P T P V U V I. INTRODUCTION. II. MODEL FORMULATION A. Cochlear fluid dynamics.

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1 Integration of outer hair cell activity in a one-dimensional cochlear model Azaria Cohen and Miriam Furst a) School of Electrical Engineering, Faculty of Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel Received 2 February 2004; revised 18 February 2004; accepted 19 February 2004 Recently, significant progress has been made in understanding the contribution of the mammalian cochlear outer hair cells OHCs to normal auditory signal processing. In the present paper an outer hair cell model is incorporated in a complete, time-domain, one-dimensional cochlear model. The two models control each other through cochlear partition movement and pressure. An OHC gain is defined to indicate the outer hair cell contribution at each location along the cochlear partition. Its value ranges from 0 to 1: 0 represents a cochlea with no active OHCs, 1 represents a nonrealistic cochlea that becomes unstable at resonance frequencies, and 0.5 represents an ideal cochlea. The model simulations reveal typical normal and abnormal excitation patterns according to the value of. The model output is used to estimate normal and hearing-impairment audiograms. High frequency loss is predicted by the model, when the OHC gain is relatively small at the basal part of the cochlear partition. The model predicts phonal trauma audiograms, when the OHC gain is random along the cochlear partition. A maximum threshold shift of about 60 db is obtained at 4 khz Acoustical Society of America. DOI: / PACS numbers: Kc, Wn, Ld WPS Pages: I. INTRODUCTION a Author to whom correspondence should be addressed. Electronic mail: mira@eng.tau.ac.il In recent years, significant progress has been made in understanding the contribution of the mammalian cochlear outer hair cells to normal auditory signal processing. The outer hair cells OHCs act as local amplifiers that are metabolically activated, and their motion effectively changes the basilar membrane mechanical response e.g., Dallos, Hearing-impairment due to acoustic trauma or aging is mainly caused by outer hair cells loss Moore, In models that describe the outer hair cell activity, it is assumed that the OHC cilia displacement generates a force that acts on the basilar membrane e.g., Allen and Neely, 1992; Mountain and Hubbard, 1994; Dallos and Evans, 1995; Rattay et al., 1998; He and Dallos, 2000; Spector, The contribution of the OHCs to psychoacoustical performance has been previously demonstrated by models that functionally include OHC activity. These models predict normal behavior and degradation in the performances due to OHC loss e.g., Kates, 1991; Goldstein, 1993; Carney, 1994; Heinz et al., Classical one-dimensional cochlear models have been modified to include the OHC activity by nonlinear damping and/or nonlinear compliance with additional delays Talmadge et al., 1998; Zweig, These models predicted various properties associated with OHCs, including cochlear otoacoustic emissions Furst et al., 1992; Talmadge et al., 1998 and two-tone-suppression e.g., Geisler, 1991, 1993; Neely, In the present paper, we develop an integrated basilar membrane outer hair cell model BM-OHC. An outer hair cell model is embedded in a complete, time-domain, onedimensional cochlear model. The two models control each other through the cochlear partition s movement and pressure. From the BM-OHC model, we derive predictions regarding normal hearing and various types of hearing impairments. II. MODEL FORMULATION A. Cochlear fluid dynamics In the simple, one-dimensional model e.g., Zwislocki, 1950; Zweig et al., 1976; Viergever, 1980; Furst and Goldstein, 1982, the cochlea is considered as an uncoiled structure with two fluid-filled compartments whose walls, separated by an elastic partition, are rigid. In the present model we include the OHC activity in the elastic partition. By applying fundamental physical principles, such as conservation of mass and the dynamics of deformable bodies, we obtain the basic equations for this activity. We define P T and P V as the pressure in scala tympaniand scala vestibuli, respectively. The fluid velocity in the scala tympani is defined as U T, and in the scala vestibuli as U V. The pressure difference across the cochlear partition is P P T P V. 1 Since both scalae, tympani and vestibuli, contain perilymph, which is presumably very nearly incompressible, and inviscid fluid, the equation of motion for each scala can be written as U T P T t x, U V P V t x, where is the perilymph density. The cochlear partition, whose mechanical properties are describable in terms of point-wise mass density, stiffness and damping, is regarded as a flexible boundary between scala 2 J. Acoust. Soc. Am. 115 (5), Pt. 1, May /2004/115(5)/2185/8/$ Acoustical Society of America 2185

