic level. By contrast, the term “macromechanics” deals with the coupling between the micromechanical motion of the system at various points along the cochlea, thus giving rise to a solution for the global response of the cochlea.

Generally, all models used to describe the BM vibration patterns or the pressure distributions along the cochlea are dealing with macromechanics, since these models concern the interaction between fluids in the two fluid chambers with the CP (all the parts of which are assumed to move in a same manner with the BM). The longitudinal coupling of the CP, for example, the phalangeal processes which longitudinally connect OHCs, is neglected. The components within the OC can move with different magnitude and phase. Nowotny and Gummer [

In the classic travelling wave model, the cochlea is taken as a hydromechanical element, determined by the physical structure of the cochlea, which provides the basis for frequency analysis. This passive, travelling wave model was first proposed by von Békésy [

(a) One-degree-of-freedom micromechanical model, in which the BM and TM are assumed to always move parallel to each other without radial motion. (b) Distribution of the magnitude and phase of the BM velocity of the passive cochlea in frequency domain calculated using one-degree-of-freedom micromechanical model.

Comparing with the cochlea

The direct physiological evidence of the active feedback process in the cochlea is the observation of sound in the ear canal caused by spontaneous oscillations, apparently of cochlear origin, retransmitted by the middle ear, which are called spontaneous otoacoustic emissions [

In the early stages of cochlear modelling, the models were formulated mechanically in the frequency domain and active undamping was assumed at a site basal to the characteristic place with only a single degree of freedom to represent the dynamics of the CP. This fixed the spatial distribution of undamping in the model, and thus the pattern of impedances was only valid for one frequency [

The distribution of the BM impedance at frequency of 1 kHz.

Alternatively, Zweig [

A number of authors have extended the mass-spring-damper representation of the passive BM mechanics, to include lumped-parameter representations of the dynamics of the OC. These micromechanical models can then include forces due to the action of the outer hair cells in an attempt to represent the cochlear amplifier. A good review of such models is provided by Patuzzi et al. [

Mechanically equivalent system for Neely and Kim’s 1986 model [

More detailed lumped-parameter micromechanical models have been proposed that have three degrees of freedom [

Figure

A lumped-parameter model of a cross-section of the physical arrangement of the cochlear partition. The transverse BM motion being driven both by the pressure difference in the fluid chambers and the pressure due to the OHC. The TM moves transversely and is radially driven by the OHC via the RL. The forces due to the radial motion can be resolved into equivalent transverse forces and the radial TM degree of freedom can be represented as an equivalent transverse degree of freedom.

Three equations of motion can be used to describe the dynamic behaviour of the three-degree-of-freedom system, shown in Figure

Many researchers have sought to refine active cochlear models by including “feedforward” [

(a) Anatomical scan of a longitudinal view of the organ of Corti of a mole rat cochlea with a representative OHC, Deiters’ rod (D), phalangeal process (PhP), and stereocilia bundle (S). (b) Schematic of the longitudinal view of the organ of Corti, showing the tilt of the OHCs based on (a) [

Apart from those relatively simply formulated active cochlear models, many researchers have sought to refine model predictions by adding more degrees of freedom to the CP [

The high dimensional cochlear models with various details of the OC are ideally suited for studying the micromechanics of the cochlea. They have the common limitation that large numbers of degrees of freedom require massive computing time and a powerful computer. Solutions in the time domain are difficult because they require analysis at thousands of time steps.

In a linear cochlear model, a sinusoidal pressure difference across the CP generates a sinusoidal BM velocity and so the relationship between them can be simply represented by an impedance function. In a nonlinear cochlear model, however, the waveform of BM velocity response to a sinusoid becomes distorted. Many efforts have been made to model the nonlinear behaviour of the cochlea, including the compressive growth at high sound levels, two-tone suppression, and DPOAEs [

To model the nonlinear behaviour of the cochlea, we need to define nonlinear functions to describe the system. In general, the input,

For a linear model, if

And, by superposition, for multiple inputs

A linear system cannot generate signal components that were not present in the stimulus spectrum, but any nonlinear system will produce harmonic distortion products in response to simple tonal stimuli, and more complex stimuli produce more complex distortion product spectra. To study cochlear nonlinear behaviour, a time domain analysis is generally necessary. To start the time domain analysis, all relevant system equations should be setup in the time domain using differentiation and integration wherever appropriate.

