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.

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.

The different models' implementations are described in Sec. III. The effect of OHC and IHC deficits is discussed in detail in Sec. IV, particularly their impact on THIQ.

The pressure in the external and middle ears is proportional to the incoming sound (Talmadge , 1998), which yields
P bones ( t ) = Γ m e P i n ( t ) ,
(1)
where Γme 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.
The OW motion is modeled as a second-order harmonic oscillator, which yields
d 2 ξ ow ( t ) d 2 t + γ OW · d ξ ow ( t ) d t + ω OW 2 ξ OW ( t ) = 1 σ OW [ P ( o , t ) + P bones ( t ) ] ,
(2)
where ξ ow ( t ) is the OW displacement, γ OW is the OW resistance, ωOW is the OW resonance frequency, and σ OW is the OW areal density.
The cochlear partition, with mechanical properties that are describable in terms of a point-wise mass density, stiffness, and damping, is regarded as a flexible boundary between scala tympani and scala vestibuli. Thus, at every point along the cochlear duct (coordinate 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
2 P ( x , t ) x 2 = 2 ρ β A 2 ξ b m ( x , t ) t 2 ,
(3)
where ξbm is the local BM displacement; 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.
Although abbreviated as the BM displacement, ξbm, the pressure profile, 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 (Pbm) is a result of both the fluid pressure difference on the scala media and the pressure caused by the OHCs (POHC). The relation between the pressures of BM, TM, and OHC is given by (Barzelay and Furst, 2011)
P b m ( x , t ) = P ( x , t ) + P OHC ( x , t ) 0 = P OHC ( x , t ) + P t m ( x , t ) .
(4)
The mechanical properties of both BM and TM are simulated as second-order oscillators, which yields
P b m ( x , t ) = K b m ξ b m ( x , t ) + R b m ξ b m ( x , t ) t + M b m 2 ξ b m ( x , t ) t 2 ,
(5)
P t m ( x , t ) = K t m ξ t m ( x , t ) + R t m ξ t m ( x , t ) t + M t m 2 ξ t m ( x , t ) t 2 ,
(6)
where the TM displacement is defined as ξtm; Kbm, Ktm, Rbm, Rtm, Mbm, Mtm are the effective stiffness, damping, and mass per unit area of BM and TM, respectively.
Since the OHCs lie between the two membranes, their displacement is
ξ OHC = ξ t m ξ b m .
(7)
The electrical potential developed on the OHC membrane (He and Dallos, 2000; Kakehata and Santos-Sacchi, 1995; Neely and Allen, 2009) causes the OHC electromotile length changes, Δ l OHC, usually described as a sigmoid,
Δ l OHC = α s e ( 2 ψ ) 1 e ( 2 ψ ) + 1 = α s  tanh ( α l ψ ) ,
(8)
where αl and αs are constants. The OHC electrical potential ψ is obtained by
ψ t + ω OHC ψ = η ξ OHC ,
(9)
where ωOHC is the OHC cutoff frequency and is a constant along the entire tonotopic axis, and η is the electromotility transduction coefficient.

Each OHC pressure, POHC, is obtained from the spring properties of the OHC. γ 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 γ OHC ( x ) = 0. On the other hand, γ OHC ( x ) = 1 represents a non-realistic cochlea whose BM motion is unstable. γ OHC ( x ) = 0.5 represents the ideal cochlea with fully active OHCs.

The OHC pressure is then given by
P OHC ( x , t ) = γ OHC ( x ) · { K OHC ( x ) · ξ OHC ( x , t ) P t m ( Δ l OHC , t ) } ,
(10)
where K OHC ( x ) is the OHC stiffness along the cochlear partition.
To completely solve Eqs. (3)–(10), boundary and initial conditions are required. The following pressure boundary conditions are
P ( x , t ) x | x = 0 = 2 ρ C o w 2 ξ o w ( t ) t 2 P ( L c o , t ) = 0 ,
(11)
where Lco is the cochlear length, and Cow is the coupling factor of the OW to the perilymph.
The initial conditions are
ξ b m ( x , 0 ) = ξ b m ( x , 0 ) t = ξ t m ( x , 0 ) = ξ t m ( x , 0 ) t = ψ ( x , 0 ) = ξ o w ( 0 ) = ξ o w ( 0 ) t = 0.
(12)
The next step in the model is calculating the motion of the fluid inside the STS. A schematic side view of the STS is shown in Fig. 1. Its upper border is the TM, and its lower border is the reticular lamina (RL). The RL motion and the organ of Corti (OC) dynamics are complex and are three-dimensional (3-D) in nature and affected by the OHC gain (Ni , 2016). Since in this study we calculated ξ ̇ bm ( x , t ) in a 1-D simplified fashion, we continue with this approach to derive the RL velocity. In Neely and Kim (1986), the OC is modeled as a rigid body; thus, the velocity of RL ( ξ ̇ rl) is relative to the BM velocity as follows:
ξ ̇ rl ( x , t ) g 0 ξ ̇ bm ( x , t ) ,
(13)
where g0 is a constant and represents the mechanical lever gain.
FIG. 1.

STS side view, adapted from the work of Prodanovic et al., Biophys. J. 108(3), 479–488 (2015). Copyright 2015 Cell Press (Prodanovic , 2015), with an enlarged diagram of the IHC HB schematics. The coordinate system is shown on the right side.

FIG. 1.

STS side view, adapted from the work of Prodanovic et al., Biophys. J. 108(3), 479–488 (2015). Copyright 2015 Cell Press (Prodanovic , 2015), with an enlarged diagram of the IHC HB schematics. The coordinate system is shown on the right side.

