Download 1 The Auditory Periphery - University of Arizona Math

Quantifying Stimulus Frequency Otoacoustic Emissions Empirically
C. Bergevin
This case study is designed with a practical emphasis on how to quantify uncertainty in experimental data. While
the focus here is on a specific type of experiment related to auditory physiology, the general principles described
will potentially be of value fin a wide range of different empirical contexts (i.e., the basic ideas discussed here apply
broadly). In terms of mathematical content, this case study explores the following topics: Fourier analysis, complex
variables and propagation of errors.
1
The Auditory Periphery
The primary purpose of the ear is convert incoming acoustic stimuli into electrical signals that can subsequently be
passed onto the central nervous system (i.e., it acts as a transducer). However, in accomplishing such a task, the ear
exhibits a remarkable frequency discrimination as well as the ability to be sensitive to stimuli encompassing a very
large range of amplitudes. For example, the range of amplitudes to which the ear normally responds over represents
roughly 12 orders of magnitude in energy!
The ear can be roughly distinguished into three different parts: the outer, middle and inner ear (Fig. 1). Sound
enters the external auditory meatus, or ear canal (outer ear), and sets the tympanic membrane (or eardrum) into motion.
On the other side coupled to the tympanic membrane, are the three small bones (ossicles) which span the middle ear.
To a first degree, these bones serve as impedance matchers between air-filled outer world and the fluid-filled inner ear.
The final bone, the stapes, is coupled to the oval window, entrance to the snail-shaped cochlea (inner ear).
The cochlea is a long coiled tube comprised of three different chambers. A flexible partition called the basilar
membrane (BM) separates the top two chambers (scala vestibuli and scala media) from the bottom (scala tympani),
except at the most apical region of the cochlea where there is a hole called the helicotrema. The width and thickness
of the BM change along the cochlear length, as does the cross-sectional area of the chambers. As the stapes footplate
moves in and out in response to sound, pressure variations in the fluid will be setup inside the cochlea.
A flexible membrane called the round window allows for the volume displacements outwards as the stapes pushes
inwards (since the cochlear fluid is largely incompressible). This creates a pressure difference across the basilar
membrane. As a result, a traveling wave propagates along the BM as shown in Fig. 1. Each point along the BM
resonates at a particular frequency due to its graded mass and stiffness. It is this tonotopic organization that allows the
cochlea to function as a spectrum analyzer (see subsequent section for further discussion). Higher frequencies excite
excite basal regions (near the stapes) while lower frequencies stimulate more apically.
Sitting on top of the BM in the scala media is a remarkable structure called the organ of corti. This structure
contains the hair cells (HCs), which effectively act as the mechano-electro transducers, mapping BM motion to action
potentials in the auditory nerve fibers (ANFs) innervating the HCs (Fig. 1). A stereociliary bundle extends out of the
epithelial surface of the HC that contains a unique set of transduction channels. As the BM is displaced and moves upwards in the transverse direction, there is a shearing between the BM and the overlying tectorial membrane (TM). This
shearing causes a deflection of the stereociliary bundle (shown by bi-directional arrow in Fig. 1), thereby stimulating
the transduction channels [Corey and Hudspeth, 1979]. The scala media is unique in that its fluid composition creates
an extremely high potential of about +80 mV (the largest resting potential in the entire body) due to the pumping
action of the stria vascularis which also causes a large K + concentration. This potential is quite high compared to the
resting potential of the hair cell, which is at about -60 mV. As a result of this large potential difference, small bundle
deflections cause appreciable changes in HC membrane potential, triggering a synaptic release of neurotransmitter to
the innervating neurons.
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Statistics Case Studies: SFOAEs
Math 363 (Spring 2009)
Figure 1: Cross-section of the human auditory periphery [A. Greene].