Generally, the relationship between the pressure difference,

Substituting (

To solve this equation in the time domain, the first step is to transform it into two coupled ODEs and then the spatial differential equation can be transformed to a difference equation. The remaining ODEs in time can be treated separately. The second-order equations can also be rewritten as sets of first-order equations, which can be solved with the numerical methods such as the Runge-Kutta method [

Different researchers define different forms for the function

One famous model is called the Van der Pol oscillator. Basically, the oscillator is described by an ODE similar to one that describes a damped mass-spring system, but the damping term is an even nonlinear term, which has a negative value for small amplitudes, which would make the oscillator unstable. In his model, the damping term has a form of the quadratic (parabolic) function. Small amplitude negative damping specifies an oscillator that generates a limit cycle oscillation if undriven and uncoupled. The relative scaling of the active part is characterized by a parameter

Duifhuis also suggested that overlap of the responses is found in the Van der Pol oscillators in the frequency domain and also a decrease of selectivity. He proposed a modified Van der Pol oscillator model [

The nonlinear model proposed by Furst and Goldstein [

Scaled damping profiles as a function of velocity.

Besides nonlinear damping in the single-degree-of-freedom model, researchers also developed various nonlinear force or pressure models to represent the effect of the OHCs motility in the more complicated micromechanical models in Section

Chadwick [

Böhnke et al. [

Lim and Steele [

Böhnke and Arnold [

Comparing 2D and 3D nonlinear cochlear models, Diependaal concludes [

In practice, it is usually impossible to analyse the nonlinear equations analytically, which requires numerical methods of time domain analysis or of linear approximation and perturbation techniques [

The state space approach is inherently set in the time-domain and can express the dynamics of a system as a set of coupled first-order differential equations and arranged in vector matrix form [

Although there are kinds of active nonlinear cochlear models built by different numerical approaches, most of them can be implemented in the state space formalism [

Mechanical vibrations in the cochlea are generated by a pressure input stimulus having a large dynamic range. These vibrations activate the hair cells (sensory organs) in the cochlea. Hair cells detect these vibrations, reduce their dynamic range, and encode them to a form that the nervous system can interpret. These processes are only possible because of electrical activities inside the cochlea. The following is a brief overview of these electrical activities, the electrical properties of the cochlea, and the mutual interaction between electrical and mechanical parts in the cochlea, as well as models of these phenomena.

Recall from Section

Vibrations of the basilar membrane deflect the stereocilia and modulate these flows of ions. Deflection of the stereocilia opens and closes pores known as MET channels, and due to the potential difference between the perilymph, endolymph, and the intracellular potential, the opening and closing of the MET channels changes the inflow of ions which results in activation of the hair cells. For investigating the properties and effects of these standing and alternating flows of ions, the cochlea can be modelled as a network of biological resistances, capacitances, and voltage and current sources, that is, an electrical model.

As previously mentioned, the organ of Corti consists of sensory receptors called the IHC and OHC. The IHC transduces mechanical vibration into neural stimulation which is sent to the brain for interpretation. The OHC nonlinearly amplifies the small basilar membrane motions, and this action consequently enhances the sensitivity of IHC to weak stimuli and compresses high-level stimuli. Thus the operation of the OHCs enormously increases the dynamic range of hearing [

At least the longest stereocilia of the OHCs are imbedded in the tectorial membrane, and hence the MET channels of the OHCs appear to be sensitive to the relative displacement of the basilar membrane and tectorial membrane [

The hair cells connect to the nervous system via the eighth cranial nerve [

The IHCs and OHCs are the primary elements of the electrical lumped model and can be individually divided into apical and basolateral parts. Each part can be modelled by membrane capacitances, variable resistances, and voltage sources.

The OHC in the mammalian cochlea is thought to use both somatic electromotility and hair bundle electromotility to provide mechanical active amplification [

More recent measurements have shown that the membrane time constant does not limit OHC amplification. The membrane time constant is approximately one order of magnitude smaller than what was previously reported [

By using some simplifications, it can be shown that the ratio of OHC length change to the membrane charge,

The electrical properties of the organ of Corti without hair cells are similar to other biological tissues of the human body and can be modelled as a passive electrical network. However, the existence of the hair cells gives special electrical properties to this sensory epithelium and affects its mechanical behaviours. The mutual interactions between mechanical and electrical parts of the organ of Corti have significant effects on the cochlear function and must be incorporated in a realistic cochlear model.