Close modal
Inside, the STS shear flow is initiated due to the difference in velocities between the TM and RL. The IHC HB is located inside the STS (Fig. 1). The fluid shear motion causes the HB stereocilia to pivot as rigid rods around an insertion point, with pivotal stiffness Ksp. The shear-flow velocity ( ξ ̇ shear) far from the IHC is given by
ξ ̇ shear ( x , t ) = g 0 ξ ̇ bm ( x , t ) ξ ̇ t m ( x , t ) .
(14)
The HB of each IHC reacts to the STS motion and develops hydrodynamic torque (τH). Billone and Raynor (1973) found that τH 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 ( ξ ̇ fs) and the IHC HB local velocity ( ξ ̇ hb), which yields
τ H ( x , y , t ) = 8 π μ c H 3 3 N cilia · ( ξ ̇ fs ( x , y , t ) ξ ̇ hb ( x , y , t ) ) ,
(15)
where c is a geometrical coefficient of the stereocilia structure response to the flow, μ is the endocochlear fluid viscosity, Ncilia is the number of stereocilia in a single IHC, and H is the height of the tallest stereocilium. The free stream velocity is assumed to be a steady state Couette flow field; thus,
ξ ̇ fs ( x , y , t ) y L ξ ̇ shear ( x , t ) ,
(16)
where L is the STS height, as shown in Fig.1.
The HB tip velocity is calculated by
ξ ̇ hb ( x , y , t ) = y H ξ ̇ st ( x , t ) ,
(17)
where ξ st ( x , t ) is the stereocilium tip displacement as indicated in Fig. 1. The stereocilia motion is modeled as a pivoting rigid body that can be defined as a first-order harmonic oscillator (Prodanovic , 2015; Shamma , 1986), which yields
C H B ξ ̇ st + K H B ( ξ st X s p ) = F H + F feedback ,
(18)
where F H = τ H / H is the effective hydrodynamic force, C H B is the motion dissipation coefficient, KHB is the HB stiffness, Xsp is the resting position of the HB tip when the tip links are cut (there is no feedback), and Ffeedback is the feedback force calculated later by Eq. (25). FH is obtained by substituting Eqs. (16) and (17) into Eq. (15), which yields
F H = 8 π μ c N cilia H 2 3 L ξ ̇ shear ( x , t ) .
(19)
Equation (18) without dissipation was derived by Gianoli (2017, 2019). The dissipation term was added based on the study of Prodanovic (2015).

The IHC stereocilium tip displacement causes ion channels to open and generate a current flow I MET ( ξ 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 ( ξ st )]. A schematic description of the elastic forces in an IHC cilia is depicted in Fig. 2.

FIG. 2.

Schematic description of IHC elastic forces.

FIG. 2.

Schematic description of IHC elastic forces.

Close modal
The ion channels are attached to the highest stereocilium at the upper end and to the adjacent shorter stereocilium at the lower end via a set of three springs. The distance between the two adjacent stereocilia is 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 lt. The length of each branch is l. The tip-link extension, xt, along with d (the branches' vertical projection) are determined by a geometrical projection of the IHC bundle tip displacement ξ st (Fig. 2),
x t = D H · ( ξ st X 0 ) d ,
(20)
where X0 is resting position of the HB.
At any time, the two ion channels can be either both open (OO) or both closed (CC) or one channel open and the other closed (OC). Each state causes a change in the elastic forces that eventually changes the tip-link spring's length by dOO, dCC, and dOC 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 (aOO, aCC, and aOC), 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,
P open , e q ( ξ st ) = P O O ( ξ st ) + P O C ( ξ st ) 2 ,
(21)
where
P O O ( ξ st ) = 1 W ( ξ st ) · exp ( E tot , O O ( ξ st ) k B T ) , P O C ( ξ st ) = 1 W ( ξ st ) · exp E tot , O C ( ξ st ) k B T , P C C ( ξ st ) = 1 W ( ξ st ) · exp E tot , C C ( ξ st ) k B T , W ( ξ st ) = P O O ( ξ st ) + 2 P O C ( ξ st ) + P C C ( ξ st ) ,
(22)
where the temperature T is the standard human body temperature; E tot , O O ( ξ st ) , E tot , O C ( ξ st ), and E tot , C C ( ξ st ) are the total energies obtained at the OO, OC, and CC states, respectively. For the calculation of those energies, along with X0 and lt, see Gianoli (2017).
The probability of the ion channels opening determines both the MET current (IMET) and the feedback force (Ffeedback). 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),
P Open ξ st + τ MET d P Open ξ st d t = P Open , e q ξ st .
(23)
Then the MET current is obtained by the Hodgkin–Huxley model (1952),
I MET ( ξ st ) = γ cilia G MET P Open ( ξ st ) ( V m E P ) ,
(24)
where EP is the endocochlear potential; Vm is the membrane potential that will be derived in Sec. II F; GMET is the total conductance when all channels are open; and, finally, γcilia indicates the relative number of cilia in a single IHC, where γ cilia = 1 represents a healthy IHC, while γ cilia < 1 represents reduction in the number of cilia in an IHC.
The feedback force is required for solving Eq. (18). This force is the elastic force of IHC HB and is given by
F feedback = N cilia k t ( E ( x t ) l t ) ,
(25)
where kt 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
E [ d ] = P O O d O O ( ξ st ) + P O C d O C ( ξ st ) + P C C d C C ( ξ st ) .
(26)
The IHC's membrane voltage is derived by modeling the IHC membrane as an electrical circuit and applying Kirchhoff's law (Altoè , 2017, 2018), which yields
C m d V m d t + I MET + I s k + I f k = 0 ,
(27)
where Cm is the membrane capacitance, IMET 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
I f / s k ( x , t ) = g f / s k ( x , t ) ( V m ( x , t ) V f / s k ) ,
(28)
where the symbol 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
g f / s k ( x , t ) + τ f / s k d g f / s k ( x , t ) d t = G f / s 1 + exp V m x , t V 0.5 k S k 1 ,
(29)
where τ f / s k is a relaxation time constant for the fast- and slow-opening channels, V 0.5 k is the potassium potential at which the conductance is half-activated, and Sk is the slope factor that defines the voltage sensitivity of current activation.

Following the solution of Eq. (29) and substituting its result in Eqs. (28) and (27) yields the non-linear equation for the IHC potential, V m ( x , t ).

The change in Vm causes the opening of calcium channels in IHC membrane that eventually triggers the exocytosis process.

