Anatomy of the cochlea
The cochlea within the inner ear is an exquisite structure containing cells that can convert sound-induced mechanical motions to electrical signals that excite afferent auditory nerve fibers. Unfortunately, the structures that make up the cochlea are hidden in a fairly inaccessible part of the skull. The bony labyrinth of the cochlea includes 3 tubes (scalae). The upper 2 are known as the scala vestibuli, which contains perilymph, and the scala media, which contains endolymph. The lower one is called the scala tympani, and contains perilymph. The scala tympani is separated from the upper tubes by parts of the osseous spiral lamina and the basilar membrane (BM).  The scala vestibuli and the scala tympani are merged together at the apical end of the cochlea via the helicotrema, which is a narrow opening. The organ of Corti (OC), as shown in Figure 1, sits on the BM along the length of the cochlea and comprises various supporting cells and sensory cells. In a cross-section of the OC, as seen in Figure 1, the sensory inner hair cells (IHCs) are located at a medial position and the outer hair cells (OHCs) are separated radially from the IHCs by pillar supporting cells. The reticular lamina (RL) is a thin, stiff lamina that extends from the OHCs to Hensen's cells. A phalangeal process links Deiters’ cells to the RL and provides a feed-forward mechanism,  which is crucial to the amplification of faint sounds. The tectorial membrane (TM) is located above the OC and is nearly parallel to the BM. During acoustic stimuli, the TM will interact with the IHCs via viscous forces from the subtectorial fluid and with the OHCs via a direct connection to the tallest stereocilia. 
In models of the cochlea, the geometrical complexity of the cochlea is often ignored or greatly simplified such that the entire OC is reduced in size and represented only by the BM. These types of models, which could be termed “macromechanical models”, are generally used to predict the responses of the BM to various stimuli.  Macromechanical models generally deal with the passive behavior of the cochlea, which means that the response does not depend on the stimulus level, except for amplitude scaling. 
In contrast to cochlear macromechanics, cochlear micromechanics focuses on supporting cells and sensory cells, and concerns their interactions or responses to either acoustic or electrical stimuli at a microscopic level. A micromechanical model is essential when characterizing the active and nonlinear amplification of the cochlea because it is the interaction between the BM and the OHCs that motivates this process. [1,4] The shearing motion between the RL and the TM is known to deflect the OHCs stereocilia when the BM moves, which opens stereociliar transduction channels to generate a change in the length and intracellular potential of the OHCs. This reaction provides a force that enhances BM motion through Deiters’ cells so that cochlear micromechanics can be described as a closed-loop feedback system. 
The goal of cochlea micromechanical models is to explain the active function. Such models were inspired by findings regarding nonlinear BM activities, the existence of otoacoustic emissions, and in vivo measurements of BM motion. [1,5–8]
The objective of the review
The cochlea plays a crucial role in the perception of sound. It acts as a frequency analyzer and an amplifier for low-level sounds. The mechanisms underlying these functions are not fully understood because its anatomical position as well as its active nonlinear behavior may be altered when observed invasively. Experimental observations have shown that the cochlea is able to compress input sound pressure with a large dynamic range into a signal with a much smaller range of responses that can be processed by hair cells. [9,10] The inaccessible and extraordinarily sensitive nature of the cochlea, however, hinders comprehensive measurements of the micromechanical behaviors inside the OC.
Many scientists have attempted to model the cochlea and mimic its function using new computing tools and advanced measuring instruments. [2,10,11] It is hoped that future cochlear models will function as effectively as the real organ in processing speech and music, and that such models will be useful in the development of hearing aids, cochlear implants, [12,13] and inner ear drug delivery.  This paper focuses on current hypotheses regarding cochlear micromechanics and associated models.
