A multi-channel adaptive nonlinear filtering structure realizing some properties of the hearing system

Computers in Biology and Medicine 35 (2005) 495 – 510 http://www.intl.elsevierhealth.com/journals/cobm

A multi-channel adaptive nonlinear (ltering structure realizing some properties of the hearing system Antanas Stasiunasa , Antanas Verikasa; d;∗ , Povilas Kemesisa , Marija Bacauskienea , Rimvydas Miliauskasb , Natalija Stasiunienec , Kerstin Malmqvistd a

Department of Applied Electronics, Kaunas University of Technology, LT-3031, Kaunas, Lithuania b Department of Physiology, Kaunas University of Medicine, LT 3000, Kaunas, Lithuania c Department of Biochemistry, Kaunas University of Medicine, LT 3000, Kaunas, Lithuania d Intelligent Systems Laboratory, Halmstad University, Box 823, S-30118, Halmstad, Sweden Received 29 July 2003; accepted 12 April 2004

Abstract An adaptive nonlinear signal-(ltering model of the cochlea is proposed based on the functional properties of the inner ear. The model consists of the cochlear (ltering segments taking into account the longitudinal, transverse and radial pressure wave propagation. On the basis of an analytical description of di8erent parts of the model and the results of computer modeling, the biological signi(cance of the nonlinearity of signal transduction processes in the outer hair cells, their role in signal compression and adaptation, the e8erent control over the characteristics of the (ltering structures (frequency selectivity and sensitivity) are explained. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Cochlear model; Hair cells; Filtering circuits with feedback

Abbreviations: BM, basilar membrane; TM, tectorial membrane; OAE, otoacoustic emission; OHC, outer hair cell; IHC, inner hair cell; CFS, cochlea (ltering segment; TFC, transverse (ltering circuit; DL, delay line; LPC, longitudinal low-pass circuit; BPC, transverse band-pass circuit; APC, adaptive low-pass circuit; MET, mechano-electrical transduction; EMT, electromechanical transduction; PG, pulse generator (initial segment of the nerve (ber); NC, nonlinear circuit; MSO, medial superior olive; SOC, superior olivary complex; LT, longitudinal–transverse (ltering; LTR, longitudinal–transverse –radial (ltering; CF, characteristic frequency. ∗ Corresponding author. Intelligent Systems Laboratory, Halmstad University, Box 823, S-30118, Halmstad, Sweden. Tel.: +46-35-167-140; fax: +46-35-216-724. E-mail addresses: [email protected], [email protected] (A. Verikas). 0010-4825/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compbiomed.2004.04.004

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1. Introduction The (ltering properties of the cochlea attracted the interest of professionals in di8erent (elds of science a long time ago. Based on the knowledge of that time, many di8erent models of biological (ltering structures of the cochlea and the (bers of the auditory nerve were proposed for the purpose of creating or improving the cochlea prosthesis and the systems of speech recognition. These models are based on the observations of the oscillations of the BM and TM [1–5], psychophysical data [6,7], (ring properties of the auditory nerve (ber [8–12], otoacoustic emission (OAE) [13,14], neurophysiology and biochemistry of the signal transduction in the hair cells [15–21]. The active elements of the (ltering structures in the cochlea are the outer hair cells (OHC) of the organ of Corti. The presence of the active elements in the cochlea was postulated more than 50 years ago [22]. Later the OHC were suggested as the source of the OAE [13], and the active elements were incorporated into a cochlear model [7]. The function of the OHC in hearing is now perceived as that of a cochlear ampli(er [23] that re(nes the sensitivity and frequency selectivity of the mechanical vibrations of the cochlea [15,16,20]. The basis of the cochlear ampli(cation is the ability of the OHC to change their length in response to changes in the membrane potential [24]. This phenomenon, called electro-motility, depends on the voltage-sensitive motor protein prestin embedded in the basolateral membrane of the OHC [25–27]. The protein is a direct voltage-to-force converter capable to operate at microsecond rates over the entire audible frequency range up to 20 kHz [17,19]. The activity of the OHC in the models of the cochlea was treated in di8erent ways: a negative resistance was introduced into the resonance circuit of the model [28,29], a nonlinear feedback was proposed [7,14,30,31] or even more complex approach was used in the phenomenological models [12,32]. In most modern models the OHC are considered as active elements responsible for the nonlinearities and are assumed to be capable of providing forces on a cycle-by-cycle basis at audio frequencies. The TM, BM and remaining cells in the organ of Corti are regarded as mechanically passive and linear. Mathematical and computer modeling of the (ltering processes taking place in the cochlea can elucidate these processes as well as functioning of the OHC. The piezoelectric models [33,34] and Jexoelectric models [19,35] were proposed for the explanation of the electro-motility. However, to our knowledge, there are no adaptive nonlinear electrical-functional models of the OHC, which could explain the essential signi(cance of the nonlinearities of the transduction processes and the e8erent control upon it. In the present paper, by applying contemporary knowledge, a multi-channel adaptive nonlinear (ltering model of the cochlea is constructed, including the signal transmission in perilymph, the transverse adaptive nonlinear (ltering in the endolymph and the impact of the (ltered signal in the radial direction on the inner hair cells (IHC). The model consists of the longitudinal–transverse–radial cochlear (ltering segments (CFS). The present model di8ers from the other ones in that the delay and (ltering of the propagating signal (i.e., changes in pressure) along the perilymph of the scala vestibuli is estimated in each CFS. The transverse (ltering circuits (TFC) in the model represent the nonlinear transduction processes making it possible to explain the signal compression, adaptation and principles of realization of the control over the characteristics of the e8erent circuits. The secondary (ltering circuit consisting of the primary (ltering circuits models the functioning of the cochlear (ltering structure at a certain approximation. It is shown that the signal compression, adaptation and

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the central e8erent control of the biological (ltering circuits and the whole multi-channel structure is possible due to the nonlinearity of the transduction processes in the OHC. The constructed model contributes to a better understanding of the nonlinear (ltering and e8erent control processes in the cochlea.