Let us define the instantaneous exocytosis rate as 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:
k j ( x , t ) = Z max [ I C a , j ( x , t ) I t h , 0 ] ,
(30)
where Ith is the resting current, and Z is a phenomenological rate constant.
The calcium current is obtained by the Hodgkin–Huxley ionic-current equation, i.e.,
I C a , j ( x , t ) = g C a , j ( x , t ) ( V m ( x , t ) E c a ) ,
(31)
where ECa 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
g C a , j ( x , t ) = G C a n C a , j ( x , t ) 2 ,
(32)
where G C a indicates the maximum calcium conductance of the channels.
The kinetics of the calcium channels are effectively described by a second-order activation process (Johnson and Marcotti, 2008), which yields
n C a , j ( x , t ) + τ C a d n C a , j ( x , t ) d t = 1 + exp V m x , t V C a , 0.5 , j S C a 1 / 2 ,
(33)
where SCa is the slope factor that defines the voltage sensitivity of current activation, and τCa is the calcium relaxation time constant. V C a , 0.5 , j is the calcium potential where conductance is half-activated. It follows that
V 0.5 , Ca , j = S Ca log ( k max , j λ spont ( j ) λ spont ( j ) ) + V rest ,
(34)
where V rest is the IHC membrane potential obtained when the cochlea is not stimulated, and k max , j is the maximum SR exocytosis rate of type j.
The instantaneous firing rate λ ( x , t ) depends on the probability of generating an action potential at the synaptic region, formally defined as
λ ( x , t ) = lim Δ t 0 1 Δ t P r ( N ( x , t , t + Δ t ) = 1 ) ,
(35)
where N ( x , t , t + Δ t ) is the number of spikes in the time interval [ t , t + Δ t ]. Verhulst (2018) modeled the probability of generating a spike as
P r ( N ( x , t , t + Δ t ) = 1 ) = α r ( x , t ) q ( x , t ) k ( x , t ) Δ t ,
(36)
where 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). α r ( x , t ) is a refractive correction function (Verhulst , 2018) that follows the equation
α r ( x , t ) = 1 t τ abs t P r ( N ( x , τ , τ + Δ t ) = 1 ) d τ t τ abs P r ( N ( x , τ , τ + Δ t ) = 1 ) · e ( τ ( t τ abs ) / τ rel ) d τ .
(37)

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).

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 , , 256. A typical input signal is S ( t ) = α sin ( 2 π f t ), where α is the amplitude of the signal and f is its frequency. The basilar and TM velocities, ξ ̇ bm ( x i , t ) and ξ ̇ 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 Δ t = 1 / F S, where F S = 20 kHz.

The next steps in the model are deriving the following variables for every xi: shear motion, ξ ̇ shear ( x i , t ); stereocilium tip displacement, ξ 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 Δ 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 (λspont) 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 λ 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.

The purpose of this study, as stated in the Introduction, is to derive THIQ as a function of AN firing rate. We assume that the listener acts as an optimal decision maker (Heinz , 2001; Siebert, 1970) and that the AN firing rate behaves like a non-homogeneous Poisson process (NHHP) (Alaoglu and Smith, 1938; Rodieck , 1962). There are two existing approaches to derive THIQ: (1) based on the instantaneous firing rate, λ j ( x , t ), called all-information (AI), and (2) based on the average rate, λ j ¯ ( x ), which is obtained by
λ ¯ j ( x ) = 1 T I 0 T I λ j ( x , t ) d t ,
(38)
where TI is the human listener integration time constant (Zwicker and Fastl, 1999).
Furst (2015) showed that for detection of tones in noise, rate information can fully describe experimental data, while AI overestimates the experimental results. We, thus, chose the rate information to derive THIQ. Since we assume that the brain acts as an optimal processor, the decision is based on the log-likelihood test. The model is motivated by the experimental setup of two signals presented to the human listener,
S 1 = α c sin ( 2 π f t ) , S 2 = ( α c + Δ α ) sin ( 2 π f t ) .
(39)
The subject's task is to determine the minimum Δ α that distinguishes between S1 and S2. The subject decision process is manifested by the parameter Y, Y ( λ ¯ 1 ( x 1 ) , , λ ¯ 3 ( x 256 ) | S ) = Y ( ( i , j ) I , λ ¯ i , j | S ), where I = [ j = 1 , 2 , 3 ; i = 1 , , 256 ], and λ ¯ i , j = λ ¯ j ( x i ). The parameter Y represents the total statistical information for the two signals, S1 and S2. For the sake of convenience, the decision variable for the two hypotheses is abbreviated as
Y 1 = Y i , j I , λ ¯ i , j S 1 , Y 2 = Y i , j I , λ ¯ i , j S 2 .
(40)
The usual figure of merit describing a detection experiment is the detectability index ( d ), which is defined as the difference between the mean responses divided by the square root of the average variance (Green and Swets, 1966), i.e.,
d = E [ Y 2 ] E [ Y 1 ] 1 2 ( Var [ Y 1 ] + Var [ Y 2 ] ) .
(41)
The maximum likelihood decision rule and the time-averaged NHPP statistics yield the following expression for the sensitivity index (see  Appendix A for details):
d = T I j = 1 3 i = 1 256 ω j · Λ i , j S 2 Λ i , j S 1 j = 1 3 i = 1 256 ω j Λ i , j | S 1 ,
(42)
where
Λ i , j | S = λ ¯ i , j | S · log λ ¯ i , j | S 2 λ ¯ i , j | S 1 ,
(43)
where the notation λ ¯ i , j | S represents the average rate for one of the two signals S1 and S2, and ωj is the relative number of AN fibers with HSR, MSR, or LSR for 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), ω 1 = 0.6 K 0, ω 2 = 0.3 K 0, and ω 3 = 0.1 K 0, where K 0 = 30 000 / 256.
While deriving THIQ, α C = 0 in both S1 and S2 [Eq. (39)], which yields for every i, λ ¯ ( i , j ) = λ spont ( j ) in response to S1. d is derived separately for every frequency f and Δ α (Heinz , 2001). A sensory threshold is defined as the stimulus level that yields d of 1.0, which corresponds to a correct response probability of 0.75 (Harvey, 1986). Thus, the threshold is obtained by
THIQ ( f ) = argmin Δ α ( | d ( Δ α , f ) 1 | ) .
(44)

The threshold in quiet is obtained by solving Eq. (44). For every frequency, d [Eq. (42)] is obtained for various values of Δ α, and the value of Δ α that yields d = 1 is defined as the threshold. An example of d as a function of Δ α is shown in Fig. 3 for two stimulus frequencies, f = 250 Hz and f = 4 kHz. The vertical dashed lines indicate the values of Δ α that were obtained for d = 1. Note that Δ α that yields d = 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).

FIG. 3.

The sensitivity index d as a function of Δ α is displayed for (A) f = 250 Hz and (B) f = 4000 Hz. The vertical dashed lines indicate the values of Δ α that were obtained for d = 1.

FIG. 3.

The sensitivity index d as a function of Δ α is displayed for (A) f = 250 Hz and (B) f = 4000 Hz. The vertical dashed lines indicate the values of Δ α that were obtained for d = 1.

Close modal

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.

FIG. 4.

Prediction of THIQ (dB SPL) as a function of frequency, compared with the threshold in quiet result from ISO 226 (ISO, 2003).

FIG. 4.

Prediction of THIQ (dB SPL) as a function of frequency, compared with the threshold in quiet result from ISO 226 (ISO, 2003).

Close modal

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 (γOHC) 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.

TABLE I.

List of simulated deficits with indication of their reference normal values.

