High level linear regime

During stimulation at moderately high levels, -6 0 dB SPL, the influence o f TW~ wave is negligible due o f the lack o f microstructure in ear canal parameters; it follows that the TW^ dominates wave propagation along the cochlea (chapters 6 and 7; Kemp, 1979ab; Kemp and Chum, 1980; Zwicker and Schloth, 1984; Shera and Zweig, 1993). From now on, specifies a stimulus level within the high level linear regime, bounded on the lower limit by cochlear nonlinearity while the threshold o f the acoustic reflex determines the upper limit.

Chapter 8; Cochlear parameters inferred from ear canal measurements

Assuming that the cochlea manifests scaling symmetry, direct measurements o f BM velocity indicate that under harmonic stimulation o f mid fi-equencies and stimulus level A„ , the phase response changes relatively slowly in the first turn o f the cochlea (Rhode, 1971; Robles et al, 1986; Zweig, 1991; Brass and Kemp, 1993). This implies that only a small decrease in the wavelength occurs as the TW^ progresses fr"om the base to the second turn, as well as a smoothly changing characteristic impedance.

Under the long wave approximation together with a slowly changing mechanical property in the basal turn and absence o f the T W ", the high level cochlear input impedance defined in equation 8.8 approximates to the characteristic impedance o f the cochlear duct (Peterson and Bogert, 1950; Zweig et al, 1976; Shera and Zweig, 1991a):-

(8.9)

From transmission line theory, the characteristic impedance at x is given by (Davidson, 1992):-

Since the cochlear fluid is assumed to be inviscid, the series impedance per unit length is a function o f the inertance reactance only:-

Z^[co,x] = j(ùM ^[x] = Jcop/S[x] (8.11)

where p is fluid density and S[x] the cross sectional area at x:-

Chapter 8: Cochlear parameters inferred from ear canal measurements

where S^[x] and S^[x] are the cross-sectional areas o f the SV and ST, respectively. With the cochlear partition impedance described by a harmonic resonator (de Boer, 1980):-

W + j ( ù M c p [ x ] + 1 /y c o Q p M (8 .1 3 )

the shunt admittance per unit length is given by an orthogonal sum o f conductance, Gcp, and susceptance, B^p, per unit length:-

Ycp G CP + jBcp [m,%] (8.14)

In the region o f the base, the stiffiiess o f the cochlear partition impedance dominates Zcp (von Békésy, 1960; Gummer et al, 1981):-

Y^,[(o.x] = (8.15)

Substituting equation 8.11 and 8.15 into 8.10 leads to:-

Z:[(ù,x]

Ccp[x] (8.16)

X IB basal turn

Compliance o f the cochlear partition, which is primarily governed by the width o f the basilar membrane, decreases approximately exponentially from the base to apex by nearly four orders o f magnitude (von Békésy, 1960; Gummer et al, 1981), therefore:-

Ccp[x] = Q 6,exp[a^% ] (8.17)

where the width o f the cochlear partition at the base is , and Q the compliance constant.

Chapter 8: Cochlear parameters inferred from ear canal measurements

Similarly, allowing the series inertance to vary exponentially with distance:-

(8.18) S „ ex p [a ,x ]

where is the scalae area constant.

In order to satisfy the observation that the TW manifests a small decrease in wavelength as it propagates through the basal turn, a ^ = - a ^. In other words Zj, and Y^p co-vary to maintain proportionality; as x increases from base to the second turn, basilar membrane width is inversely proportional to scalae area. From equations 8.17 and 8.18 it follows

z ;[c o ,x ]«

S A C ,

(8.19)

x»basal turn

With parameter values, p = 1000 kg m '\ 3 , = 1 0 ^ r n ^ , b ^ = \ Q ^ m , Cg = 10"'° kg ' m^ s^, the cochlear input impedance is approximately 3.2 x 10" MKS acoustic ohms (110 dB re M MKS acoustic ohm).

Equation 8.19, which is real and independent o f frequency, is equivalent to the asymptotic expression o f the exact analytical solution for the cochlear input impedance (de Boer, 1980). To reiterate, the approximate expression given in 8.19 is valid for mid frequencies, during stimulation levels corresponding to A^j.

For reasons o f convenience the cross-sectional area o f the cochlear duct is often set independent o f axial distance, %, i.e., = 0, leading to a ‘box-model’ approximation (de Boer, 1980). Under the box-model approximation, variations in the mechanical properties o f the cochlear duct per unit length are carried in the cochlear partition impedance, Z^p, only, since the fluid channels are isotropic. Shera and Zweig (1991a) argue that the box-model violates the slowly changing wavelength constraint upon the

Chapter 8: Cochlear parameters inferred from ear canal measurements

TW propagation; for the box-model design, since a ^ equation 8.19 then becomes a function o f x.

Zwislocki (1962) noted that the cochlear input impedance, Z^, is dominated by a resistive part, a condition which is favourable, since ideally the function o f the cochlear is to absorb power, concurring with experimental observations in the cat (Nedzelnitsky, 1980; Lynch et al, 1982). The study by Zwislocki (1962) was prior to any discussion o f the existence o f a basalward TW and its role in modifying ear canal input impedance, and is therefore consistent for moderate to high stimulus levels. Taking into account the ME transformation, Zwislocki (1962) used a figure o f 2.9 x 10^® MKS acoustic ohms (89 dB re 1 M MKS acoustic ohms) to characterise the resistive part o f Z^. Von Békésy (1960) identified a small mass component in Z^ which Zwislocki (1962) attributed to the bulging volume known as the vestibule located immediately interior to the OW and noted that its effect in the analogue model proved negligible.

Data on the cochlear input impedance in the cat indicates that between 0.5 and 5 kHz, 1ZJ« 10” MKS acoustic ohms (100 dB re 1 M MKS acoustic ohms) with an argument non-systematic variation about zero o f roughly 20 degrees (Lynch et al, 1982).