Chinchilla middle ear transmission matrix model and middle-ear flexibility

Abstract

The function of the middle ear (ME) in transforming ME acoustic inputs and outputs (sound pressures and volume velocities) can be described with an acoustic two-port transmission matrix. This description is independent of the load on the ME (cochlea or ear canal) and holds in either direction: forward (from ear canal to cochlea) or reverse (from cochlea to ear canal). A transmission matrix describing ME function in chinchilla, an animal commonly used in auditory research, is presented, computed from measurements of forward ME function: input admittance Y_TM, ME pressure gain G_MEP, ME velocity transfer function H_V, and cochlear input admittance Y_C, in the same set of ears [Ravicz and Rosowski (2012b). J. Acoust. Soc. Am. 132, 2437–2454; (2013a). J. Acoust. Soc. Am. 133, 2208–2223; (2013b). J. Acoust. Soc. Am. 134, 2852–2865]. Unlike previous estimates, these computations require no assumptions about the state of the inner ear, effectiveness of ME manipulations, or measurements of sound transmission in the reverse direction. These element values are generally consistent with physical constraints and the anatomical ME “transformer ratio.” Differences from a previous estimate in chinchilla [Songer and Rosowski (2007). J. Acoust. Soc. Am. 122, 932–942] may be due to a difference in ME flexibility between the two subject groups.

I. INTRODUCTION

The middle ear (ME) plays an important role in hearing: to collect, transform, and deliver sound power from the ear canal to the inner ear (IE). Determining sound transmission parameters of the ME is helpful to explore how structural differences in normal ears among different species, pathological changes in ME structure and function, and reconstructive techniques that modify the ME affect ME function and hearing sensitivity. Knowledge of chinchilla sound transmission parameters is important because chinchilla has been used in studies of noise damage and hearing loss (e.g., Patterson et al., 1993). Furthermore, this knowledge would help validate animal models of human hearing.

The ME as usually defined comprises the tympanic membrane (TM), the three ME ossicles (malleus, incus, stapes) and their ligaments, and the air space [middle ear cavity (MEC)] within which the ossicles reside—see Fig. 1(A). In forward sound transmission, sound pressure in the ear canal (EC) induces volume velocity of the TM, which is transduced to force and motion of the ossicles and transduced again to volume velocity of the stapes and sound pressure in the cochlea. The TM and the geometry and dynamics of ossicular motion act to transform the relatively low sound pressure and high volume velocity in the ear canal to relatively high sound pressure and low stapes volume velocity at the oval window (OW), the entrance to the cochlea, which better matches the low cochlear input admittance (high input impedance, e.g., Wever and Lawrence, 1954).

FIG. 1.

(Color online) (A) Schematic of the mammalian ME showing relevant structures and locations of measurements used in this paper. For forward sound transmission, ME inputs are sound pressure in the ear canal near the TM P_TM and TM volume velocity U_TM; ME outputs are stapes volume velocity U_S and sound pressure in the cochlear vestibule just inside the OW P_V. The computations described in this paper use measurements of P_TM with a microphone, ME input admittance Y_TM = U_TM/P_TM with a calibrated acoustic source, stapes linear velocity with a laser vibrometer, and P_V with a miniature pressure sensor. The air-filled ear canal and MEC are shown in white, while the inner-ear fluid is shown shaded. (B) ME transmission matrix T_ME shown diagrammatically for forward sound transmission. The load on the ME is the cochlear input admittance Y_C. (C) Reverse ME transmission is described by the reverse transmission matrix ${\mathbf{T}}_{\mathbf{ME}}^{\mathbf{\prime }}$ computed from T_ME that relates the reverse ME input admittance ${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}$ to the ME load in this case, the ear canal output impedance ${\mathbf{Y}}_{\mathbf{E}\mathbf{C}}^{\mathbf{R}}$.

In a previously-published conceptual framework, Shera and Zweig (1991, 1992) outlined a method to develop a two-port transmission-matrix description of ME function from measurements of ME inputs and outputs. In general, two-port matrices provide relationships among four state variables that describe the inputs and/or outputs of a system. Transmission two-port matrices (sometimes called ABCD matrices) are useful for studies of multistage systems in that they relate two state variables at its input to analogous state variables at its output. In this way, a collection of transmission matrices that each describes a separate stage of a system can be cascaded to arrive at a relationship between the inputs and outputs of the entire system. A transmission-matrix description of the ME function can aid the development of a lumped- or continuum-element ME model (e.g., Lemons and Meaud, 2016). We use a single transmission matrix T_ME to describe the ME from the TM to the OW.

In acousto-mechanical systems like the ME, sound pressure P and volume velocity U are commonly-used state variables.¹ For forward (normal) sound transmission from the ear canal through the ME into the cochlea, with the ME input evaluated at the TM and the ME output evaluated just inside the OW, we use the sound pressure at the TM P_TM and the TM volume velocity U_TM as inputs and sound pressure in the cochlear vestibule just inside the OW P_V and stapes volume velocity U_S as outputs—see Fig. 1(B).² The cochlear input admittance Y_C is the ratio of U_S to P_V, where U is analogous to electrical current and P is analogous to voltage.

A transmission matrix provides a particularly useful representation of the ME function: (1) It can predict inputs from outputs or, by inverting the matrix, predict outputs from inputs. (2) By transforming inputs to outputs (or vice versa), it can be used in series (cascade) with other transmission matrices that describe the function of other parts of the ear (e.g., the ear canal and IE, see Shera and Zweig, 1991, 1992). (3) The matrix elements that describe the transformation are independent of the load on the system, so a description developed from measurements in one loading condition can be used to describe inputs or outputs in another loading condition. (4) The reverse transmission matrix (${\mathbf{T}}_{\mathbf{ME}}^{\mathbf{\prime }}$) easily computed from T_ME provides a description of transmission in the opposite direction [Fig. 1(C)]. As examples, ME input admittance Y_TM = U_TM/P_TM can be estimated from the ME volume velocity transfer function H_U = U_S/P_TM, or the cochlear input admittance Y_C = U_S/P_V can be estimated from Y_TM; the ME pressure gain G_MEP in the intact IE can be estimated from measurements made with a sensor in a hole in the cochlear vestibule (that changes Y_C); and the ME loading for sound generated within the cochlea (an intracochlear otoacoustic emission) can be estimated for different ear canal conditions.

Transmission-matrix descriptions of the ME function have been developed for chinchillas (Songer and Rosowski, 2007; Lemons and Meaud, 2016) as well as for humans (Puria, 2004) and cats (Voss and Shera, 2004; Miller, 2006). Transmission-matrix ME descriptions in humans and cats relied on measurements of both forward (inner-ear response to sound in the ear canal) and reverse sound transmission (ear-canal sound in response to cochlear stimulus or activity, such as otoacoustic emissions). Previous two-port descriptions in chinchilla (Songer and Rosowski, 2007; Lemons and Meaud, 2016) used only forward sound transmission measurements but relied on assumptions concerning manipulation-induced changes in the load on the ME (e.g., immobilizing the stapes in the OW or draining the cochlea).

Though the previous chinchilla two-port transmission matrix elements are internally consistent with the data set used to compute them, they are not completely consistent with a more recent set of ME transfer function measurements (Ravicz and Rosowski, 2012b, 2013a,b). To the best of our knowledge, no ME transmission matrix in any species has been checked for consistency with basic anatomical or physical constraints (as outlined in Shera and Zweig, 1992). These potential inconsistencies were a motivation for this study.

We present a new two-port transmission matrix description of the chinchilla ME using only measurements of forward sound transmission in which we need not rely on assumptions about the ME loading. We rely on only the previously-verified assumption that the ME is passive, reciprocal, and time-invariant (Shera and Zweig, 1992).³ Matrix elements are computed from measurements of ME input and output variables with forward sound transmission in two conditions: with the IE intact, and with a pressure sensor in a small hole in the cochlear vestibule near the OW [Fig. 1(A)]. It should be noted that in our preparation two large holes were made in the bulla, thereby removing the effect of the MEC volume at low frequencies (Rosowski et al., 2006); therefore, the two-port ME descriptions presented here include the effects of this manipulation. We assess the degree to which these new estimates of the matrix elements are consistent with (a) other ME measurements not used in computations, (b) physical constraints, and (c) anatomical parameters. We predict T_ME for the case where the MEC is intact. These new estimates are used to predict a ME transmission parameter in the reverse direction, from cochlea to ear canal: the reverse ME input admittance at the OW. The new estimates and data imply greater flexibility in the chinchilla ME than previous estimates.

II. METHODS

A. Source of data: Preparation and experiments

The analyses in this paper use data from experiments that were performed in accordance with guidelines published by the U.S. Public Health Service and were approved by the Massachusetts Eye & Ear Infirmary Animal Care Committee. The preparation of chinchilla ears, equipment, calibration, and experiments have been described previously (Rosowski et al., 2006; Ravicz and Rosowski, 2012b, 2013a,b). ME input admittance was measured with a custom acoustic source (Rosowski et al., 2006), sound pressure was measured in the ear canal with a probe tube microphone and in the cochlear vestibule with a custom fiber-optic pressure sensor (Olson, 1998), and stapes velocity was measured at a single point on the stapes footplate or posterior crus with a commercial laser-Doppler vibrometer [Fig. 1(A)]. Animals remained alive throughout the experiments. All measurements were made with the MEC opened widely by two ∼4-mm-diameter holes.

