In recent years, experimental studies have demonstrated that malfunction of the inner-hair cells and their synapse to the auditory nerve is a significant hearing loss (HL) contributor. This study presents a detailed biophysical model of the inner-hair cells embedded in an end-to-end computational model of the auditory pathway with an acoustic signal as an input and prediction of human audiometric thresholds as an output. The contribution of the outer hair cells is included in the mechanical model of the cochlea. Different types of HL were simulated by changing mechanical and biochemical parameters of the inner and outer hair cells. The predicted thresholds yielded common audiograms of hearing impairment. Outer hair cell damage could only introduce threshold shifts at mid-high frequencies up to 40 dB. Inner hair cell damage affects low and high frequencies differently. All types of inner hair cell deficits yielded a maximum of 40 dB HL at low frequencies. Only a significant reduction in the number of cilia of the inner-hair cells yielded HL of up to 120 dB HL at high frequencies. Sloping audiograms can be explained by a combination of gradual change in the number of cilia of inner and outer hair cells along the cochlear partition from apex to base.

## I. INTRODUCTION

Hearing loss (HL) is a common phenomenon in modern society. It can occur due to aging (presbycusis), noise exposure, or genetic or other pathological reasons. Thresholds of pure tones in quiet (THIQ) is the standard method of diagnosing HL. Thus, HL is partly manifested in the form of permanent threshold shift (PTS). Although quantification and measurement of HL are feasible up to a certain degree, there is still ongoing research on physical causes of HL (Stenfelt and Rönnberg, 2009). A main contributor of noise-induced hearing loss (NIHL) is outer hair cell (OHC) damage or loss. OHCs act as cochlear amplifiers that increase locally the basilar membrane (BM) motion, especially at high frequencies (Chen and Fechter, 2003; Robles and Ruggero, 2001). OHCs seem to be highly vulnerable to noise exposure (Liberman , 1986). Thus, for many years it was presumed that mild-to-moderate HLs are primarily caused by OHC loss. In the past, animal experiments (Lobarinas , 2013; Shi , 2016) have shown that inner hair cell (IHC) loss does not cause a significant PTS until a significant hair cell loss amount is reached (about 80% loss, mainly at high frequencies). IHC loss, however, is only one measure of the IHC's participation in creating PTS. The IHCs' signal transmission to the auditory nerve (AN) can be damaged in other possible ways (Kujawa and Liberman, 2019; Schmiedt, 2010). According to Schmiedt's study, a decline in the endocochlear potential (EP) caused presbyacusis-induced PTS. The recent study of Wu (2020) challenges this long-standing paradigm for human subjects and re-opens the question of hair cell participation in presbycusis, hence, indicating its source as sensory rather than predominately metabolic. They showed from autopsy analysis that both IHC and OHC loss can largely explain the audiometric patterns in aging. Before IHC death or loss, damage to its stereocilia occurs over a wide tonotopic area (Wang , 2002). Such non-local damage to IHCs may create significant PTS (Kujawa and Liberman, 2019). Calcium homeostasis of the IHCs might also contribute to PTS (Kidd and Bao, 2012).

The purpose of this study is to explore the effect of IHC dysfunction on THIQ from a theoretical perspective. We have chosen an end-to-end computational approach. Its input is an acoustic signal presented to the external ear, and its output is human listener performance in a threshold-task-based AN firing rate. This framework was developed by Siebert (1970), followed by Heinz (2001) and Furst (2015). The computational model includes the following steps: (a) The pressures in the outer and middle ear are calculated, along with the oval window (OW) motion in response to an acoustic signal; (b) a time-domain non-linear cochlear model is created that calculates the BM, tectorial membrane (TM), and OHC motions in response to the OW displacement (Barzelay and Furst, 2011; Cohen and Furst, 2004; Sabo , 2014); (c) a shear-flow model is created that calculates the sub-tectorial space (STS) fluid velocity in response to the BM and TM motions (Billone and Raynor, 1973; Neely and Kim, 1986); (d) a model is created of the IHC mechanotransduction (MET) process that is stimulated by hair bundle (HB) displacement due to the shear flow in the STS and its output is the probability of open calcium ion channels in the IHC membrane (Gianoli , 2017); (e) IHC membrane voltage is derived as a function of the open ion channels (Altoè , 2017, 2018; Lopez-Poveda and Eustaquio-Martín, 2006); (f) AN firing rate due to exocytosis (Verhulst , 2018; Westerman and Smith, 1988) is derived; and finally (g) the THIQ is derived from the AN response by assuming optimal decision (Heinz , 2001).

In Sec. II, each step of this framework is described in detail. Some of the models are up to date and highly detailed, while others are coarse yet represent the essential physical features of the relevant process. Unless stated otherwise, the equations for each model are given for the cochlea cross sections along its tonotopic axis, denoted by *x*. In each of these sections, the biophysical processes are described in a two-dimensional (2-D) fashion.

## II. THE COMPUTATIONAL MODEL

### A. The external and middle ear

_{me}is the mechanical gain, $ P i n ( t )$ is the incoming free field sound wave pressure,

*t*is time, and $ P bones ( t )$ is the pressure that causes the motion of the OW.

*ω*

_{OW}is the OW resonance frequency, and $ \sigma OW$ is the OW areal density.

### B. The BM and the OHCs

*x*, also known as the tonotopic axis), the pressure difference

*P*across the partition drives the partition's velocity. By applying fundamental physical principles, conservation of mass and the dynamics of deformable bodies (Cohen and Furst, 2004), the differential equation for

*P*is obtained by

*ξ*is the local BM displacement;

_{bm}*A*represents the cross-sectional area of scala tympani and scala vestibuli,

*β*is the BM width; and

*ρ*is the density of the fluid in both scalae, vestibuli and tympani.

*ξ*, the pressure profile,

_{bm}*P*, embodies the pressure difference that falls on the whole scala media, which is approximated to move simultaneously with BM. The pressure on the BM (

*P*) is a result of both the fluid pressure difference on the scala media and the pressure caused by the OHCs (

_{bm}*P*

_{OHC}). The relation between the pressures of BM, TM, and OHC is given by (Barzelay and Furst, 2011)

*ξ*;

_{tm}*K*,

_{bm}*K*,

_{tm}*R*,

_{bm}*R*,

_{tm}*M*,

_{bm}*M*are the effective stiffness, damping, and mass per unit area of BM and TM, respectively.