Parameter name Symbol Normal value ( P Normal ) Units
OHC gain  γOHC  0.5  # 
Number of cilia  Ncilia  50  # 
Hair bundle stiffness  KHB  0.65 × 1 0 3  N/m 
Endocochlear potential  EP  90  mV 
Exocytosis rate constant  Z  2.5 × 1 0 11  #/A·s 
Parameter name Symbol Normal value ( P Normal ) Units
OHC gain  γOHC  0.5  # 
Number of cilia  Ncilia  50  # 
Hair bundle stiffness  KHB  0.65 × 1 0 3  N/m 
Endocochlear potential  EP  90  mV 
Exocytosis rate constant  Z  2.5 × 1 0 11  #/A·s 
In this work, HL( f) is defined as the difference between THIQ( f) of a damaged ear and a normal ear, i.e.,
H L ( f ) = THI Q damage ( f ) THI Q normal ( f ) .
(45)
Each biophysical deficit in a parameter P is defined as
P d = % P Normal ,
(46)
where PNormal is the value of the parameter 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., Pd, 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 Hz, with a maximum at the middle part where 2000 < C F < 4000 Hz. 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.

FIG. 5.

HL as a function of frequency for different percentages of γOHC.

FIG. 5.

HL as a function of frequency for different percentages of γOHC.

Close modal
On the other hand, let us assume a reduction in OHC efficiency, γOHC [see Eq. (10)], along the cochlear partition from the base to the apex, according to the equation
γ OHC = 0.5 e α OHC ( x 3.5 ) ,
(47)
where αOHC is a constant. Figure 6 represents HL predictions for different values of αOHC. 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 % γ OHC = 0 %, which represents γ OHC ( x ) = 0, is similar to the audiogram in Fig. 6 that was obtained with α OHC = 0.9, which represents γ OHC = 0.02 at the apex and γ OHC = 0.49 at the base.
FIG. 6.

HL as a function of frequency for variable γ OHC = 0.5 · e α OHC ( x 3.5 ).

FIG. 6.

HL as a function of frequency for variable γ OHC = 0.5 · e α OHC ( x 3.5 ).

Close modal

2. IHC HB stiffness deficit

The IHC KHB 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 KHB is shown in Fig. 7. Decrease of KHB to 80% causes HL of about 35 dB at low frequencies and 20 dB at high frequencies. Decreasing KHB 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 KHB via the feedback force [Eq. (25)] (Gianoli , 2017).

FIG. 7.

HL as a function of frequency for different percentages of KHB in a cochlea with fully active OHCs [ γ OHC = 100 % (upper panel] and for a cochlea without active OHCs [ γ OHC = 0 % (lower panel)].

FIG. 7.

HL as a function of frequency for different percentages of KHB in a cochlea with fully active OHCs [ γ OHC = 100 % (upper panel] and for a cochlea without active OHCs [ γ OHC = 0 % (lower panel)].

Close modal

The effect of deficits in KHB in a cochlea without active OHCs ( γ 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 KHB (upper panel of Fig. 7).

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 KHB deficit (Fig. 7), with a decrease in HL with the increase in frequency. Combination of deficits of EP and inactive OHCs ( γ OHC = 0 %) yields superposition of the two types of deficits. Both parameters, KHB and EP, eventually affect the IHC membrane potential, Vm, which is derived by solving Eq. (27) and yields a low-pass type of frequency selectivity.

FIG. 8.

HL as a function of frequency for different percentages of EP in a cochlea with fully active OHCs [ γ OHC = 100 % (upper panel)] and for a cochlea without active OHCs [ γ OHC = 0 % (lower panel)].

FIG. 8.

HL as a function of frequency for different percentages of EP in a cochlea with fully active OHCs [ γ OHC = 100 % (upper panel)] and for a cochlea without active OHCs [ γ OHC = 0 % (lower panel)].

Close modal

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 (Ncilia) 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 Ncilia 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 Hz) 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 Popen [Eq. (21)] as a function of frequency and by the fact that Vm at high frequencies becomes a direct current (dc) signal (Altoè , 2018). This result is demonstrated in Fig. 10, where the Vm, 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 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.

FIG. 9.

HL as a function of frequency for different percentages of Ncilia in a cochlea with fully active OHCs [ γ OHC = 100 % (upper panel)] and for a cochlea without active OHCs [ γ OHC = 0 % (lower panel)].

FIG. 9.

HL as a function of frequency for different percentages of Ncilia in a cochlea with fully active OHCs [ γ OHC = 100 % (upper panel)] and for a cochlea without active OHCs [ γ OHC = 0 % (lower panel)].

Close modal
FIG. 10.

The Vm signal at the steady state for the healthy (continuous line) and IHC damaged ear ( % N cilia = 20, dashed line). Each panel represents a different input frequency and level.

FIG. 10.

The Vm signal at the steady state for the healthy (continuous line) and IHC damaged ear ( % N cilia = 20, dashed line). Each panel represents a different input frequency and level.

Close modal

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 γ OHC = 100 %, and for all Z values when γ OHC = 0 %.

FIG. 11.

HL as a function of frequency for different percentages of Z in a cochlea with fully active OHCs [ γ OHC = 100 % (upper panel)] and for a cochlea without active OHCs [ γ OHC = 0 % (lower panel)].

FIG. 11.

HL as a function of frequency for different percentages of Z in a cochlea with fully active OHCs [ γ OHC = 100 % (upper panel)] and for a cochlea without active OHCs [ γ OHC = 0 % (lower panel)].

Close modal

6. Tonotopic variation of Ncilia and γOHC

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 NCilia for the IHC and γOHC for the OHC is not a trivial task. For the sake of simplicity, we assumed a linear dependence between the suggested indices and NCilia and γOHC. 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.

FIG. 12.

Comparison between Wu and Liberman (2022) data and model prediction. (A) Percentage of IHC cells (○), percentage of IHC cilia (solid line), percentage of normalized IHC cilia (dashed line). (B) Percentage of OHC cells (○), percentage of OHC cilia (solid line), percentage of normalized OHC cilia (dashed line). (C) Measured audiogram (○) and simulated audiogram (dashed line).

FIG. 12.

Comparison between Wu and Liberman (2022) data and model prediction. (A) Percentage of IHC cells (○), percentage of IHC cilia (solid line), percentage of normalized IHC cilia (dashed line). (B) Percentage of OHC cells (○), percentage of OHC cilia (solid line), percentage of normalized OHC cilia (dashed line). (C) Measured audiogram (○) and simulated audiogram (dashed line).

Close modal

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 KHB 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.

FIG. 13.

HL as a function of different IHC deficits for 250 and 4000 Hz.

FIG. 13.

HL as a function of different IHC deficits for 250 and 4000 Hz.