Database search strategy
The articles used in this modeling cochlear micromechanics review were retrieved by replicating the search terms of Ni et al.  The authors used the following inclusion criteria: studies that discussed modeling methods and models of the cochlea, mechanism of the cochlea, experiments that provide in vivo measurements of the cochlea. English language and full-text articles published between January 1986 and March 2019 were included in this review. The authors searched the PubMed, ASA and Sciencedirect database to identify relevant publications. The literature search strategy was conducted as follows: the synonymous phrase, that is, cochlear, was combined with each of: (a) model, (b) modeling, (c) micromechanics, (d) mechanics, for example, “cochlear model”, viz. (1) + (a); “cochlear micromechanics”, viz. (1) + (c), etc. Four queries were obtained. The authors screened the reference list of included studies to identify other potentially useful studies. Firstly, the authors screened the titles and abstracts, then, the full texts for keywords, such as “cochlear model”, “cochlear mechanics” to find those that were potentially suitable. The data extraction process focused on the information about simulation, prediction, validation and hypotheses of cochlear models.
Hypotheses and simplifications
The most basic cochlear mechanics can be illustrated by the “travelling wave” theory, which was first described by von Békésy while analyzing postmortem cochleae. [1,15,16] When the cochlea is stimulated by a pure tone, the fluid-solid interaction induces a wave that travels along the BM with gradually increased amplitude until it reaches a maximum at a frequency-dependent location, termed the characteristic place, and then quickly decays. Thus, sounds at different frequencies exhibit maximum responses at different positions along the cochlear length, where the basal region corresponds to high frequencies and the apical region corresponds to low frequencies. This behavior is one of the most important criteria for evaluating cochlear models. 
Effects of coiling
The spiral coil structure of the cochlea is believed to be an adaptation that permits the cochlea to house a relatively long BM, which enables hearing at low frequencies within a limited headspace. In some models, straight-edged box shapes have been used to represent the cochlea, and coiling was not considered to be an essential factor.  However, numerical investigations have shown that a major contribution of coiling is the reduction in the fluid impedance at the apical end of the cochlea, which helps enhance the reception of sounds at low frequencies. 
Effects of viscosity
In some models, fluid viscosity is ignored and energy dissipation is accounted by BM resistive damping only.  When cochlear micromechanics are considered, however, viscosity of the fluid is important with respect to the small fluid gaps. For example, hair bundle movement and rest during stimulation largely benefit from viscosity over the range of audible frequencies, which was investigated using a finite element (FE) model of hair bundles. 
Feed-forward and feed-backward forces
Forces that are positive and generated by tilted OHCs apically toward the helicotrema are termed as “feed-forward”.  These greatly amplify the travelling wave amplitude close to the characteristic place. In contrast, forces that are negative and generated by phalangeal process basally toward the stapes are termed as “feed-backward”,  as shown in Figure 2. This mechanism has been used in many types of cochlear models, including 1-dimensional, 2-dimensional, and 3-dimensional  models. However, in vivo experiments have shown that the predicted BM velocities do not match well in terms of phase lag. 
Somatic electromotility and hair bundle motility in outer hair cells
The types of cellular processes that underlie cochlea active nonlinear amplification have not been discerned. One general view is that OHC electromotility is a main component of the amplification process, and this has been supported by repeatable in vivo measurements. [1,19] This view has been widely incorporated in active cochlear models by adding a feedback gain determined by the relative shearing motion between the TM and the RL. [9,23] Another view is that active hair bundle motility is an important contributor, and this has been observed in the frog and turtle. [19,24] Sul and Iwasa  developed a theoretical hair bundle model to study the contribution of hair bundle motility in cochlear amplification. In their model, they assumed that hair bundle energy was sufficient to counteract viscous drag in subtectorial space.
Modeling cochlear micromechanics
The mammalian cochlea is complex and sealed in a bony structure, making it difficult to study, especially in vivo. [9–11,26] Moreover, the multi-level arrangement, which includes components that range in dimension from centimeters down to nanometers, is very challenging to characterize using existing instruments. Models of the cochlea, however, can be used to carry out “numerical experiments”. These enable predictions of cochlear responses under different conditions, which can then be compared with available experimental observations.  This modeling can provide guidance regarding experiments that cannot currently be performed due to technological limitations. Theoretical or numerical models of the cochlea are also helpful for gaining physical insight regarding experimental observations.