_{tm}*α*and

_{l}*α*are constants. The OHC electrical potential

_{s}*ψ*is obtained by

*ω*

_{OHC}is the OHC cutoff frequency and is a constant along the entire tonotopic axis, and

*η*is the electromotility transduction coefficient.

Each OHC pressure, *P*_{OHC}, is obtained from the spring properties of the OHC. $ \gamma OHC ( x )$ is defined as the OHC effective index on the tonotopic axis *x*, representing the effective distribution of OHCs along the tonotopic axis. Cohen and Furst (2004) showed that a cochlea with no active OHC is given by $ \gamma OHC ( x ) = 0$. On the other hand, $ \gamma OHC ( x ) = 1$ represents a non-realistic cochlea whose BM motion is unstable. $ \gamma OHC ( x ) = 0.5$ represents the ideal cochlea with fully active OHCs.

*L*is the cochlear length, and

_{co}*C*is the coupling factor of the OW to the perilymph.

_{ow}### C. RL-TM shear motion

*g*

_{0}is a constant and represents the mechanical lever gain.

*K*. The shear-flow velocity ( $ \xi \u0307 shear$) far from the IHC is given by

_{sp}### D. The IHC hydrodynamic torque

*τ*). Billone and Raynor (1973) found that

_{H}*τ*is proportional to the endocochlear fluid velocity. We further assume that the IHC torque is proportional to the relative velocity between the free-stream velocity ( $ \xi \u0307 fs$) and the IHC HB local velocity ( $ \xi \u0307 hb$), which yields

_{H}*c*is a geometrical coefficient of the stereocilia structure response to the flow,

*μ*is the endocochlear fluid viscosity,

*N*is the number of stereocilia in a single IHC, and

_{cilia}*H*is the height of the tallest stereocilium. The free stream velocity is assumed to be a steady state Couette flow field; thus,

*L*is the STS height, as shown in Fig.1.

*K*is the HB stiffness,

_{HB}*X*is the resting position of the HB tip when the tip links are cut (there is no feedback), and

_{sp}*F*is the feedback force calculated later by Eq. (25).

_{feedback}*F*is obtained by substituting Eqs. (16) and (17) into Eq. (15), which yields

_{H}### E. MET current and feedback force

The IHC stereocilium tip displacement causes ion channels to open and generate a current flow $ I MET ( \xi st )$. The channel opening scheme is determined by statistical mechanics. Recently, Gianoli (2017) developed an analytic model for deriving the probability of ion-channel opening [ $ P open ( \xi st )$]. A schematic description of the elastic forces in an IHC cilia is depicted in Fig. 2.

*D*, and the height of the long stereocilium is

*H*. An enlarged diagram of the tip-link complex is seen in the right panel of Fig. 2. The two ion channels are attached by two adaptation springs, and each of them is attached by two rigid branches to the tip-link spring, whose resting length is

*l*. The length of each branch is

_{t}*l*. The tip-link extension,

*x*, along with

_{t}*d*(the branches' vertical projection) are determined by a geometrical projection of the IHC bundle tip displacement $ \xi st$ (Fig. 2),

*X*

_{0}is resting position of the HB.

*d*,

_{OO}*d*, and

_{CC}*d*for the states OO, CC, and OC, respectively. For a given bundle tip displacement, the tip-link length is fully determined by the adaptation spring extensions (

_{OC}*a*,

_{OO}*a*, and

_{CC}*a*), since for every state, the geometry constraint $ l 2 = a 2 + d 2$ holds, as can be seen in Fig. 2. Boltzmann curves in equilibrium are considered when referring to the probability of openings of bilayer channels (Gianoli , 2017; Howard and Hudspeth, 1988). Thus,

_{OC}*T*is the standard human body temperature; $ E tot , O O ( \xi st ) , \u2009 E tot , O C ( \xi st )$, and $ E tot , C C ( \xi st )$ are the total energies obtained at the

*OO*,

*OC*, and

*CC*states, respectively. For the calculation of those energies, along with

*X*

_{0}and

*l*, see Gianoli (2017).

_{t}*I*) and the feedback force (

_{MET}*F*). We first consider a quasi-static first-order relaxation of the MET channels to the equilibrium open probability, based on the work of Verhulst (2018),

_{feedback}*V*is the membrane potential that will be derived in Sec. II F;

_{m}*G*is the total conductance when all channels are open; and, finally,

_{MET}*γ*indicates the relative number of cilia in a single IHC, where $ \gamma cilia = 1$ represents a healthy IHC, while $ \gamma cilia < 1$ represents reduction in the number of cilia in an IHC.

_{cilia}*k*is the tip-link stiffness, and $ E [ x t ]$ is the average value of the tip-link extension of all the different cilia of the HB. According to Eq. (20), this average value is calculated by considering $ E [ d ]$. This value is obtained by

_{t}### F. The membrane voltage

*C*is the membrane capacitance,

_{m}*I*is obtained by Eq. (24), and $ I f k$ and $ I s k$ are the fast and slow potassium currents of the basolateral channels inside the cell body, respectively. We can express those currents by using the conductance of the fast and slow potassium channels and voltage differences ( $ g f / s k$), which yields

_{MET}*f*/

*s*indicates the fast and slow channels, and $ V f / s k$ are the potassium Nernst potentials. Both the fast and slow potassium conductance are dynamically changing according to the first-order differential equation (Altoè , 2017), which yields

*S*is the slope factor that defines the voltage sensitivity of current activation.

_{k}### G. Exocytosis

The change in *V _{m}* causes the opening of calcium channels in IHC membrane that eventually triggers the exocytosis process.

*k*(

*x*,

*t*), which has an approximately straight-line relationship with the calcium current $ I C a , j ( x , t )$ (Goutman and Glowatzki, 2007; Johnson , 2005). The

*j*index indicates the type of fiber according to its spontaneous rate: (1) high spontaneous rate (HSR), (2) medium spontaneous rate (MSR), and (3) low spontaneous rate (LSR). Altoè (2017) modeled the exocytosis rate by the following quasi-linear synaptic transfer function:

*I*is the resting current, and

_{th}*Z*is a phenomenological rate constant.