B. Technique for estimating 2-port transmission matrix elements

The ME transmission matrix T_ME relates ME inputs P_TM and U_TM to outputs P_V and U_S by

$$\left\{\begin{array}{c}{\mathbf{P}}_{\mathbf{T}\mathbf{M}}\\ {\mathbf{U}}_{\mathbf{T}\mathbf{M}}\end{array}\right\}={\mathbf{T}}_{\mathbf{M}\mathbf{E}}\left\{\begin{array}{c}{\mathbf{P}}_{\mathbf{V}}\\ {\mathbf{U}}_{\mathbf{S}}\end{array}\right\}=\left[\begin{array}{cc}{\mathbf{T}}_{11}& {\mathbf{T}}_{12}\\ {\mathbf{T}}_{21}& {\mathbf{T}}_{22}\end{array}\right]\left\{\begin{array}{c}{\mathbf{P}}_{\mathbf{V}}\\ {\mathbf{U}}_{\mathbf{S}}\end{array}\right\}=\left[\begin{array}{cc}\mathbf{A}& \mathbf{B}\\ \mathbf{C}& \mathbf{D}\end{array}\right]\left\{\begin{array}{c}{\mathbf{P}}_{\mathbf{V}}\\ {\mathbf{U}}_{\mathbf{S}}\end{array}\right\},$$ (1a)

where T_ME is exactly analogous to ${{}^e\widehat{T}}_{\mathit{o}\mathit{w}}$ in Shera and Zweig (1992). The elements T_ij of T_ME are commonly labeled A, B, C, D, and we continue that convention here.

In general, the transmission matrix of a passive mechanical or acoustic system is reciprocal (e.g., Desoer and Kuh, 1969). If the ME transmission matrix is assumed reciprocal (e.g., Shera and Zweig, 1992; Voss and Shera, 2004), its determinant is unity [det(T_ME) = AD − BC = 1; Desoer and Kuh, 1969], which greatly simplifies computations. ME outputs are then related to inputs [Fig. 1(B)] by

$$\left\{\begin{array}{c}{\mathbf{P}}_{\mathbf{V}}\\ {\mathbf{U}}_{\mathbf{S}}\end{array}\right\}={\mathbf{T}}_{\mathbf{M}\mathbf{E}}^{-1}\left\{\begin{array}{c}{\mathbf{P}}_{\mathbf{T}\mathbf{M}}\\ {\mathbf{U}}_{\mathbf{T}\mathbf{M}}\end{array}\right\}=\left[\begin{array}{cc}\mathbf{D}& –\mathbf{B}\\ –\mathbf{C}& \mathbf{A}\end{array}\right]\left\{\begin{array}{c}{\mathbf{P}}_{\mathbf{T}\mathbf{M}}\\ {\mathbf{U}}_{\mathbf{T}\mathbf{M}}\end{array}\right\}.$$ (1b)

1. Physical significance of matrix elements

Each of the transmission matrix elements is representative of a physically measurable ME transfer function under certain conditions (Shera and Zweig, 1991, 1992). This representation can be seen easily from Eq. (1a) by setting one of the output quantities (P_V or U_S) to zero. If U_S = 0 (infinite load on ME; OW blocked or immobilized), P_TM = A P_V and U_TM = C P_V, so

2.

$$\mathbf{A}={\left({\mathbf{P}}_{\mathbf{T}\mathbf{M}}/{\mathbf{P}}_{\mathbf{V}}\right)|}_{\mathbf{U}\mathbf{s}=0}\hspace{1em}\text{and}\mathbf{C}={\left({\mathbf{U}}_{\mathbf{T}\mathbf{M}}/{\mathbf{P}}_{\mathbf{V}}\right)|}_{\mathbf{U}\mathbf{s}=0}.$$ (2a)

If P_V =0 (no load on ME), P_TM = B U_S and U_TM = D U_S, so

$$\hspace{1em}\mathbf{B}={\left({\mathbf{P}}_{\mathbf{T}\mathbf{M}}/{\mathbf{U}}_{\mathbf{S}}\right)|}_{\mathbf{P}\mathbf{v}=0}\hspace{1em}\text{and}\mathbf{D}={\left({\mathbf{U}}_{\mathbf{T}\mathbf{M}}/{\mathbf{U}}_{\mathbf{S}}\right)|}_{\mathbf{P}\mathbf{v}=0}.$$ (2b)

2. Previous studies: Modifications to ME

Previous studies in chinchillas have used these relationships to estimate ME transmission matrix elements from measurements with stapes mobility reduced with glues to approximate the no-cochlear-motion condition or the cochlea drained to approximate the no-ME-load condition (Shera and Zweig, 1992; Songer and Rosowski, 2007). The degree to which the actual measurement conditions deviated from the ideal no-motion or no-load conditions has an effect on the accuracy of matrix element estimates (Songer and Rosowski, 2007), discussed further in Sec. IV E.

3. This study: Measurements in different inner-ear conditions

In this study, the transmission matrix is computed solely from forward measurements under two conditions, using assumptions of reciprocity and time-invariance and without relying on assumptions about the effects of ME manipulations on the ME loading, as outlined below.

The basic transformations are obtained by expanding Eqs. (1a) and (1b) above:

3.

$${\mathbf{P}}_{\mathbf{T}\mathbf{M}}=\mathbf{A}{\mathbf{P}}_{\mathbf{V}}+\mathbf{B}{\mathbf{U}}_{\mathbf{S}},$$ (3a)

$${\mathbf{P}}_{\mathbf{V}}=\mathbf{D}{\mathbf{P}}_{\mathbf{T}\mathbf{M}}-\mathbf{B}{\mathbf{U}}_{\mathbf{T}\mathbf{M}},$$ (3b)

$${\mathbf{U}}_{\mathbf{S}}=\mathbf{A}{\mathbf{U}}_{\mathbf{T}\mathbf{M}}-\mathbf{C}{\mathbf{P}}_{\mathbf{T}\mathbf{M}}.$$ (3c)

Since we have used T_ME, its inverse, and the reciprocity assumption to formulate these equations, the fourth equation of this set,

$${\mathbf{U}}_{\mathbf{T}\mathbf{M}}=\mathbf{C}{\mathbf{P}}_{\mathbf{V}}+\mathbf{D}{\mathbf{U}}_{\mathbf{S}},$$ (3d)

is redundant.

Equations (3a)–(3c) can be rearranged to get three equations expressing measureable transfer functions in terms of matrix elements. Note that all of these transfer functions are measured with a sensor in the cochlear vestibule, which alters the load on the ME.

4.

$$\text{Rearrange}\hspace{0.17em}(3\mathrm{a}):\mathbf{A}{\mathbf{G}}_{\mathbf{M}\mathbf{E}\mathbf{P}}+\mathbf{B}{\mathbf{H}}_{\mathbf{U}}=1,\hspace{0.17em}\text{where}\hspace{0.17em}\hspace{0.17em}{\mathbf{G}}_{\mathbf{M}\mathbf{E}\mathbf{P}}={\mathbf{P}}_{\mathbf{V}}/{\mathbf{P}}_{\mathbf{T}\mathbf{M}}\hspace{1em}\text{and}\hspace{1em}{\mathbf{H}}_{\mathbf{U}}={\mathbf{U}}_{\mathbf{S}}/{\mathbf{P}}_{\mathbf{T}\mathbf{M}}\hspace{0.17em}\hspace{0.17em}\text{as}\hspace{0.17em}\text{above};$$ (4a)

$$\text{Rearrange}\hspace{0.17em}(3\mathrm{b}):\hspace{0.17em}{\mathbf{G}}_{\mathbf{M}\mathbf{E}\mathbf{P}}=\mathbf{D}-\mathbf{B}{\mathbf{Y}}_{\mathbf{T}\mathbf{M}},\hspace{0.17em}\text{where}\hspace{0.17em}\hspace{0.17em}{\mathbf{Y}}_{\mathbf{T}\mathbf{M}}={\mathbf{U}}_{\mathbf{T}\mathbf{M}}/{\mathbf{P}}_{\mathbf{T}\mathbf{M}}\hspace{0.17em}\text{as}\hspace{0.17em}\text{above};$$ (4b)

$$\text{Rearrange}\hspace{0.17em}(3\mathrm{c}):\hspace{0.17em}{\mathbf{H}}_{\mathbf{U}}=\mathbf{A}{\mathbf{Y}}_{\mathbf{T}\mathbf{M}}-\mathbf{C}.$$ (4c)

The fourth equation is the same as Eq. (4c), but with the IE intact (before opening the IE to introduce the pressure sensor), denoted by superscript ^I:

$${\mathbf{H}}_{\mathbf{U}}^{\mathbf{I}}=\mathbf{A}{\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{I}}-\mathbf{C},\hspace{1em}\text{where}\hspace{1em}{\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{I}}={\mathbf{U}}_{\mathbf{T}\mathbf{M}}^{\mathbf{I}}/{\mathbf{P}}_{\mathbf{T}\mathbf{M}}^{\mathbf{I}}.$$ (4d)

Equations (4c) and (4d) are solved for A:

5.

$$\mathbf{A}=\Delta {\mathbf{H}}_{\mathbf{U}}/\Delta {\mathbf{Y}}_{\mathbf{T}\mathbf{M}},\hspace{1em}\text{where}\hspace{1em}\Delta {\mathbf{H}}_{\mathbf{U}}={\mathbf{H}}_{\mathbf{U}}-{\mathbf{H}}_{\mathbf{U}}^{\mathbf{I}}\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\Delta {\mathbf{Y}}_{\mathbf{T}\mathbf{M}}={\mathbf{Y}}_{\mathbf{T}\mathbf{M}}–{\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{I}}.$$ (5a)

Once A is known, the other elements can be found easily:

$$\mathbf{B}={\mathbf{H}}_{\mathbf{U}}^{–1}–\mathbf{A}/{\mathbf{Y}}_{\mathbf{C}},$$ (5b)

$$\mathbf{C}=\mathbf{A}{\mathbf{Y}}_{\mathbf{T}\mathbf{M}}–{\mathbf{H}}_{\mathbf{U}},\hspace{1em}\text{and}$$ (5c)

$$\mathbf{D}=\mathbf{B}{\mathbf{Y}}_{\mathbf{T}\mathbf{M}}+{\mathbf{G}}_{\mathbf{M}\mathbf{E}\mathbf{P}},$$ (5d)

where Y_C is the cochlear input admittance as described above.

