Quantitative Evaluation of Limit-Cycle Oscillator ...

Talmadge, C .L., Long, G. R., Murphy, W. J. & Tubis, A. (1990). Quantitative evaluation of limitcycle oscillator models of spontaneous otoacoustic emissions. In P. Dallos, C. D. Geisler, J. W. Matthews, M. Ruggero & C. R. Steele (Eds.), Mechanics and Biophysics of Hearing (pp. 235-242). Springer Verlag, New York.

Quantitative Evaluation of Limit-Cycle Oscillator Models of Spontaneous Otoacoustic Emissions Carrick L. Talmadge,1 Glenis R. Long,2 William J. Murphy,1 and Arnold Tubis1 1 Department

of Physics; Purdue University, West Lafayette, IN 47907 2Department of Audiology and Speech Sciences; Purdue University, West Lafayette, IN 47907

Introduction Single limit-cycle-oscillator models of spontaneous otoacoustic emissions (SOAE’s) axe based on the assumption that the pattern of interactions between spontaneous emissions and external tones in the ear canal may be partially de scribed by the gross compaction of a full cochlear model to a single nonlinear dif ferential equation such as that of a free (or driven) Van der Pol oscillator. Such an equation incorporates, in a highly idealized way, the type of nonlinear-active damping which, if assumed to be present over certain portions of the cochlear par tition, would produce stabilized cochlear self-oscillations and lead to measurable spontaneous emissions in the ear-canal. These models have been used successfully by our group and by Wit and collaborators to account for a number of features of the emission data including: a) statistical properties of emissions (e.g. Bialek and Wit, 1984; Wit, 1986; van Dijk, 1990); b) suppression of emissions (Long and Tubis, 1990) and synchronization (phase locking) of emissions by external tones (e.g., van Dijk, 1990; Long, et al., 1990); and c) reduction of the level of emissions by aspirin consumption (e.g., Long and Tubis, 1988a,b). In this paper, we describe several studies which are designed to probe the validity of these models. In particular we obtain two independent estimates of the active damping parameter of a Van der Pol model of an SOAE from: 1) data on the dynamical interactions of SOAE’s with external tones during suppression and the release from suppressions (Schloth and Zwicker, 1983; Dallmayr, 1983, 1985); and 2) the details of the frequency-locking or synchronization tuning curves of these SOAE’s.

Theoretical Results The single Van der Pol oscillator model of a spontaneous emission, in isolation or in interaction with an external tone, is described by the differential equation, y + { - r i + r 2 y 2 )y + u ly = E cos u t, 235

(1)

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Quantitative Evaluation of Limit-Cycle Oscillator Models

where y is interpreted as the incremental pressure in the ear canal, the approximate limit cycle frequency uo is associated with the angular frequency of a spontaneous emission, ri and r2 are respectively the active and nonlinear damping components, t is time, u is the angular frequency of the external tone and E is proportional to the external tone transducer voltage. In order to have a limit-cycle solution of Eq. (1), both r\ and T2 must be positive, in which case the approximate solution in the absence of external driving (E = 0) is (Nayfeh, 1973, pp. 245-248) y = a0 cos(u;of + 0 0),

(2)

where o is a constant phase and the amplitude ao is given by

-

=

V

5

-

( 3 )

To the extent that Eq. (1) is a reasonable descriptor of an SOAE, it provides several independent means for obtaining estimates of the active parameter rx. One of these is based on the observation of the temporal behavior of the emission’s level during external-tone suppression and following release from suppression (Zurek and Clark, 1981; Schloth and Zwicker, 1983). The other is based on an analysis of the frequency locking tuning curves (Zwicker and Schloth, 1984; Long and Tubis, 1988a; 1990).

Relaxation Dynamics of the Van der Pol Oscillator In order to study the relaxation dynamics of the Van der Pol oscillator, we consider the application of a square-wave modulated external tone with frequency u sufficiently far away from u>o and with a duty cycle sufficiently long for fully developed suppression and recovery from suppression to occur. By applying the Krylov-Bogoliubov method of averaging (c.f., Nayfeh and Mook, 1979; Hanggi and Riseborough, 1983) to Eq (1), we obtain the approximate results, y(t) = a(t) cos(u;o£ + ), a(t) = - r j a ^ ) 1 _ A =

2£

(4) -

a(t)2l

------ j , Uq —u>1

(5) (6)

—

where is a constant. We consider first the case of suppression of an emission by an external tone [a(0) = ao, and a(t —* 00) = kao], for which the solution of Eq. (5) is \ a (0 = k2 =

/CC&0 / 2 yj 1 + (k2 — l1)\e x p (( - r i K ^2+T’ )

/m\ ^^

1 - 2A2/ajj.