*E*is the reversal potential of the calcium channels, and $ g C a , j ( x , t )$ is the conductance of the calcium channels with respect to the relevant spontaneous rate (SR) type. It follows the equation

_{Ca}*S*is the slope factor that defines the voltage sensitivity of current activation, and

_{Ca}*τ*is the calcium relaxation time constant. $ V C a , 0.5 , j$ is the calcium potential where conductance is half-activated. It follows that

_{Ca}*j*.

### H. AN spiking rate

*k*(

*x*,

*t*) is the exocytosis rate obtained by Eq. (30), and

*q*(

*x*,

*t*) is the immediate pool of neurotransmitters near the synaptic regions.

*q*(

*x*,

*t*) is obtained by the three-store diffusion model of AN adaptation (Westerman and Smith, 1988). $ \alpha r ( x , t )$ is a refractive correction function (Verhulst , 2018) that follows the equation

According to the model, the vesicles are located in three different pools: The first pool is the immediate pool located near the synaptic regions, the second is local, and the third is global. The third pool replenishes the second pool with a constant amount of vesicles. A diffusion flux exists between the global and the local pools and between the local and immediate pools (Verhulst , 2018).

## III. THE MODEL IMPLEMENTATION

The model was implemented by dividing the cochlear length into 256 sections. Therefore, the solutions for the different variables are given for $ x i , i = 1 , \u2026 , 256$. A typical input signal is $S(t)=\alpha sin(2\pi ft)$, where *α* is the amplitude of the signal and *f* is its frequency. The basilar and TM velocities, $ \xi \u0307 bm ( x i , t )$ and $ \xi \u0307 tm ( x i , t )$, were obtained by numerically solving Eqs. (3)–(12). In the present study, we used the code developed by Koral (2018) using parallel programming written in cuda for the Nvidia (Santa Clara, CA) GTX 1080 TI graphics processing unit (GPU) (Furst, 2015; Koral, 2018; Sabo , 2014). The time step was dynamically changed to ensure convergence of the differential equations. The actual solution was stored at a time step of $ \Delta t = 1 / F S$, where $ F S=20\u2009kHz$.

The next steps in the model are deriving the following variables for every *x _{i}*: shear motion, $ \xi \u0307 shear ( x i , t )$; stereocilium tip displacement, $ \xi s t ( x i , t )$; MET current, $ I MET ( x i , t )$; and IHC membrane voltage, $ V m ( x i , t )$. Those variables were obtained by solving Eqs. (13)–(29) in the time domain. The differential equations were solved in matlab by using iterative Euler method, and the solutions were stored at a time step of $ \Delta t$.

For deriving the exocytosis [Eqs. (30)–(34)] and the AN firing rate [Eqs. (35)–(36)], we used the open-source python code of the Verhulst (2018) auditory periphery model.

The three types of ANs (*λ _{spont}*) are differentiated by their half-activation potential ( $ V C a , 0.5 , j$) and the peak exocytosis discharge rate (Verhulst , 2018). Those parameters were adjusted to match the SR for the different fibers and their corresponding rate-level curves. Thus, we indicate the exocytosis as $ k j ( x i , t )$ and the instantaneous rate (IR) as $ \lambda j ( x i , t )$, where

*j*= 1, 2, and 3 represent the HSR, MSR, and LSR, respectively. The parameters used in the computational model are listed in Appendix B. The parameter values were taken from the theoretical models described in Sec. II. They were all based on experimental physiological measurements.

## IV. THE OPTIMAL DECISION MODEL

*T*is the human listener integration time constant (Zwicker and Fastl, 1999).

_{I}*S*1 and

*S*2. The subject decision process is manifested by the parameter

*Y*, $ Y ( \lambda \xaf 1 ( x 1 ) , \u2026 , \lambda \xaf 3 ( x 256 ) | S ) = Y ( \u2200 ( i , j ) \u2208 I , \lambda \xaf i , j | S )$, where $ I = [ j = 1 , 2 , 3 ; i = 1 , \u2026 , 256 ]$, and $ \lambda \xaf i , j = \lambda \xaf j ( x i )$. The parameter

*Y*represents the total statistical information for the two signals,

*S*1 and

*S*2. For the sake of convenience, the decision variable for the two hypotheses is abbreviated as

*S*1 and

*S*2, and

*ω*is the relative number of AN fibers with HSR, MSR, or LSR for

_{j}*j*= 1, 2, or 3, respectively. The total number of AN fibers is about 30 000 (Spoendlin and Schrott, 1989). The distribution of AN fibers among the HSR, MSR, and LSR is based on a physiological study in cats (Liberman, 1978). We adopted those values for the human AN; thus, in Eq. (42), $ \omega 1 = 0.6 K 0$, $ \omega 2 = 0.3 K 0$, and $ \omega 3 = 0.1 K 0$, where $ K 0=30\u2009000/256$.

*S*1 and

*S*2 [Eq. (39)], which yields for every

*i*, $ \lambda \xaf ( i , j ) = \lambda spont ( j )$ in response to

*S*1. $ d \u2032$ is derived separately for every frequency

*f*and $ \Delta \alpha $ (Heinz , 2001). A sensory threshold is defined as the stimulus level that yields $ d \u2032$ of 1.0, which corresponds to a correct response probability of 0.75 (Harvey, 1986). Thus, the threshold is obtained by

## V. RESULTS

### A. Prediction of normal ear threshold in quiet

The threshold in quiet is obtained by solving Eq. (44). For every frequency, $ d \u2032$ [Eq. (42)] is obtained for various values of $ \Delta \alpha $, and the value of $ \Delta \alpha $ that yields $ d \u2032 = 1$ is defined as the threshold. An example of $ d \u2032$ as a function of $ \Delta \alpha $ is shown in Fig. 3 for two stimulus frequencies, *f* = 250 Hz and *f* = 4 kHz. The vertical dashed lines indicate the values of $ \Delta \alpha $ that were obtained for $ d \u2032 = 1$. Note that $ \Delta \alpha $ that yields $ d \u2032 = 1$ for 250 Hz is 23 dB sound pressure level (SPL), while for 4 kHz, it is 0 dB SPL. This difference between high and low frequencies is obtained because of the variability in BM response as a function of frequency (Cohen and Furst, 2004).