2. Materials and methods 2.1. The biological background of the model There is a general agreement that the sound signal delivered to the cochlea by the stapes footplate is transmitted mainly via pressure waves in the cochlear Juids. A di8erential Juid pressure between scala vestibuli and scala tympani acts on the Jexible BM producing a traveling wave. The site of the maximal displacement of the BM is related to the corresponding frequency of the input sound signal [36–40]. In the functioning of the cochlea, three directions of the signal transmission and three (ltering stages can be distinguished: 1. Longitudinal signal transmission in the perilymph along the scala vestibuli from the base to the apex of the cochlea with a delay and the narrowing band pass. 2. Transverse parallel transmission of the (ltered and delayed signal through the Reissner membrane from the perilymph of the scala vestibuli to the endolymph of the cochlear duct and the adaptive nonlinear primary transverse (ltering performed by the local passive cellular structures of the TM, BM and the organ of Corti. 3. The secondary radial (ltering and transmission of the pressure changes between the TM and BM to the cilia of the IHC produced by the transverse (ltering systems including OHC. The functional model of the (ltering system of the cochlea has to include all the above-mentioned directions of the transmission of changes in pressure and the corresponding (ltering circuits. In the scheme of the model (Fig. 1), CFS consists of the longitudinal delay and (ltering circuit, the primary transverse (ltering circuit TFC and the secondary radial (ltering circuit. The longitudinal delay and (ltering circuit of every CFS consists of delay elements (DE) and a second-order low-pass circuit (LPC). Since the OHC and IHC are arranged in a hexagonal array [4,41,42], it may be considered that the odd TFC consists of one OHC and the even TFC consists of two OHC. In addition to these cells, the transverse local passive cellular structures of the TM, BM, and the organ of Corti should be included into the TFC. The feedback circuit of the OHC is closed through these structures, represented by the band pass circuit (BPC). Therefore, the TFC can be considered as a model of the functioning OHC in the cochlea. Thus, the investigation of the properties of the TFC can contribute to a better understanding of the role of the OHC in the (ltering system of the cochlea. The OHC of the cochlea are located on the BM and the tips of their longest hairs are embedded in the TM. The initial stimulation of the OHC is produced by a shearing motion between the TM and BM evoked by the longitudinal–transverse pressure changes. These pressure changes are represented as the signal x(t) in the scheme of the TFC. The total local change in position of the TM and BM in the transverse direction in the TFC is the mechanical output signal y(t). The deJection

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A. Stasiunas et al. / Computers in Biology and Medicine 35 (2005) 495 – 510 Efferent control

To CN

sa

se

sa

Transverse filtering

From MSO

sa

se

Transverse filtering S

PG

S

dz (t ) z0

APC

z (t )

AMP

da (t )

se

Transverse filtering

c(t )

a (t )

y e (t ) EMT

TFC

TFC

MET

TFC

z m (t ) BPC

x k −1 (t )

y k −1 (t )

x k (t )

Longitudinal

LPC From Base

p(t − t k −1 )

x k +1 (t ) y k +1 (t )

y k (t ) filtering

and

LPC DE ∆t k

p(t − t k )

DE ∆t k +1

Radial

filtering

C k −1 C k −2 To IHC

delay

LPC

Ck

∑

p(t − t k +1 )

DE ∆t k + 2

To Apex

C k +1 Ck +2 rk (t )

Fig. 1. The functional model of the adaptive nonlinear cochlea (ltering structure: DE—delay element; LPC—longitudinal low-pass circuit; TFC—transverse (ltering circuit (BPC—band-pass circuit; AMP—nonlinear ampli(er; APC—adaptive low-pass circuit); EMT—electromechanical transducer; MET—mechano-electrical transducer; PG—pulse generator (initial segment); S—synapses; CN—cochlear nucleus; MSO—medial superior olive; sa —a8erent nerve (ber; se —e8erent nerve (ber.

of hairs back and forth opens and closes K + ion channels of the hairs resulting in synchronous changes of the cell membrane potential. The changes in the membrane potential in the OHC (the signal z(t) in the model) evoke molecular changes in the membrane resulting in shortening of the cell when depolarized and lengthening when hyperpolarized. The change in length of the OHC, or electro-motility [24], a8ects, as a mechanical feedback signal (the signal zm (t) in the model), the oscillations of the TM and BM supposedly amplifying them [19,20,43]. The mechano-electrical transduction (MET) process and the electro-mechanical transduction (EMT) process in the OHC are nonlinear and very fast [15,17,19,44–46]. In the model (Fig. 1), the deviations d z(t) of the membrane potential z(t) from the resting potential z0 are transduced into the nerve pulses by the synapse S at the initial segment (PG) of the a8erent nerve (ber sa . These pulses, together with the pulses generated by the neighboring OHC are