2 tympani and scala vestibuli. Thus at every point along the cochlear duct, the pressure difference (P) across the partition drives the partition s velocity. From the conservation of mass principle we can derive the relation between the fluid velocity and the basilar membrane displacement, BM, A U V x BM, A U T t x BM, 3 t where A represents the cross-sectional area of scala tympani and scala vestibuli, and is the basilar membrane wih. Combining Eqs. 1 3 yields the differential equation for P, 2 P x BM A t 2. 4 If we assume that the oval and round window areas are equal to A, then the boundary conditions are U V 0,t U T 0,t f t, P L,t 0, 5 where f (t) is the stapes velocity, and L is the cochlear length. In previous one-dimensional cochlear models e.g., Zwislocki 1950; Viergever, 1980; Furst and Goldstein, 1982, the pressure difference is only obtained by the basilar membrane mechanical properties. In the present model, we include the pressure produced by the OHCs, therefore P BM P P OHC, 6 where P BM m x 2 BM t 2 r x BM s x t BM, and m, r, and s represent basilar membrane mass, resistance, and stiffness per unit area, respectively. The next section presents the model for the outer-hair cell and the corresponding equations for P OHC. 7 FIG. 1. An equivalent electrical circuit of the outer-hair cell model. The outer-hair cells OHC are epithelial cells that are located in two different electrochemical environments. The apical part of the cell faces the endolymph of the scala media, while the basolateral part faces the supporting cells and the perilymph of the scala tympani. The OHC s stereocilia are bent due to basilar membrane motion; because their tip links are stretched and physically open ion channels, potassium and calcium ions flow from the endolymph into the cells. As a result, an electrical voltage drop is introduced on the basolateral of the cell s membrane Dallos, These electrical changes change the shape of the OHCs Jerry et al., 1995; Brownell et al., The somatic elongationcontraction process operates on a cycle-by-cycle basis at the frequency stimulus and provides a mechanical feedback to the basilar membrane motion. This process underlies the high sensitivity and frequency selectivity of the mammalian cochlea. The outer hair cell membrane acts as a low-pass filter with a cutoff frequency of less than 1 khz. Dallos and Evans 1995 proposed that the extracellular potential gradient across the hair cell provides electro-motility at high frequencies. The following model is a modification of this proposal. Figure 1 presents an equivalent electrical circuit model for the OHC. The cell membrane is divided into two parts, the apical and basolateral, each represented by its conductance (G AP or G BAS ), and capacitance (C AP or C BAS ). We assume that the stereocilia movement affects both the capacitance and conductance of the OHC s apical part. The electrical potential across the cell membrane is presumably equal to the potential of the scala media, V SM, and the potential difference across the basolateral part the receptor potential is presented by in Fig. 1. A battery (E BAS ), in series with the basolateral conductance, sets the resting OHC potential at roughly 70 mv Mountain and Hubbard, The stereocilia motion causes a current to flow through the apical part of the OHC which yields apical current I AP, I AP V SM G AP d C AP V SM. 8 B. The outer-hair cell model The basolateral current I BAS, on the other hand, is given by I BAS G BAS E BAS d C BAS. According to Kirchhoff s current law, the current that flows through the apical segment is equal to the current that flows through the basolateral segment. Thus I AP I BAS yields the differential equation for, C BAS C AP d G AP G BAS dc BAS V SM G AP dc AP dc AP 9 dv SM C AP. G BAS E BAS. 10 In order to simplify Eq. 10, we apply the following assumptions: (i) The capacitance and the conductance of the OHC membrane are proportional to the membrane surface area. The basolateral membrane area is larger than the apical part. According to Housley and Ashmore 1992 the ratio between 2186 J. Acoust. Soc. Am., Vol. 115, No. 5, Pt. 1, May 2004 A. Cohen and M. Furst: Cochlear model with OHCs