In these computations we use ME input admittances ${\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{I}}$ and Y_TM [Ravicz and Rosowski, 2012b, Figs. 5(A) and 5(C)] and stapes velocity transfer functions ${\mathbf{H}}_{\mathbf{V}}^{\mathbf{I}}$ and H_V (Ravicz and Rosowski, 2013b, Fig. 4) measured before and after placing a pressure sensor into the IE, together with ME pressure gain G_MEP [Ravicz and Rosowski, 2013a, Fig. 5(B)] and cochlear input admittance Y_C [Ravicz and Rosowski, 2013b, Fig. 5(A)] measured only after placing the sensor. All of these measurements were made in the same five ears. We compute H_U from H_V and the stapes footplate area A_FP = 2 mm² (Vrettakos et al., 1988) by H_U = H_V A_FP. Implicit in this computation are the assumptions that (a) the footplate is rigid, (b) stapes velocity in three dimensions is described adequately by V_S, and (c) the angle between the measurement direction of V_S and the piston direction (∼45°) is sufficiently small that U_S ≈ A_FP V_S. (G_MEP, H_U, and ${\mathbf{H}}_{\mathbf{U}}^{\mathbf{I}}$ are included in Fig. 8 below.) We use Y_C instead of (H_U/G_MEP) where possible because (a) Y_C was computed from simultaneous measurements of H_U and G_MEP and (b) Y_C is subject to physical constraints that were used to omit data possibly contaminated by errors (Ravicz and Rosowski, 2013b). In this way, we compute the ME transmission matrix using only measurements of forward sound transmission from the ear canal to the cochlea.

FIG. 8.

(Color online) Comparison of the reciprocals of mean ME transmission matrix elements ±1 s.d. (thick black solid line and shaded area), which represent ME transfer functions with Y_C = 0 (A and C) or Z_C = 0 (B and D), to measurements of those transfer functions in real ears with the IE intact (long-dash-dotted lines) or with a sensor in the vestibule (dashed lines). Panels (A) and (D) also show the anatomical TR or its inverse, respectively (thin dotted-dashed lines). All panels: Top: magnitude; bottom: phase.

A benefit of the method described above is that no assumption of ideal conditions is necessary; however, the accuracy of our new element estimates does depend on the size of the changes in Y_TM and H_U when the IE is opened in order to place the pressure sensor (see below). It should be kept in mind that, because all quantities were measured with the MEC opened by two holes, this matrix describes the ME function with the MEC open.

C. Estimation of effects of uncertainty in measured quantities

The accuracy of the derived transmission matrix elements depends on the relative size of the uncertainties in the quantities that go into the computations. All other elements depend on A directly or indirectly, where the computation of A depends on the effect of inserting the IE sensor on H_U and Y_TM. The new estimates of A are most robust in frequency ranges where ΔH_U and ΔY_TM were large, and were more uncertain in frequency ranges where these differences were small. Inserting the sensor produced a change in |Y_TM| of at least 30% at most frequencies in most ears below 2.5 kHz and between 6 and 10 kHz [Fig. 2(A)], so estimates of T_ME in these frequency ranges should be fairly robust. Between 3 and 6 kHz and above 10 kHz, the change in |Y_TM| was less than 30% in nearly all ears, so estimates of T_ME are less certain in these frequency ranges. By comparison, the variability between successive Y_TM measurements [comparable to the standard deviation (s.d.) in $\left|{\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{I}}\right|$; Ravicz and Rosowski, 2012b] is 10%–20% at these frequencies. The effect of opening the IE on |H_U| was larger than the effect on |Y_TM| over nearly the entire frequency range [Fig. 2(B)]: at least 50% in all ears at nearly all frequencies. In general, we have greater confidence in our new transmission matrix element computations below 2.5 kHz and between 6 and 10 kHz where Δ|Y_TM| and Δ|H_U| are relatively large, and less confidence at other frequencies where Δ|Y_TM| is comparable to the inter-measurement variability.

FIG. 2.

(Color online) Estimates of the magnitude of uncertainties in Y_TM and H_U for computation of transmission matrix element A by Eq. (5 a): in ears ch13, ch15, ch16, and ch17 (thin solid and dotted-dashed lines) and the mean of these four ears (thick black line). (A) Ratio of the magnitude of the change in ${\mathbf{Y}}_{\mathbf{M}\mathbf{T}}^{\mathbf{I}}$ upon opening the IE Δ|Y_TM| = |Y_TM - ${\mathbf{Y}}_{\mathbf{M}\mathbf{T}}^{\mathbf{I}}$| to $\left|{\mathbf{Y}}_{\mathbf{M}\mathbf{T}}^{\mathbf{I}}\right|$. (B) Ratio of the change in ${\mathbf{H}}_{\mathbf{U}}^{\mathbf{I}}$ upon opening the IE Δ|H_U| = |H_U - ${\mathbf{H}}_{\mathbf{U}}^{\mathbf{I}}$| to $\left|{\mathbf{H}}_{\mathbf{U}}^{\mathbf{I}}\right|$. At frequencies where either of these ratios is less than 0.3 in magnitude (double-dot-dashed horizontal line), uncertainties in the measured quantities are likely to generate large errors in A.

III. RESULTS

A. ME transmission matrix elements

The two-port transmission matrix elements computed from Eq. (5) and measurements in 5 ears (ch11, 13, 15, 16, 17) are shown in Fig. 3, along with the logarithmic means and s.d. (expressed as fractional variations; thick line and shading). The computed elements in all ears vary substantially with small changes in frequency, but general frequency-dependent trends are apparent in most ears for all elements. The mean values reflect the common trends and show less irregularity with frequency.

FIG. 3.

(Color online) ME transmission matrix elements A, B, C, and D computed from data in 4–5 individual ears (thin lines); also, the (logarithmic) mean values of A, B, C, and D in all ears ±1 s.d. (thick solid line and shaded area). Element estimates $\widehat{\mathbf{A}}$, $\widehat{\mathbf{B}}$, $\widehat{\mathbf{C}}$, and $\widehat{\mathbf{D}}$ computed from mean components are shown by thick dashed lines. All panels: Top: magnitude; bottom: phase.

Element A in four individual ears (ch13, 15, 16, 17) is shown in Fig. 3(A), along with the mean $\left(\overline{\mathbf{A}}\right)$ ±1 s.d. (${\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{I}}$ and ${\mathbf{H}}_{\mathbf{U}}^{\mathbf{I}}$ were not measured in ear ch11.) The magnitude |A| among ears is about 0.03 and generally flat with frequency between 100 Hz and 3 kHz; the phase ∠A is near zero at low frequencies and increases to about 1 cycle at 3 kHz and 1.7 cycles at higher frequencies. We have unwrapped ∠A in each ear to minimize differences between ears, keeping in mind that the uncertainty in ∠A is large at frequencies where |ΔY_TM| is smallest (between 2.5 and 6 kHz and above 10 kHz). The s.d. of $\left|\overline{\mathbf{A}}\right|$ is about a factor of 2 between 100 Hz and 3 kHz, and the s.d. of ∠$\overline{\mathbf{A}}$ is about 0.1–0.2 cycles. Above 3 kHz the s.d. increases sharply, consistent with the greater uncertainty in A due to the smaller change in Y_TM and H_U in this frequency range upon inserting the sensor in the IE (discussed in Sec. II C above). The s.d. of ∠$\overline{\mathbf{A}}$ is also small between 6 and 10 kHz where the change in Y_TM and H_U by the sensor was robust, but the s.d. of $\left|\overline{\mathbf{A}}\right|$ in this range is large. Though both |A| and ∠A show differences between ears, the frequency dependencies of these quantities are consistent among ears and well represented by the means.

Elements B, C, and D in all five ears are shown in Figs. 3(B), 3(C), and 3(D), respectively. For ear ch11, B, C, and D were computed by using the mean $\overline{\mathbf{A}}$ in the other four ears in Eqs. (5b) and (5c), as no value of A could be computed for ear ch11. As for ∠A above, we have unwrapped the phase in each ear in a manner that minimizes differences between ears.

Both |B| and ∠B increase with frequency above 250 Hz [Fig. 3(B)]. The slope of |B| with frequency above 250 Hz is about 0.5 on this log-log scale. The s.d. of $\left|\overline{\mathbf{B}}\right|$ is smaller than that of $\left|\overline{\mathbf{A}}\right|$, especially below 3 kHz; and the s.d. of ∠$\overline{\mathbf{B}}$ is smaller between 0.6 and 3 kHz and above 7 kHz than at other frequencies and smaller than the s.d. of ∠$\overline{\mathbf{A}}$. ∠$\overline{\mathbf{B}}$ is about −0.25 cycles at the lowest frequencies and increases sharply to slightly positive at about 250 Hz, the frequency of the minimum in $\left|\overline{\mathbf{B}}\right|$, consistent with a resonance.

Element C magnitude [Fig. 3(C)] decreases slightly with frequency above 250 Hz, and ∠C is near zero at low frequencies and increases about 0.5 cycles between 0.5 and 10 kHz. The s.d. of $\left|\overline{\mathbf{C}}\right|$ is comparable to that of $\left|\overline{\mathbf{A}}\right|$, while the s.d. of ∠$\overline{\mathbf{C}}$ is larger than the s.d. of ∠$\overline{\mathbf{A}}$ below 500 Hz and above 3 kHz. |C| decreases by a factor of about 10 between 0.3 and 10 kHz.

Element D magnitude [Fig. 3(D)] is about 70 and approximately flat with frequency between 0.1 and 10 kHz. ∠D is approximately zero at low frequencies and increases about 0.5 cycles by 10 kHz and about 1 cycle by 20 kHz. The s.d. of $\left|\overline{\mathbf{D}}\right|$ is comparable to that of $\left|\overline{\mathbf{B}}\right|$ at most frequencies above 300 Hz, and the s.d. of ∠$\overline{\mathbf{D}}$ is smaller than the s.d. of ∠$\overline{\mathbf{B}}$ below 800 Hz.

In general, the s.d. of $\overline{\mathbf{B}}$ and $\overline{\mathbf{D}}$ (expressed as fractional variations) is smaller than the s.d. of $\overline{\mathbf{A}}$ and $\overline{\mathbf{C}}$, and the s.d. of $\overline{\mathbf{C}}$ is largest among the elements.

