Methods of Frequency Selectivityand Synchronization Measurement in
Single Auditory Nerve Fibers: Application to the Alligator Lizard
by
Christopher Rose
B.S.E.E. Massachusetts Institute of Technology
(1979)
M.S.E.E. Massachusetts Institute of Technology
(1981)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May, 1985
) Christopher Rose, 1985
Signature of Author
%- -q-
Department of Electrical Engineering
& Computer Science
May 17, 1985
Certified by
Prof.Thomas F. Weiss
ThesjSpervisor
__
Accepted by
Archives
MASSACiUSETTS iNST' E
OF TECHNOLOGY
JUL 1 9 1985
IDACir.'
___
Prof. Arthur C. Smith
Chairman, Department Committee
METHODS OF FREQUENCY SE CIICVITYMEASUREMENT IN SINGLE AUDITORY
NERVE FIBERS: APPLICATION TO THE ALLIGATOR LIZARD
by
CHRISTOPHER ROSE
Submitted to the Department of Electrical Engineering and Computer Science
on May 17, 1985 in partial fulfillment of the requirements for the Degree
of Doctor of Philosophy in Electrical Enginee
ABSTRACT
This work is composed of two separate studies: 1) an analysis of an iso-response contour estimation
algorithm used in auditory physiological studies and 2) the dependence of neural synchronization upon frequency. The first study is an in-depth analysis of an automated algorithm used by many auditory physiologists to measure the frequency selectivity of auditory neurons (Kiang, Moxon & Levine, 1970). Specifically,
we use a Markov model to specify the behavior of the algorithm. We then compare the properties of the
model to the properties exhibited by the automated algorithm with auditory neurons. We present performance bounds as well as suggestions for improving the effectiveness of this algorithm. In the second study
we measure neural synchronization decline with increasing frequency and make comparisons to a previously
reported study of hair cell receptor potential (Holton & Weiss, 1983a). We show that the process which
relates hair cell receptor potential and the neural response is low-pass in nature. We also show that the frequency at which neural synchronization begins to decline in a given fiber does not depend upon the characteristic frequency of the fiber. In addition, we present two new methods to study neural .synchronization.
The first method allows the study of synchronized comanponentvariation with frequency separate from the frequency variation of the average firing rate response; we measure synchronization at constant average rate.
The second method uses a new synchronization estimation technique which allows the rapid measurement of
iso-synchronization (iso-fundamental) contours in much the same way that iso-rate contours are presently
measured.
THESIS SUPERVISOR: Thomas F. Weiss
TTILE: Professor of Electrical Engineering
-2-
ACKNOWLEDGEMENTS
My three year stay at the Eaton-Peabody Laboratory has been a pleasant one. For this I have many people
to thank. Ruth Anne Eatock brought a sunny biologists perspective into this engineer's professional life (as
of this writing a tadpole sits metamorphosing in a tank on her desk). Denny Freeman, although he may not
realize it, was a very important peer for an analytic type like myself. He was one of the few people to whom
I could say "Bessel Function" without fear of his fainting. Discussions with Denny about the "wall", "rod" and
numerical solutions for problems in fluid dynamics were a lot of fun (I still think you should build a scale
model in a viscous fluid, Denny!). John Rosowski was always available to answer the basic questions; John
where's the power switch? John, how does the sweep system work? John, do snakes have cars? John, what
three rivers flow by Three River Stadium? I will miss John's humor and warmth. Chris Brown provided a
much needed wildness outlet in an otherwise staid atmosphere (MAKE 'EM SCREAM! MAKE 'EM
SHOUT!) in addition to very good-naturedly introducing me to the guinea pig ear in the middle of his experiments. Ca Williams performed several excellent lizard surgeries for me as well as making sure I did not forget the simpler things like personal hygiene after crawling under a desk in search of an escaped lizard (Chris,
do you realize that you have dust all over one side of your big head?).
The engineering staff, Ishmael
Stefanov-Wagner, Mark Curby and Bob Brown provided continual technical support. It would have been a
very hard row to hoe without them.