Mammalian Cochlea Uncoiled
to
Vestibular
System
Stapes
to
Middle
Ear
Helicotrema
Acoustic Energy
pliant &
massive
stiff &
thin
Round
Window
Cochlear
Partition
C.D. Geisler (modified)
Figure 2: Schematic showing BM traveling wave along a straightened (uncoiled) cochlea [Geisler, 1990].
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Ca2+ ions
K+ ions
+80 mV
Inner
Hair Cell
-60 mV
Afferent Auditory
Nerve Fiber (ANF)
Figure 3: Schematic showing that the hair cells act as mechano-electro transducers that convert BM displacement into electrical signals that trigger
the synapsed neurons.
There are two distinct types of HCs in the mammalian inner ear: inner hair cells (IHCs) and outer hair cells
(OHCs). There are roughly three OHCs for every IHC. IHCs receive the bulk of the afferent innervation (going to
the brain) while OHCs receive the bulk of the efferent innervation (coming back down from the brain). Mammalian
OHCs appear unique in that they exhibit somatic cell motility, a process by which the cell changes its length in
response to mechanical or electrical stimulation [Brownell et al., 1985]. It is commonly believed somatic motility
plays an important role in cochlear mechanics, acting in some way to provide an amplification mechanism and boost
the response of the BM to low-level signals.
2
Otoacoustic Emissions
In addition to being responsive to sound, the inner ear also emits sound [Kemp, 1978]. These otoacoustic emissions
(OAEs) are believed to be a by-product of processes occurring in the inner ear that allow it to achieve its exquisite
sensitivity and frequency selectivity. OAEs can arise both with or without an evoking stimulus and can be measured
non-invasively. An example of spontaneous emissions (SOAEs) are shown in Fig. 2. A time waveform was recorded
from a sensitive microphone placed in the ear canal of an adult with normal hearing. Taking the Fourier transform of
the time waveform (see subsequent section), the spectrum in Fig. 2 shows the various frequency components present
in the signal. The tall narrow-peaks indicate almost tone-like sounds that are spontaneously (i.e., no stimulating sound
is present) coming from the ear.
Given the difficulty of direct physiological measurements (e.g., the fragile mammalian cochlea is completely encased by the hardest bone in the body), OAEs provide a direct and objective measure of the processes of the inner
ear. In general, healthy ears emit while impaired ones (i.e., individuals with some form of hearing loss) do not. As
a result, OAEs have been utilized extensively in clinical settings (e.g. diagnostic hearing screening for adults and
newborns, monitoring intra-cranial pressure in head trauma patients). However, further applications are limited by
our incomplete understanding of the actual mechanisms in the inner ear that give rise to the emissions. Thus, there is
significant motivation to further study OAEs and characterize their underlying generation mechanisms.
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Measured Level (dB SPL)
40
30
20
10
0
-10
-20
-30
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Frequency (Hz)
Figure 4: Spontaneous otoacoustic emissions (SOAEs) from an individual human ear.
One type of evoked emission that is of significant interest is a stimulus-frequency emission (SFOAE). This is
an emission that is evoked by a single tone. While relatively straight-forward to interpret1 , SFOAEs are somewhat
difficult to measure in that they occur at the same frequency as the stimulus being used to evoke them, but typically
at a much lower level. Thus the microphone signal in the ear canal is typically dominated by the stimulus tone. At
lower stimulus levels, interference between the SFOAE and the stimulus can be observed as ripples in the measured
microphone response (Fig. 2).
3
3.1
Empirical Determination of SFOAEs
Measuring Time Waveforms From the Ear Canal
In order to measure SFOAEs, the proper equipment and acoustic environment are required. As schematized in Fig. 3.1,
a subject sits in an acoustic isolation booth (to minimize the effects of external noise) with a sensitive probe placed
gently in their ear (similar to an insert earphone one uses for an iPod). This probe contains a microphone (to measure
the emission) and two earphones (to provide a stimulus to evoke the emission). Further technical details with regard
to measuring SFOAEs are provided in the appendix. An example of an SFOAE frequency sweep in a normal hearing
adult is shown in Fig. 3.1.