(8)

A.

For recovery from suppression by the external tone of the emission [a(0) = Acao, a(f —►00 ) = ao], we have do 236

Talmadge et al.

Quantitative Evaluation of Limit-Cycle Oscillator Models

FIGURE 1 Level of the Van der Pol oscillator during one modulation cycle (dashed line) of the suppressing tone for various levels of the ex ternal tone.

10

20

30

40

50

60

70

80

90

100

M The behaviors given by Eqs. (7) and (9) are plotted in Fig. 1. It can be shown that the exponential factor r\ in the recovery behavior of Eq. (9) is unchanged if the dynamical model of SOAE’s given by Eq. (5) is replaced by ji+ ( - n + Ti\y\r )y -y

( 10)

= E co su t,

with p > 0. On the other hand, the exponential factor n 2 r\ in the suppression behavior of Eq. (7) depends directly on the form of the nonlinear component of the damping. We conclude that fits to data on the recovery from suppression are insensitive to the specific form of the nonlinear damping of the underlying limit-cycle oscillator model, in contrast to the case of fits to onset-of-suppression data. So far we have only considered models which incorporate linear stiffness. We relax this assumption by modeling SOAE’s with the differential equation, y + ( - n + ^2y2)y + u l(y + ey3) = E cosu t.

( 11)

Applying the Krylov-Bogoliubov method, we find y(t) m a(t) cos(u0t -f (t)),

(12)

a(t) = - r i a(t)

(13)

Ssoqujo 8

A2 2—T + On

fln 21 a(t)

(14)

at,

We see from Eq. (13) that nonlinear stiffness does not affect the amplitude relaxation process to first order. There is, however, a transient effect on phase during the process of suppression or relaxation of the oscillator. The solution of Eq. (14), when a suppressing tone is turned on at t = 0 and 0(0) = o is given by 3 0(f) = -eagu/o (1 - n2)t + — log ri 237

, n r 1t

+ K —1

+ 00-

(15)

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Quantitative Evaluation of Limit-Cycle Oscillator Models

Finally, the solution during relaxation, when the suppressor is turned off at t = 0 and again 0(0) = o, is =1| £ £ o ^ i 0g (K2er,t + i _ k2) + a ri

^

Frequency Locking Tuning Curves We consider frequency locking tuning curves which are obtained from the re sponse of the SOAE to the application of an external tone, which is swept slowly in frequency through a region containing the frequency of the emission. The fre quency at which the locking(unlocking) occurs for a Van der Pol oscillator is given by the approximate expression (Nayfeh, 1973, pp. 250-253; Hanggi and Riseborough, 1983), 2

(w

u/o)2 +

^

-

2

E2

2f l r

(17)

M)u0

for |w —wo | ^ 0.28864ri and by

r\

(18) 2

-

Ii - 12( ^ - ^ o ) M ri

27E 2 2r?uf o l '

for |w — wo | ^ 0.28864ri. In Eqs. (17) and (18), E is interpreted as the force on the SOAE oscillator due to the external tone. Unfortunately, E is not a directly measurable quantity, but instead is a function of Pe, the instantaneous pressure in the ear canal. For this preliminary study, we assume linearity of E(Pe), so that we can rewrite Eq. (17), for example, as (w —wo)2 +

= k P 2,

(19)

where \/& is the assumed constant of proportionality. Eq. (19) and the analogous equation to Eq. (18) can be fit to the tuning curve data for the parameters wo, r\ and k.

Methods The levels of spontaneous otoacoustic emissions and external tones in the ear canal were detected by an Etymotic Research ER-10 microphone and evaluated by a Wavetek 5820A Spectrum Analyzer. An Etymotic Research ER-2 tubephone attached to the ER 10 assembly was used to transduce the external tones. In order to model the suppression and recovery of a spontaneous otoacoustic emission, the transient behavior of an SOAE was measured in response to a pulsed (168.34 msec on/off - 2 msec rise/fall) sinusoid generated by a Hewlett Packard 3325A function generator. The ear canal signal was amplified and bandpassed filtered (Q = 1) by an Ithaco 3961 Lock-in amplifier centered at the frequency of the emission before being 238

Talmadge et al.

Quantitative Evaluation of Limit-Cycle Oscillator Models 1100

FIGURE 2 Effect of a pulsed external tone on the level of an SOAE for subject LN. Here (a), (b), and (c) were obtained with different levels of the external tone (see Table 1).