The prediction of *THIQ*( *f*) along with the standard threshold given by ISO 226 (ISO, 2003) are shown in Fig. 4. It is apparent from Fig. 4 that the model developed in this study is providing a good fit to the ISO measurement, with deviation of less than ±5 dB for frequencies between 1000 and 6000 Hz. For 8000 Hz, there is a deviation of 11.4 dB, while at 250 Hz, the deviation is 6.6 dB. It is important to note that the derived threshold in Fig. 4 was mainly obtained from the HSR. The contributions of the other fiber types, LSR and MSR, were negligible. In several cases, thresholds were determined by either HSR only or all types of AN fibers. The resulting audiograms differ by less than 1% at all frequencies.

### B. Deficits in cochlear sensory cells

The purpose of this study is to predict the HL that might develop due to different damages in either OHC or IHC. Deficits can occur in many stages of the transduction process. Table I lists different types of possible deficits that are tested in Secs. V B 1–V B 6. For OHC deficits, the present study is based on the model of Cohen and Furst (2004) and includes a single parameter (*γ*_{OHC}) that describes OHC efficiency. Most HLs are caused by damages in both IHCs and OHCs (Liberman and Dodds, 1987). Thus, the combined damages were tested as well.

Parameter name . | Symbol . | Normal value ( $ P Normal )$ . | Units . |
---|---|---|---|

OHC gain | γ_{OHC} | 0.5 | $#$ |

Number of cilia | N _{cilia} | 50 | $#$ |

Hair bundle stiffness | K _{HB} | $0.65\xd71 0 \u2212 3$ | N/m |

Endocochlear potential | EP | 90 | mV |

Exocytosis rate constant | Z | $2.5\xd71 0 11$ | #/A·s |

Parameter name . | Symbol . | Normal value ( $ P Normal )$ . | Units . |
---|---|---|---|

OHC gain | γ_{OHC} | 0.5 | $#$ |

Number of cilia | N _{cilia} | 50 | $#$ |

Hair bundle stiffness | K _{HB} | $0.65\xd71 0 \u2212 3$ | N/m |

Endocochlear potential | EP | 90 | mV |

Exocytosis rate constant | Z | $2.5\xd71 0 11$ | #/A·s |

*f*) is defined as the difference between

*THIQ*(

*f*) of a damaged ear and a normal ear, i.e.,

*P*is defined as

*P*is the value of the parameter

_{Normal}*P*in a normal ear as listed in Table I.

In the present study, for sake of simplicity, the deficits are applied uniformly along the cochlear partition, i.e., *P _{d}*, is independent of

*x*, with the exception of Sec. V B 6.

#### 1. OHC deficits

The contribution of OHC to the BM motion is limited to the upper part of the cochlear partition that is characterized by CF (characteristic frequency) $>1000\u2009Hz$, with a maximum at the middle part where $2000<CF<4000\u2009Hz$. Therefore, any damage to the OHC yields reduction in the BM motion in the middle part of the cochlea, and, thus, different functions of OHC loss can yield similar HL (Cohen and Furst, 2004). The predicted audiograms resemble NIHL, which usually yields a loss of sensitivity in the 4-kHz range. Human studies showed that this type of audiogram is obtained independently of the type of noise exposure (Moore, 2007; Saunders , 1985), which eventually might produce a different distribution of OHC loss along the cochlear partition as was shown in animal studies (Maison and Liberman, 2000).

The effect of a uniform OHC deficit is shown in Fig. 5. Dysfunction of OHC, in particular, affects the mid-frequencies with a maximum HL of 40 dB at 4 kHz.

*γ*

_{OHC}[see Eq. (10)], along the cochlear partition from the base to the apex, according to the equation

*α*

_{OHC}is a constant. Figure 6 represents HL predictions for different values of

*α*

_{OHC}. The predicted audiograms in Fig. 6 are similar in their shape to the predicted audiograms presented in Fig. 5. For example, the audiogram in Fig. 5 that was obtained with $ % \gamma OHC = 0 %$, which represents $ \gamma OHC ( x ) = 0$, is similar to the audiogram in Fig. 6 that was obtained with $ \alpha OHC = 0.9$, which represents $ \gamma OHC = 0.02$ at the apex and $ \gamma OHC = 0.49$ at the base.

#### 2. IHC HB stiffness deficit

The IHC *K _{HB}* can be affected by any mechanical damage to the IHC (Duncan and Saunders, 2000), which eventually affects the stereocilium tip displacement [Eq. (18)]. The predicted HL caused by the reduction of

*K*is shown in Fig. 7. Decrease of

_{HB}*K*to 80% causes HL of about 35 dB at low frequencies and 20 dB at high frequencies. Decreasing

_{HB}*K*to a minimal value reveals a convergence in HL, about 40 dB at low frequencies and about 30 dB at high frequencies. The 40 dB limit in HL is probably due to the total tip-link stiffness, which effectively adds a constant value to the

_{HB}*K*via the feedback force [Eq. (25)] (Gianoli , 2017).

_{HB}The effect of deficits in *K _{HB}* in a cochlea without active OHCs ( $ \gamma OHC = 0$) is shown in the lower panel of Fig. 7. The resulting audiograms are approximately a superposition between the two types of deficits: (1) no active OHCs (Fig. 5) and (2) deficits in

*K*(upper panel of Fig. 7).

_{HB}#### 3. EP deficit

Physiological studies indicate that due to aging, a reduction in the EP often occurs (Schmiedt, 2010). In the present model, EP mainly affects the MET current [Eq. (24)]. Figure 8 represents the effect of deficits in EP on HL. A minor increase in HL (maximum of 30 dB) was obtained with a decrease in $ % E P$. The shapes of the resulting audiograms in Fig. 8 are very similar to those obtained due to *K _{HB}* deficit (Fig. 7), with a decrease in HL with the increase in frequency. Combination of deficits of EP and inactive OHCs ( $ \gamma OHC = 0 %$) yields superposition of the two types of deficits. Both parameters,

*K*and EP, eventually affect the IHC membrane potential,

_{HB}*V*, which is derived by solving Eq. (27) and yields a low-pass type of frequency selectivity.