B. Consistency checks

1. With elements computed from the means of components

We also computed an estimate of each matrix element (denoted by a circumflex) from the means of the quantities in Eq. (5). For example, $\widehat{\mathbf{A}}$ was computed from the mean Y_TM, mean ${\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{I}}$, mean H_U, and mean ${\mathbf{H}}_{\mathbf{U}}^{\mathbf{I}}$ by $\widehat{\mathbf{A}}$ = (${\overline{\mathbf{Y}}}_{\mathbf{T}\mathbf{M}}$ − ${\overline{\mathbf{Y}}}_{\mathbf{T}\mathbf{M}}^{\mathbf{I}}$) /(${\overline{\mathbf{H}}}_{\mathbf{U}}$ − ${\overline{\mathbf{H}}}_{\mathbf{U}}^{\mathbf{I}}$), and $\widehat{\mathbf{C}}$ was computed from $\widehat{\mathbf{A}}$ by $\widehat{\mathbf{C}}$ = $\widehat{\mathbf{A}}$${\overline{\mathbf{Y}}}_{\mathbf{T}\mathbf{M}}$ − ${\overline{\mathbf{H}}}_{\mathbf{U}}$. Generally, these element estimates computed from mean components are quite consistent with the means of the elements computed in individuals. The phase accumulation of ∠$\widehat{\mathbf{A}}$ is about 1 cycle less than ∠A above 2 kHz, which suggests that the nearly 2-cycle phase accumulation is due to noisy A estimates affecting phase unwrapping around 2 kHz (see Sec. II C).

2. With other data not used in computations

To check the consistency of these matrix element estimates, we used the transmission matrix to relate two sets of data not used in the matrix computations: the ME input admittance ${\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{O}}$ and stapes volume velocity transfer function ${\mathbf{H}}_{\mathbf{U}}^{\mathbf{O}}$ with the vestibule hole open and no sensor in place (Ravicz and Rosowski, 2012a). Figure 4 compares the measured ${\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{O}}$ with estimates computed from ${\mathbf{H}}_{\mathbf{U}}^{\mathbf{O}}$ by

$${\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{O}}=\left(\mathbf{C}+{\mathbf{H}}_{\mathbf{U}}^{\mathbf{O}}\right)/\mathbf{A}.$$ (6)

Note that this computation uses elements A and C, which have the largest s.d. and show the largest difference from previous estimates (see Sec. IV A below). As such, it provides a stringent test of the consistency of the new data set and new ME transmission matrix.

FIG. 4.

(Color online) Consistency check of elements A and C of the ME transmission matrix in Fig. 4. Mean ME admittance with the vestibule hole open ${\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{O}}$ in four ears, computed from measured ${\mathbf{H}}_{\mathbf{U}}^{\mathbf{O}}$ by Eq. (6), ±1 s.d. (dotted line and cross-hatched area), compared with the mean of ${\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{O}}$ measured in the same ears ±1 s.d. (thick solid line and shaded area). Top: magnitude; bottom: phase.

Figure 4 shows that the mean ${\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{O}}$ estimated from ${\mathbf{H}}_{\mathbf{U}}^{\mathbf{O}}$ is very similar to the mean of measured ${\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{O}}$ in the four ears (the difference is less than the s.d.), and the two s.d.'s are similar also. Data from individuals (not shown) also match closely except for a single ear at isolated frequencies; the exception is in ear ch16 between 5 and 10 kHz, where the estimated $\left|{\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{O}}\right|$ is a factor of 3–15 higher than the measured $\left|{\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{O}}\right|$. Overall, the close match demonstrates that these two matrix elements (A and C) are internally consistent.

C. Effect of variations in stapes footplate area

The computation of stapes volume velocity transfer function H_U from the point velocity transfer function H_V depends on the footplate area A_FP. Since we did not measure A_FP in our experimental animals, we rely on a previously-published mean value ± s.d. of 2.0 ± 0.13 mm² (Vrettakos et al., 1988). We repeated the computations of the elements of T_ME using values of A_FP two standard deviations (0.26 mm² = 13%) above and below the mean value. The effects of these variations in footplate area on element magnitude were less than the s.d. of the element means, and the effect on element phase was negligible: Variations in ∠B and ∠D were <0.003 cycles except 0.01–0.03 at a few isolated frequencies; variations in ∠A and ∠C were <0.003 cycles at all frequencies.

IV. DISCUSSION

We have presented a new chinchilla ME transmission matrix based on previously published data (Ravicz and Rosowski, 2012b, 2013a,b) and have demonstrated that the new matrix elements A (which depends on relatively small differences in measured quantities, and on which the other elements depend) and C are consistent with other data not used in their calculation (Fig. 4). In this section we compare the new matrix elements to previous element estimates in chinchilla, check the elements against physical and anatomical constraints, and investigate sources of variability between the two matrices.

A. Comparison to a previously-described chinchilla transmission matrix

The new mean ME transmission matrix computed in this study shows some similarities and differences from a previous estimate computed from ME input admittance and stapes velocity measured with the cochlea fluids drained or with the stapes footplate mobility reduced with cement (Songer and Rosowski, 2007; “S&R”). (This previous estimate also describes ME function with the MEC open.) Figure 5 compares the new mean matrix elements ±1 s.d. (smoothed by a 3-point moving triangle) to the S&R mean and 95% confidence interval (c.i.), where the c.i. is 1.96 × the standard error of the mean. For convenience, we drop the overbar notation, and all further references to the new matrix elements refer to their means. For the size of the subject population in S&R (5 animals), the standard error is about 0.5 × the s.d.; so the S&R c.i. is approximately equivalent to ±1 s.d. The statistical significance of differences can be inferred by comparing the new mean values to S&R's 95% c.i.'s: At frequencies where the new mean is outside S&R's c.i., the means differ at a 5% significance level.

FIG. 5.

(Color online) Mean smoothed ME transmission matrix elements from Fig. 3 ±1 s.d. (thick solid line and shaded area) compared with the mean elements ±1 s.d. from Songer and Rosowski (2007, Fig. 6; dashed line and hatched area). The anatomical TR = 80 is shown by the thick dotted-dashed line in panel D (magnitude); its inverse is shown by the dotted-dashed line in panel A (magnitude). All panels: Top: magnitude; bottom: phase.

The new transmission matrix elements A, B, and D are fairly similar to S&R's across the frequency range of measurement, but the new C is markedly different at most frequencies (Fig. 5). Note that we have offset ∠A by 1 cycle above 2 kHz, as we believe the extra cycle of phase accumulation is artifactual (see Sec. III B 1). The new A and S&R's A have a similar frequency dependence, but the new |A| is a factor of 3 to 5 higher than S&R's at most frequencies [Fig. 5(A)], and the lack of overlap between the S&R 95% c.i. and the new estimate below 2 kHz suggests that these differences are statistically significant (p < 0.05). The new ∠A is higher than S&R's above 0.5 kHz. The s.d. of the new |A| is larger than S&R's c.i. below 2 kHz, but the s.d. of the new ∠A is smaller than S&R's c.i. over most of the frequency range.

The new B is similar to S&R's over nearly the entire frequency range 0.1–8 kHz [Fig. 5(B)], but there are differences. The new |B| is slightly but significantly lower than S&R's below 250 Hz (by about a factor of 2), and the new ∠B is slightly lower than S&R's (by less than 0.25 cycle) above 6 kHz. The s.d. of the new B is similar to S&R's c.i.

Element C shows the largest difference between the new estimate and S&R [Fig. 5(C)]. The new |C| is significantly higher than S&R's (by as much as a factor of 30) at frequencies below 2 kHz. The frequency dependence also differs substantially: the new |C| decreases with frequency between 300 Hz and 2 kHz, while S&R's |C| increases approximately proportionally to frequency below 6 kHz. The new ∠C increases from about 0 below 1 kHz to approximately +0.5 cycles at 1.5 kHz and above, while S&R's ∠C is roughly constant at +0.2 cycles across frequency, though the difference in the phase angles is statistically significant only between 1 and 2.5 kHz.

The estimates of D from this study and S&R are similar, though the new |D| is a factor of 2 lower than S&R's (∼70 vs ∼150) below 2 kHz [Fig. 5(D)]. This difference is larger than the c.i. and s.d. and is statistically significant (p < 0.05). The phase of the new D and S&R's are comparable except above 6 kHz.

In summary, the estimates of B from the two studies are very similar, and the estimates of D and A show many similarities, but the estimates of C are quite different in frequency dependence and exhibit magnitude differences as large as a factor of 10–30. We investigate possible reasons for these differences below.

B. Matrix elements and ME transfer functions

As mentioned in Sec. II B 1 above, each transmission matrix element represents a physically measurable ME transfer function when one of the output quantities (P_V or U_S) is set to zero (Shera and Zweig, 1992); see Eq. (2). In Secs. IV C and IV D below, we consider the reciprocal of each matrix element, which under these specific conditions is a transfer function between an output and an input. For U_S = 0 (Y_C = 0; infinite cochlear load on ME; OW blocked or immobilized), P_TM = A P_V|_Us=0 and C = (U_TM/P_V)|_Us=0 [Eq. (2)], so

$${\mathbf{A}}^{–1}={\left({\mathbf{P}}_{\mathbf{V}}/{\mathbf{P}}_{\mathbf{T}\mathbf{M}}\right)|}_{\mathbf{U}\mathbf{s}=0}={{\mathbf{G}}_{\mathbf{M}\mathbf{E}\mathbf{P}}|}_{\mathbf{U}\mathbf{s}=0},$$ (7a)

the ME pressure gain with no cochlear motion. For P_V = 0 (no cochlear load on middle ear; cochlear input impedance Z_C = Y_C⁻¹ = 0),

$${\mathbf{B}}^{–1}={\left(\mathbf{U}\mathbf{s}/{\mathbf{P}}_{\mathbf{T}\mathbf{M}}\right)|}_{\mathbf{P}\mathbf{v}=0}={{\mathbf{H}}_{\mathbf{U}}|}_{\mathbf{P}\mathbf{v}=0},$$ (7b)

the ME volume velocity transfer function with a short-circuited cochlear load. Similarly, for U_S = 0,

$${\mathbf{C}}^{–1}={\left({\mathbf{P}}_{\mathbf{V}}/{\mathbf{U}}_{\mathbf{T}\mathbf{M}}\right)|}_{\mathbf{U}\mathbf{s}=0},$$ (7c)

the ME pressure transfer function with no cochlear motion and, for P_V = 0,

$${\mathbf{D}}^{–1}={\left(\mathbf{U}\mathbf{s}/{\mathbf{U}}_{\mathbf{T}\mathbf{M}}\right)|}_{\mathbf{P}\mathbf{v}=0}={{\mathbf{G}}_{\mathbf{M}\mathbf{E}\mathbf{U}}|}_{\mathbf{P}\mathbf{v}=0},$$ (7d)

the ME volume velocity gain with no ME loading (e.g., Voss and Shera, 2004).