For help with the preparation of this document I must thank my two thesis readers, Larry Frishkopf
and Bill Peake. The speed and accuracy with which they reviewed this mathematically dense manuscript on
short notice is amazing. They have both greatly improved the quality of this work through their suggestions
and coments.
M.I.T.
Extra thanks, however, is due to Bill Peake who served as a patron saint during my stay at
I am greatly indebted to him for his counsel during my early wide-eyed-young-graduate-student
years.
More than just thanks are due to the two men who had a major influence upon this study. Without
Dave Strffens, none of the work on synchronization or the new and improved TCA 3 would exist. Dave
wrote all the software (in assembly language!) usually within two days of my initial request and sometimes
-3-
before I even asked. He was even able to program around one of my hardware blunders.
(Rose, the
hardware has to dear the ready flag!). Dave was also always available for help on any computer related topic
and he always got the message across (even to the unix-uninitiated and the C-unseasoned). I greatly appreciate his help.
Tom Weiss, my thesis supervisor, has provided continual guidance and support. When I thought I had
all my ducks in a row he would invariably find the stragglers which I had missed. His contribution to this
work is immense. Tom has made my professional development enjoyable. Incredibly enough, he managed
to do all this without overpowering me. He even managed to avoid laughing outright when I would come to
him in the early going of this work with a particularly outlandish idea. He'd simply bite his tongue and let
me come to my senses later (Rose, let me get this straight. You want to record from the auditory nerve in
the alligator lizard using a multichannel electrode which you will fabricate on a semiconductor wafer? By
next week?). Tom, you will be happy to know that although I have resisted your every attempt to transform
me into a died-in-the-wool auditory physiologist you have in some sense succeeded. In the noble tradition of
Alexander Graham Bell who was initially an audiologist and Bekesy who was initially a telecommunications
engineer, I will be working for the telephone company.
Very special mention must go to Stephanie and Stephen (Boomer) Rose who made it all worthwhile.
Stephanie, a Harvard-trained attorney, never bargained on becoming a hausfrau and having to cook, dean
and care for two babies during the last few months of my graduate studies. She has come through it all with
very good humor (Chris, it's five in the morning and those dicks and beeps are driving me crazy. Log out or
die!). Stephanie and Stephen were a joy to come home to, and on those many late nights at home in front of
the computer terminal it was greatly reassuring to know they were there with me. In fact, Stephen, an occasion stayed up to give me moral support. I therefore thought it only fitting that he should write the dosing
comments to these acknowledgements.
c p.g
foc mmbvc,/.k bf p,. off?:
polmn l,m&%6
r:
b/w=p
;,. ,cd .n,
gv cd 9 hb 976mn hg 89, m!
,> I xki z?
. Look at all those numbers and symbols! Already a flair for the technical!
4-
The Boomer Seal of Approval
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-5-
TABLE OF CONTENTS
TitlePage.
..........................................................
.
.....................................................
I
Abstract
.............................................................................
2
Acknowledgements ...
3
Seal of Approval.
Table of Contents
.
.....................................................................................................
............................................
S
..............................................................
....................................................................................................
6
10
List of Figures............................................................................................................
List of Tables .............................................................................................................
CHAPTER 1: ntroduction ............................................................
14
CHAPTER 2: ISO-RESPONSE CONTOUR ESTIMATION
An Analysis of the Tuning Curve Algorithm ............................................................
16
INTRODUCTION
............................................................
16
ISO-RESPONSE AND ISO-STIMULUS METHODS IN THE MEASUREMENT OF
FREQUENCY RESPONSE CHARACTERISTICS...........................................................
16
FORMALIZATION OF A BASIC IRC ESTIMATION PROCEDURE ........................................
17
PREVIOUS RIGOROUS STUDIES OF IRC ESTI77MATION
ALGORITHMS:
to
Measurements
on
Cochlear
Neurons
.........................................