_{m}#### 4. Deficits in number of IHC stereocilia

The most common deficit of IHCs is a reduction in the number of stereocilia of the IHC HB as observed in histological studies (Lim, 1986). According to the present model, the number of active stereocilia in the IHC HB (*N _{cilia}*) affects both the IHC hydrodynamic torque [Eqs. (15), (18), and (25)] and the MET current [Eq. (24)]. Its deficit is depicted in Fig. 9. The upper panel of Fig. 9 represents HL as a result of reduction in

*N*for a fully active OHC. For $ % N cilia$ greater than 32.5%, HL increases for all frequencies. There is a slight difference between low and high frequencies: A faster increase occurs for low frequencies ( $f<1000\u2009Hz$) than for high frequencies. For $ % N cilia < 32.5$, a significant increase in HL occurs at high frequencies, with a maximum between 4 and 6 kHz. This result is potentially explained by the saturation of

_{cilia}*P*[Eq. (21)] as a function of frequency and by the fact that

_{open}*V*at high frequencies becomes a direct current (dc) signal (Altoè , 2018). This result is demonstrated in Fig. 10, where the

_{m}*V*, the electrical potential of the IHC membrane, was derived as a response to stimuli with frequencies of 250 and 4000 Hz at different input levels at their correspondent

_{m}*CF*along the cochlear partition, for healthy cochlea and a damaged cochlea whose $ N cilia = 20 %$. For both the healthy and damaged cochleae, a dominant alternating current (ac) component was obtained for the low-frequency signal (250 Hz), and a dc component was obtained for the high-frequency signal (4000 Hz). For both high and low frequencies, the dc component was lower in the damaged ear than in the healthy cochlea. Therefore, the threshold was mainly affected at the high frequencies, since the membrane potential dc component was insufficient to initiate the exocytosis. The minimal potential required is about –58 mV, as was obtained by Eq. (27), for healthy cochlea without an acoustic stimulus. At low frequencies, on the other hand, the ac component was enough to excite the AN synapse. A combination of OHC and IHC deficits is presented in the lower panel of Fig. 9. The resulting curves seem similar to commonly measured audiograms, in which high frequencies are significantly attenuated relative to low frequencies.

#### 5. IHC-AN exocytosis

A possible IHC deficit is in the exocytosis rate (Boero , 2021), denoted by the parameter *Z* [Eq. (30)]. Figure 11 represents the audiograms obtained with deficits in *Z*, with and without deficits in OHC. For low frequencies (less than 1 kHz), HL converges to 40 dB when $ % Z$ decreases to 0.3. For high frequencies, a *U* shape audiogram is obtained with a maximum HL at 3–5 kHz for $ % Z$ of less than 1.5%, when $ \gamma OHC = 100 %$, and for all *Z* values when $ \gamma OHC = 0 %$.

#### 6. Tonotopic variation of N_{cilia} and γ_{OHC}

To demonstrate the simulation abilities with a variable number of IHCs and OHCs along the cochlear partition, an audiogram of a 71-year-old person presented in Wu and Liberman (2022) was derived and is shown in Fig. 12. The data for the IHC and OHC were obtained from Figs. 1 and 5 of Wu and Liberman (2022). The percentages of the surviving IHC and OHC cells are represented by the open circle symbols and are taken from Fig. 1 in Wu and Liberman (2022). Figures 5(A)–5(C) of Wu and Liberman (2022) include indices that represent the remaining cilia of the IHC and OHC along the cochlear partition. Finding the exact relationship between this index and the simulated parameter of *N _{Cilia}* for the IHC and

*γ*

_{OHC}for the OHC is not a trivial task. For the sake of simplicity, we assumed a linear dependence between the suggested indices and

*N*and

_{Cilia}*γ*

_{OHC}. Those values are presented by the solid lines in Figs. 12(A) and 12(B). Since some of the cilia used for the index derivation include cells that have no cilia at all [see Figs. 5(D)–5(G) of Wu and Liberman (2022)] and, therefore, do not contribute to the transduction process, we normalized the cilia indices by taking into account only hair cells that have more than one active cilia and by setting the indices to 0 near the apex, where there are no surviving OHCs or IHCs. The resulting normalized number of cilia ( $ N . cilia$) is presented by the dashed lines in Figs. 12(A) and 12(B). The audiogram (dashed line) in Fig. 12(C) was obtained by substituting $ N . cilia$ for both IHC and OHC. The measured audiogram [taken from Fig. 5 in Wu and Liberman (2022)] is presented in Fig. 12(C) by the open circle symbols. The similarity between the two audiograms is conspicuous, yet it is clear that future work is required to yield a better correlation.

## VI. DISCUSSION

The computational model presented in this paper allows us to compute the outcome audiogram due to different types of deficits. A detailed description of the IHC transduction process is given in the present study, yielding a list of various possible deficits. In particular, we have tested the PTS caused by damage to the HB stiffness, the EP, the number of cilia in IHC HB, and the IHC exocytosis. It is clear from the results that each deficit affects the HL at low and high frequencies to a different degree.

To summarize the effect of the different parameters on the HL, a quantitative comparison is shown in Fig. 13, for *f* = 250 Hz and *f* = 4000 Hz. The HL is depicted for four different percentages of each parameter relative to its normal value. For a relatively mild deficit (80%), a maximum HL (35 dB) is obtained for the deficit in the *K _{HB}* at 250 Hz. The effects of all other parameters are minor (less than 20 dB) for both frequencies. For a moderate deficit ( $ 50 % )$, the obtained HL is moderate for all parameters and both frequencies, with a slightly greater HL for 250 Hz compared to 4000 Hz. Only a severe deficit, less than 20% of remaining cilia in the IHC HB, dramatically affects the HL for 4000 Hz. For 250 Hz, even a severe deficit of 1% yields HL of maximum 40 dB in all the tested parameters. These theoretical results are most likely consistent with the observation of Lobarinas (2013) that threshold shift in behaviorally trained chinchillas was only evident when IHC loss exceeded 80%, hence, when great damage was caused to the IHC parameters.

The limit of 40 dB in HL of low frequencies is most likely a result of a strong ac component of the IHC membrane voltage (Altoè , 2018) (see Sec. II F). Even with large deficits, a strong ac component persists. It is sufficient to excite the AN response to low-frequency tones (see Fig. 10).