A. Stasiunas et al. / Computers in Biology and Medicine 35 (2005) 495 – 510

H (s )

499

c(t )

dz (t ) z0

APC da (t ) K (Y ) a (t )

z (t ) NC BPC x k (t ) L(s ) p(t )

DL tk

y k (t )

W (s )

LPC p(t − t k )

∑ rk (t )

Fig. 2. The functional electrical scheme of the adaptive nonlinear CFS: DL—delay line; H (s)—transfer function of the APC; W (s)—transfer function of the BPC; L(s)—transfer function of the LPC; K(Y )—transfer coeLcient of the NC; tk —delay time.

transmitted by the e8erent (bers to the cochlear nuclei. The e8erent (bers se from the MSO branch and make up synapses (S) with several OHC acting on the nonlinear (ltering circuit by their signal c(t) [47–51]. In this way the central (e8erent) control circuit is completed. It was shown that the OHC could change their length both rapidly and slowly [15,17,24,43,52–54]. Fast movements, which are most likely associated with the nonlinear (ltering, increase the sensitivity and selectivity (quality) of the local passive structures of the TM, BM and organ of Corti represented by the BPC in the model (Fig. 2). Slow movements are probably associated with the adaptation. The adaptation low-pass circuit (APC) in the model (Fig. 1) isolates the slowly changing component of the adaptation da(t) from the change in the membrane potential d z(t). This component reJects long-lasting changes in the intensity of the input signal and acting as a feedback signal evokes changes in the membrane potential of the OHC. The distribution of the hair cells in the organ of Corti suggests that the hairs of the IHC can be displaced by the changes in pressure of the endolymph between TM and BM produced by several OHC arranged in the radial direction. Thus the local pressure changes between TM and BM are transformed into the radial changes in pressure between the OHC and IHC resulting in displacement of the hairs of the IHC. In the scheme of the model (Fig. 1), the (ve weighted (weight Ck ) output signals of neighboring TFC are summated in the element . The resulting signal r(t) is the output signal of the CFS. The signals altogether comprise the output signals of the (ltering system of the cochlea. The number of the signals equals to the number of the IHC in the cochlea, and the number of the TFC is by one third less than the number of the OHC.

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2.2. Functional model of the adaptive nonlinear cochlea 7ltering segment (CFS) A simpli(ed functional electrical scheme of the adaptive nonlinear CFS of the model (Fig. 1) is shown in Fig. 2. In this scheme, the nonlinearities of the MET and EMT processes are modeled by the nonlinear circuit (NC). The BPC represents the local transverse passive elements of the cochlea. The APC responds to the slow components of changes d z(t) in the membrane potential z(t). The LPC is a longitudinal (ltering circuit. The signal of the e8erent control c(t) and the signal da(t) change the operating point of the nonlinear characteristic of NC in respect to the output signal y(t) of the BPC. 2.2.1. Longitudinal 7ltering and delay The pressure changes p(t) produced by the movements of the oval window in the perilymph of the scala vestibuli travels at the speed of several meters per second [40] and are (ltered. The LPC models a mechanical circuit consisting of the mass, sti8ness and viscosity of the perilymph and the sti8ness of Reisner’s membrane. The mass of the perilymph progressively increases while sti8ness decreases from the base to the apex, therefore the band-pass of every next segment of the LPC becomes narrower. It can be suggested that the (lters are of the second order and of the low-pass type. Therefore it is possible to model the structure of every CFS by the longitudinal transmission and (ltering of the signal by a serial connection of the delay line (DL) and the second order linear LPC. The output signal of the serial connections DL and LPC is expressed by a convolution of functions: x(t) = p(t) ∗ l(t − tk );

(1)

where l(t) is a weight function of the LPC. The equivalent transfer function of the DL and LPC connections is given by [55] Le (s) = L(s)exp(stk ); where s =  + j! is the complex variable, and tk is the delay time of the DL. If the delay time tk depends on the parameters of the LPC in the following way:
1 − 2k 1 arctg tk =
k fk 1 − 2k

(2)

(3)

then the serial connection of the DL and LPC (Fig. 2) is equivalent to the band-pass circuit. Here fk and k are the natural frequency (Hz) and damping factor of the LPC, respectively. Using the cochlea frequency map and (3) it is possible to (nd the velocity of the transmission of the signal spectral components: √ dk fk 1 − k √ 2 dk ; vk = = (4) 1−  tk arctg k k where dk is a distance from the base of the cochlea to the point being considered. As can be seen from (4), the velocity decreases with the decrease of both the frequency of the spectral components (according to the nonlinear cochlear frequency map), and the damping factor of the LPC. This decrease is nonlinear.