Inapplicability
18
TCA HISTORICAL PERSPECTIVE ......
19
...
.................................................................
TOOLS USED TO ASSESS TCA PERFORMANCE
SIMULI
......................................
......
........................................................
...
NEURAL RESPONSE DEFINITIONS .........
.................................................................
............................................................
TEST CIRCUIT.......................................................................................................
.....
24
.................................25
STEADY STATE BEHAVIOR OF THE MODEL ................................................................
Theoretical Considerations .....................................................
.........
Derivationf SteadyStateEquations
..
+6
21
............................24
.............................................................
MODEL FORMULATION .....................................................
21
22
A MARKOV MODEL OF TCA BEHAVIOR: Theory and Experiment .....
A DESCRIPTION OF THE TCA .........
21
...........................
27
27
................ 27
TheAverageLevelsE(l) andE( ) ...........................................
30
NumericalDetrminations ofp (l) and p, ( Isuccess)...................................................
The TCA Does NOT NecessarilySatisfya Rate Criterion............................................
33
34
Experiments to Examine TCA behavior in the Steady State
........................................... 36
PredictedValuesof E (I) andE (I,) Using
Measurements
of g (1).
...........................................................
Measurementprocedure ..
Measurementson the test ircu
.............
...
..
it ........................................
36
.......
............36
37
Measurements on cochlear neurons ...................................
..................................
Variation of AA with Spontaneous Rate, X0 ...............................................................
Prediction of AX vL X0 assuming a
homogeneous Poisson spike generation proces .
......................................
37
39
Measurementson the test crcuit ......................................................................
Measurementson codlear neurons .............
.......
...............
Variationof A(123) with Frequency.......................................................................
40
39
.40
Analytic
description
. . ................................. 4242
Measurements on codilear neuromns.................................................................
43
Summary of Steady State Properties ........................................................................
43
TRANSIENT BEHAVIOR OF THE MODEL AND TCA IRC ESTIMATION ERROR .....................
Theoretical Considerations ...........................................................
44
44
Experiments to Examine TCA Transient Behavior ......................................................
46
Topical Outline..................................................................................................
46
TCA Errors in Estimating Neural IRCs ..............................................................
Qualitative descriptions and simulations.............................47
47
trackingof a frequency-invariantIRC ................................................................
trackingof a linearIRC .................................................................................
trackingof a V-shapedIRC.............................................................................
Measurement of TCA errors for cochlear neurons ...........
47
48
49
.........
.....50
50.........
effects of initial startinglevel............
............
.....................................
... 50
TCA error and IRC slope.......................................................................
51
TCA IRC parameterestimationerror ...........................................
52
TCA Error Caused by Correctionfor the Acoustic System Frequency Characteristic
............
Qualitativedescription........................................................................
Measurementson codlear neurons ..................
...................
................
Miscellaneous Observations....
........
..............
...........
TCA errors caused by nonmonotonic g(l) ................................................
measurements on a cochlear neuron .........
TCA errors caused by neural adaptation ...............................................................
.........................................
measurements on cochlear neurons .....
SUM MARY .
.........
........
INTRODUCTION ........
....
...................................................
-7-
54
54.................................54
55
............55..........
55
.....
...........
..................................
56
...................................................................................................
A CASE HISTORY OF TCA IRC ESTIMATION ERROR ..........
53
.............. 54
.........54
qualitativedescription
...................................................................................
qualitativedescription.....
53
.... 53
.........
57
...............................
.................................
8 8.....
OF RECEPTORPOTENTIALg () .....................................................
RECONSTRUCTION
..
EFFECTSUPONREPORTEDRESULTS.......
....................................
..............
IMPROVING TCA ACCURACY ........
....
MODIFICATION OF TCA PARAMETERS
.....................
...........
59
60
61
61
................................................
COMBAT7NG TCA ERRORS CAUSED BY CORRECTION FOR THE ACOUSTIC
SYSTEM FREQUENCY CHARACTERISTIC ......
.................................................