1000

900 800 700

'cO 500 400 300 200

100 0

50

100

150

200

250

300

350

t (ms) TABLE 1 Results for fitting the experimentally measured response of the level of an SOAE for subject LN to a pulsed external tone, as described in the text.

Data Set

Level of Suppressor

(a) (b) (c)

37 dB SPL 38 39

—

—

dB Suppression -3.2 dB -6 .4 -11.2

i/n (suppression) 3.12 ms 2.48 1.28

i/n (recovery) 5.39 ms 5.41 4.87

(2.29 ± 0.54) ms (5.22 ± 0.18) ms

—

sampled at 100 kHz by a Data Translation DT2827 16 bit A/D board installed in a Zenith 386/16 PC compatible computer. Each data set was analyzed on a NeXT computer by digital filtering using a time domain finite response band pass filter centered at the emission frequency and averaged over 62 repetitions. The level and frequency of an external tone needed to synchronize a sponta neous emission was determined by recording the RMS level of the sound pressure in the ear canal in the presence of a sweeping external tone generated by a Hewlett Packard 3325A function generator. The frequency range of the sweep was chosen to be 30 Hz, and the time of sweep was 50 seconds. The ear canal signal was filtered and analyzed by a Bruel and Kjaer 2010 heterodyne analyzer. A DC voltage propor tional to the log RMS level within a filter (31.6 Hz) centered around the frequency of the sweeping stimulus was stored on a Tektronix 2230 digital storage oscilloscope and transferred to the Zenith 386 computer and the NeXT computer for further analysis. The frequencies of the sweeping tone at which the RMS level changed from regular beating to stability and back to regular beating were determined for each level of the external tone. The results presented here were obtained from two human subjects. The emis239

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Quantitative Evaluation of Limit-Cycle Oscillator Models

FIGURE 3 Effect of a pulsed external tone on the level of an SOAE for subject SS. Here (a), (b), (c), and (d) were obtained with dif ferent levels of the ex ternal tone [see Table

2]-

200

250

350

t (ms)

TABLE 2 Results for fitting the experimentally measured response of the level of an SOAE for subject SS to a pulsed external tone, as described in the text.

Data Set

Level of Suppressor

(a) (b) (c) (d)

39 dB SPL 40 41 42

—

—

dB Suppression -4 .0 dB -5 .5 -9 .0 -11.5

1/ri (suppression) 3.21 ms 2.66 2.99 2.08

1A i (recovery) 5.66 ms 5.32 6.06 6.31

(2.74 ± 0.25) ms (5.84 ± 0.22) m

—

sion studied from subject LN’s left ear had a typical frequency of 4285-4310 Hz, and an amplitude of 13-15 dB SPL. The emission studied from subject SS’s left ear had a typical frequency of 3445-3465 Hz and an amplitude of 11-14 dB SPL.

Results Fig. 2 shows the suppression and relaxation behavior of an SOAE for one subject (LN) for 3 different levels of a pulsed external tone. We observe that the qualitative behavior of these curves is similar to those for an isolated Van der Pol [see Fig. 1], These data were fit to Eq. (7) during suppression and Eq. (9) during recovery from suppression, and the results of these fits are presented in Table 1. We also show the results for another subject (SS) in Fig. 3 and Table 2. In a preliminary analysis, an independent estimate of r\ determined by fitting Eqs. (17) and (18) to the frequency locking tuning curve shown in Fig. 4. We obtained the average value of 1 /n = (9.8 ±2.7) ms using only the lowest three levels 240

Talmadge et al.

Quantitative Evaluation of Limit-Cycle Oscillator Models TTTTTTTTTTTTTTTTTTTT-

I I I I

t

2 dB

r l1,,ir f OdB

r

—

I T I I

1^ =9.812.7 018

4 dB

J........

I I I I I I I I I I | |

SS Left Ear 6-5-90

50 second sweeps

___

.. T

*n

t

•

t

-2 dB

-4dB

I

.6 dB

f

FIGURE 4 Frequency locking tuning curve obtained from a sweep ing external tone. Here the arrows show the values chosen for the point of entrainment or loss of entrainment.

£ t

i ii ii ii ■ i t> i* i' t* i* i*_i i_i i_— i «—i >i *i ii *i ii *i —i *—i •—i *— i *—i ■—i *—i —i ■— i i i i j i *i i i ii ii »i i * ■

3450

3455

3460

3465

34703475

Frequency (Hz)

3480

shown. There was a marked systematic increase with level of the external tone (if ail values were used, a values of Xjffi 2* 40.3 ms was obtained). This systematic increase may be due, in part, to the inadequacy of the first-order theory for high levels of the external tone (as has been demonstrated in numerical simulations). We are currently studying this along with the effects of noise on the frequency locking tuning curve.