3.2
Fourier Transform
It is likely worthwhile to briefly provide a brief review of the concept of the Fourier transform. The basic idea is
that a signal can be expressed as a sum of linearly-independent basis functions. Similar to a Taylor series expansion
(which describes a function, within some region of convergence, as the sum of polynomials), the Fourier transform
uses sinusoids as the underlying basis functions. Given that many acoustic sounds in nature derive from some sort of
oscillatory behavior (e.g., vocal fold vibration, oscillation of a piano string), it intuitively makes sense that sinusoids
make an appropriate choice for the underlying basis function.
In our specific case of measuring OAEs, the Fourier transform allows for one to go back and forth between describing a signal in either the time or frequency domain. As indicated in Fig. 2, a time waveform from the probe
1 Many other types of evoked emissions use mutli-tone (DPOAEs) or broadband (CEOAEs) stimuli. Furthermore, these emissions are thought
to be composed of different generator regions throughout the cochlea while SFOAEs are thought to arise in a single localized region.
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Figure 5: Frequency dependence of ear-canal pressure and its variation with sound level in a normal human ear. The curves, offset vertically
from one another for clarity, represent the normalized amplitude of the ear-canal pres- sure, Pec , measured with a constant voltage applied to
the earphone driver [Shera and Zweig, 1993]. The approximate sensation level of the stimulus tone at 1300 Hz is indicated on the right. At the
highest level the pressure amplitude varies relatively smoothly with frequency. As the stimulus level is lowered, sound generated within the cochlea
combines with the stimulus tone to create an oscillatory acoustic interference pattern that appears super- posed on the smoothly varying background
seen at high levels. Near 1500 Hz, the frequency spacing fOAE between adjacent spectral maxima is ap- proximately 100 Hz. [figure and caption
taken from Shera and Guinan, 1999]
OAE Measurement System
ER-10C
(amplifier)
to
earphones
from
mic
24 bit A/D and D/A
COMPUTER
- all acquisition/analysis
software coded
manually in C
human subject
OAE probe contains both a microphone
and two earphones (to minimize system
distortion)
probe coupled
tightly to ear
noise reduction chamber
Figure 6: Schematic showing setup for measuring SFOAEs.
5
Phase [cycles]
Magnitude [dB SPL]
Statistics Case Studies: SFOAEs
Math 363 (Spring 2009)
10
0
-10
-20
-30
-40
0.5
1
2
3
4
5
0.5
1
2
3
4
5
0
-10
-20
-30
Probe Frequency [kHz]
Figure 7: SFOAE frequency sweep from a single individual human ear. Both magnitude (top) and phase (bottom) are shown. Noise floor is shown
by the dashed lines. Error bars indicate standard error of the mean across the 35 measurements taken at a given frequency. [Lp = 40 dB SPL,
Ls = Lp + 15 dB, fs = fp + 40 Hz]
microphone was transformed into the spectral representation, thereby allowing us to see the different frequency components making up the signal. Each component has a unique frequency, amplitude and phase2 . In the most general
case, we can express a time waveform f (t) as
f (t) = a0 +
∞
X
an cos (nt) +
n=0
∞
X
bn sin (nt)
(1)
n=0
where the coefficients (ai and bi ) can be explicitly expressed as definite integrals related to f (t). In the continuous
case, f (t) can be expressed as
Z ∞
f (t) =
F(ω)eiωt dω
(2)
−∞
where F(ω) is the (complex) Fourier coefficient (describing the amplitude and phase of the sinusoidal component at
frequency ω = 2πf ). Similarly, the transform also goes back from the frequency domain to the time domain
Z ∞
F(ω) =
f (t)e−iωt dt
(3)
−∞
One technique, called the fast fourier transform (FFT) allows for a rapid transformation between the time- and
frequency domain representations of a signal. In this case, one uses the discrete Fourier transform (since the time
waveform is obviously sampled at a finite rate over a certain interval). As is shown in Fig. 2, the FFT was used to
determine F(ω) from the time waveform f (t), of which the magnitude is plotted.