...........
62
SUGGESTION FOR FURTHER STUDY ........................................................................
64
..........................................................
Figures
65
CHAPTER 3: DEPENDENCE OF NEURAL SYNCHRONIZATION UPON FREQUENCY ...... 129
129
INTRODUCTION .............................................................................................
METHODS
...
STIMULI. ..
........................................................................................ 131
...
.................................................
131
....... . ...............................................
132
132
NEURAL RESPONSE COMPONENTS .........................................................................
Average Rate .....................................................................................................
.................................................................................
Synchronization
............. 132
Measurement of Iso-fundamental Contours ..............................................................
RESULTS ...............................................................................................................
ISO-FUNDAMENTAL CONTOURS: Dependence Upon Fiber CF .........................................
134
135
135
SYNCHRONIZATION AT CONSTANT AVERAGE RATE DIFFERENCE:
Dependenceupon CF for Low and High-CFFibers....................................................
136
SYNCHRONIZATION AT CONSTANT AVERAGE RATE DIFFERENCE:
...................................................................
Dependenceupon AX for High-CFFibers .
DISCUSSION
.
.
137
.................................................139
NOVEL SYNCHRONIZATION MEASUREMENT SCHEMES ................................................
139
RELATION OF NEURAL SYNCHRONIZATION TO RECEPTOR
........................................
POTENTIAL A.C. COMPONENT: Free-Standing Region
139
COMPARISON OF SYNCHRONIZATION FOR FREE-STANDING AND TECTORIAL FIBERS ..... 140
COMPARISON TO THE CAT
....................
-8-
141.................
143
...................................................................................................................
Figures
APPENDIX A: Existence of E(I) .................................................................................
157
APPENDIX B: Estimates for Average Rate and Synchronization ........................................
160
POISSONMODEL.....
..
............
..............................................
AVERAGE RATE ESTIMATE ......
......
SYNCHRONIZATION
ESTIMATE
..
. .
..
..............
160
.................................................................. 160
.......
.....
..............................................
161
Estimation of A.................................................................................................161
Estimation of 1/A0 .....................................................
................................................................................
Synchrony estimate synthesis .
163
164
SYNCHRONIZATION MEASURES FOR IRC ESTIMATION ................................................
165
APPENDIX C: Tests of Iso-fundamental Contour Estimation Algorithm
Accuracy Using Single Cochlear Neurons in the Alligator Lizard ........................................
167
Figure....................................................................................................................
168
APPE.NDIX D: Justification of Comparisons between Synchronization Measured
with Tone Burst Stimuli and Synchronization Measured with Continuous Tone
Stimuli .......
...........................................................................................................
169
AVERAGE RATE MEASUREMENTS ............................................................................
169
SYNCHRONIZATION MEASUREMENTS .......................................................................
170
Figures...............................................................................................
REFERENCES.........................................................................................................174
BIOGRAPHY
............
.....................................................................
177
-9-
171
LIST OF FIGURES
65
..........................................................................................
Figure 2.1: Cascaded system
66
..................................................................................
Figure 2.2: The toneburst stimulus
Figure 2.3: Block diagram of test circuit ...........................................................................
67
Figure 2.4: Frequency response characteristic LTI filter in figure 2.3 .......................................
68
Figure 2.5: Spike generation by threshold crossings of Gaussian noise
69
waveform......................................................................................................
Figure 2.6: PST histograms of toneburst response for test drcuit ............................................
70
Figure 2.7: Schematic representation of the neural response to a
71
tone-burst.....................................................................................................
Figure 2.8: Discrete-time discrete-state Markov models of the tuning
curve algorithm at fixed frequency .....................................................................
Figure 2.9: Demonstration that E(l) and E ()
72
... 73
lie near 12,3 ......................................
Figure 2.10: TCA threshold history and g (l) for test circuit and
nerve fiber ...................................................................................................
75
Figure 2.11: Off-window rate vs. spontaneous rate .................................