The battery or variable resistance model by Davis [

Battery or variable resistance model; +80 mv and −60 mV are the endolymph and intercellular potential, respectively. From [

Strelioff’s network model [

In [

Even though electrical coupling has been rarely amalgamated into the cochlear models, some works can be seen in the literature.

In Ramamoorthy et al. [

Nonlinear saturation behaviour of the MET channel has been incorporated in Liu and Neely [

Electrical properties of hair cells

A circuit diagram for the inner and outer hair cells. Superscripts

The electrical network model of the organ of Corti [

The electrical network model of the organ of Corti with dependent current sources instead of variable resistors [

An electrical model of the inner hair cell [

Experimental recordings from the cochlea show that the cochlear amplifier results in sharply tuned cochlear mechanical responses. These tuning curves indicate that auditory nerve excitation fully matches mechanical tuning [

A comparison between the hair cells and basilar membrane tuning curves. The tuning curves are for IHC (○), OHC (△), and basilar membrane displacement (

A comparison between the neural and basilar membrane displacement/velocity tuning curves. From [

Using realistic electrical coupling in cochlear models will facilitate opportunities to gain further information about mechanical and electrical interactions.

Modelling the voltage distribution in the cochlea not only helps to understand cochlear function but also may be used to design new strategies for delivering acoustical signal information to auditory nerves which can be used in cochlear implants [

In cochlear implants the number of physical electrodes is limited to 12–22 in contemporary devices. This limited number of electrodes can only stimulate a small number of fairly broad fixed regions along the cochlea. Frequency resolution can be improved by using virtual channel techniques in which additional places can be stimulated by the available electrodes [

Two techniques have been developed to create a virtual channel. The current steering technique uses superposition of the electrical fields of two simultaneously activated electrodes to trigger the intermediate auditory neurons. In this approach, independent current sources are required. Another technique uses fast consecutive pulses to activate two electrodes. The impedance between these two electrodes completes the circuit and results in the stimulation of neurons between the electrodes [

Deformations of the BM cause deflection of the stereocilia which open and close the MET channels. This alternation causes a varying electrical current through the biological resistances and capacitances in the organ of Corti and generates the potentials both inside and outside of the hair cells. This potential outside of the hair cells is referred to as the cochlear microphonic (CM). The CM was first observed by Wever and Bray in cats [

The cochlear microphonic is a by-product of the cochlear activities in response to a sound stimulus. The CM can be observed by placing an electrode on the surface of the cochlea [

The CM is mainly generated by the OHC [

Integrated mechanical and electrical models have the potential to shed light on the source of these discrepancies and notches.

Modelling work largely depends on available data from experimental measurements, but material properties of some components in the cochlea are difficult or impossible to obtain, especially for humans. Assumptions and data fitting are always used in modelling the cochlea, especially for the models of cochlear micromechanics. Great effort has been put into finding reasonable values for some modelling parameter, but unfortunately, many others are still largely empirical, making some models difficult to validate.

The sources of the nonlinearities in the cochlear amplifier are still not well identified. Possible sources include material, geometrical (the dependence of the stiffness on the displacement), and state-dependent nonlinearities of mechanical structures; the nonlinear mechanoelectrical transduction process of auditory hair cells; and nonlinear neural coding of information, for example, rate intensity functions. A mixture of all these nonlinear factors is effective

The

Another important area related to nonlinearity is the method of solving the models in the time domain. Ku [

An alternative approach for implementing a model in the time domain is to use a circuit analogy of the model, and a powerful nonlinear circuit simulator such as SPICE (simulation program with integrated circuit emphasis) to solve it [

The discussion of multimode motion in the OC has been raised by some researchers [

Besides replicating experimental finding in laboratory animals, the other important goal of modelling the mammalian (human) cochlea is to predict hearing defects [

One goal of modelling cochlear mechanics is to replicate results observed in experiments. A complete quantitative cochlear model is currently not feasible, since it requires extensive geometrical and material properties, which are difficult to measure. In most cochlear models, parameters have to be tuned to match observations from experiments, which is time consuming. It is quite often that good agreement only occurs for a specific experimental observation but cannot replicate all cochlear behaviours.