Close modal

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 ( % γ 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 Ncilia and KHB 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).

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.

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

Using the parametric notation of the optimal decision model, the NHHP distribution with average rate (38) is
P Rate ( ( i , j ) I | S ) = j = 1 3 i = 1 256 ( T I λ ¯ i , j ) k i , j k i , j ! · exp ( λ ¯ i , j T I ) ,
(A1)
where k i , j is the number of spikes generated by the i, j AN fiber in the time window TI.
The maximum likelihood decision rule equation is
log ( P Rate ( ( i , j ) I | S 2 ) P Rate ( ( i , j ) I | S 1 ) ) S 2 S 1 0.
(A2)
Inserting (A1) into (A2) yields
j = 1 3 i = 1 256 k i , j log λ ¯ i , j | S 2 λ ¯ i , j | S 1 S 2 S 1 T I j = 1 3 i = 1 256 ( λ ¯ i , j | S 2 λ ¯ i , j | S 1 ) .
(A3)
The left-hand side of (A3) is defined as the decision variable Y ( ( i , j ) I , λ ¯ i , j ) (Heinz , 2001). Then Eq. (40) can be rewritten as
Y 1 = j = 1 3 i = 1 256 k i , j | S 1 log λ ¯ i , j | S 2 λ ¯ i , j | S 1 , Y 2 = j = 1 3 i = 1 256 k i , j | S 2 log λ ¯ i , j | S 2 λ ¯ i , j | S 1 ,
(A4)
where k i , j | S is the number of spikes generated by the i, j AN fiber in the time window TI for one of the two signals S1 and S2.
Using the NHHP distribution known mean and variance equivalence and their respective formulas, one gets
E k i , j | S  log λ ¯ i , j | S 2 λ ¯ i , j | S 1 = Var k i , j | S  log λ ¯ i , j | S 2 λ ¯ i , j | S 1 = T I Λ i , j | S ,
(A5)
where Λ i , j | S is defined as in Eq. (43). Near the auditory threshold, one gets that Λ i , j | S 1 Λ i , j | S 2. Using this first-order approximation for the denominator in Eq. (41) mean and variance relations in Eq. (A5), one gets Eq. (42).

See Tables II–IV for lists of model parameters.

TABLE II.

BM parameters.

Parameter Description Value Units
Lco  Cochlear length  3.5  cm 
A  Cross-sectional area of the cochlea scalae  0.5  cm2 
ρ  Perilymph density  g/cm3 
β  Width of the BM  3 × 1 0 3  cm 
Kbm  BM stiffness per unit area  1.28 × 1 0 4 e 1.5 x  g/cm2/s2 
Rbm  BM damping per unit area  0.25 e 0.06 x  g/cm2/s2 
Mbm  BM mass per unit area  1.28 × 1 0 6 e 1.5 x  g/cm2 
Ktm  TM stiffness per unit area  4 × 1 0 5 e 3 x  g/cm2/s2 
Rtm  TM mass per unit area  g/cm2 
Mtm  TM damping per unit area  0.25 e 0.6 x  g/cm2/s2 
αs  Peak-to-peak electromotility displacement  10 6  cm 
αl  Reference electromotility voltage  2 × 1 0 6 
KOHC  OHC membrane's stiffness  400 e 3 x  g/s2 
ωOHC  OHC cutoff frequency  2 · π · 1000  rad/s 
ψ0  Perilymph resting potential  70 × 1 0 3 
ωow  OW cutoff frequency  2 · π · 1500  Hz 
σow  OW aerial density  0.5  g/cm2 
γow  OW resistance  2 × 1 0 4  1/s 
Cow  Coupling of OW to perilymph  6 × 1 0 3  [None] 
Γme  Mechanical gain of ossicles  21.4  [None] 
η  Electromotility transduction coefficient  3.14 × 1 0 9  V/cm·s 
Parameter Description Value Units
Lco  Cochlear length  3.5  cm 
A  Cross-sectional area of the cochlea scalae  0.5  cm2 
ρ  Perilymph density  g/cm3 
β  Width of the BM  3 × 1 0 3  cm 
Kbm  BM stiffness per unit area  1.28 × 1 0 4 e 1.5 x  g/cm2/s2 
Rbm  BM damping per unit area  0.25 e 0.06 x  g/cm2/s2 
Mbm  BM mass per unit area  1.28 × 1 0 6 e 1.5 x  g/cm2 
Ktm  TM stiffness per unit area  4 × 1 0 5 e 3 x  g/cm2/s2 
Rtm  TM mass per unit area  g/cm2 
Mtm  TM damping per unit area  0.25 e 0.6 x  g/cm2/s2 
αs  Peak-to-peak electromotility displacement  10 6  cm 
αl  Reference electromotility voltage  2 × 1 0 6 
KOHC  OHC membrane's stiffness  400 e 3 x  g/s2 
ωOHC  OHC cutoff frequency  2 · π · 1000  rad/s 
ψ0  Perilymph resting potential  70 × 1 0 3 
ωow  OW cutoff frequency  2 · π · 1500  Hz 
σow  OW aerial density  0.5  g/cm2 
γow  OW resistance  2 × 1 0 4  1/s 
Cow  Coupling of OW to perilymph  6 × 1 0 3  [None] 
Γme  Mechanical gain of ossicles  21.4  [None] 
η  Electromotility transduction coefficient  3.14 × 1 0 9  V/cm·s 
TABLE III.

IHC parameters.

Parameter Description Value Units
g0  RL gain  [None] 
μ  Endocochlear fluid viscosity  1.2 × 1 0 3  [Pa·s] 
c  Cilia flow response coefficient  [None] 
L  STS height  6 × 1 0 6 
H  Stereocilium height  4 × 1 0 6 
CHB  HB dissipation coefficient  1.95 × 1 0 7  kg/s 
KHB  HB total stiffness  0.65 × 1 0 3  N/m 
Xsp  HB tip resting position without tip links  100  nm 
Ncilia  Number of stereocilium in the HB  50  [None] 
l  Tip-link branch length  13  nm 
kt  Tip-link spring stiffness  0.57  N/m 
ka  Adaptation springs' stiffness  N/m 
τ e q , MET  MET equilibrium time constant  50  μ
EP  Endocochlear apical potential  90  mV 
GMET  Maximal MET channel conductance  30  nS 
Cm  IHC membrane capacitance  12.5  pF 
Parameter Description Value Units
g0  RL gain  [None] 
μ  Endocochlear fluid viscosity  1.2 × 1 0 3  [Pa·s] 
c  Cilia flow response coefficient  [None] 
L  STS height  6 × 1 0 6 
H  Stereocilium height  4 × 1 0 6 
CHB  HB dissipation coefficient  1.95 × 1 0 7  kg/s 
KHB  HB total stiffness  0.65 × 1 0 3  N/m 
Xsp  HB tip resting position without tip links  100  nm 
Ncilia  Number of stereocilium in the HB  50  [None] 
l  Tip-link branch length  13  nm 
kt  Tip-link spring stiffness  0.57  N/m 
ka  Adaptation springs' stiffness  N/m 
τ e q , MET  MET equilibrium time constant  50  μ
EP  Endocochlear apical potential  90  mV 
GMET  Maximal MET channel conductance  30  nS 
Cm  IHC membrane capacitance  12.5  pF 
TABLE IV.