77
Figure 2.12: Examination of probability/rate shift; g(l) vs.
AX at different frequencies ...............................................................................
78
Figure 2.13: Expected value + s.d. of the first success level
starting from level k for various values of Lm .....................................................
82
Figure 2.14: Expected value ± s.d. of the first success level starting
from level k using the function g (l) for various
values of Lmx, gmin and g . .........................................................................
85
Figure 2.15: Schematic and simulation of TCA error dependence upon IRC
starting initial level for a frequency invariant IRC .................................................
88
Figure 2.16: Schematic of TCA error dependence upon IRC slope
. .....................
90
...................
92
Figure 2.17: Simulated average TCA error vs. normalized IRC slope ...........
Figure 2.18: Schematic of TCA estimation error for V-shaped IRCs
....................................
96
Figure 2.19: Simulated low-high & high-low TCA estimates of
neural IRC .
ardchetypical
..................................................................................
Figure 2.20: Three TCA IRC estimates for a cochlear never fiber for
different initial starting levels ..........................................................
-10-
..........
97
98
Figure 2.21: TCA estimate compasm to the TCA3 estimate in
alligator lizard nerve fibers ...................
........
........................
99
Figure 2.22: Comparison of parameters measured from TCA estimates
and TCA 3 esimates for cochlear neurons in the
alligator l ard
..............................................................................................
102
Figure 2.23: Schematic of error caused by correction for the
acoustic system frequency characteristic ...................................................
107
Figure 2.24: Simulation of artifact caused by correction for the
acoustic system frequency characteristic..............................................................
108
Figure 2.25: Error caused by acoustic system correction upon a
TCA-estimated IRC in an alligator lizard cochlear neuron .....................................
109
Figure 2.26: Schematic of a pathological nonmonotonic g (1)................................................
110
Figure 2.27: Effects of nonmonotonic g(1) on neural IRC estimation....................................
111
Figure 2.28: AX vs. level for suprathreshold
and subthreshold stimuli ...................................................
112
Figure 2.29: Reconstructed hair-cell receptor potential g() .................................................
114
Figure 2.30: Simulation of TCA estimation of iso-D.C. hair-cell receptor
potential contours
...........................................................................
116
Figure 2.31: Expected value - s.d. of the second success level
starting from level k for various values of Lnm ...................................................
117
Figure 2.32: Expected value -+ s.d. of the third success level
starting from level k for various values of L. ...................................................
120
Figure 2.33: Expected value ± s.d. of the fifth success level
starting from level k for various values of LmR= .
123
.......................................
Figure 2.34: Comparison of TCA and TCA3 performane ..................................................
126
Figure 3.1: The tone burst stimulus...................................................
143
Figure 3.2: Tone burst stimulus divided into M equal time intervals ......................................
143
Figure 3.3: Instantaneous rate response of a cochlear nerve fiber in the
alligator lizard to different frequency tones at 80 dB SPL.......................................
144
Figure 3.4: Iso-rate and Iso-synchrony contours for two low and two
high CF fibrs ...
.............................................. ............................................
Figure 3.5: Scatter plot of iso- A1 12 contour CF versus
iso-rate contour CF for low and high CF units in 12
alligator lizards ..................................................
Figure 3.6: Synchronization at constant rate (=20
sp/sec) for
-11-
145
147
148
low- and high-CF units ..................................................................................
Figure 3.7: Synchroizafion versus frequency for different
constant Ax criterion .......................................
............
..............
Figure 3.8: Comparison, for the free-standing region, of hair cell
........................................
receptor potential and neural instantaneous rate
151
153
Figure 3.9: Comparison of the dependence of neural syncrhonization
upon frequency for cochlear nerve fibers in the alligator
lzard and the cat.
.....................................................
.................................... 156
Figure C.1: Measurement of All at points on an
iso-fundamental contour .
.....................................................
......
168
Figure D.1: Neural instantaneous rate response to a tone-burst .............................................