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501

2.2.2. Transverse 7ltering and e8erent control The TFC (Fig. 2) has three control loops: the loop of nonlinear (ltering, adaptation loop and e8erent control loop. The loop of the lowest hierarchical level of the TFC, which consists of the BPC and NC, implements the nonlinear (ltering of the signal. The properties of this loop were analyzed elsewhere [56]. The adaptation contour consists of the NC and APC, and the circuit of e8erent control consists of the APC and the higher nervous centers. As can be seen from Fig. 2, the output signal of the higher hierarchical level loop inJuences the control parameter of the lower hierarchical level. Let us admit that nonlinearities in the NC of the TFC (Fig. 2) reJect the overall nonlinear processes of the signal transduction in the OHC, and the NC characteristic is a sigmoid z = k tanh(y − a);

(5)

where a is the shift of the NC characteristic and k is the constant ampli(cation coeLcient. To avoid the shift of the operating point of the nonlinear characteristic by the low-frequency (direct) component, the BPC should not pass this component. To suppress the higher-order harmonic, the BPC should have a suLcient quality (the frequency selectivity). It follows that the BPC should be of a band-pass type. Let us admit, that the transfer function of the BPC is of the second order: !p s W (s) = 2 (6) s + 2p !p s + !p2 where !p is the natural frequency, and p is the damping factor. When p ¡ 1, the poles of the (6) are complex. Let us introduce the transfer coeLcient of the NC of the TFC for the (rst harmonic K(Y ) = kK(Y; a);

(7)

where k is the constant multiplier and K(Y; a) is the normalized transfer coeLcient of the NC. The K(Y; a) is de(ned by computer simulations and is presented in Fig. 3b. Having introduced the transfer coeLcient (7) of the NC and having assumed s = j! in expression (6), the properties of the TFC (Fig. 2) can be determined by employing a frequency transfer function We (j!; Y ) =

!p2

−

!2

j!!p ; + j!!p [2p − K(Y )]

(8)

where (for the stability of the TFC) K(Y ) ¡ 2p :

(9)

When exciting the TFC by the harmonic signal x(t)=X sin !0 t the frequency of which is !0 =!p , the amplitude Y of the harmonic output signal y(t) depends on the value We (j!p ; Y ) of the frequency transfer function (8). The value is found by inserting ! = !p into it. Then, at this input signal frequency the amplitude characteristic of the TFC can be expressed as follows: Y = Qe X;

(10)

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A. Stasiunas et al. / Computers in Biology and Medicine 35 (2005) 495 – 510

where

1 (11) 2p − K(Y ) is the equivalent circuit quality. For a low-level input signal of the NC, it can be considered that the transfer coeLcient K(Y ) is constant. In this case, the TFC is linear and the frequency transfer function (8) and its inverse Fourier transform (the weight function) fully describe the properties of the circuit. The adaptation takes place when the amplitude of the signal changes and remains changed for a longer time since the inert elements of the adaptation circuit (certain proteins) need some time to change their con(guration in response to slow changes in the membrane potential. Due to such inertia the rapid changes in the membrane potential have no signi(cant contribution to the adaptation. The APC in the CFS (Fig. 2) is an inert circuit and passes only the slow component da(t) of the change d z(t) of the NC output signal z(t). The slow component da(t) changes the parameter of the shift a(t) of the characteristic (5) of the NC in accordance to the input signal y(t). The output signal y(t) of the TFC (Fig. 2) can be expressed by the following equations: Qe =

y(t) = (x(t) + z(t)) ∗ w(t);

(12)

z(t) = tgh(y(t) − a(t));

(13)

a(t) = −(da(t) + c(t));

(14)

da(t) = d z(t) ∗ h(t);

(15)

d z(t) = z(t) − z0 ;

(16)

z0 = tgh(−c);

(17)

c(t) = "(d z(t); ?);

(18)

where x(t) is the input signal (1), z(t) is the feedback signal, a(t) is the general control signal, d z(t) is the change in the receptor potential, da(t) is the adaptation signal, c(t) is the e8erent control signal, z0 is the resting membrane potential, w(t) is the weight function of the BPC, and h(t) is the weight function of the APC. The dependence of the adaptation component (15) on the amplitude Y of the harmonic input signal of the NC at the initial shift of the transfer characteristic (5), a(t) = c = 0:725, is explored in the experimental section (Fig. 4b). The dependence was obtained by iterative computer modeling. The mechanism of the e8erent control over the characteristics of the TFC is similar to that of the adaptation loop. Its action, however, is signi(cantly faster, since the signal formed by this mechanism c(t) shifts the working point of the NC directly and changes the characteristics of the TFC. The input signal of the e8erent control loop is the signal d z(t), which is the change of the output signal z(t) from the resting potential z0 . On the basis of d z(t) and the functional logic of the higher auditory centers, the e8erent control signal c(t) is produced. How this signal is formed in response to d z(t) and what is the speci(c logic of the higher auditory centers is not known. It is believed that the e8erent signals protect biological (ltering structures of the cochlea (by reducing the sensitivity for several milliseconds) from being damaged by short strong sounds [57]. We can assume, therefore, that e8erent signals are able to alter characteristics of the transduction processes of the cells.