3 the basolateral and apical capacitance is more than 16, and the ratio of the conductance is more than 7. We thus assume C AP C BAS and G AP G BAS. (ii) The changes in the basolateral capacitance are relatively small Lukashkin and Russell, 1997, thus dc BAS (iii) We shall further assume that dc AP / is at most in the order-of-magnitude of G AP. Thus from assumption (i) we can conclude that dc AP G BAS. 12 (iv) The OHC membrane acts as a low pass filter with a fixed cutoff frequency of less than 1 khz Dallos and Evans, The OHC cutoff frequency ( OHC ) can be derived from OHC G AP G BAS G BAS const. 13 C AP C BAS C BAS (v) The electrical potential across the scala media is presumably constant Mountain and Hubbard, 1995, thus dv SM Substituting Eqs in Eq. 10 yields a simplified differential equation for : d OHC E BAS G AP dc AP, 15 where V SM / C BAS C AP V SM /C BAS const. The apical membrane s capacitance and conductance change due to ion channels actively opening in the apical part of the outer hair cell. The OHC stereocilia are shallowly but firmly embedded in the under-surface of the tectorial membrane. Since this membrane is attached on one side to the basilar membrane, a shear motion arises between the tectorial membrane and the organ of Corti, when the basilar membrane moves up and down Pickles, The arch of Corti maintains the rigidity of the organ during the movement. As part of our one-dimensional assumption, we further assume that the tectorial membrane along with Dieters cells is a rigid structure. Thus it is reasonable to assume that the OHC bundle displacement is a function of the basilar membrane vertical displacement Nobili et al., We, therefore, assume that both G AP and C AP are functions of BM the basilar membrane vertical displacement. Their functional dependences are best described by second order Boltzmann functions e.g., Sachs and Lecar, 1991; Lukashkin and Russel, The voltage variation across the basolateral part of the OHC causes a length change ( l OHC ) in the OHC as Brownell et al first showed. Thus the force F OHC that an OHC exhibits due to voltage change is derived by F OHC K OHC BM l OHC, 16 where l OHC l OHC ( ( BM )). The pressure that the OHCs contribute to the basilar membrane pressure is derived from P OHC x F OHC, 17 where (x) is the relative density of healthy OHCs per unit area along the cochlear duct. In the rest of the paper we will refer to (x) as the OHC gain, whose value ranges from 0 to 1. C. Linear approximation The contribution of the OHCs to the basilar membrane motion is particularly effective in low level stimuli. Therefore, the second order Boltzmann functions can be approximated by linear functions for the capacitance, conductance, length change, and OHC stiffness. In the following section we shall assume linear dependence, i.e., G AP G 0 G BM, C AP C 0 C BM, l OHC l 0 l, 18 where G, C, and l are all positive numbers. Substituting the linear assumptions 18 in Eqs yields the differential equation for P OHC : where P OHC t OHC P OHC 1 BM 2 BM t 0, 19 0 K OHC OHC l 0 l E BAS l G 0, 1 K OHC OHC l G, 2 K OHC 1 l C. 20 When 0, the OHCs are not contributing to the basilar membrane motion, and thus P OHC 0. On the other hand, 1 means that the basilar membrane displacement can reach infinity at the location whose characteristic frequency is equal to the stimulus frequency. In order to solve Eq. 19, all the parameters in Eq. 20 should be determined. Without losing the generality, we can assume that 0 0 this term contributes to the nonhomogeneous part of Eq. 19 which can be accounted by the initial conditions. The values of 1 and 2 were obtained from the frequency domain solution. For a steady-state sinusoidal input, the frequency domain representation of Eq. 19 is given by P OHC,x x 1 x j 2 x OHC j BM,x. 21 Substituting Eq. 21 in the frequency domain representation of Eqs. 6 and 7 yields P x, j Z x,w BM x,, 22 J. Acoust. Soc. Am., Vol. 115, No. 5, Pt. 1, May 2004 A. Cohen and M. Furst: Cochlear model with OHCs 2187

4 where Re Z x, r x x 2 x OHC 1 x 2 2, OHC Im Z x, m x s x 23 x 1 x OHC 2 x OHC We further assume that for every location along the cochlear partition, when the stimulus frequency is equal to CF (x) s(x)/m(x), a resonance is produced. The resonance frequency in a cochlea with no active OHCs (x) 0 was obtained when Im Z(x, CF ) 0 but Re Z(x, CF ) 0. When (x) 1, we require that at the resonance frequency both Re Z(x, CF ) 0 and Im Z(x, CF ) 0. Substituting those requirements in Eq. 23 reveals the values for 1 (x) and 2 (x), 1 x r x s x, m x 2 x r x OHC. 24 Comparison between Eqs. 24 and 20 yields some constraints on the parameters G, C, and l ; specifically, l G OHC and l C 1, which implies that C G / OHC. This result might explain why changes in the OHC s apical conductance were more frequently detected than its capacitance changes. III. SIMULATION METHODS A. Frequency domain