Discussion In this paper we have shown that the frequency locking tuning curve and the dynamics of the recovery from suppression of an SOAE are correlated in the way implied by a simple Van der Pol model of the emission. In particular, we find that the relaxation parameter ri determined from the preliminary analysis of the fre quency locking tuning curve data using Eq. (17) is in agreement with that obtained from the recovery function given by Eq. (9). Additionally, using Eq. (9) and the SOAE relaxation data of Schloth and Zwicker (1983), we find 1 /ri = 6.8 ms, which is also in agreement with these two estimates. However, the value of r\ extracted from the onset-of-suppression data using Eq. (7) appears to be in disagreement with the values obtained from the frequency locking and relaxation from suppression data. Using Eq. (7) and the Schloth and Zwicker SOAE suppression data, in which the emission was suppressed by —6.4 dB, we find 1 /n £ 4.4 ms, which is also characteristically lower than the value obtained from their recovery-from-suppression data. As was mentioned in Theoretical Re sults, the discrepancy between the value of ri measured in the suppression data and the value obtained from the recovery-from-suppression data may be evidence that the nonlinear component of damping of the underlying dynamical model is different from that of the Van der Pol oscillator. 241

Quantitative Evaluation of Limit-Cycle Oscillator Models

Talmadge et al.

It is, of course, expected that otoacoustic emissions recorded from a human ear canal may not be fully modeled by a single limit-cycle oscillator. Even in the case of a nonlinear active damping of the Van der Pol type highly localized in the cochlear partition, there are several effects which are not accounted for by a simple oscillator model of an SOAE. First, there will be a time delay in the response of the emission, which reflects the time for the traveling wave to arrive at the site of the emission in the cochlea. Secondly, there may be filtering of both the external tone and of the emission from both the cochlea and the middle ear. The results of studies of these effects in the context of full cochlear model simulations will be reported elsewhere.

Acknowledgements We would like to thank Hero Wit and Savithri Sivaramakrishnan for helpful discussions. This work was supported in part by NIH grant NIDCD-DC00307.

References Bialek, W.S., and Wit, H.P. (1984) Quantum limits to oscillator stability: Theory and experiments on acoustic emissions from the human ear. Phys. Lett. 104A, 1973-1978. Dallmayr, C. (1985) Spontane oto-akustische Emissionen: Statistik und Reaktion auf akustische Stortone. Acustica 59, 67-75. Dallmayr, C. (1987) Stationary and dynamical properties of simultaneous evoked otoa coustic emissions (SEOAE). Acustica 63, 243-255. Hanggi, P. and Riseborough, P. (1983) Dynamics of nonlinear dissipative oscillators. Am. J. Physics 51, 347-351. Long, G. R. and Tubis, A. (1988a) Investigations into the nature of the association between threshold microstructure and otoacoustic emissions. Hear. Res. 36, 125-138. Long, G. R. and Tubis, A. (1988b) Modification of spontaneous and evoked otoacoustic emissions and associated psychoacoustic microstructure by aspirin consumption. J. Acoust. Soc. Am. 84, 1343-1353. Long, G. R., Tubis, A., and Jones, K. L. (1990) Changes in synchronization and suppression tuning curves of spontaneous oto-acoustic emissions when the levels of the emissions are modified by aspirin consumption. Submitted to J. Acoust. Soc. Am. Nayfeh, A. H. Perturbation Methods. John Wiley, New York (1973) Nayfeh, A. H. and D. T. Mook, D. T. Nonlinear Oscillations. John Wiley, New York, (1979). Schloth, E. and Zwicker, E. (1983) Mechanical and acoustical influences on spontaneous oto-acoustic emissions. Hear. Res. 11, 285-293. Van Dijk, P. (1990) Characteristics and Mechanisms of Spontaneous Otoacoustic Emis sions. Thesis, University of Groningen, The Netherlands. W it, H.P. (1986) Statistical properties of a strong spontaneous otoacoustic emission. In: Peripheral Auditory Mechanics (Eds: by J.B. Allen, J.L. Hall, A.E. Hubbard, S.T. Neely, and A. Tubis) Springer Verlag, Berlin, pp. 221-228. Zurek P. M. and Clark, W. W. (1981) Narrow-band acoustic signals emitted by chinchilla ears after noise exposure. J. Acoust. Soc. Am. 70, 446-450. Zwicker E. and Schloth, E. (1984) Interrelation of different oto-acoustics emissions. J. Acoust. Soc. Am. 75, 1184-1154. 242