A FE model is an elemental representation of a real continuous subject. FE models have been widely used to study cochlear mechanism including waves, [19,27] fluid coupling,  dynamics,  power dissipation,  electrical behavior,  bone conduction,  and cochlear implants.  Although FE cochlear models are discrete, they are flexible, and can include fine details and complicated structures inside the cochlea, such as the OC. [20,32,33] However, optimization of FE models requires a combination of modeling related physics, mathematics, and computing science.
Although the structure inside the cochlea is complicated, in terms of mechanics, it can be viewed as a fluid-solid coupled system. The essential independent variables of the fluid element are the pressure, which can be approximated using linear shape functions with nodes at the corners, and the displacement, which couples the fluid and the solid domains at the interface and is usually approximated by quadratic shape functions with nodes at the corners and at the mid-points of the edges. [2,5,20] Implementation of the FE method should follow the 2-field model of incompressible elasticity proposed by Zienkiewicz et al.  A harmonic time dependence, eiωt, in which i is the imaginary unit and ω is the angular frequency, can be used for the time derivatives, since the mechanical problem of interest can be assumed to be linear. Thus, the cochlear model can be formulated in the frequency domain. The equation for motion of the coupled fluid-solid system can be given in matrix form as
where M stands for the inertial term, C for system damping, and K for stiffness. The degrees of freedom of displacement, pressure, and voltage (if the OHC walls are modeled with piezoelectrical elements  ) are included in the vector w, and the external excitations are included in the vector f.
Kolston and Ashmore  conducted pioneering work on modeling cochlear micromechanics using the FE method. They simplified the cellular and membrane components of the OC and introduced a force to represent the OHC activity. The force was determined by the shearing displacement between the RL and TM, acting at the 2 ends of the OHC.
With the development of the FE method, computing algorithms, and hardware, researchers have been able to model the micromechanics of the cochlea in both passive and active conditions. Passive micromechanical models of the cochlea have enabled effective analysis of fluid-solid coupling  and detailed responses within the OC to static pressure loading.  An important use of the micromechanical model was the investigation of active response to input sound stimuli, in which mechanical effects of hair bundle motility or somatic OHC motility were represented by a dependent force or velocity source. [4,25,33] Studies have indicated that the Y-shaped geometry (phalangeal process) is crucial to the correct representation of the somatic contribution of the OHC to the wide sensitivity and frequency selectivity of the mammalian cochlea. [2,33]
Alternatively, the active amplification process in the cochlea can also be studied using lumped-parameter models, [23,30] in which the detailed structures of the OC are simplified and represented by a limited number of degrees of freedom.
Discussion and conclusion
The level of detail
Anatomically, the OC consists of a multitude of different cells, hair cells, and supporting cells. When considering the micromechanical behavior of the cochlea, the first step is to determine the level of detail to include. A model is not necessarily more accurate if more details are included, even though the advantage of a numerical model, such as a FE model, is its capacity to model structures with complex geometry. The complexity of the model should depend on the investigation objective and computation efficiency. If the research goal is to simulate the BM response to a sound, a simple 2-degree of freedom model may be sufficient to represent the passive, active, and nonlinear conditions  such that detailed structures inside the OC can be ignored. However, if the objective is to study the functions of each component (eg, the RL, OHCs, IHCs, BM, and TM in the OC) and the interaction between them, a more detailed model would likely be required.  For instance, to study mechano-electrical transduction, a finer model that considers elements at the order of nanometers may be necessary to detect the molecular details of the myosin motors. 