solution For low-level, steady-state stimuli, we consider a linear model that can be solved in the frequency domain. The model Eq. 4 for the pressure difference, P(x, ), is written in the frequency domain as d 2 P x, dx 2 K 2 x, P x, 0, 25 where K 2 (x, ) 2 j /AZ(x, ); Z(x, ) is obtained by substituting Eq. 24 in Eq. 23. The frequency domain boundary conditions are obtained by transforming Eq. 5 to the frequency domain and substituting the result in the frequency domain representation of Eq. 2, which yields TABLE I. List of model parameters. Parameter Value/definition Description L 3.5 cm Cochlear length 1 g/cm 3 Density of perilymph cm Wih of the basilar membrane A 0.5 cm 2 Cross-sectional area of the cochlea scalae m e 1.5 X g/cm 2 Basilar membrane mass per unit area s e 1.5 X g/cm 2 s 2 Basilar membrane stiffness per unit area r 0.25 e 0.06 X g/cm2 s Basilar membrane resistance per unit area OHC 1 khz OHC cutoff frequency x cm Distance from stapes f (t) cm/s Stapes velocity dp x, 2 j F, P L, 0, 26 dx x 0 where F( ) is Fourier transform of f (t). The most common method used for solving the cochlear boundary value problem Eqs. 25 and 26 is the WKB method e.g., Viergever, In this approximation, one assumes that K(x) is varying slowly with x. However, near CF, this assumption does not hold, especially when we assume a significant existence of OHC population (x) 1. We chose to solve the boundary value problem using a finite difference method, one of the simplest ways to solve these sort of problems e.g., Burden and Faires, Although more sophisticated methods such as finite elements might provide better approximations to the boundary value problem, the finite difference method is significantly more efficient for a dense net Diependaal, A detailed description of the frequency domain solution is included in Appendix A. B. Time domain solution The time domain solution is performed in two stages Furst and Goldstein, In the first stage, the boundary value differential equation is solved when the time is held as a parameter. In the second stage, the set of the initial condition time dependent equations is solved. In order to solve the boundary value differential equation Eq. 4, we rewrite the equation by substituting Eqs. 6 and 7 in Eq. 4, which yields 2 P x 2 x P g x,t x, 27 where (x) 2 /ma, and g(x,t) r BM / t s BM P OHC. The initial value differential equations to be solved are Eqs. 6, 7 and 19 with the initial conditions BM x,t 0 0, P OHC x,t 0 0. BM x,t t 0, t 0 28 The actual simulation method is described in Appendix B. The parameters that were used in the simulation are listed in Table I. IV. SIMULATIONS RESULTS Figure 2 represents relative basilar membrane velocities (BM V ) along the cochlear length as obtained by various sinusoidal inputs. Both time and frequency solutions yielded similar results for those steady-state conditions. Each set of BM V was derived with a different value of. In these calculations, a fixed was considered along the whole cochlear 2188 J. Acoust. Soc. Am., Vol. 115, No. 5, Pt. 1, May 2004 A. Cohen and M. Furst: Cochlear model with OHCs

5 FIG. 3. Simulated audiograms derived by various values of. FIG. 2. Simulation of relative basilar membrane velocity for different input frequencies and various values of, the OHC gain. partition. It is obvious from Fig. 2 that with the increase of, the location of the resonance for each input frequency moves towards the helicotrema, and the peak at the resonance becomes more significant, especially for frequencies above 1 khz. We chose (x) 0.5 to represent the ideal cochlea. Let us define loudness (L d ) as the energy acquired by the whole cochlea due to the basilar membrane velocity Furst et al., 1992, i.e., L d 1 T 0 L 0 T 2 BM t dx, 29 where T is the stimulus duration. Human cochlear activity is best described by its audiogram, which is the threshold obtained in different stimulus frequencies relative to an ideal cochlea. Thus we can use the model simulation in order to estimate the audiograms that will be obtained with different values of. Let us define, as the threshold obtained by the model for an input frequency and a cochlea with an OHC gain,, L d,0.5 L d,. 