C. Physical constraints on matrix elements: Causality

An important physical constraint on the elements of T_ME (or the transfer functions from which they are computed) is that they be causal: Outputs should not occur before inputs (Shera and Zweig, 1991, Sec. I B; Oppenheim and Schafer, 2010). As defined, T_ME computes inputs from outputs; for example, A = (P_TM/P_V)|_Us=0 [see Eq. (2a)]. Consequently, element reciprocals ${\mathit{T}}_{\mathit{ij}}^{–1}$ define outputs as a function of inputs and must be causal, e.g., A⁻¹ = (P_V/P_TM)|_Us=0 as outlined above. We test each element for causality by comparing the phase of ${\mathit{T}}_{\mathit{ij}}^{–1}\hspace{0.17em}$ to the minimum-phase response⁴ corresponding to |${\mathit{T}}_{\mathit{ij}}^{–1}$|, as described in the appendix. ∠${\mathit{T}}_{\mathit{ij}}^{–1}$ more negative than minimum-phase is consistent with causality and suggests only the presence of additional delay somewhere in the ME, which would not be surprising given that delay is known to accumulate through the ME of other rodents (de La Rochefoucauld et al., 2010). ∠${\mathit{T}}_{\mathit{ij}}^{–1}$ more positive than minimum-phase implies that some response occurs before the stimulus, which in turn suggests that ${\mathit{T}}_{\mathit{ij}}^{–1}$ is non-causal and that there is some error in measurements or computations.

The phases of B⁻¹ and D⁻¹ are nearly identical to the minimum-phase predictions. ∠A⁻¹ and ∠C⁻¹ [solid lines in Fig. 6(a)] deviate noticeably from their minimum-phase predictions (dotted lines), but these deviations are generally negative and generally less than ±1 s.d. (shading; about 0.2 cycles). ∠A⁻¹ is significantly more positive than its minimum-phase estimate above 10 kHz, and ∠C⁻¹ is more positive than its minimum-phase prediction by more than 1 s.d. above 15 kHz. As mentioned in Sec. II C above, small differences in the quantities used to compute element A at frequencies between 2.5 and 6 kHz and above 10 kHz imply that errors in element values are likely larger in these frequency ranges than in others. Such errors may have caused elements A and C to deviate from their minimum-phase estimates in these frequency ranges.

FIG. 6.

(Color online) Comparison of the phase of the reciprocals of mean ME transmission matrix elements A and C to a minimum-phase estimate (dotted line, computed as described in the text). (a) Means from this study ±1 s.d. (thick solid line and shaded area); (b) means from Songer and Rosowski (2007; solid line).

We performed the same causality analysis on the S&R matrix elements. As with the new elements, S&R's ∠B⁻¹ and ∠D⁻¹ are nearly identical to their minimum-phase predictions, while deviations of S&R's ∠A⁻¹ and ∠C⁻¹ are somewhat larger at higher frequencies. Figure 6(b) shows that S&R's ∠A⁻¹ and ∠C⁻¹ (solid lines) are nearly identical or more negative than their minimum-phase predictions (dotted lines) below 4 kHz. S&R's ∠A⁻¹ and ∠C⁻¹ are more positive than the minimum-phase predictions at several of the higher frequencies, but phase unwrapping and the large c.i.'s at these frequencies (Songer and Rosowski, 2007) make interpretation difficult.

In general, the phases of the matrix elements computed in this study and in the S&R study are consistent with causality and at many frequencies are indistinguishable from a minimum-phase prediction.

D. Comparison of matrix elements to anatomical and functional constraints

1. Comparison to an anatomical transformer ratio

As described above, A⁻¹ represents the ME pressure gain when the cochlear impedance is infinite (OW is immobilized), and element D⁻¹ represents the ME volume velocity gain when cochlear impedance (and pressure) is zero. In our two-port description of the ME (following Shera and Zweig, 1992), both A and D are bounded by the anatomical transformer ratio (TR) defined by the product of (a) the “area ratio” of TM area to OW area and (b) the “lever ratio” of malleus manubrium length to incus long process length (e.g., Wever and Lawrence, 1954).⁵ Using data from Vrettakos et al. (1988), the chinchilla area ratio is about 30 and the lever ratio is about 2.84, so TR ≈ 80.

If the TM was a rigid plate and the ossicles were rigid rods coupled by inflexible joints, the ratio of U_TM to U_S would be exactly equal to the anatomical TR: |D| = TR (Shera and Zweig, 1992). If parts of the TM move independently of the ossicles (e.g., Shaw and Stinson, 1983; Rosowski, 1994) and/or the ossicular chain has some flexibility (e.g., Willi et al., 2002; Nakajima et al., 2005; Mason and Farr, 2013), some volume velocity would be lost between the TM and the stapes, and

8.

$$\left|\mathbf{D}\right|>\mathit{\text{TR}}.$$ (8a)

Using this same line of reasoning, the ratio of P_TM to P_V with a rigid TM and ossicular chain would be exactly equal to the reciprocal of the anatomical TR: |A| = TR⁻¹. Any lost motion would result in a lower |P_V| for a given |P_TM|, and

$$\left|\mathbf{A}\right|>{\mathit{\text{TR}}}^{–1}.$$ (8b)

Therefore, TR represents a lower bound on |D|, and TR⁻¹ represents a lower bound on |A|.

The new |D| is about 0.8 × TR at frequencies below 2 kHz [Fig. 5(D)], inconsistent with Eq. (8a), but |D| + 1 s.d. is greater than TR at nearly all frequencies, and a one-tailed t-test indicates that |D| is not significantly below TR (p generally >0.11) except near 500 Hz. Songer and Rosowski's (2007) |D| is greater than TR by about a factor of 2 (except near 5 kHz), consistent with the constraint of Eq. (8a).⁶

The new |A| is greater than TR⁻¹ at nearly all frequencies [Fig. 5(A)], and |A| + 1 s.d. is greater than TR⁻¹ at all frequencies, consistent with the constraint of Eq. (8b). In contrast, Songer and Rosowski's (2007) |A| is significantly lower than TR⁻¹ by a factor of 2–3 at nearly all frequencies below 5 kHz (the upper bound of the 95% c.i. is lower than TR⁻¹). These results imply that a TR consistent with Eq. (8) and the S&R matrix elements must be at least a factor of 3 larger than the product of the anatomical area and lever ratios, while a TR consistent with Eq. (8) and the new matrix elements need be only a factor of 1.2 larger than that product.

2. Relationship between elements A and D in a non-rigid ME

As noted by Shera and Zweig [1992, Eq. (51)], an additional constraint on a real ME in which the TM and/or ossicular chain are flexible is that |AD|⁻¹ < 1: The product of the no-motion ME pressure gain and no-load volume velocity gain must be less than unity [seen by multiplying Eqs. (8a) and (8b) above]. The mean |AD|⁻¹ from the new data is <1 at nearly all frequencies (at no frequency is |AD|⁻¹ significantly >1, p > 0.14), while the product of S&R's mean |A|⁻¹ and |D|⁻¹ is higher than ours (Fig. 7) and even slightly >1 at many frequencies (though the difference between the S&R |AD|⁻¹ and 1 is well within the variance of the estimates). The somewhat higher value of the S&R |AD|⁻¹ is consistent with a lower ME flexibility in that study population.

FIG. 7.

(Color online) Magnitude of the mean of |AD|⁻¹ from this study (solid line) ±1 s.d. (shaded area) and from Songer and Rosowski (2007; dashed line). Also shown is ${\left|\widehat{\mathbf{A}}\widehat{\mathbf{D}}\right|}^{-1}$ (dotted line).

3. Comparison to measurements in real ears

In the real ear, mechanical flexibility in the ME may cause additional lost motion (Shera and Zweig, 1992; Willi et al., 2002; Nakajima et al., 2005). This can be seen by referring to Eqs. (3) and (8) above. For instance, dividing Eq. (3a) by P_V and rearranging, A = G_MEP⁻¹ − B Y_C; so, using a Taylor expansion

$${\mathbf{G}}_{\mathbf{M}\mathbf{E}\mathbf{P}}={\mathbf{A}}^{–1}/\left(1+{\mathbf{A}}^{–1}\mathbf{B}\hspace{0.17em}{\mathbf{Y}}_{\mathbf{C}}\right)\approx {\mathbf{A}}^{–1}\left(1-{\mathbf{A}}^{–1}\mathbf{B}\hspace{0.17em}{\mathbf{Y}}_{\mathbf{C}}\right).$$ (9a)

Similarly

$${\mathbf{H}}_{\mathbf{U}}\approx {\mathbf{B}}^{–1}\left(1-\mathbf{A}{\mathbf{B}}^{–1}{\mathbf{Z}}_{\mathbf{C}}\right),$$ (9b)

and, by rearranging Eq. (3d),

$$\left({\mathbf{P}}_{\mathbf{V}}/{\mathbf{U}}_{\mathbf{T}\mathbf{M}}\right)={\mathbf{C}}^{–1}/\left(1+{\mathbf{C}}^{–1}\mathbf{D}\hspace{0.17em}{\mathbf{Y}}_{\mathbf{C}}\right)\approx {\mathbf{C}}^{–1}\left(1-{\mathbf{C}}^{–1}\mathbf{D}\hspace{0.17em}{\mathbf{Y}}_{\mathbf{C}}\right)\hspace{0.17em}\text{and}$$ (9c)

$${\mathbf{G}}_{\mathbf{M}\mathbf{E}\mathbf{U}}\approx {\mathbf{D}}^{–1}\left(1-\mathbf{C}{\mathbf{D}}^{–1}{\mathbf{Z}}_{\mathbf{C}}\right).$$ (9d)