The audiograms obtained by moderate and severe damage to the OHC or to the cilia of the IHC (Figs. 5 and 9) yield similar qualitative results as measured by Liberman (1986). For example, Fig. 1 in Liberman (1986) depicts the tuning curve of the cat's ear, where the IHCs remain intact and a subtotal loss of OHC is present. This curve is juxtaposed with the normal hearing tuning curve of the cat. It is apparent that there is an about 45 dB SPL dip between 3 and 4 kHz in the OHC damaged tuning curve. Hence, this curve is qualitatively tantamount to the notch type OHC damaged audiogram presented in Fig. 5. In Fig. 9, bottom panel, severe damage to the IHC ( $ % N Cilia = 20$) and OHC ( $ % \gamma OHC = 0$) HL is presented. For the sake of discussion, we refer to it as the critical damage curve. Qualitatively, a decreasing slope with increasing frequency until 4000 Hz is apparent, and a slight incline is observed afterward. Figure 3 in the work of Liberman (1986) depicts the tuning curve of the cat's ear, where the IHC and OHC are severely damaged. It is juxtaposed with the normal hearing tuning curve of the cat. By subtracting the normal ear tuning curve from the damaged ear tuning curve, a curve similar to an audiogram is yielded for this severe damage condition. A qualitative comparison of this yielded curve with the critical damage curve yields a similar notch decline-incline frequency profile.

Most types of HL involve deficits in both IHCs and OHCs (Schmiedt, 2010; Vaden , 2022; Wu , 2020). In this paper, we have used our previous model for OHC damage (Barzelay and Furst, 2011; Cohen and Furst, 2004; Furst, 2015; Sabo , 2014). In this model, the OHC deficit is defined by a single parameter, which is mainly correlated to the number of active and healthy OHCs and along the cochlear partition. According to our current biophysical study, the most prominent damage of the IHCs is a reduction in the number of cilia. This type of damage was correlated with hair cell count in certain experiments (Liberman and Dodds, 1987; Wang , 2002); it was shown that once most of the cilia are missing, hair cells die. A decrease in IHC count is also equivalent to a reduction in the total number of active AN fibers. However, the threshold derivation that depends on the number of AN fibers yields only a minor increase in HL, probably due to redundancy in the number of AN fibers at a specific location along the cochlear partition (Wu , 2020). A recent statistical analysis (Wu , 2020) of post-mortem cochlea demonstrated a high correlation between counts of OHCs and IHCs and audiograms that were obtained close to the time of death. From a theoretical and biophysical perspective, it, thus, seems that those observations can be best explained by the effective reduction in cilia of both types of hair cells, as also shown experimentally in Wu and Liberman (2022). This is consistent with the PTS correlated with IHC cilia disarray and fusion (Wang , 2002) and with the audiogram comparison between the experimental results of Wu and Liberman (2022) and the simulation in Fig. 12.

Changes in the EP (Saremi and Stenfelt, 2013) can change OHC and IHC metabolic properties. Vaden (2022) differentiate between metabolic and sensory types of HL. They define metabolic HL as a result of a malfunction of stria vascularis, which eventually causes a reduction in EP, whereas sensory HL occurs when there is a decrease in the number of OHCs. According to the statistical analysis of Wu (2020), there is no correlation between deficits in stria vascularis and HL. According to the biophysical analysis presented in the current study, the HL followed by EP deficits is limited to 35 dB and is non-local. Thus, our results show that decrease in EP cannot be correlated to the HL measured at high frequencies. Nevertheless, our analysis was limited to IHC metabolism from a certain simulation perspective, without the possible effect of potassium concentration deficit on the MET conductance [see Eq. (1) and Table I of O'Beirne and Patuzzi (2007)]. This might be included in future work, as potassium concentration deficit and EP deficit might be correlated (Melichar , 1980). It is most likely that a mild impact on the HL can be obtained by applying EP deficit in the OHC model as well. Systematic future research is required to determine the effect of EP and OHC HB stiffness on the outcome audiogram.

The model presented in this paper clearly demonstrated the difference between OHC and IHC loss. The effect of a gradual change in OHC gain along the cochlear partition reveals an audiogram similar to that obtained by uniform OHC loss (see Figs. 5 and 6). On the other hand, the model for the IHC was solved for each point along the cochlear partition with no relation to other areas in the cochlea. Therefore, if we cause damage to the IHC only in a specific location, it will cause an increase in the HL at the frequencies close to the characteristic frequency of solely that location. Since the OHC contribution to the auditory threshold is mainly at the high mid-frequencies (2–4 kHz), any deficit along the cochlear partition will cause HL at those frequencies. This type of audiogram is frequently found in young adults who suffer from noise exposure (Hannula , 2011). In aged audiograms, the HL is often increased with frequency (Hannula , 2011). Such audiograms can be predicted by our model due to a combination of deficits in both IHCs and OHCs, where a decrease in the number of hair cells is more often near the apex than close to the base (as depicted in Fig. 12). While the variation of IHC and OHC cilia along the tonotopic axis in Fig. 12 yielded similar audiograms to an extent, future work can improve this result. For example, it is possible to examine the influence of other possible deficit variation on the audiogram or to examine the tonotopic differences of IHC along the cochlear partition (Dierich , 2020).

In conclusion, the present work presents the potential for biophysical implications and understanding from a simulation study. Along with experiments, this can assist in unveiling HL fundamentals. Moreover, the model can be used also to test an intensity difference task and evaluate how the different OHC and IHC deficits affect human performance in such a task.

It is important to note that the models used in this paper are not the only possible choice to develop a computational model for deriving human thresholds on basis of physiological studies. The models we used in this paper were chosen because they were time-domain models that we could solve numerically with reasonable complexity. Other models could have been chosen for different stages of the computational models (e.g., Dierich , 2020; Lopez-Poveda and Meddis, 2001; Zilany , 2014). A future study can compare the prediction of those models with the computational model presented in this paper.