30 It is obvious then from Eq. 30 that (,0.5) 0, for every. Figure 3 represents the estimated audiograms for different values of. For cochleae with 0.5, the estimated audiograms reveal thresholds that are better than that for the ideal cochlea. The difference in the estimated thresholds for different values of is less than 20 db for input frequencies below 1 khz. However, there is a significant difference in the estimated threshold for higher frequencies. For 0.2, each of the estimated audiograms has a maximum threshold at a frequency between 4 and 6 khz. Those types of audiograms resemble typical phonal trauma audiograms. Higher values of ( 0.5), however, reveal audiograms with decreased threshold as a function of frequency. These types of audiograms are not realistic, since they are not found among human audiograms. We, therefore, define a normal healthy cochlea as one whose OHC gain is equal to 0.5, and a cochlea with OHC loss is presented by 0.5. Most studies that measured OHC loss on humans or animals e.g., Liberman and Dodds, 1987; Saunders et al., 1985 demonstrated that OHC loss typically starts at the basal part of the cochlear partition and gradually expands toward the helicotrema. In order to test the model prediction for such a case, we assumed a gradual change in (x) and calculated the corresponding audiograms. Figure 4 a presents a set of OHC gains obtained by (x) 0.5 (1 e x ), for different values of. The corresponding audiograms are presented in Fig. 4 b. The estimated audiograms resemble hearing-impaired human audiograms that are typically found among the elderly. While there is almost no hearing loss at low frequencies below 1 khz, there is a gradual decrease in hearing for higher frequencies. The OHC gain can also be interpreted as the density of the OHC population along the cochlear partition. It is reasonable to assume that this density is not fixed, but randomly varies along the cochlea. Let us assume that (x) isa Gaussian random variable with a mean of 0.5 and a standard deviation. Figure 5 shows the excitation patterns obtained for different values of while the corresponding audiograms appear in Fig. 6. Each BM V in Fig. 5 was obtained from 50 runs of the model; for each run, a different (x) was generated. It is clear from Fig. 5 that low values of yielded very similar responses obtained by a fix as shown in Fig. 2. However, for larger values of, the different BM V s lose their significant peaks and resemble the responses in Fig. 2 that were obtained for 0. Figure 6 presents the correspondent mean and standard deviation thresholds. An obvious maximum threshold was obtained at 4 khz when A maximum threshold of 60 db was obtained for 0.1. The J. Acoust. Soc. Am., Vol. 115, No. 5, Pt. 1, May 2004 A. Cohen and M. Furst: Cochlear model with OHCs 2189

6 FIG. 4. Simulated audiograms b for different types of OHC hearing loss as characterized by the OHC gain a. estimated audiograms resemble typical, noise-induced hearing loss audiograms. V. DISCUSSION The model presented in this paper is a modification of the classical one-dimensional cochlear model, where the FIG. 5. Simulation of basilar membrane velocity for different input frequencies and normally distributed with a mean of 0.5 and different values of variance 2. Each BM V was obtained from 50 runs; mean and standard deviation are shown. FIG. 6. Simulated audiograms obtained for normally distributed with a mean of 0.5 and different values of variance 2. OHC activity is integrated in the basilar membrane motion. The OHCs generate forces due to their electro-motility. These forces act simultaneously along with the transpartition pressure that the cochlear fluids produce. In the vicinity of the characteristic frequency CF, these forces are in phase for a stimulus at the frequency of the CF and yield a significant enhancement in the basilar membrane motion. The response to stimuli that are equal to CF is reduced, because these forces are not in phase. The OHCs are known to be sensitive to metabolism, efferent stimulation, etc. Ulfendahl, In the present model their mode of activity is characterized by one parameter, which can vary along the cochlear partition. We refer to a cochlea whose (x) 0 as a passive cochlea, i.e., it has no active OHCs. On the other hand, (x) 1 represents a nonrealistic cochlea whose basilar membrane motion is unstable. An ideal cochlea was defined as one with (x) 0.5. The main effect of OHC activity is to enhance cochlear displacement and velocity in the region where the stimulus frequency is close to the local characteristic frequency. Moreover, this enhancement is mostly significant for frequencies above 1 khz the cutoff frequency of the OHC membrane. Any random change in the OHC gain, even a very fine one, causes a decrease in the enhanced basilar membrane motion. On the basis of the basilar membrane velocity, we have simulated audiograms for normal ears and ears with OHC loss. The simulation results revealed the following observations: 2190 J. Acoust. Soc. Am., Vol. 115, No. 5, Pt. 1, May 2004 A. Cohen and M. Furst: Cochlear model with OHCs