It can be seen easily that setting Y_C = 0 in Eq. (9a) yields G_MEP = A⁻¹ [Eq. (7a)]; similarly for Eq. (9c) and for setting Z_C = 0 in Eqs. (9b) and (9d) to yield Eqs. (7b) and (7d). As |Y_C| increases, as in a real ear, the contribution of the second term increases, and |G_MEP| decreases relative to |A⁻¹| [similarly for (P_V / U_TM) and |C⁻¹|]; and as |Z_C| increases in Eqs. (9b) and (9d), H_U and G_MEU decrease relative to |B⁻¹| and |D⁻¹|, respectively. Therefore

$$\left|{\mathbf{A}}^{–1}\right|>\left|{\mathbf{G}}_{\mathbf{M}\mathbf{E}\mathbf{P}}\right|,$$ (10a)

$$\left|{\mathbf{B}}^{–1}\right|>\left|{\mathbf{H}}_{\mathbf{U}}\right|>\left|{\mathbf{H}}_{\mathbf{U}}^{\mathbf{I}}\right|,$$ (10b)

$$\left|{\mathbf{C}}^{–1}\right|>\left|\left({\mathbf{P}}_{\mathbf{V}}/{\mathbf{U}}_{\mathbf{T}\mathbf{M}}\right)\right|,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{and}$$ (10c)

$$\left|{\mathbf{D}}^{–1}\right|>\left|{\mathbf{G}}_{\mathbf{M}\mathbf{E}\mathbf{U}}\right|>\left|{\mathbf{G}}_{\mathbf{M}\mathbf{E}\mathbf{U}}^{\mathbf{I}}\right|.$$ (10d)

These relationships can be seen in Fig. 8. |A⁻¹| and |C⁻¹| are larger than |G_MEP| and |(P_V / U_TM)|, respectively, at nearly all frequencies, generally by a factor of 2–3. Both ∠A⁻¹ and ∠C⁻¹ show greater accumulation than the corresponding measurements. B⁻¹ and D⁻¹ are nearly identical to H_U and G_MEU, which suggests that, with a sensor in the vestibule, |Z_C| is fairly low. |B⁻¹| and |D⁻¹| are larger than $\left|{\mathbf{H}}_{\mathbf{U}}^{\mathbf{I}}\right|$ and |${\mathbf{G}}_{\mathbf{M}\mathbf{E}\mathbf{U}}^{\mathbf{I}}$| as predicted below 2 kHz.

Figure 8(A) also shows that |G_MEP| is about a factor of 2.5 lower than TR at all frequencies, consistent with previous observations in chinchilla and other species (Puria et al., 1997). Figure 8(D) shows that |G_MEU| is comparable to TR⁻¹ at all frequencies and that |${\mathbf{G}}_{\mathbf{M}\mathbf{E}\mathbf{U}}^{\mathbf{I}}$| is somewhat lower, especially below 2 kHz. All these results show that the new transmission matrix elements are consistent with measurements of ME transfer functions and with anatomical constraints.

E. Effect of methodological differences in computing matrix elements

Are the differences between Songer and Rosowski's (2007) and the new matrix elements due to different computational methods? In S&R, B is the reciprocal of the drained-cochlea stapes velocity transfer function [($H_P^D$)⁻¹; S&R Eq. (4)], and D is computed from B and the ME admittance with cochlea drained [$Y_{\text{ME}}^D$; S&R Eq. (5)]. S&R's B and D are fairly similar to ours (though there are small differences), as are the data that went into the computations (see Ravicz and Rosowski, 2012b, 2013b).

The greatest differences between our transmission matrix and the previous estimate are in elements A and C, which S&R computed from B, D, and measurements with stapes mobility reduced with cement (“fixed”). To check the effect of computational method, we used S&R's measurements of input admittance and stapes velocity transfer function in intact ears (Y_ME and H_P) or with the stapes fixed ($Y_{\text{ME}}^F$ and $H_P^F$; Songer, 2011) in Eqs. (5a) and (5c) to compute a third estimate of A and C. The A and C computed by our method were very similar to the A and C reported by S&R, though the estimates by our method were noisier. This similarity suggests that the different techniques used to compute A and C in this report and S&R yield similar results, and that the differences between S&R's matrix elements and ours are due to differences in the data sets used in the computations.

F. Predicted transmission matrix for an intact MEC

The ME transmission matrix presented here describes the ME function with the MEC opened by two holes, one into the superior bulla and a second into the posterior bulla. The usefulness of a ME transmission matrix model would be improved by an estimate of the ME function with an intact MEC.

A ME transmission matrix that includes the effect of an intact MEC can be computed from estimates of the effect of the open MEC on the measurements used to compute the matrix elements. The few previous data on the effect of opening or closing the MEC on ME transmission in chinchilla suggest that closing the MEC should (a) reduce |Y_TM| below 500 Hz, by a factor of 10 or more below 250 Hz, and increase ∠Y_TM by ∼0.2 cycles below 1.5 kHz [Rosowski et al., 2006; Ravicz and Rosowski, 2012b, Fig. 7(a)]; (b) reduce |H_V| (and therefore |H_U|) by as much as a factor of 5 below 1 kHz and increase ∠H_V by ∼0.2 cycles below 1 kHz (Dallos, 1970; Ruggero et al., 1990); and (c) have the same effect on G_MEP as on H_U, as the cochlear input admittance Y_C = H_U/G_MEP should be unaffected by the state of the MEC (Huang et al., 1997; Ravicz and Rosowski, 2013a). T_ME elements for a ME with an intact MEC could then be computed from modified measurements by Eq. (5).

The transmission-matrix ME representation facilitates a simpler estimate. Most ME circuit or lumped-element models assume that the primary effect of closing the MEC is to add an acoustic compliance C_MEC in series with the TM and the rest of the ME (e.g., Zwislocki, 1962; Dallos, 1970; Wilson and Johnstone, 1975; Ruggero et al., 1990; Huang et al., 1997). As mentioned in Sec. I, an advantage of a transmission-matrix ME model is that the contribution of additional elements in series can be described by a cascade of transmission matrices. We model the closed MEC as a simple impedance Z_MEC computed from its volume V_MEC = 2000 mm³ (Rosowski et al., 2006)⁷ by

$${\mathbf{Z}}_{\mathbf{M}\mathbf{E}\mathbf{C}}=1/\left(j\hspace{0.17em}\omega \hspace{0.17em}{C}_{\text{MEC}}\right)=\rho c^2/\left(j\hspace{0.17em}\omega \hspace{0.17em}{V}_{\text{MEC}}\right),$$ (11)

where ρ = 1.19 kg m⁻³ is the density and c = 345 m s⁻¹ is the speed of sound in moist air at 37  °C, j is the imaginary operator, and ω is the radian frequency [Beranek, 1986, Eqs. (5.37) and (5.38)]. The transmission matrix T_MEC of Z_MEC in series is

$${\mathbf{T}}_{\mathbf{M}\mathbf{E}\mathbf{C}}=\left[\begin{array}{cc}\mathbf{1}& {\mathbf{Z}}_{\mathbf{M}\mathbf{E}\mathbf{C}}\\ \mathbf{0}& \mathbf{1}\end{array}\right]$$ (12a)

(e.g., Lampton, 1978; Shera and Zweig, 1991). The matrices are cascaded to get the matrix describing the ME with MEC intact

$${\mathbf{T}}_{\mathbf{M}\mathbf{E}}^{\mathbf{M}\mathbf{E}\mathbf{I}}={\mathbf{T}}_{\mathbf{M}\mathbf{E}\mathbf{C}}{\mathbf{T}}_{\mathbf{M}\mathbf{E}}=\left[\begin{array}{cc}\mathbf{A}+\mathbf{C}{\mathbf{Z}}_{\mathbf{M}\mathbf{E}\mathbf{C}}& \mathbf{B}+\mathbf{D}{\mathbf{Z}}_{\mathbf{M}\mathbf{E}\mathbf{C}}\\ \mathbf{C}& \mathbf{D}\end{array}\right].$$ (12b)

This procedure predicts that closing the MEC affects only A and B.

The predictions of the effects on A and B by these two methods are shown in Fig. 9. The series matrix method predicts an increase in |A| of as much as a factor of 5 below 0.5 kHz and a decrease in ∠A below 1.5 kHz [Fig. 9(A)], while the prediction from the effects on measurements is smaller and over a small frequency range. Both methods predict changes in B [Fig. 9(B)] similar to each other and to the changes in A. Equation (5) predicts a change in C approximately opposite to those in B, whereas the series matrix method predicts no effect on C as described above. The prediction of a similar effect on only A and B is consistent with a series impedance and with the model of Lemons and Meaud [2016, Eq. (13)], in which only A and B depend on Z_MEC.

FIG. 9.

Effect of closing the ME cavities (result: ME intact) on elements A and B, predicted by a cascaded transmission matrix of the ME cavities (T_MEC, solid line) and by the measured effects on Y_TM (Rosowski et al., 2006) and H_V (Ruggero et al., 1990; dashed line). All panels: Top: magnitude; bottom: phase.

G. Evidence for and implications of ME flexibility

It is surprising that two estimates of the chinchilla ME transmission matrix from the same laboratory (this study and Songer and Rosowski, 2007) could have substantial differences in two of the elements (A and C). These differences could have several causes, including differences in the MEs of the subject populations, deviations of S&R's measurement conditions from the no-motion or no-load cases, differences in methodology, or a change in ME properties between the first and last of the measurements—a violation of the reciprocity assumption that the ME is statistically stationary. Great care was taken during both sets of experiments to minimize unplanned changes in the condition of the ME, and the consistency check of our data (Fig. 4) supports this point. It is possible that making the hole near the OW to place the P_V pressure sensor may have changed the properties of the annular ligament or incudostapedial joint, but G_MEP in this study is very similar to that in a later study in which access to the vestibule was through the braincase (Chhan et al., 2013) instead of near the OW. The similarity between estimates of C from S&R's data using their technique vs ours suggests that their reduction of stapes mobility with glue was a sufficiently good approximation of immobilizing the OW.