Synapse parameters.

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 
Sk  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 
τ f k  Fast K channel time constant  0.3  ms 
τ s k  Slow K channel time constant  ms 
Z  Current-spike rate constant  2.5 × 1 0 11  spikes/A·s 
ECa  Calcium channel Nernst potential  45  mV 
GCa  Calcium channel maximal conductance  4.1  nS 
Ith  Exocytosis calcium threshold current  spikes/A·s 
SCa  Calcium channel sensitivity  1.5  mV 
τCa  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 
λ spont ( 1 )  HSR  60  spikes/s 
λ spont ( 2 )  MSR  10  spikes/s 
λ spont ( 3 )  LSR  spikes/s 
wH  Relative weight of high rate fibers  0.6  [None] 
wM  Relative weight of medium rate fibers  0.23  [None] 
wL  Relative weight of low rate fibers  0.17  [None] 
TI  Integration time of the ear  0.2 
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 
Sk  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 
τ f k  Fast K channel time constant  0.3  ms 
τ s k  Slow K channel time constant  ms 
Z  Current-spike rate constant  2.5 × 1 0 11  spikes/A·s 
ECa  Calcium channel Nernst potential  45  mV 
GCa  Calcium channel maximal conductance  4.1  nS 
Ith  Exocytosis calcium threshold current  spikes/A·s 
SCa  Calcium channel sensitivity  1.5  mV 
τCa  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 
λ spont ( 1 )  HSR  60  spikes/s 
λ spont ( 2 )  MSR  10  spikes/s 
λ spont ( 3 )  LSR  spikes/s 
wH  Relative weight of high rate fibers  0.6  [None] 
wM  Relative weight of medium rate fibers  0.23  [None] 
wL  Relative weight of low rate fibers  0.17  [None] 
TI  Integration time of the ear  0.2 
1.
Alaoglu
,
L.
, and
Smith
,
N. M.
, Jr.
(
1938
). “
Statistical theory of a scaling circuit
,”
Phys. Rev.
53
(
10
),
832
836
.
2.
Altoè
,
A.
,
Pulkki
,
V.
, and
Verhulst
,
S.
(
2017
). “
Model-based estimation of the frequency tuning of the inner-hair-cell stereocilia from neural tuning curves
,”
J. Acoust. Soc. Am.
141
(
6
),
4438
4451
.
3.
Altoè
,
A.
,
Pulkki
,
V.
, and
Verhulst
,
S.
(
2018
). “
The effects of the activation of the inner-hair-cell basolateral K+ channels on auditory nerve responses
,”
Hear. Res.
364
,
68
80
.
4.
Barzelay
,
O.
, and
Furst
,
M.
(
2011
). “
Time domain one-dimensional cochlear model with integrated tectorial membrane and outer hair cells
,”
AIP Conf. Proc.
1403
(
1
),
79
84
.
5.
Billone
,
M.
, and
Raynor
,
S.
(
1973
). “
Transmission of radial shear forces to cochlear hair cells
,”
J. Acoust. Soc. Am.
54
(
5
),
1143
1156
.
6.
Boero
,
L. E.
,
Payne
,
S.
,
Gómez-Casati
,
M. E.
,
Rutherford
,
M. A.
, and
Goutman
,
J. D.
(
2021
). “
Noise exposure potentiates exocytosis from cochlear inner hair cells
,”
Front. Synaptic Neurosci.
13
,
740368
.
7.
Bruce
,
I. C.
,
Erfani
,
Y.
, and
Zilany
,
M. S.
(
2018
). “
A phenomenological model of the synapse between the inner hair cell and auditory nerve: Implications of limited neurotransmitter release sites
,”
Hear. Res.
360
,
40
54
.
8.
Chen
,
G.-D.
, and
Fechter
,
L. D.
(
2003
). “
The relationship between noise-induced hearing loss and hair cell loss in rats
,”
Hear. Res.
177
(
1–2
),
81
90
.
9.
Choe
,
Y.
,
Magnasco
,
M. O.
, and
Hudspeth
,
A.
(
1998
). “
A model for amplification of hair-bundle motion by cyclical binding of Ca2+ to mechanoelectrical-transduction channels
,”
Proc. Natl. Acad. Sci. U.S.A.
95
(
26
),
15321
15326
.
10.
Cohen
,
A.
, and
Furst
,
M.
(
2004
). “
Integration of outer hair cell activity in a one-dimensional cochlear model
,”
J. Acoust. Soc. Am.
115
(
5
),
2185
2192
.
11.
Dierich
,
M.
,
Altoè
,
A.
,
Koppelmann
,
J.
,
Evers
,
S.
,
Renigunta
,
V.
,
Schäfer
,
M. K.
,
Naumann
,
R.
,
Verhulst
,
S.
,
Oliver
,
D.
, and
Leitner
,
M. G.
(
2020
). “
Optimized tuning of auditory inner hair cells to encode complex sound through synergistic activity of six independent K+ current entities
,”
Cell Rep.
32
(
1
),
107869
.
12.
Duncan
,
R.
, and
Saunders
,
J.
(
2000
). “
Stereocilium injury mediates hair bundle stiffness loss and recovery following intense water-jet stimulation
,”
J. Comp. Physiol. A
186
(
11
),
1095
1106
.
13.
Freeman
,
D. M.
(
1987
). “
Hydrodynamic study of stereociliary tuft motion in hair cell organs
,” Ph.D. thesis,
Massachusetts Institute of Technology
,
Cambridge, MA
.
14.
Freeman
,
D. M.
, and
Weiss
,
T. F.
(
1990
). “
Superposition of hydrodynamic forces on a hair bundle
,”
Hear. Res.
48
(
1–2
),
1
15
.
15.
Furst
,
M.
(
2015
). “
Cochlear model for hearing loss
,” in
Update on Hearing Loss
(
IntechOpen
,
London
).
16.
Gianoli
,
F.
,
Risler
,
T.
, and
Kozlov
,
A. S.
(
2017
). “
Lipid bilayer mediates ion-channel cooperativity in a model of hair-cell mechanotransduction
,”
Proc. Natl. Acad. Sci. U.S.A.