Integration with other models
In addition to mechanical models of the cochlea, electrical  and neural  models have also been developed. These have been used to examine the mechanisms underlying different types of hearing ability, as well as in the development of new strategies for transforming acoustic information in the auditory nerve, which has applications in hearing prostheses such as the cochlear implant.  Cochlear models based on different physical fields have led to useful insights regarding different processes inside the cochlea. However, few models are able to represent all direct connections between each physical field. This is complicated by the finding that physical processes inside the cochlea interact with one another. For example, BM deformations generate stereocilia deflections that open and close the mechanoelectrical transduction channels. This changes the electrical current through the biological capacitances and resistances in the OC, and generates potentials both inside and outside of the OHCs, which, in turn, affects the BM motions. This is a typical example of mechano-electrical coupling. As another example, the IHCs convert mechanical vibrations into neural stimulation that is sent to the brain for perception. Thus, this information is first translated from mechanical signals to electrical signals and is then converted to neural spikes. Conversely, efferent nerve fibers that contact with the OHCs will carry information from the auditory system to affect the OHCs. 
Some reported models have proposed the integration of multiple physical fields. For example, Ramamoorthy et al  proposed a model integrating the mechanical, electrical, and acoustical components of the cochlea. This model was used to predict and reproduce cochlear responses that were comparable to those obtained via experimental measurement. Further, Ni et al  proposed a FE model of the active OC that includes fluid-structure coupling and mechano-electrical coupling using piezoelectrical elements for the OHC walls. This model was able to predict detailed motions within the OC in response to either acoustic or electrical stimuli.
Variation in micromechanics along the length of the cochlea
The active amplification of the cochlea does not remain unchanged along the length of the cochlea. [32,41] Conversely, the behavior at the apex is quite different from that near the base. Amplification in the cochlear basal region has been found to produce compressive nonlinearity, and to enhance both BM and hair bundle displacement. [1,9,10] However, amplification in the apical region of the cochlea is fairly unidirectional and enhances hair bundle displacement only.  The other discrepancy between the 2 regions is the motion direction of the BM when IHC excitation is highest. At the apex, IHC excitation is highest when the BM approaches the maximum displacement and velocity toward the scala vestibuli, while the opposite has been observed at the base. It is possible that the IHC excitation phase, which relates to the tension in the tip-links of the tallest stereocilia on the IHC, could be “arbitrary” depending on the elastic properties of the TM. 
Although a significant feature of cochlear models is their use in numerical manipulations and testing assumptions about cochlear mechanisms, concerns and debates about model accuracy remain. With the development of new measurement instruments, especially those that can be used to measure responses to acoustic or electrical stimuli at the nanometer scale, [9,10,44,45] cochlear micromechanical models can be validated via comparison with experimental data in terms of amplitude, phase, relative phase variation between 2 components, the BM and RL for example, in response to either acoustic or electrical stimulus. [2,5,30] In most cases, experimental data are measured at a given position along the cochlea at different frequencies. Whereas, models are usually stimulated at one given frequency and measured at different positions, which is fast and requires less computational resources. In this way, 2 results cannot be directly compared. A clever way to indirectly validate the model is to use local scaling symmetry  by defining a dimensionless variable that depends on both frequency and position. Comparison can be made by plotting spatial-domain and frequency-domain measurements against this univariate description. It needs to be noted that this local scaling is more accurate nearby the peak response.
Simplifications and assumptions are unavoidable in any cochlear model, depending on the modeling objectives. Thus, it is necessary to find a balance between computational efficiency and the source data. As more tools for in vivo measurement are developed, especially with respect to 3-dimensional motion within the OC, increased model accuracy will improve the quality of physical insight gained from their use.
GN and DM conceived and designed the manuscript. GN and JP wrote the manuscript with contributions from all of the authors. All authors approved the final version of the manuscript.
This work was supported by Tianjin Key Laboratory of Brain Science and Neural Engineering, Beijing-Tianjin-Hebei Basic Research Cooperation Project of China (No.18JCZDJC45300), and Tianjin Plan of Funding Outstanding Science and Technology Projects Launched by Talents Returning from Studying Overseas of China (No. 2018004).
Conflicts of interest
The authors declare that they have no conflicts of interest.