1. Evidence of ME flexibility from changes in Y_TM and H_U

The results of Fig. 7 suggest that one possible explanation for the differences in matrix elements A and C is a difference in the flexibility of the ME between the two subject populations. Insight into the degree of ME flexibility can be gained by examining the relative changes in stapes velocity and ME input admittance when the cochlear load is changed. We first examine the effect of reducing the cochlear load by opening or draining the IE on Y_TM and H_U. (Increasing the cochlear load or, equivalently, fixing the stapes, provides a more sensitive assay of ME flexibility than reducing the cochlear load, as reducing Z_C toward 0 tends to “short out” the flexibility shunt paths, but the data set in this study contains no data with an increased cochlear load.) If the TM and ossicular chain were rigid, any change in stapes mobility should cause an equal change in TM mobility (Shera and Zweig, 1992). With some flexibility or lost motion in the TM and ossicular chain, the changes in TM mobility may be substantially smaller than changes in stapes mobility.

Figure 10(A) compares the increase in |Y_TM| to the increase in |H_U| resulting from reductions in cochlear load, using measurements of Y_TM and H_U with a sensor in the vestibule (Ravicz and Rosowski, 2012b, 2013b) or with the vestibule hole open (Fig. 4), and an estimate of Y_TM with Z_C = 0, computed from H_U|_Pv=0 = B⁻¹ using Eq. (7). In each case, the increase in |Y_TM| of a factor of 2 or less is substantially smaller than the increase in |H_U| of a factor of 3–4 below 1 kHz. In contrast, Fig. 10(B) shows that the increases in S&R's |Y_ME| and |H_P| upon draining the cochlea (Songer and Rosowski, 2007, Figs. 3 and 4) are nearly identical below 3 kHz. These results are consistent with the MEs of our experimental animals having a higher TM and/or ossicular flexibility than those used in S&R's experiments: changes in ME load produce larger changes in |U_S| than in |U_TM|.

FIG. 10.

(Color online) (A) Comparison of the effects of reducing the cochlear load on |Y_TM| and |H_U|. Ratio of mean measured |H_U| with a sensor in the vestibule (long-dashed-dotted line) and with the vestibule hole open (double-dotted-dashed line) to $\left|{\mathbf{H}}_{\mathbf{U}}^{\mathbf{I}}\right|$ with the IE intact; also, ratio of |B⁻¹| = predicted |H_U| with Z_C = 0 (solid line) to $\left|{\mathbf{H}}_{\mathbf{U}}^{\mathbf{I}}\right|$. Also shown are the ratios of mean measured |Y_TM| with a sensor in the vestibule (long-dashed line) and with the vestibule hole open (dotted-dashed line) to $\left|{\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{I}}\right|$ with the IE intact; also shown is the ratio of predicted |Y_TM| with the cochlea drained (dotted line) to $\left|{\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{I}}\right|$. (B) Ratio of Songer and Rosowski (2007) stapes velocity transfer function ($\left|H_P^D\right|$; solid line) and ME input admittance magnitudes ($\left|Y_{\text{ME}}^D\right|$; dotted line) with the cochlea drained to values with the IE intact (from S&R Figs. 3 and 4).

Similar conclusions can be drawn by comparing an estimate of the reduction in |Y_TM| with the OW blocked [computed from Eq. (4c) by setting H_U = 0 and therefore U_S = 0] to S&R's reduction with the stapes fixed (Songer and Rosowski, 2007, Fig. 3)—see Fig. 11. S&R saw a reduction in ME input admittance of roughly an order of magnitude below 500 Hz when stapes velocity was reduced by at least a factor of 10–20 (also Songer, 2005), which implies that the TM and ossicular chain are fairly rigid. In contrast, our prediction of the reduction in |Y_TM| is less than a factor of 2. This smaller predicted reduction in |Y_TM| is again consistent with higher ME flexibility in the data set presented here.

FIG. 11.

(Color online) Predicted effect of blocking the OW (increasing the cochlear load) on $\left|{\mathbf{Y}}_{\mathbf{T}\mathbf{M}}^{\mathbf{I}}\right|$ (solid line), compared with the effect measured by Songer and Rosowski (2007; ratio of $\left|Y_{\text{ME}}^F\right|$ to |Y_ME| from their Fig. 3; dashed line). Also shown are the effects from measurements in cat (Lynch, 1981) and the mean ratio of umbo velocities in human temporal bones with the stapes fixed with dental cement versus normal (from Nakajima et al., 2005, Fig. 6; dotted line).

Our predicted reduction is similar to the effect on umbo velocity observed in human temporal bones (Fig. 11), in which the incudomallelar joint is known to be flexible (Willi et al., 2002): Reducing stapes velocity by a factor of 30–100 reduced umbo velocity by only a factor of 2–3 (Nakajima et al., 2005). (Note that this does not include any effect of TM compliance.) In contrast, the effect on ME input admittance of fixing the stapes in cat (Lynch, 1981, Chap. III, Fig. 8), which appears to have a much lower ossicular flexibility (Guinan and Peake, 1967), is more similar to S&R's results than to ours.

2. Implications for model predictions of reverse ME transmission

ME flexibility also influences ME sound transmission in reverse, from cochlea to ear canal. For example, the ME input admittance at the OW ${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}$ is affected both by the impedance looking out the ear canal from the TM ${\mathbf{Z}}_{\mathbf{E}\mathbf{C}}^{\mathbf{R}}$ (or its reciprocal, the output admittance ${\mathbf{Y}}_{\mathbf{E}\mathbf{C}}^{\mathbf{R}}$) and by the properties of the ME itself. As mentioned in Sec. I, an advantage of a transmission-matrix ME model is that it can also be used to predict ME transmission in reverse. In this case, the reverse transmission matrix ${\mathbf{T}}_{\mathbf{ME}}^{\mathbf{\prime }}$ is used, which is computed easily from T_ME by

$${\mathbf{T}}_{\mathbf{ME}}^{\mathbf{\prime }}=\left[\begin{array}{cc}{\mathbf{A}}^{\mathbf{\prime }}& {\mathbf{B}}^{\mathbf{\prime }}\\ {\mathbf{C}}^{\mathbf{\prime }}& {\mathbf{D}}^{\mathbf{\prime }}\end{array}\right]=\left[\begin{array}{cc}\mathbf{D}& \mathbf{B}\\ \mathbf{C}& \mathbf{A}\end{array}\right]/\text{det}\left({\mathbf{T}}_{\mathbf{M}\mathbf{E}}\right)$$ (13)

(e.g., Lampton, 1978), where det(T_ME) = 1 because T_ME is assumed reciprocal, as mentioned above. So, in the reverse transmission case, where the outwardly-directed volume velocities are defined as the negative of the forward case [Fig. 1(C)],

$$\left\{\begin{array}{c}{\mathbf{P}}_{\mathbf{V}}\\ –{\mathbf{U}}_{\mathbf{S}}\end{array}\right\}={\mathbf{T}}_{\mathbf{ME}}^{\mathbf{\prime }}\left\{\begin{array}{c}{\mathbf{P}}_{\mathbf{T}\mathbf{M}}\\ –{\mathbf{U}}_{\mathbf{T}\mathbf{M}}\end{array}\right\}=\left[\begin{array}{cc}\mathbf{D}& \mathbf{B}\\ \mathbf{C}& \mathbf{A}\end{array}\right]\left\{\begin{array}{c}{\mathbf{P}}_{\mathbf{T}\mathbf{M}}\\ –{\mathbf{U}}_{\mathbf{T}\mathbf{M}}\end{array}\right\}.$$ (14a)

The reverse ME input admittance ${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}$ = −U_S/P_V is computed for various values of ${\mathbf{Y}}_{\mathbf{E}\mathbf{C}}^{\mathbf{R}}$ = −U_TM/P_TM by expanding Eq. (14a):

$${\mathbf{P}}_{\mathbf{V}}=\mathbf{D}\hspace{0.17em}{\mathbf{P}}_{\mathbf{T}\mathbf{M}}–\mathbf{B}\hspace{0.17em}{\mathbf{U}}_{\mathbf{T}\mathbf{M}},$$ (14b)

$$–{\mathbf{U}}_{\mathbf{S}}=\mathbf{C}\hspace{0.17em}{\mathbf{P}}_{\mathbf{T}\mathbf{M}}–\mathbf{A}\hspace{0.17em}{\mathbf{U}}_{\mathbf{T}\mathbf{M}}.$$ (14c)

Then, dividing Eq. (14c) by Eq. (14b),

$${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}=\frac{\mathbf{A}{\mathbf{Y}}_{\mathbf{E}\mathbf{C}}^{\mathbf{R}}+\mathbf{C}}{\mathbf{B}{\mathbf{Y}}_{\mathbf{E}\mathbf{C}}^{\mathbf{R}}+\mathbf{D}}$$ (14d)

(e.g., Voss and Shera, 2004; Songer and Rosowski, 2007).

The EC output impedance is computed for two extremes: low ${\mathbf{Y}}_{\mathbf{E}\mathbf{C}}^{\mathbf{R}}$, modeled from the impedance of the acoustic source used to measure EC input admittances (Z_src, Rosowski et al., 2006; Songer and Rosowski, 2006); and high ${\mathbf{Y}}_{\mathbf{E}\mathbf{C}}^{\mathbf{R}}$, modeled from the radiation impedance Z_E looking out the ear canal from the TM (Dear, 1987; Rosowski, 1991). These extreme values of ${\mathbf{Y}}_{\mathbf{E}\mathbf{C}}^{\mathbf{R}}$ are shown in Fig. 12(A). Note that the magnitudes differ by a factor of 1000 at 100 Hz and are similar above 2 kHz.

FIG. 12.

(A) Ear canal output admittance ${\mathbf{Y}}_{\mathbf{E}\mathbf{C}}^{\mathbf{R}}$ for two extreme conditions: Open (dotted line) and blocked with an admittance acoustic source (solid line; Songer and Rosowski, 2007). (B) Reverse ME input admittance ${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}$ predicted for the two EC loads by our model (open: black dashed line; blocked: black solid line) and the S&R model (open: gray dotted-dashed line; blocked: gray long-dashed-dotted line). Both panels: Top: magnitude; bottom: phase.