7 i ii iii Typical audiograms with high frequency loss were obtained when the OHC gain decreased at the basal part of the cochlear partition. Phonal trauma typical audiograms were obtained when the OHC gain was random along the cochlear partition. The maximum threshold shift that was obtained due to OHC loss was about 60 db at 4 khz. Noise-induced hearing loss is typically presented by an audiogram whose maximum threshold is obtained at 4 khz, similar to the simulated audiograms with random. The model therefore indicates that random loss of OHCs along the entire cochlear partition characterizes phonal trauma. This description of noise-induced cochleae is incompatible with animal studies on noise trauma. These studies Liberman and Mulroy, 1982; Maison and Liberman, 2000 show that among animals exposed to narrow-band noise signals, the OHC loss was especially found at cochlear locations whose characteristic frequencies were equal to or an octave above the noise exposure. Such simulation of OHCs loss will also produce similar audiograms. However, the assumption of random loss of OHCs along the cochlea seems more realistic, since in humans the loss of sensitivity in the 4-kHz range was found independently of the type of noise exposure Saunders et al., 1985; Moore, Hearing-impairment due to advanced age is typically characterized by high frequency hearing loss. The typical audiogram for this sort of hearing loss is predicted by the model when OHC loss is considered to be primarily occurring at the basal part of the cochlea. Indeed, studies on cadaver cochlea indicate that that OHC loss in the basal part is more frequent than in the apical part of the cochlea e.g., Saunders et al., The model predicts a maximum of 60 db hearing loss, when in practice hearing loss can be as high as 120 db. Since the model deals only with OHC loss, it is very reasonable to assume that loss of inner hair-cells that typically follow OHC loss are the cause of the profound hearing loss. The time-domain solution presented in this paper allows us to test the model s predictions for known nonlinear phenomena such as two-tone suppression, combination tones, and cochlear otoacoustic emissions. Hearing-impaired people who suffer from OHC loss show a significant degradation in these nonlinear phenomena Moore, Infuture studies, we intend to demonstrate the model s predictions for these nonlinear phenomena. ACKNOWLEDGMENTS We thank two unnamed reviewers whose comprehensive and detailed comments were very helpful. APPENDIX A: FREQUENCY DOMAIN SOLUTION Let us define a uniform grid of N points in the interval 0,L, so that x l lh and P l P(x l, ), where h L/N, and l 0,1,...,N 1. Applying linear combinations of the Taylor expansions for P l 1 and P l 1 yields d 2 P dx 2 x x l P l 1 2P l P l 1 h 2 o h 2, 1 l N 2 d 2 P dx 2 x 0 2 P 1 P 0 hp 0 h 2 o h, A1 where P 0 2 j F( ) is derived from the boundary values Eq. 26. Note that for 1 l N 2, the derivatives of P are approximated by the order of h 2, while at the border point, l 0, P is approximated only by the order of h. Since we assume that P is a smooth function, o(h) approximation at the edges might be sufficient. Thus Eq. A1 can be expressed as a set of linear equations: P, A2 where P P 0,P 1,...,P N 1 T, and 2hj F 1,0,...,0 T. The matrix is a tridiagonal square matrix of the size N N, whose values on the diagonal ( ll ) are obtained by 1 h2 K2 x 0 /2, l 0, ll 2 h 2 K 2 x l, 1 l N 2,, A3 1, l N 1, and whose bordering values are l,l 1 1 for l 0,...,N 2, l,l 1 1 for l 1,...,N 2, and N 1,N 2 0. If the matrix is regular, then a unique solution exists and can be easily obtained. Although we were not able to prove that for any choice of h, det( ) 0, in practice we have always been able to calculate 1. APPENDIX B: TIME DOMAIN SOLUTION The time domain solution is performed in two stages Furst and Goldstein, In the first stage, the boundary value differential equation is solved when the time is held as a parameter. In the second stage, the set of the initial condition time dependent equations is solved. The boundary-value problem Eq. 27 can be expressed as a set of linear equations P Y, B1 where P P 0,P 1,...,P N 1 T is the same as in the frequency domain solution, but it is obtained for every time step t. The boundary values Eq. 5 are expressed as P 0 2 f (t) and P N 1 0, thus 1 f t 2 Y 2h h2 g x 0,t x 0 h 2 g x 1,t x 1. B2 h 2 g x N 2,t x N 2 0 The matrix is a tridiagonal square matrix of the size N N, whose values on the diagonal ( ll ) are obtained by J. Acoust. Soc. Am., Vol. 115, No. 5, Pt. 1, May 2004 A. Cohen and M. Furst: Cochlear model with OHCs 2191

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