. Robles L, Ruggero MA. Mechanics of the mammalian cochlea
. Physiol Rev 2001;81:1305–1352.
. Motallebzadeh H, Soons JAM, Puria S. Cochlear amplification and tuning depend on the cellular arrangement within the organ of Corti. Proc Natl Acad Sci U S A 2018;115:5762–5767.
. Ni G, Elliott SJ, Ayat M, et al Modelling cochlear mechanics. Biomed Res Int 2014;2014:150637.
. Nam JH. Microstructures in the organ of Corti help outer hair cells form traveling waves along the cochlear coil. Biophys J 2014;106:2426–2433.
. Ni G, Elliott SJ, Baumgart J. Finite-element model of the active organ of Corti. J R Soc Interface 2016;13:20150913.
. Zweig G. Nonlinear cochlear mechanics. J Acoust Soc Am 2016;139:2561.
. Lin NC, Fallah E, Strimbu CE, et al Scanning optical coherence tomography probe for in vivo imaging and displacement measurements in the cochlea
. Biomedical Optics Express 2019;10:1032–1043.
. Dong W, Xia A, Raphael PD, et al Organ of Corti vibration within the intact gerbil cochlea
measured by volumetric optical coherence tomography and vibrometry. J Neurophysiol 2018;120:2847–2857.
. Ren T, He W, Kemp D. Reticular lamina and basilar membrane vibrations in living mouse cochleae. Proc Natl Acad Sci U S A 2016;113:9910–9915.
. Lee HY, Raphael PD, Xia A, et al Two-dimensional cochlear micromechanics
measured in vivo demonstrate radial tuning within the mouse organ of Corti. J Neurosci 2016;36:8160–8173.
. He W, Kemp D, Ren T. Timing of the reticular lamina and basilar membrane vibration in living gerbil cochleae. Elife 2018;7:e37625.
. Ceresa M, Mangado N, Andrews RJ, et al Computational models for predicting outcomes of neuroprosthesis implantation: the case of cochlear implants. Mol Neurobiol 2015;52:934–941.
. Saba R, Elliott SJ, Wang S. Modelling the effects of cochlear implant current focusing. Cochlear Implants Int 2014;15:318–326.
. El Kechai N, Agnely F, Mamelle E, et al Recent advances in local drug delivery to the inner ear. Int J Pharm 2015;494:83–101.
. Prodanovic S, Gracewski SM, Nam JH. Power dissipation in the cochlea
can enhance frequency selectivity. Biophys J 2019;116:1362–1375.
. Chen F, Zha D, Yang X, et al Hydromechanical structure of the cochlea
supports the backward traveling wave in the cochlea
in vivo. Neural Plast 2018;2018:7502648.
. Campbell L, Bester C, Iseli C, et al Electrophysiological evidence of the basilar-membrane travelling wave and frequency place coding of sound in cochlear implant recipients. Audiol Neurootol 2017;22:180–189.
. De Paolis A, Bikson M, Nelson JT, et al Analytical and numerical modeling of the hearing system: Advances towards the assessment of hearing damage. Hear Res 2017;349:111–128.
. Reichenbach T, Hudspeth AJ. The physics of hearing: fluid mechanics and the active process of the inner ear. Rep Prog Phys 2014;77:076601.
. Kozlov AS, Baumgart J, Risler T, et al Forces between clustered stereocilia minimize friction in the ear on a subnanometre scale. Nature 2011;474:376–379.
. Yoon YJ, Steele CR, Puria S. Feed-forward and feed-backward amplification model from cochlear cytoarchitecture: an interspecies comparison. Biophys J 2011;100:1–10.
. Steele CR, Lim KM. Cochlear model with three-dimensional fluid, inner sulcus and feed-forward mechanism. Audiol Neurootol 1999;4:197–203.
. Neely ST, Kim DO. A model for active elements in cochlear biomechanics. J Acoust Soc Am 1986;79:1472–1480.