3.3
Two-Tone Suppression
Given that SFOAEs tend to be small relative to the stimulus used to evoke them (called the probe tone), they are
typically hard to measure. However, several techniques have been developed that make use of the nonlinearity present
2 Conversely, amplitude and phase can be considered as the sum of a cosine and sine. For example, consider Euler’s formula: Aeiθ =
A cos (θ) + iA sin (θ).
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Statistics Case Studies: SFOAEs
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in the ear in order to measure SFOAEs. We will focus on one known as two-tone suppression (2TS), schematized in
Fig.3.3 [Brass and Kemp 1993; Shera and Guinan 1999].
The basic idea is as follows. In present the probe tone in two different conditions. For condition 1, the probe is
presented alone, the microphone picking up both the stimulus and the evoked SFOAE. In condition 2, an additional
tone (called the suppressor tone) is also simultaneously presented. In this case, the SFOAE has been effectively
removed (due to nonlinearities in the inner ear) and thus the microphone only picks up the stimulus at the probe
frequency. By comparing the signal at the probe frequency across the two conditions, the stimulus can effectively be
subtracted out and the SFOAE revealed.
Let’s flesh this out in further detail. During condition 1, you measure a time waveform. Taking the fast Fourier
transform (FFT) of this signal, you can extract the magnitude and phase at the probe frequency (Pn and θn respectively). We can express this n’th measurement as the complex quantity P¯n = Pn eiθn for the probe alone condition.
You repeat this process a total of N times to minimize random background noise. Thus, across your N averages, you
end up with mean values for the magnitude and phase (P and θ), such that
P̄ = P eiθ
(4)
Furthermore, from the averages you have estimates of the standard deviations (σP and σθ ) such that P ±σP and θ±σθ .
Similarly for condition 2, you will end up with the magnitude and phase for the probe plus suppressor condition that
can be expressed as S̄ = Seiφ and also have values for σS and σφ .
We now define the emission as
P̄OAE ≡ P̄ − S̄
(5)
That is, the complex difference between the probe alone and probe and suppressor conditions. As shown in Fig.3.3,
this essentially amounts to subtracting two vectors in the complex plane. Let
α ≡ P̄OAE β ≡ arg (P̄OAE )
(6)
and let P̄ = a + ib and S̄ = c + id, then
α=
p
(a − c)2 + (b − d)2
(7)
and
β = arctan
b−d
a−c
(8)
So now based upon our two (time averaged) measurements from both stimulus conditions, we have an expression for
the magnitude and phase of the SFOAE.
4
Quantifying Uncertainty in SFOAE Measurements
The basis for determining uncertainty in SFOAE measurements stems from two considerations. First, there is some
degree of inherent uncertainty in any particular measurement, most notably due to the influence of background noise.
Thus, there is some degree of random fluctuation (which we assume to be normally distributed) from one measurement to another for a given set of stimulus conditions. This is the main reason why we average: to minimize the
confounding factor of random error/fluctuations. Averaging also allows the opportunity to quantify how big said fluctuations are (i.e., determine standard deviation over the course of averages). Second, SFOAEs are determined through
a suppression paradigm (as outlined in the previous section) that involves a functional composition across two distinct
conditions. Thus, we will need to consider propagation of error in order to quantify the desired uncertainty. Keep in
mind that our goal is to determine σα and σβ (i.e., how do we determine the error bars as is shown in Fig. 3.1).