114
(
51
),
E11010
E11019
.
17.
Gianoli
,
F.
,
Risler
,
T.
, and
Kozlov
,
A. S.
(
2019
). “
The development of cooperative channels explains the maturation of hair cell's mechanotransduction
,”
Biophys. J.
117
(
8
),
1536
1548
.
18.
Goutman
,
J. D.
, and
Glowatzki
,
E.
(
2007
). “
Time course and calcium dependence of transmitter release at a single ribbon synapse
,”
Proc. Natl. Acad. Sci. U.S.A.
104
(
41
),
16341
16346
.
19.
Green
,
D. M.
, and
Swets
,
J. A.
(
1966
).
Signal Detection Theory and Psychophysics
(
Wiley
New York
), Vol. 1 .
20.
Hannula
,
S.
,
Bloigu
,
R.
,
Majamaa
,
K.
,
Sorri
,
M.
, and
Mäki-Torkko
,
E.
(
2011
). “
Audiogram configurations among older adults: Prevalence and relation to self-reported hearing problems
,”
Int. J. Audiol.
50
(
11
),
793
801
.
21.
Harvey
,
L. O.
(
1986
). “
Efficient estimation of sensory thresholds
,”
Behav. Res. Methods Instrum. Comput.
18
(
6
),
623
632
.
22.
He
,
D. Z.
, and
Dallos
,
P.
(
2000
). “
Properties of voltage-dependent somatic stiffness of cochlear outer hair cells
,”
J. Assoc. Res. Otolaryngol.
1
(
1
),
64
81
.
23.
Heinz
,
M. G.
,
Colburn
,
H. S.
, and
Carney
,
L. H.
(
2001
). “
Evaluating auditory performance limits: I. one-parameter discrimination using a computational model for the auditory nerve
,”
Neural Comput.
13
(
10
),
2273
2316
.
24.
Howard
,
J.
, and
Hudspeth
,
A.
(
1988
). “
Compliance of the hair bundle associated with gating of mechanoelectrical transduction channels in the bullfrog's saccular hair cell
,”
Neuron
1
(
3
),
189
199
.
25.
ISO
(
2003
). “
Acoustics—Normal equal-loudness-level contours
” (
International Organization for Standardization
,
Geneva, Switzerland
).
26.
Johnson
,
S. L.
, and
Marcotti
,
W.
(
2008
). “
Biophysical properties of CaV1. 3 calcium channels in gerbil inner hair cells
,”
J. Physiol.
586
(
4
),
1029
1042
.
27.
Johnson
,
S. L.
,
Marcotti
,
W.
, and
Kros
,
C. J.
(
2005
). “
Increase in efficiency and reduction in Ca2+ dependence of exocytosis during development of mouse inner hair cells
,”
J. Physiol.
563
(
1
),
177
191
.
28.
Kakehata
,
S.
, and
Santos-Sacchi
,
J.
(
1995
). “
Membrane tension directly shifts voltage dependence of outer hair cell motility and associated gating charge
,”
Biophys. J.
68
(
5
),
2190
2197
.
29.
Kidd
,
A. R.
, III
, and
Bao
,
J.
(
2012
). “
Recent advances in the study of age-related hearing loss: A mini-review
,”
Gerontology
58
(
6
),
490
496
.
30.
Koral
,
J.
(
2018
). “
An efficient simulation tool for the auditory system by parallel processing
,” Master's thesis,
Tel-Aviv University
,
Tel-Aviv, Israel
.
31.
Kujawa
,
S. G.
, and
Liberman
,
M. C.
(
2019
). “
Translating animal models to human therapeutics in noise-induced and age-related hearing loss
,”
Hear. Res.
377
,
44
52
.
32.
Liberman
,
M. C.
(
1978
). “
Auditory-nerve response from cats raised in a low-noise chamber
,”
J. Acoust. Soc. Am.
63
(
2
),
442
455
.
33.
Liberman
,
M. C.
, and
Dodds
,
L.
(
1987
). “
Acute ultrastructural changes in acoustic trauma: Serial-section reconstruction of stereocilia and cuticular plates
,”
Hear. Res.
26
(
1
),
45
64
.
34.
Liberman
,
M. C.
,
Dodds
,
L.
, and
Learson
,
D.
(
1986
). “
Structure-function correlation in noise-damaged ears: A light and electron-microscopic study
,” in
Basic and Applied Aspects of Noise-Induced Hearing Loss
(
Springer
,
New York
), pp.
163
177
.
35.
Lim
,
D. J.
(
1986
). “
Functional structure of the organ of corti: A review
,”
Hear. Res.
22
(
1–3
),
117
146
.
36.
Lobarinas
,
E.
,
Salvi
,
R.
, and
Ding
,
D.
(
2013
). “
Insensitivity of the audiogram to carboplatin induced inner hair cell loss in chinchillas
,”
Hear. Res.
302
,
113
120
.
37.
Lopez-Poveda
,
E. A.
, and
Eustaquio-Martín
,
A.
(
2006
). “
A biophysical model of the inner hair cell: The contribution of potassium currents to peripheral auditory compression
,”
J. Assoc. Res. Otolaryngol.
7
(
3
),
218
235
.
38.
Lopez-Poveda
,
E. A.
, and
Meddis
,
R.
(
2001
). “
A human nonlinear cochlear filterbank
,”
J. Acoust. Soc. Am.
110
(
6
),
3107
3118
.
39.
Maison
,
S. F.
, and
Liberman
,
M. C.
(
2000
). “
Predicting vulnerability to acoustic injury with a noninvasive assay of olivocochlear reflex strength
,”
J. Neurosci.
20
(
12
),
4701
4707
.
40.
Melichar
,
I.
,
Syka
,
J.
, and
Úlehlová
,
L.
(
1980
). “
Recovery of the endocochlear potential and the K+ concentrations in the cochlear fluids after acoustic trauma
,”
Hear. Res.
2
(
1
),
55
63
.
41.
Moore
,
B. C.
(
2007
).
Cochlear Hearing Loss: Physiological, Psychological and Technical Issues
(
Wiley
,
New York
).
42.
Neely
,
S. T.
, and
Allen
,
J. B.
(
2009
). “
Retrograde waves in the cochlea
,” in
Concepts and Challenges in the Biophysics of Hearing—Proceedings of the 10th International Workshop on the Mechanics of Hearing