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503

2.2.3. Longitudinal–transverse 7ltering The frequency transfer function of the transverse–longitudinal (ltering is given by serial connection of the (ltering functions of the longitudinal and the transverse parts of the CFS (Fig. 2): T (j!; Y ) = L(j!)We (j!; Y )exp(j!tk );

(19)

where L(j!) and We (j!; Y ) are frequency transfer functions of the LPC and the TFC, and tk is the delay time of DL (3). At the low-level of the input signals of the NC the function We (j!; Y ) does not depend on Y . 2.2.4. Longitudinal–transverse–radial 7ltering The CFS are distributed from the base to the apex of the cochlea. Their characteristic frequency decreases approximately according to the logarithmic law [1,4,36,40,58]. We consider that the signal acting on the hairs of the IHC consists of the sum of the longitudinal –transverse output signals (Fig. 2) of (ve neighboring CFS segments rk (t) =

i=2 

Ck+i yk+i (t);

(20)

i=−2

where k = 3; 5; 7; : : : ; n − 2, n is number of the CFS, and the following weight coeLcients for the summation are chosen c 2c Ck −2 = Ck+2 = ; Ck −1 = Ck+1 = − ; Ck = c; (21) 6 3 where c is a positive value, and the sum of the weight coeLcients is zero. In (21), the opposite signs of the weight coeLcients of the summation of the neighboring TFC represent the lateral inhibition. After introducing the transfer coeLcient (7) and the frequency transfer function (8), the dependence of the amplitude characteristic (10) and the quality (11) on the amplitude of the input signal was obtained by iterative computer modeling. These dependencies obtained during the adaptation and without adaptation are shown in the next section (Fig. 3). For the low level input signals of the NC, the methods of analysis of linear systems are applicable for (nding the characteristics (19) and the output signals (20) of the (ltering system of the cochlea. The characteristics are given in the next section (Fig. 4). 3. Experimental tests The main circuits of the multi-channel (ltering system are the TFC. As can be seen from (10) and (11) the sensitivity and selectivity of these circuits depend nonlinearly on the amplitude of the input signal due to nonlinearity of the transduction processes in the OHC. The adaptation and control of the (ltering characteristics by a8erent signals from the higher nerve centers are possible due to the same reason. The dependence of the amplitude characteristics and quality of the TFC in the nonlinear steady state on the input signal amplitude is calculated using the previously obtained relation of the transfer coeLcient of the NC (7) and Eqs. (10) and (11). The characteristics of the longitudinal–transverse (LT) part of the CFS without adaptation (Figs. 3a, c and e) and with adaptation (Figs. 3b, d and f)

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A. Stasiunas et al. / Computers in Biology and Medicine 35 (2005) 495 – 510 K(Y,a)

K(Y,a)

Without adaptation 0.6

0.4

a(t)=c+da(t) c=0.725 da(t)=0

0.2 0

−20

(a)

Amplitude

Amplitude

0.6 k=1

0

20

3

1 2 −60

TFC quality

−40

Amplitude

a(t)=c+da(t) 1. k=1.571 c=0.725 da(t)=0

−40

(c)

2. k=1.547 3. k=1.476 4. k=0.476

4 −20 0 X, dB

4

4

2 0

0

20

40

Y, dB

20

k=1.547 k=1.476 5

−20

(d)

15

−60

−40

2. k=1.547 3. k=1.475 4. k=0.476

3 2 −60

40

a(t)=c+da(t) c=0.725 da(t)=0

10

20 a(t)=c+da(t) c=0.725 0 1. k=1.571

−40

4 −20 0 X, dB

20

40

k=1.571 k=1.547

10 a(t)=c+da(t) c=0.725

k=1.476 5 k=0.476

k=0.476

(e)

4

1,2,3

−20

−20

k=1.571

15

0

6

40

TFC quality

Amplitude

0

0.2

(b)

Y, dB

−20

0.4

da(t)

a(t)=c+da(t) c=0.725 1. k=1.571 2. k=1.547 3. k=1.476 4. k=0.476

0

40

40 20

With adaptation

−20 X, dB

0

20

0

40

(f)

−60

−40

−20

0

20

40

X, dB

Fig. 3. Characteristics of the transverse part of the CFS operating in the adaptive nonlinear mode: Transfer coeLcient of the NC without adaptation (a) and with adaptation (b). TFC amplitude characteristics without adaptation (c) and with adaptation (d). TFC quality without adaptation (e) and with adaptation (f). Adaptation signal da(t) (b, the right-hand side).

were obtained by computer modeling. To calculate the characteristics, the following values of the transfer function parameters of the BPC (6) were used in (8) and (11): !p = 2&95; p = 0:5. Fig. 3 shows that the amplitude characteristics of the TFC are linear and the quality is constant at low input amplitudes Y of the NC. With increasing amplitude, there is a transition to the nonlinear (compressive) mode. By shifting the operating point a in the nonlinear part of the transfer characteristic of the NC (5) it is possible to change the sensitivity and selectivity of the TFC. On the basis of the MET characteristic of the hair cells [19], it is reasonable to assume that the initial operating point is in the range a = (0:7–0:8). If the amplitude Y is changed for a longer time, the adaptation begins, because the APC responds to the slow change in signal amplitude and produces the adaptation signal da(t) (15). This signal summates with the initial shift a and evokes changes in the sensitivity and selectivity of the TFC. The adaptation time and the stepwise change in signal amplitude are determined by the duration of the transient response of the APC. Fig. 3b (right-hand side) shows how the adaptation signal da(t) depends on the amplitude Y of the NC input signal. This dependence was obtained by computer simulation using Eqs. (12)–(18) and taking a = c = 0:725. Rapid changes of the signal do not inJuence the adaptation since the APC does not pass them. Therefore nonlinear (ltration proceeds at a new slowly changing shifted value of the operating point (14).