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Step 1. Present two different conditions: A. Probe tone alone
B. Probe and Suppressor tones
Step 2. Take Fourier transform of both (time-averaged) waveforms
to obtain magnitude and phase
Magnitude
Magnitude
Condition 2:
Probe +
Suppressor
Condition 1:
Probe Alone
Frequency
Frequency
[consider as vectors in the complex plane at probe frequency only]
Im
Im
Condition 1
ma
Condition 2
P)
g. (
θ
Re
Re
Step 3. Subtract phasors to obtain SFOAE (nonlinear suppression)
Im
SFOAE (=
revealed!!
)
Re
Figure 8: Schematic outlining the basic 2TS paradigm. Note that the phasors in the complex plane represent the magnitude and phase at the probe
frequency.
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Statistics Case Studies: SFOAEs
4.1
Math 363 (Spring 2009)
Propagation of Error
Let us first consider the general case3 . Suppose that we wish to determine a quantity x that depends upon at least two
measured values, u and v, such that
x = f (u, v, · · ·)
(9)
Suppose that you make N measurements of u and v, then
x̄ ≈ f (ū, v̄, · · ·)
(10)
PN
where ū = N1 i=1 ui (and similarly for v̄). From our N measurements, you also have empirical estimates for the
variances σu2 and σv2 . What we now wish to determine is the variance σx2 . In the case that N → ∞,
#
"
N
1 X
2
2
σx = lim
(xi − x̄)
(11)
N →∞ N
i=1
where xi = f (ui , vi , · · ·). Now we wish to express our deviations in xi as deviations in ui and vi (i.e., the things we
actually measure). We then have
∂x
∂x
+ (vi − v̄)
···
(12)
xi − x̄ ≈ (ui − ū)
∂u
∂v
Combining Eqns.11 and 12, one can show that
σx2
≈
σu2
∂x
∂u
h
2
= limN →∞
where the covariance is σuv
propagation formula
4.2
2
+
1
N
σv2
∂x
∂v
2
+
2
σuv
∂x
∂u
∂x
∂v
+ ···
(13)
i
(u
−
ū)(v
−
v̄)
. Equation 13 is commonly known as the error
i
i
i=1
PN
SFOAEs
For simplicity in deriving the uncertainty in our SFOAE measurements, we assume that there is no covariance, that is
no correlation amongst the various measured quantities4 .
Combining Eqns.7 and 13, one can show that
σα2 =
1 2
σa (a − c)2 + σb2 (b − d)2 + σc2 (c − a)2 + σd2 (d − b)2
2
α
Similarly, combining Eqns.8 and 13
2
1
2
σβ =
σa (b − d)2 + σb2 (a − c)2 + σc2 (b − d)2 + σd2 (a − c)2
2
2
(a − c) + (b − d)
(14)
(15)
So now we can express the SFOAE magnitude as α ± σα and phase as β ± σβ based solely off of our measured
quantities (i.e., we know the appropriate error bars to plot)!
Also note that in a lot of contexts, the standard error (sometimes also referred to as the standard error of the mean)
is commonly plotted to visualize the uncertainty. The standard error is simply the variance divided by the square root
of the number of trials
σ
SE =
(16)
N
3 See
Data Reduction and Error Analysis by P.R. Bevington and D.K Robinson for further reference.
needs to be careful here. In our present case with regard to SFOAE, this simplifying assumptions turns out to be fairly reasonable.
However, in many other cases this assumption is not valid and the covariation across the measured quantities needs to be accounted for.
4 One
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Math 363 (Spring 2009)
Summary
Using a specific example relevant to auditory physiology and incorporating aspects of Fourier transforms and complex
numbers, we were able to use the propagation of errors to derive the appropriate expression for the uncertainty associated with a specific type of measurement. Since different intervals (i.e., two conditions in the suppression paradigm)
and thus the SFOAE depends upon multiple variables, we saw how the uncertainty in the various measured quantities
propagates through to an uncertainty in the final, desired quantity5 . Hopefully it should be apparent how easily this
routine can be automated by the use of a computer (i.e., you have to do the initial coding, but after that, the computer
does all the work for you!).