A future study is required to overcome some of the simplifications used in the present computational model, e.g., (1) a detailed biophysical model of OHC is needed; (2) a more comprehensive model for the IHC cilia motion is needed, especially for low frequencies (Freeman, 1987; Freeman and Weiss, 1990); (3) the exact analytic relation of simultaneous damage of the *N _{cilia}* and

*K*should be determined; (4) the quasi-static assumption of utilizing the Boltzmann distribution for the MET channels opening scheme might be inaccurate for high frequencies, as it is described in equilibrium (Gianoli , 2017) [a different possible approach is to include a kinetic model for the cilia channel opening (Choe , 1998)]; and (5) a more modern synaptic model is needed, for example, a model that is found to be more appropriate to sounds with time-varying amplitudes (Bruce , 2018; Zilany , 2009).

_{HB}## ACKNOWLEDGMENTS

This research was partially supported by Israel Science Foundation Grant No. 563/12. The research was motivated by Baruch Frenkel, who inspired one of the authors to research HL.

### APPENDIX A: SENSITIVITY INDEX DERIVATION

To derive Eq. (37), a maximum likelihood decision is assumed for the human listener.

*i*,

*j*AN fiber in the time window

*T*.

_{I}*i*,

*j*AN fiber in the time window

*T*for one of the two signals

_{I}*S*1 and

*S*2.

### APPENDIX B: LIST OF MODEL PARAMETERS

See Tables II–IV for lists of model parameters.

Parameter . | Description . | Value . | Units . |
---|---|---|---|

L _{co} | Cochlear length | 3.5 | cm |

A | Cross-sectional area of the cochlea scalae | 0.5 | cm^{2} |

ρ | Perilymph density | 1 | g/cm^{3} |

β | Width of the BM | $3\xd71 0 \u2212 3$ | cm |

K _{bm} | BM stiffness per unit area | $1.28\xd71 0 4 e \u2212 1.5 x$ | g/cm^{2}/s^{2} |

R _{bm} | BM damping per unit area | $0.25 e \u2212 0.06 x$ | g/cm^{2}/s^{2} |

M _{bm} | BM mass per unit area | $1.28\xd71 0 \u2212 6 e 1.5 x$ | g/cm^{2} |

K _{tm} | TM stiffness per unit area | $4\xd71 0 5 e \u2212 3 x$ | g/cm^{2}/s^{2} |

R _{tm} | TM mass per unit area | 0 | g/cm^{2} |

M _{tm} | TM damping per unit area | $0.25 e \u2212 0.6 x$ | g/cm^{2}/s^{2} |

α _{s} | Peak-to-peak electromotility displacement | $ 10 \u2212 6$ | cm |

α _{l} | Reference electromotility voltage | $2\xd71 0 \u2212 6$ | V |

K_{OHC} | OHC membrane's stiffness | $400 e \u2212 3 x$ | g/s^{2} |

ω_{OHC} | OHC cutoff frequency | $ 2 \xb7 \pi \xb7 1000$ | rad/s |

ψ_{0} | Perilymph resting potential | $70\xd71 0 \u2212 3$ | V |

ω _{ow} | OW cutoff frequency | $ 2 \xb7 \pi \xb7 1500$ | Hz |

σ _{ow} | OW aerial density | 0.5 | g/cm^{2} |

γ _{ow} | OW resistance | $2\xd71 0 4$ | 1/s |

C _{ow} | Coupling of OW to perilymph | $6\xd71 0 \u2212 3$ | [None] |

Γ_{me} | Mechanical gain of ossicles | 21.4 | [None] |

η | Electromotility transduction coefficient | $3.14\xd71 0 \u2212 9$ | V/cm·s |

Parameter . | Description . | Value . | Units . |
---|---|---|---|

L _{co} | Cochlear length | 3.5 | cm |

A | Cross-sectional area of the cochlea scalae | 0.5 | cm^{2} |

ρ | Perilymph density | 1 | g/cm^{3} |

β | Width of the BM | $3\xd71 0 \u2212 3$ | cm |

K _{bm} | BM stiffness per unit area | $1.28\xd71 0 4 e \u2212 1.5 x$ | g/cm^{2}/s^{2} |

R _{bm} | BM damping per unit area | $0.25 e \u2212 0.06 x$ | g/cm^{2}/s^{2} |

M _{bm} | BM mass per unit area | $1.28\xd71 0 \u2212 6 e 1.5 x$ | g/cm^{2} |

K _{tm} | TM stiffness per unit area | $4\xd71 0 5 e \u2212 3 x$ | g/cm^{2}/s^{2} |