A. Stasiunas et al. / Computers in Biology and Medicine 35 (2005) 495 – 510

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Amplitude frequency responses

15 LT filtering 10 5 0

100

110

120

(a)

130

140

150

f, Hz LTR filtering

6 4 2 0

100

110

120

(b)

130

140

150

f, Hz

functions

1

Inverse LTR filtering

−1 0

0.1

0.2

0.3

t, s

(d)

0

1 LTR filtering −50

−100

Weight

Exitatio level, dB

50

LT filtering

0

2

2.05

2.1

(c)

0 −1

2.15

(e)

0

0.1

0.2

0.3

t, s

Fig. 4. Frequency characteristics of the LT (a) and LRT (b) parts of the CFS. Inverse frequency characteristics of the LRT parts (c). Transient responses of the LT part (d) and the LTR part (e) of the corresponding characteristics shown by bold lines in (a) and (b).

During the a8erent control, the signal c(t) changes the shift a(t) directly and without the inertia. This evokes changes in the characteristics and the amplitude of the output signal of the TFC (the input signal of the NC). The TFC response consists of a fast and slow component. The fast component is a direct response of the TFC to the e8erent signal. The slow component is a result of the TFC adaptation to the e8erent signal. The graphs show that at low input signals the TFC acts as a linear circuit. Therefore the properties of the TFC are fully characterized by the transfer and weight functions. By assigning a constant value to the transfer coeLcient K(Y; a) at low NC input signals and on the basis of Eqs. (8), (19) and (20), the transfer function of the CFS can be expressed as a sum Rk (s) =

i=2 

Ck+i Lk+i (s)Wk+i (s):

(22)

i=−2

The inverse Laplace transformation of this function [55] gives the transient characteristics (weight functions) of the CFS in the linear mode.

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Assuming s = j!; p = 0:5; k = 0:5, and !p = !k = 2&95 in Eq. (22) the frequency characteristics and the transient responses of the longitudinal fragment of the cochlea (ltering system were found in a linear mode by computer modeling (Fig. 4). Fig. 4a shows a family of frequency characteristics of the longitudinal–transverse (LT) (ltering structures of the CFS. Fig. 4b shows a family of frequency characteristics of the radial output (LTR) of the (ltering structures of the CFS. Inverted characteristics (Fig. 4c) of the radial output (Fig. 4b) resemble the selectivity curves of the a8erent nerve (bers of the IHC [8,19]. Due to the inJuence of the neighboring TFC, the LTR frequency characteristics are signi(cantly more selective than the LT characteristics. Some frequency characteristics of the CFS are shown in Figs. 4a and b by bold lines. Corresponding transient responses for LT (ltering are shown in Fig. 4d and for the LTR (ltering in Fig. 4e. Due to the interaction of the neighboring TFC, the envelopes of the transient responses of the radial output (LTR (ltering) as well as the frequency characteristics of the CFS are similar to Gaussian. It is necessary to stress that the product of the width of the frequency characteristic of such band-pass (lters and the duration of the transient response is close to the theoretical minimum. 4. Discussion The proposed 3D-model of the (ltering system of the cochlea is represented by a serial-parallel (ltering structure consisting of the basic elements, the CFS. The longitudinal part of the CFS of the model assesses changes in pressure produced in the perilymph by the mechanical oscillations of the oval window, the delay of the transmission of the changes to the endolymph and the low-pass (ltering. The transverse part of the CFS, the transverse (ltering circuit (TFC), represents the adaptive nonlinear (ltering structure in the cochlear duct consisting of local passive and one or two active elements (OHC) controlled by the e8erent signals. The oscillations in the endolymph between the TM and the organ of Corti produced by the central TFC (containing two OHC) in the radial direction summate with the oscillations produced by two neighboring TFC and move the hairs of the IHC forcing them to transduce the mechanical oscillations into nerve impulses. Therefore the proposed 3D-model of the (ltering system of the cochlea di8ers from the cochlear partition models, which do not include adaptation and e8erent control processes. The longitudinal (ltering structure of the CFS in the model consists of the DL and LPC of the second order. The delay time depends on the transmission velocity of changes in pressure in the perilymph and the distance of the TFC from the base of the cochlea. The mass of perilymph increases and the sti8ness decreases with the distance from the base of the cochlea, therefore the higher frequencies are progressively suppressed. This kind of (ltering is performed by the second-order LPC in every CFS of the model. If the delay of transmission of changes in pressure in the perilymph is matched with the damping factor of the LPC (3), then the serial connection of the DL and LPC (Fig. 2) becomes the band-pass circuit. Therefore during the transmission of changes in pressure in the perilymph through Reissner’s membrane the frequency components are distributed along the cochlear frequency map. Obviously, a rough band-pass (ltering is performed. Using the cochlear frequency map and (4) it is possible to (nd the dependence of the velocity of transmission of the spectral components along the cochlea on the CF. The law of the distribution of di8erent frequencies