Another objective of modelling the cochlea is to predict phenomena yet to be observed. If the model works well on replicating certain behaviour of the cochlea, it is important to test predictions from the model, which have not been observed in practice, in new experiments.

Many authors [

A multiscale model of the cochlea is always desirable, since it can incorporate detailed components in the cochlear partition. The question is how deep a model needs to be. A 2-DOF model can reasonably well represent the passive, active, and nonlinear behaviours of the cochlea. Detailed micromechanics, however, are ignored. To understand functions of each element, such as the RL, OHCs, IHCs, BM, and TM, in the OC, a detailed model including these elements is needed. Moreover, to explore the mechanism of mechanoelectrical transduction, which is believed to be the reason for cochlear active process, modelling on a scale down to nanometre may be required to show the molecular details of the myosin motors that maintain the tension in the tip links that connect the individual stereocilia within the hair bundle.

The argument is whether a model should cover every scale of the cochlea, namely, from millimetre for the SV and ST down to nanometre for tip-links, or, different scales could be considered separately and incorporate each individual response together to give an overall response. The former would require an extremely fine mesh for those tiny components, which leads to enormous number of elements that may cause convergence and computation difficulties.

Nonlinearity is an elaborate feature in the cochlea and could be modelled by different mechanisms, such as nonlinear damping, nonlinear OHC force, or nonlinear geometry. Different nonlinear mechanisms need different scales of modelling and it is difficult to say which scale is appropriate in any given situation.

In the mammalian cochlea, the amplification is a nonlinear active process providing extraordinary sensitivity and selectivity along with a large dynamic range and sharp frequency tuning. Although it is generally agreed that the amplification results are from active force generated by hair cells, there is a debate about the cellular processes behind the nonlinear amplification. One suggestion is that the outer hair cells electromotility underlies the cochlear amplification and another is that it is due to the active hair bundle motility.

The OHC electromotility is supported by a somatic motor in the OHCs body, which has the ability to elongate or contract axially, due to changes in membrane potential, and could provide positive feedback to reduce viscous damping and provide active amplification. Santos-Sacchi et al. [

A finite element model of the organ of Corti, in which the fluid loading and fluid longitudinal coupling were excluded, was used by Nam and Fettiplace [

It is still a particular challenge to model the mechanism of the cochlea in the apex, since only a few experimental measurements are available from

Measurements show that there is a strong compressive nonlinearity in the cochlear basal region and that hair bundle and BM displacements are both amplified. However, the amplification in the cochlear apical region is somewhat unidirectional and only the hair bundle displacement is amplified [

In conclusion, then, cochlear models are still limited by both a lack of detail in the models and experimental data on material properties and

OAEs were initially observed by Kemp more than three decades ago [

SOAEs are one of the major classes of OAEs which can be detected in the ear canal without any acoustic stimulus. Existence of these emissions is an explicit manifestation of the active mechanism in the cochlea [

In a “

Ghaffari et al. [

Apart from the 1D fluid coupling accounted for in most models, there are many other forms fluid and mechanical longitudinal coupling in the cochlea. It is not currently clear which of these types of longitudinal coupling are important to the proper function of the cochlea and which are not.

Basilar membrane

Cochlear amplifier

Click-evoked otoacoustic emission

Characteristic frequency

Cochlear implant

Cochlea microphonic

Cochlear partition

Degree of freedom

Distortion product

Distortion product otoacoustic emission

Finite element

finite element analysis

Fast Fourier transform

Inverse fast Fourier transform

Inner hair cell

Otoacoustic emission

Organ of Corti

Ordinary differential equation

Outer hair cell

Oval window

Partial differential equation

Phalangeal processes

Reticular lamina

Reissner’s membrane

Round window

Scala medium

Spontaneous acoustic emission

Sound pressure level

Scala vestibuli

Scala tympani

Tectorial membrane

Wave finite element

Wentzel-Kramers-Brillouin approximation.

The authors declare that they have no conflict of interests regarding the publication of this paper.

The authors are supported by EU project SIFEM on the multiscale modelling and 3D visualization for the inner ear. Grant FP7-600933.

_{1}-f

_{2}distortion product otoacoustic emission within a micromechanical cochlear model