R _{tm} | TM mass per unit area | 0 | g/cm^{2} |

M _{tm} | TM damping per unit area | $0.25 e \u2212 0.6 x$ | g/cm^{2}/s^{2} |

α _{s} | Peak-to-peak electromotility displacement | $ 10 \u2212 6$ | cm |

α _{l} | Reference electromotility voltage | $2\xd71 0 \u2212 6$ | V |

K_{OHC} | OHC membrane's stiffness | $400 e \u2212 3 x$ | g/s^{2} |

ω_{OHC} | OHC cutoff frequency | $ 2 \xb7 \pi \xb7 1000$ | rad/s |

ψ_{0} | Perilymph resting potential | $70\xd71 0 \u2212 3$ | V |

ω _{ow} | OW cutoff frequency | $ 2 \xb7 \pi \xb7 1500$ | Hz |

σ _{ow} | OW aerial density | 0.5 | g/cm^{2} |

γ _{ow} | OW resistance | $2\xd71 0 4$ | 1/s |

C _{ow} | Coupling of OW to perilymph | $6\xd71 0 \u2212 3$ | [None] |

Γ_{me} | Mechanical gain of ossicles | 21.4 | [None] |

η | Electromotility transduction coefficient | $3.14\xd71 0 \u2212 9$ | V/cm·s |

Parameter . | Description . | Value . | Units . |
---|---|---|---|

g_{0} | RL gain | 1 | [None] |

μ | Endocochlear fluid viscosity | $1.2\xd71 0 \u2212 3$ | [Pa·s] |

c | Cilia flow response coefficient | 4 | [None] |

L | STS height | $6\xd71 0 \u2212 6$ | m |

H | Stereocilium height | $4\xd71 0 \u2212 6$ | m |

C _{HB} | HB dissipation coefficient | $1.95\xd71 0 \u2212 7$ | kg/s |

K _{HB} | HB total stiffness | $0.65\xd71 0 \u2212 3$ | N/m |

X _{sp} | HB tip resting position without tip links | 100 | nm |

N _{cilia} | Number of stereocilium in the HB | 50 | [None] |

l | Tip-link branch length | 13 | nm |

k _{t} | Tip-link spring stiffness | 0.57 | N/m |

k _{a} | Adaptation springs' stiffness | 1 | N/m |

$ \tau e q , MET$ | MET equilibrium time constant | 50 | μs |

EP | Endocochlear apical potential | 90 | mV |

G _{MET} | Maximal MET channel conductance | 30 | nS |

C _{m} | IHC membrane capacitance | 12.5 | pF |

Parameter . | Description . | Value . | Units . |
---|---|---|---|

g_{0} | RL gain | 1 | [None] |

μ | Endocochlear fluid viscosity | $1.2\xd71 0 \u2212 3$ | [Pa·s] |

c | Cilia flow response coefficient | 4 | [None] |

L | STS height | $6\xd71 0 \u2212 6$ | m |

H | Stereocilium height | $4\xd71 0 \u2212 6$ | m |

C _{HB} | HB dissipation coefficient | $1.95\xd71 0 \u2212 7$ | kg/s |

K _{HB} | HB total stiffness | $0.65\xd71 0 \u2212 3$ | N/m |

X _{sp} | HB tip resting position without tip links | 100 | nm |

N _{cilia} | Number of stereocilium in the HB | 50 | [None] |

l | Tip-link branch length | 13 | nm |

k _{t} | Tip-link spring stiffness | 0.57 | N/m |

k _{a} | Adaptation springs' stiffness | 1 | N/m |

$ \tau e q , MET$ | MET equilibrium time constant | 50 | μs |

EP | Endocochlear apical potential | 90 | mV |

G _{MET} | Maximal MET channel conductance | 30 | nS |

C _{m} | IHC membrane capacitance | 12.5 | pF |

Parameter . | Description . | Value . | Units . |
---|---|---|---|

$ G k , f / s$ | Maximal K channel conductance | 12.5 | pF |

$ V 0.5 k$ | K channel half-activation potential | −31 | mV |

S _{k} | K channel voltage sensitivity | 10.5 | mV |

$ V f , 0.5 k$ | Fast K channel reversal potential | 71 | mV |

$ V s , 0.5 k$ | Slow K channel reversal potential | 78 | mV |

$ \tau f k$ | Fast K channel time constant | 0.3 | ms |

$ \tau s k$ | Slow K channel time constant | 8 | ms |

Z | Current-spike rate constant | $2.5\xd71 0 11$ | spikes/A·s |

E _{Ca} | Calcium channel Nernst potential | 45 | mV |

G _{Ca} | Calcium channel maximal conductance | 4.1 | nS |

I _{th} | Exocytosis calcium threshold current | 0 | spikes/A·s |

S _{Ca} | Calcium channel sensitivity | 1.5 | mV |

τ _{Ca} | Calcium channel time constant | 0.25 | ms |

$ k max , H$ | HSR peak exocytosis rate | 3000 | spikes/s |

$ k max , M$ | MSR peak exocytosis rate | 1000 | spikes/s |

$ k max , L$ | LSR peak exocytosis rate | 800 | spikes/s |

$ t abs / rel$ | AN refractory period | 0.6 | ms |

$ \lambda spont ( 1 )$ | HSR | 60 | spikes/s |

$ \lambda spont ( 2 )$ | MSR | 10 | spikes/s |

$ \lambda spont ( 3 )$ | LSR | 1 | spikes/s |

w _{H} | Relative weight of high rate fibers | 0.6 | [None] |

w _{M} | Relative weight of medium rate fibers | 0.23 | [None] |

w _{L} | Relative weight of low rate fibers | 0.17 | [None] |

T _{I} | Integration time of the ear | 0.2 | s |

Parameter . | Description . | Value . | Units . |
---|---|---|---|

$ G k , f / s$ | Maximal K channel conductance | 12.5 | pF |

$ V 0.5 k$ | K channel half-activation potential | −31 | mV |

S _{k} | K channel voltage sensitivity | 10.5 | mV |

$ V f , 0.5 k$ | Fast K channel reversal potential | 71 | mV |

$ V s , 0.5 k$ | Slow K channel reversal potential | 78 | mV |

$ \tau f k$ | Fast K channel time constant | 0.3 | ms |

$ \tau s k$ | Slow K channel time constant | 8 | ms |

Z | Current-spike rate constant | $2.5\xd71 0 11$ | spikes/A·s |

E _{Ca} | Calcium channel Nernst potential | 45 | mV |

G _{Ca} | Calcium channel maximal conductance | 4.1 | nS |

I _{th} | Exocytosis calcium threshold current | 0 | spikes/A·s |

S _{Ca} | Calcium channel sensitivity | 1.5 | mV |

τ _{Ca} | Calcium channel time constant | 0.25 | ms |

$ k max , H$ | HSR peak exocytosis rate | 3000 | spikes/s |

$ k max , M$ | MSR peak exocytosis rate | 1000 | spikes/s |

$ k max , L$ | LSR peak exocytosis rate | 800 | spikes/s |

$ t abs / rel$ | AN refractory period | 0.6 | ms |

$ \lambda spont ( 1 )$ | HSR | 60 | spikes/s |

$ \lambda spont ( 2 )$ | MSR | 10 | spikes/s |

$ \lambda spont ( 3 )$ | LSR | 1 | spikes/s |

w _{H} | Relative weight of high rate fibers | 0.6 | [None] |

w _{M} | Relative weight of medium rate fibers | 0.23 | [None] |

w _{L} | Relative weight of low rate fibers | 0.17 | [None] |

T _{I} | Integration time of the ear | 0.2 | s |

## REFERENCES

^{+}channels on auditory nerve responses

^{2+}to mechanoelectrical-transduction channels

^{+}current entities

*Update on Hearing Loss*

*Signal Detection Theory and Psychophysics*

_{V}1. 3 calcium channels in gerbil inner hair cells

^{2+}dependence of exocytosis during development of mouse inner hair cells

*Basic and Applied Aspects of Noise-Induced Hearing Loss*

^{+}concentrations in the cochlear fluids after acoustic trauma

*Cochlear Hearing Loss: Physiological, Psychological and Technical Issues*

*The Aging Auditory System*