(
World Scientific
,
Singapore
), pp.
62
67
.
43.
Neely
,
S. T.
, and
Kim
,
D.
(
1986
). “
A model for active elements in cochlear biomechanics
,”
J. Acoust. Soc. Am.
79
(
5
),
1472
1480
.
44.
Ni
,
G.
,
Elliott
,
S. J.
, and
Baumgart
,
J.
(
2016
). “
Finite-element model of the active organ of Corti
,”
J. R. Soc. Interface
13
(
115
),
20150913
.
45.
O'Beirne
,
G. A.
, and
Patuzzi
,
R. B.
(
2007
). “
Mathematical model of outer hair cell regulation including ion transport and cell motility
,”
Hear. Res.
234
(
1–2
),
29
51
.
46.
Prodanovic
,
S.
,
Gracewski
,
S.
, and
Nam
,
J.-H.
(
2015
). “
Power dissipation in the subtectorial space of the mammalian cochlea is modulated by inner hair cell stereocilia
,”
Biophys. J.
108
(
3
),
479
488
.
47.
Robles
,
L.
, and
Ruggero
,
M. A.
(
2001
). “
Mechanics of the mammalian cochlea
,”
Physiol. Rev.
81
(
3
),
1305
1352
.
48.
Rodieck
,
R.
,
Kiang
,
N.-S.
, and
Gerstein
,
G.
(
1962
). “
Some quantitative methods for the study of spontaneous activity of single neurons
,”
Biophys. J.
2
(
4
),
351
368
.
49.
Sabo
,
D.
,
Barzelay
,
O.
,
Weiss
,
S.
, and
Furst
,
M.
(
2014
). “
Fast evaluation of a time-domain non-linear cochlear model on GPUs
,”
J. Comput. Phys.
265
,
97
112
.
50.
Saremi
,
A.
, and
Stenfelt
,
S.
(
2013
). “
Effect of metabolic presbyacusis on cochlear responses: A simulation approach using a physiologically-based model
,”
J. Acoust. Soc. Am.
134
(
4
),
2833
2851
.
51.
Saunders
,
J. C.
,
Dear
,
S. P.
, and
Schneider
,
M. E.
(
1985
). “
The anatomical consequences of acoustic injury: A review and tutorial
,”
J. Acoust. Soc. Am.
78
(
3
),
833
860
.
52.
Schmiedt
,
R. A.
(
2010
). “
The physiology of cochlear presbycusis
,” in
The Aging Auditory System
(
Springer
,
New York
), pp.
9
38
.
53.
Shamma
,
S. A.
,
Chadwick
,
R. S.
,
Wilbur
,
W. J.
,
Morrish
,
K. A.
, and
Rinzel
,
J.
(
1986
). “
A biophysical model of cochlear processing: Intensity dependence of pure tone responses
,”
J. Acoust. Soc. Am.
80
(
1
),
133
145
.
54.
Shi
,
L.
,
Chang
,
Y.
,
Li
,
X.
,
Aiken
,
S.
,
Liu
,
L.
, and
Wang
,
J.
(
2016
). “
Cochlear synaptopathy and noise-induced hidden hearing loss
,”
Neural Plast.
2016
,
6143164
.
55.
Siebert
,
W. M.
(
1970
). “
Frequency discrimination in the auditory system: Place or periodicity mechanisms?
Proc. IEEE
58
(
5
),
723
730
.
56.
Spoendlin
,
H.
, and
Schrott
,
A.
(
1989
). “
Analysis of the human auditory nerve
,”
Hear. Res.
43
(
1
),
25
38
.
57.
Stenfelt
,
S.
, and
Rönnberg
,
J.
(
2009
). “
The signal-cognition interface: Interactions between degraded auditory signals and cognitive processes
,”
Scan. J. Psychol.
50
(
5
),
385
393
.
58.
Talmadge
,
C. L.
,
Tubis
,
A.
,
Long
,
G. R.
, and
Piskorski
,
P.
(
1998
). “
Modeling otoacoustic emission and hearing threshold fine structures
,”
J. Acoust. Soc. Am.
104
(
3
),
1517
1543
.
59.
Vaden
,
K. I.
,
Eckert
,
M. A.
,
Matthews
,
L. J.
,
Schmiedt
,
R. A.
, and
Dubno
,
J. R.
(
2022
). “
Metabolic and sensory components of age-related hearing loss
,”
J. Assoc. Res. Otolaryngol.
23
(
2
),
253
272
.
60.
Verhulst
,
S.
,
Altoè
,
A.
, and
Vasilkov
,
V.
(
2018
). “
Computational modeling of the human auditory periphery: Auditory-nerve responses, evoked potentials and hearing loss
,”
Hear. Res.
360
,
55
75
.
61.
Wang
,
Y.
,
Hirose
,
K.
, and
Liberman
,
M. C.
(
2002
). “
Dynamics of noise-induced cellular injury and repair in the mouse cochlea
,”
J. Assoc. Res. Otolaryngol.
3
(
3
),
248
268
.
62.
Westerman
,
L. A.
, and
Smith
,
R. L.
(
1988
). “
A diffusion model of the transient response of the cochlear inner hair cell synapse
,”
J. Acoust. Soc. Am.
83
(
6
),
2266
2276
.
63.
Wu
,
P.-Z.
, and
Liberman
,
M. C.
(
2022
). “
Age-related stereocilia pathology in the human cochlea
,”
Hear. Res.
422
,
108551
.
64.
Wu
,
P.-Z.
,
O'Malley
,
J. T.
,
de Gruttola
,
V.
, and
Liberman
,
M. C.
(
2020
). “
Age-related hearing loss is dominated by damage to inner ear sensory cells, not the cellular battery that powers them
,”
J. Neurosci.
40
(
33
),
6357
6366
.
65.
Zilany
,
M. S.
,
Bruce
,
I. C.
, and
Carney
,
L. H.
(
2014
). “
Updated parameters and expanded simulation options for a model of the auditory periphery
,”
J. Acoust. Soc. Am.
135
(
1
),
283
286
.
66.
Zilany
,
M. S.
,
Bruce
,
I. C.
,
Nelson
,
P. C.
, and
Carney
,
L. H.
(
2009
). “
A phenomenological model of the synapse between the inner hair cell and auditory nerve: Long-term adaptation with power-law dynamics
,”
J. Acoust. Soc. Am.
126
(
5
),
2390
2412
.
67.
Zwicker
,
E.
, and
Fastl
,
H.
(
1999
).
Facts and Models
(
Springer
,
New York
).
Published open access through an agreement with Tel-Aviv University Department of Psychology