. Kennedy HJ, Crawford AC, Fettiplace R. Force generation by mammalian hair bundles supports a role in cochlear amplification. Nature 2005;433:880–883.
. Sul B, Iwasa KH. Effectiveness of hair bundle motility as the cochlear amplifier
. Biophys J 2009;97:2653–2663.
. Nuttall AL, Fridberger A. Instrumentation for studies of cochlear mechanics: from von Bekesy forward. Hear Res 2012;293:3–11.
. Elliott SJ, Ni G, Mace BR, et al A wave finite element analysis of the passive cochlea
. J Acoust Soc Am 2013;133:1535–1545.
. Ni G, Elliott SJ. Comparing methods of modeling near field fluid coupling in the cochlea
. J Acoust Soc Am 2015;137:1309–1317.
. Wang Y, Steele CR, Puria S. Cochlear outer-hair-cell power generation and viscous fluid loss. Sci Rep 2016;6:19475.
. Ramamoorthy S, Deo NV, Grosh K. A mechano-electro-acoustical model for the cochlea
: response to acoustic stimuli. J Acoust Soc Am 2007;121:2758–2773.
. Kim N, Homma K, Puria S. Inertial bone conduction: symmetric and anti-symmetric components. J Assoc Res Otolaryngol 2011;12:261–279.
. Liu Y, Gracewski SM, Nam JH. Consequences of location-dependent organ of Corti micro-mechanics. PLoS One 2015;10:e0133284.
. Nam JH, Fettiplace R. Force transmission in the organ of Corti micromachine. Biophys J 2010;98:2813–2821.
. Zienkiewicz OC, Taylor RL. The Finite Element Method. London, UK: McGraw-Hill; 1994.
. Kolston PJ, Ashmore JF. Finite element micromechanical modeling of the cochlea
in three dimensions. J Acoust Soc Am 1996;99:455–467.
. Steele CR, Boutet de Monvel J, Puria S. A multiscale model of the organ of corti. J Mech Mater Struct 2009;4:755–778.
. Johnson SL, Beurg M, Marcotti W, et al Prestin-driven cochlear amplification is not limited by the outer hair cell membrane time constant. Neuron 2011;70:1143–1154.
. Frijns JH, Kalkman RK, Vanpoucke FJ, et al Simultaneous and non-simultaneous dual electrode stimulation in cochlear implants: evidence for two neural response modalities. Acta Otolaryngol 2009;129:433–439.
. Hanekom T, Hanekom JJ. Three-dimensional models of cochlear implants: a review of their development and how they could support management and maintenance of cochlear implant performance. Network 2016;27:67–106.
. Liberman MC, Guinan JJ Jr. Feedback control of the auditory
periphery: anti-masking effects of middle ear muscles vs olivocochlear efferents. J Commun Disord 1998;31:471–482. quiz 483; 553.
. Recio-Spinoso A, Oghalai JS. Unusual mechanical processing of sounds at the apex of the Guinea pig cochlea
. Hear Res 2018;370:84–93.
. Reichenbach T, Hudspeth AJ. Shera CA, Olson ES. Unidirectional amplification as mechanism for low-frequency hearing in mammals. What Fire is in Mine Ears: Progress in Auditory
Biomechanics Massachusetts, USA: AIP; 2011;507–512.
. Steele CR, Puria S. Force on inner hair cell cilia. Int J Solids Struct 2005;42:5887–5904.
. Cooper NP, Vavakou A, van der Heijden M. Vibration hotspots reveal longitudinal funneling of sound-evoked motion in the mammalian cochlea
. Nat Commun 2018;9:3054.
. Dewey JB, Applegate BE, Oghalai JS. Amplification and suppression of traveling waves along the mouse organ of Corti: evidence for spatial variation in the longitudinal coupling of outer hair cell-generated forces. J Neurosci 2019;39:1805–1816.
. Zweig G. Finding the impedance of the organ of Corti. J Acoust Soc Am 1991;89:1229–1254.