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507

in the cochlea is approximately logarithmic. It is linear in our model (Fig. 4a). However, the main structure of the selective (ltering controlled by the e8erent signals is located in the endolymph. This parallel structure in our model is composed of separate TFC (Fig. 1). The TFC simulates the nonlinear interaction of the local transverse passive structures of the cochlea with the active OHC. The OHC, as a part of the positive feedback circuit of the passive structures, increases the frequency selectivity and amplitude sensitivity. These (ltering characteristics are under control due to a8erent connections with the CN and e8erent connections with the MOC. The main role in the control process and adaptation is played by the nonlinearity of the signal transduction in the OHC, and the position of the initial operating point in the nonlinear characteristic of the NC. In choosing this point we relied on the resting potential of the cell membrane. The dependence of the operating point on the input signal amplitude is the basis of the compression and adaptation. The dependence on the e8erent signal is the basis of the central control. This is con(rmed by the experimental investigation of the nonlinear mode of the model (Fig. 3). The experimental results show that the TFC can be considered as linear at low input signals. A transition to the compression mode occurs with the increase in amplitude of the signal. The adaptation ensues when the amplitude is changed for a longer time. Then component (14) appears in the output of the APC, which shifts the operating point and the (ltering characteristics of the TFC. In the linear mode and for fast changes of the signal the adaptation does not take place. A very fast EMT response produced in the OHC by a protein prestin [25,26] is essential for the nonlinear (ltering. The slow motor proteins are needed for the adaptation (Fig. 2). We do not know which proteins are responsible for this and where they are located (in the lateral cell membrane or in the hairs). At low amplitudes of the TFC input (i.e. in the linear mode) the properties of the longitudinal– transverse part of the CFS are characterized by the transfer function and its inverse Laplace transformation (the weight function). By using these functions, the frequency and temporal characteristics of the CFS comprising a fragment of the cochlea were obtained (Fig. 4). The frequency characteristics of the longitudinal–transverse parts or the CFS are undistinguished in selectivity (Fig. 4a). However, the characteristics of the radial outputs of the CFS (20) a8ecting the IHC are signi(cantly more selective (Fig. 4b). Their inverse frequency characteristics (Fig. 4c) are very similar to the selectivity curves of the nerve (bers [4.8]. The results of our mathematical-computer modeling of (ltering in the cochlea show that the essential role is played by the OHC. Due to asymmetrical nonlinearity of the transduction processes in the OHC, the compression occurs at higher input amplitudes. At longer lasting changes in signal intensity the adaptation occurs. The e8erent control over the nonlinear asymmetry (the shift of the operating point) of transduction processes in the OHC can change the selectivity and sensitivity of the (ltering system. The presented model contributes to a better understanding of the (ltering processes in the cochlea. 5. Summary An adaptive nonlinear signal (ltering model of the cochlea is proposed based on the established functional properties of the inner ear and on some assumptions. The model consists of longitudinal –transverse–radial CFS. A signal delay and low-pass (ltering in the perilymph of the scala vestibuli

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and Reissner’s membrane are estimated in the longitudinal part of the CFS. Nonlinear transduction processes in the OHC, the local (ltering properties of the tectorial membrane (TM), basilar membrane (BM) and passive cells of the organ of Corti, in addition to the inJuence of a8erent and e8erent connections of the OHC with the higher auditory centers on the characteristics of the CFS, are estimated in the transverse part of the CFS. The output of the radial part of the CFS is given by the weighted sum of the output signals of the (ve adjacent longitudinal–transverse parts of the CFS. This signal inJuences hairs of the inner hair cells (IHC). The number of longitudinal–transverse– radial CFS is equal to the number of the IHC in the cochlea. On the basis of an analytical description of di8erent parts of the model and the results of computer modeling, the biological signi(cance of the nonlinearity of signal transduction processes in the OHC, their role in signal compression and adaptation, the e8erent control over the characteristics of the (ltering structures (frequency selectivity and sensitivity) are explained. Acknowledgements We gratefully acknowledge the support of the Royal Swedish Academy of Sciences. References [1] G. Von Bekesy, Experiments in Hearing, McGraw-Hill, New York, 1960. [2] A.W. Gummer, J.W. Smolders, R. Klinke, Basilar membrane motion in the pigeon measured with the Mossbauer technique, Hear. Res. 29 (1987) 63–92. [3] A.W. Gummer, W. Hemmert, H.P. Zenner, Resonant tectorial membrane motion in the inner ear: its crucial role in frequency tuning, Proc. Natl. Acad. Sci. USA 93 (16) (1996) 8727–8732. [4] J.P. Kelly, Hearing, in: E.R. Kandel, J.H. Schwartz, T.M. Jessell (Eds.), Principles of Neural Science, 3rd Edition, McGraw-Hill, New York, 1991, pp. 481–499. [5] S.M. Khanna, M. Ulfendahl, C.R. Steele, Vibration of reJective beads placed on the basilar membrane, Hear. Res. 116 (1998) 71–85. [6] E. Zwicker, Subdivision of the audible frequency range into critical band (Frequenzgruppen), J. Acoust. Soc. Am. 33 (1961) 248. [7] E. Zwicker, A model describing nonlinearities in hearing by active processes with saturation at 40 dB, Biol. Cybernetics 35 (1979) 243–250. [8] N.Y. Kiang, E.C. Moxon, Tails of tuning curves of auditory-nerve (bers, J. Acoust. Soc. Am. 55 (1974) 620–630. [9] G.K. Yates, I.M. Winter, D. Robertson, Basilar membrane nonlinearity determines auditory nerve rate-intensity functions and cochlear dynamic range, Hear. Res. 45 (1990) 203–220. [10] G.K. Yates, Basilar membrane nonlinearity and its inJuence on auditory nerve rate-intensity functions, Hear. Res. 50 (1990) 145–162. [11] D.C. Mountain, A.E. Hubbard, Computational analysis of hair cell and auditory nerve processes, in: H.L. Hawkins, T.A. Mullen, A.N. Popper, R.R. Fay (Eds.), Auditory Computation, Springer, Berlin, 1993, pp. 121–156. [12] X. Zhang, M.G. Heinz, I.C. Bruce, L.H. Carney, A phenomenological model for the responses of auditory-nerve (bers: I. Nonlinear tuning with compression and suppression, J. Acoust. Soc. Am. 109 (2001) 648–670. [13] D.T. Kemp, Stimulated acoustic emissions from within the human auditory system, J. Acoust. Soc. Am. 64 (1978) 1386–1391. [14] E. Zwicker, “Otoacoustic” emissions in a nonlinear cochlear hardware model with feedback, J. Acoust. Soc. Am. 80 (1986) 154–162. [15] J.F. Ashmore, A fast motile response in guinea-pig outer hair cells: the cellular basis of the cochlear ampli(er, J. Physiol. 388 (1987) 323–347.