Estimates of ${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}$ computed from Eq. (14d) by the new model and by the S&R model of a lower-flexibility ME are shown in Fig. 12(B). Our predicted ${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}$ is nearly invariant with ${\mathbf{Y}}_{\mathbf{E}\mathbf{C}}^{\mathbf{R}}$, even though the EC loads vary over several orders of magnitude below 2 kHz. The contribution of the ME alone (not shown), computed by setting ${\mathbf{Y}}_{\mathbf{E}\mathbf{C}}^{\mathbf{R}}$ infinite, is very similar to the other ${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}$ estimates. The invariance with the EC load predicted by our model is consistent with a relatively high ME flexibility, as nearly all of ${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}$ is due to the properties of the ME and little is due to the terminating impedance.

Our estimate of $\left|{\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}\right|$ is less than $\left|{\mathbf{Y}}_{\mathbf{E}\mathbf{C}}^{\mathbf{R}}\right|$ with the EC closed at low frequencies by about a factor of 100 and less than $\left|{\mathbf{Y}}_{\mathbf{E}\mathbf{C}}^{\mathbf{R}}\right|$ with the EC open by about a factor of 10⁵ at 100 Hz. This result suggests that, even in this flexible ME, ${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}$ is controlled by the ME, probably by the stapes annular ligament.

In contrast, S&R's estimate of $\left|{\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}\right|$ with the EC closed is less than $\left|{\mathbf{Y}}_{\mathbf{E}\mathbf{C}}^{\mathbf{R}}\right|$ by about a factor of 10⁴ at 100 Hz, and an estimate of $\left|{\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}\right|$ with the EC open computed from S&R's transmission matrix (Lemons and Meaud, 2016) is less than $\left|{\mathbf{Y}}_{\mathbf{E}\mathbf{C}}^{\mathbf{R}}\right|$ with the EC open by nearly a factor of 10⁶ at 100 Hz. S&R's model predicts a substantial effect of the EC termination on ${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}$: about a factor of 10 in magnitude below 500 Hz, with a phase change consistent with the magnitude slope. Even with the EC open, S&R's estimate of $\left|{\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}\right|$ is lower than ours by about a factor of 5. The greater sensitivity of ${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}$ to ${\mathbf{Y}}_{\mathbf{E}\mathbf{C}}^{\mathbf{R}}$ in S&R's model is consistent with lower ME flexibility, but the lower ${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}$ than ours even with the EC open suggests that the overall admittance of the MEs in S&R's study was lower than ours.

At frequencies above 2 kHz, ${\mathbf{Y}}_{\mathbf{E}\mathbf{C}}^{\mathbf{R}}$ is similar with the EC open or closed, and consequently our estimates of ${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}$ and S&R's estimates are the same for both terminating admittances.

The predictions of ${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}$ also suggest similarities and differences in the ME function in the forward and reverse directions. The relatively low sensitivity of our $\left|{\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}\right|$ to the EC terminating impedance is consistent with the relatively small effect on |Y_TM| of opening or draining the cochlea (Fig. 10) or of blocking the OW (Fig. 11) compared to the effect on |H_U|; see Fig. 2. A comparison of $\left|{\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}\right|$ to |Y_C| in Fig. 13 shows that the admittances looking into and out of the OW are similar, especially below about 0.8 kHz where the magnitudes are within a factor of 2 or 3 and the phases are similar and near 0.

FIG. 13.

Comparison of reverse ME input admittance magnitude $\left|{\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}\right|$ predicted for an open EC (thick solid line) with the cochlear input admittance |Y_C| (long-dashed-dotted line).

It should be noted that our estimates of the phase of ${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}$ [Fig. 12(B)] are not always within the −0.25 to +0.25 cycle range associated with a passive admittance, especially between 2 and 6 kHz and above 14 kHz. In these frequency ranges, errors in matrix elements due to the small effect of the pressure sensor on Y_TM and H_U could have been substantial. Nevertheless, our ME transmission matrix model provides an estimate of the approximate magnitude of ${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}$ and an indication of how $\left|{\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}\right|$ is affected by the EC load. The sensitivity of ${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}$ to the EC load has implications for the study of otoacoustic emissions and perception of bone-conducted sound as well as active ME devices such as bone-anchored hearing aids.

3. Possible sources and causes

To date, there has not been much investigation of ME flexibility [see Mason and Farr (2013) for a review]. Signs of ossicular flexibility include an increase in the ossicular lever ratio and a phase accumulation between umbo and stapes motion (e.g., Guinan and Peake, 1967, in cat; Manley and Johnstone, 1974, in guinea pig; Willi et al., 2002, in human; de La Rochefoucauld et al., 2010, in gerbil). Though the chinchilla has a freely-mobile ME (Rosowski, 1992), the chinchilla incudo-mallelar joint is generally considered to be fused (Mason and Farr, 2013), as in the guinea pig (Amin and Tucker, 2006) and gerbil (Fleischer, 1978); yet G_MEP and H_V [Ravicz and Rosowski (2013a,b)] show a high-frequency magnitude decrease and phase accumulation consistent with TM or ossicular flexibility.

However, most investigations of ME flexibility have focused on the ossicular chain; very few have included the effects of the TM (Mason and Farr, 2013). In fact, there is substantial evidence that, over various frequency ranges, much of the TM moves relatively independently of the umbo. It has been known for some time that the motion of most of the TM has a much higher amplitude than that of the umbo (Tonndorf and Khanna, 1972 in human; Khanna and Tonndorf, 1972, in cat), and modeling human TM input to the ME has necessitated considering the TM to be divided into coupled and uncoupled portions (Shaw and Stinson, 1983). More recent studies have demonstrated that, at moderate frequencies and above, the TM motion is complex and much of the TM moves in opposition to the umbo at these frequencies (Rosowski et al., 2013; Cheng et al., 2013). In chinchilla, this frequency range extends to much lower frequencies (∼300 Hz; Rosowski et al., 2013).

The question remains why the ME flexibility implied by our transmission matrix is higher than what Songer and Rosowski's (2007) matrix implies. We see no apparent flaws in the technique nor the measurements used in this or the previous study. Have these studies used examples from opposite ends of a flexibility distribution or continuum in the normal chinchilla population? The populations used in the two studies came from different sources. Have some of these relatively small captive breeding populations of chinchillas in the U.S.A. evolved a more (or less) flexible ME over the last few years? Further work is clearly necessary to resolve this issue.

V. SUMMARY AND CONCLUSIONS

• (1)We present a new estimate of a chinchilla ME transmission matrix T_ME = [A B; C D] based solely on measurements of ear-canal and vestibule sound pressure, ME input admittance, and stapes velocity in the same animals with the ME open and a pressure sensor in the IE (Ravicz and Rosowski, 2012b, 2013a,b). Admittance and stapes velocity were also measured before inserting the pressure sensor in the IE. This new transmission matrix extends the frequency range to 18 kHz (Fig. 3).
• (2)The new values of elements A and C are consistent with independent measurements (Fig. 4).
• (3)The new element B is virtually identical to that estimated in a previous study (Songer and Rosowski, 2007; Fig. 5), and the new element D is similar to the previous estimate. The new elements A and C differ substantially from the previous estimates.
• (4)Element values were checked against physical, anatomical, and functional constraints. The element reciprocals (A⁻¹, B⁻¹, C⁻¹, D⁻¹) in both studies are causal (Fig. 6) and are therefore consistent with physical constraints. |A⁻¹| in this study is more consistent with the anatomical TR of a factor of 80 than the previous study (Fig. 5); and |AD|⁻¹ < 1, as required for a flexible TM-ME (Fig. 7). The magnitudes of element reciprocals are larger than corresponding measured transfer functions in real ears (Fig. 8).
• (5)The substantial difference in C between the studies, and in the data sets used in its calculation, suggests that ME flexibility differs between the animal subject populations used in the two studies (Figs. 10 and 11).
• (6)An estimate of the reverse ME input admittance ${\mathbf{Y}}_{\mathbf{M}\mathbf{E}}^{\mathbf{R}}$ from the transmission matrix model in this study has a higher magnitude and a substantially smaller variation with ear-canal load than an estimate from the previous study, consistent with higher ME flexibility in the population in this study (Figs. 12 and 13).
• (7)The effect of the open ME in the measurements used to compute T_ME is estimated by a cascaded transmission matrix of a series impedance. Closing the ME is predicted to increase |A| and |B| by up to a factor of 5 below 0.5 Hz and decrease ∠A and ∠B below 1.5 kHz (Fig. 9), similar to estimates from previous measurements, with no effect on C or D.
• (8)We have shown that the ME transmission matrix can be computed solely from measurements of forward sound transmission without relying on modifications to the ME or assumptions of the state of the stapes (fixed or not) or the cochlea (drained or not). Applying this technique to the data set used in the previous study (Songer and Rosowski, 2007) produced element values similar to those presented previously.

APPENDIX

We assessed causality of transmission matrix elements by comparing the phase of their reciprocals to a minimum-phase estimate computed as follows. For a minimum-phase system, the complex cepstrum is causal (Randall, 1987; Oppenheim and Schafer, 2010), and the complex cepstrum is related to the power cepstrum by a factor of 2 (Randall, 1987). The Matlab function below (adapted from minphase.m; Ru, 1997) computes the power cepstrum from the real part RealH of the spectrum of the reciprocal of the matrix element, computes the complex cepstrum, and enforces the condition that the complex cepstrum is causal (by setting it to zero for all quefrencies <0). This causal cepstrum is then converted back into a spectrum which is therefore minimum-phase.

function [H, imagH] = causality (realH)

L = length (realH); % Must have uniform frequency spacing

R1 = [real1; real1(L); real1(L:-1:2)]; % pure even magnitude function

fftR1 = fft(R1);

fftR1(2:L) = fftR1(2:L) * 2; % select single side band

fftR1(L+1:2*L) = fftR1(L+1:2*L) * 0; % enforce causality

R1 = ifft(fftR1);

imag1 = -imag(R1);

imagH = [0; (imag1(2:L)-imag1(2*L:-1:L+2))/2]; % pure odd phase function

H = realH(:) + 1i*imagH;