5 Note
that since acoustic sound pressure is typically plotted on a logarithmic scale, the error bars will not be symmetric, as is shown in Fig. 3.1.
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Potential Questions to Explore
• So if we measure the magnitude and phase at the probe frequency for both stimulus conditions, then we measured
values P ±σP , θ ±σθ , S ±σS , and φ±σφ . However, our expression for the SFOAE and the uncertainty (Eqns.7
and 8) are in terms a ± σa , b ± σb , etc. What is the connection between these two sets of representations and
how does that specifically relate in the numerical estimates for the uncertainties?
• Measure SFOAEs in your own ear [contact C. Bergevin about this ([email protected])].
• Why might the OAE probe used to measure the emissions have two earphones (in stead of just one)?
• Explicitly derive Eqn.13 from Eqns.11 and 12.
• Explicitly derive Eqns.14 and 15.
• Consider Eqn.14. At what values of α, a, b, etc... is the equation evaluated at? Can you explain why?
• In many contexts, data are reported with their 95% confidence intervals. Explain what these intervals represent
and (clearly) explain how they relate to the variance and standard error.
• Explain the difference between a sample distribution and parent distribution. Go one step further to explain how
the standard error is connected to both of these.
Appendix - Technical Details On Measuring SFOAEs
For those interested in more specific details, one possible paradigm for measuring SFOAEs is as follows (see subsequent section first for description of the suppression paradigm). A desktop computer houses a 24-bit soundcard (Lynx
TWO-A, Lynx Studio Technology), whose synchronous input/output is controlled using a custom data-acquisition
system. A sample rate of 44.1 kHz is used to transduce signals to/from an Etymotic ER-10C (+40 dB gain). The
microphone signal is high-pass filtered with a cut-off frequency of 0.41 kHz to minimize the effects of noise.
The probe earphones are calibrated in-situ using flat-spectrum, random-phase noise. Calibrations are typically
verified repeatedly throughout the experiment. Re-calibration is performed if the level presented differed by more that
3 dB from the specified value. The stimulus frequency range employed (fp for SFOAEs) is typically 0.5–5 kHz. The
suppressor stimulus parameters are commonly as follows: fs = fp + 40 Hz Ls = Lp + 15 dB. One earphone produces
a sinusoidal signal over a 464 ms time window at the probe frequency fp , ramped on/off over 12 ms at the ends of the
window. The other earphone also produces a 464 ms signal, but at the suppressor frequency fs , which was ramped
on only during the latter half of the window (the first half was silence). The microphone response is extracted from
two 186 ms segments from the total waveform, one from the probe alone and one with the probe+suppressor. These
segments are extracted at least 20 ms after the end of the ramping-on periods to allow any transient behavior to decay.
Thus, the measurements are for the steady-state condition. The Fourier transform of each segment is then computed
and the complex difference of the two Fourier coefficients at fp was defined to be the SFOAE.
For SFOAEs, 35 waveforms are typically averaged, excluding any flagged by an artifact-rejection paradigm [Shera
and Guinan, 1999]. Furthermore, all stimulus frequencies are quantized so that an integral number of stimulus periods
fit in the response segment. Frequency step-size during sweeps should be small enough to avoid ambiguity during the
phase unwrapping. Delays associated with the measurement system (such as the analog/digital converters on the sound
card) can readily be corrected for in the fp phase before taking the complex difference in the SFOAE suppression
paradigm. The noise floor can be defined as the average sound-pressure level centered about (but excluding) the
frequency of interest. It is found by averaging the Fourier amplitudes in the ±3 adjacent frequency bins centered on
the OAE frequency. For the stimulus levels commonly used to measure, artifactual system distortionis small compared
to the signal levels of the SFOAEs (≈70–80 dB below the evoking stimulus levels and typically beneath the acoustic
noise floor).
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