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[47] J.J. Art, A.C. Crawford, R. Fettiplace, P.A. Fuchs, E8erent regulation of hair cells in the turtle cochlea, Proc. R. Soc. London B. Biol. Sci. 216 (1982) 377–384. [48] H. Spoendlin, Anatomy of cochlear innervation, Am. J. Otolaryngol. 6 (1985) 453–467. [49] P. Bon(ls, M.C. Remond, R. Pujol, E8erent tracts and cochlear frequency selectivity, Hear. Res. 24 (1986) 277–283. [50] R. Pujol, Lateral and medial e8erents: a double neurochemical mechanism to protect and regulate inner and outer hair cell function in the cochlea, Br. J. Audiol. 28 (1994) 185–191. [51] J.J. Guinan, Psysiology of olivocochlear e8erents, in: P. Dallos, A.N. Popper, R.R. Fay (Eds.), The Cochlea, Springer, Berlin, 1996, pp. 435–502. [52] A.C. Crawford, R. Fettiplace, The mechanical properties of ciliary bundles of turtle cochlear hair cells, J. Physiol. (London) 385 (1985) 207–242. [53] Y.C. Wu, A.J. Ricci, R. Fettiplace, Two components of transducer adaptation in auditory hair cells, J. Neurophysiol. 82 (1999) 2171–2181. [54] P. Martin, A.J. Hudspeth, Compressive nonlinearity in the hair bundles active response to mechanical stimulation, Proc. Natl. Acad. Sci. USA 98 (25) (2001) 14386–14391. [55] M.F. Gardner, J.L. Barnes, Transients in Linear Systems, Wiley, New York, 1942. [56] A. Stasiunas, A. Verikas, P. Kemesis, M. Bacauskiene, R. Miliauskas, N. Stasiuniene, K. Malmqvist, A nonlinear circuit for simulating OHC of the cochlea, Med. Eng. Phys. 25 (2003) 591–601. [57] R.B. Patuzzi, M.L. Thompson, Cochlear e8erent neurones and protection against acoustic trauma: protection of outer hair cell receptor current and interanimal variability, Hear. Res. 54 (1991) 45–58. [58] R.B. Patuzzi, Cochlear micromechanics and macromechanics, in: P. Dallos, A.N. Popper, R.R. Fay (Eds.), The Cochlea, Springer, Berlin, 1996, pp. 186–257. Antanas Stasiunas received his Ph.D. degree in signal processing in 1971 from Kaunas University of Technology, Lithuania. He became Associate Professor at the same university (Department of Control Systems) in 1978. His research interests include signal transmission and processing, process control, speech and hearing system analysis. Antanas Verikas is currently holding a professor position at both Halmstad University Sweden and Kaunas University of Technology, Lithuania. His research interests include image processing, pattern recognition, neural networks, fuzzy logic, and visual media technology. He is a member of the International Pattern Recognition Society, European Neural Network Society, and a member of the IEEE. Povilas Kemesis is professor emeritus at Kaunas University of Technology, Lithuania. The (elds of his research interests are speech recognition, image analysis, process control, and neural networks. Marija Bacauskiene is a senior researcher in the Department of Applied Electronics at Kaunas University of Technology, Lithuania. Her research interests include neural networks, image processing, pattern recognition, and fuzzy logic. Rimvydas Miliauskas received his Ph.D. degree in Neurophysiology in 1970 from Kaunas University of Medicine, Lithuania. He became Associate Professor at the same university in 1993. The main (eld of his scienti(c interests is the electrical activity of the brain and nerve cells. Natalija Stasiuniene received her Ph.D. degree in Biochemistry in 1969 from Kaunas University of Medicine, Lithuania. She became Associate Professor at the same university (Department of Biochemistry) in 1976. The main (elds of her scienti(c interests are molecular processes in the cells. Kerstin Malmqvist received a Ph.D. degree in Applied Mathematics and Computer Science in 1980 at UmeaV University, Sweden. Since 2002 she is head of the Intelligent Systems Laboratory at Halmstad University, Sweden where her research is focused on methods and tools for colour measurements in multicolour printing. She is a member of the board of VISIT, a national program for research in Visual Information Technology.