171
Figure D.2: Average rate measured over the tone-burst on-window plotted
against the average rate measured over the last 80% of the
tone-burst on-window......................................................
Figure D.3: Neural rate response averaged over tone-burst on-window plotted
against the response averaged over the final 80% of the
tone-burst on-window .....................................................................................
-12-
............ 172
173
LIST OF TABLES
Table 2.1: Success State History......................................................................................26
Table 2.2: TCA steady state threshold determination (800 trials/point) .....................................
37
Table 2.3: TCA steady state threshold determination on auditory
nerve fibers (14 units in two animals) (150 trials/point) .............................................
38
Table 2.4: A? for Prob(N
T =Tff
-No I f >C)=2/3
for
50 ms ...........................................................................................
40
Table 2.5: Average rate difference, AX, vs. off-window
rate, offI, measured at stimulus level,
12 3 for the test circuit.......................................................................................40
Table 2.6: 19 average rate differences, AX, vs. off-window
rate, koff, measured at stimulus level,
23 for 14 units in two aligator liards ................................................................
-13-
42
CHAPTER 1: Introduction
The present form of this thesis is the result of a scientific inquiry whose pursuit required the development of special techniques. Originally, this thesis was to be an experimental study of the neural firing rate
response to sound stimuli. Both the average rate and synchronized responses were to be measured and the
results were then to be compared with previous work on hair cell receptor potential. For the study of synchronized neural response, a specific experiment was formulated: measurement of neural synchronization as a
function of frequency with average firing rate held constant. To perform this experiment it was envisioned
that an automated algorithm would be used to hold average firing rate constant as a function of frequency.
Measurement of such an "iso-rate contour" in the frequency-level plane using this algorithm would have been
followed by measurements of synchronization at points along the iso-rate contour. It became immediately
apparent, however, that the algorithm made errors in the estimation of iso-rate contours that rendered it
inappropriate for our use. Thus began the analysis of the automated algorithm in order to ascertain how it
made errors and how these errors might be corrected. The culmination of this analysis led to the large part
of this work in Chapter 2 where we explore iso-response contour estimation. The study of iso-response contour estimation led to the development of a new means to study synchronization: measurement of isosynchronization contours. Use of this measurement technique forms a large part of Chapter 3 wherein we
explore neural synchronization. Thus, what started as a strictly physiological thesis rapidly became an amalgam of technical development and physiological inquiry.
Since this work addresses both technical issues and physiological issues it has been written in a modular
manner so as to keep the two relatively separate. The two chapters of which this work is composed are
independent of each other (except for a few tangential references).
It was felt that this form would most
benefit the reader; to understand the physiology one need not wade through the technical details and conversely, to understand the technical analysis one need not wade through the physiology. We now introduce
each chapter in order to acquaint the reader with the salient features of this work.
Chapter 2 is an in-depth analysis of an automated algorithm used in auditory physiology to measure the
frequency selectivity of auditory neurons (Kiang, Moxon & Levine, 1970). Specifically, we use a Markov
model to specify the behavior of the algorithm and compare the properties of the model to the properties
-14-
exhibited by the automated algorithm with auditory neurons. The type of model used is applicable to many
sequential threshold estimation algorithms. We present performance bounds as well as suggestions for
improving the effectiveness of this algorithm.
In Chapter 3 we study the decline of neural synchronization with increasing frequency. Our results are
compared to a previously reported study of hair cell receptor potential (Holton & Weiss, 1983a). We show
that the process which relates hair cell receptor potential and the neural response is low-pass in nature. We
also show that the frequency at which neural synchronization begins to decline in a given fiber does not
depend upon the characteristic frequency of the fiber. In addition, we present two new methods to study
neural synchronization. The first method allows the study of synchronized component variation with frequency separate from the frequency variation of the average firing rate response; we measure synchronization
at constant average rate. The second method uses a new synchronization estimation technique which allows
the rapid measurement of iso-synchronization (iso-fundamental) contours in much the same way that iso-rate
contours are presently measured.
-15-
CHAPTER 2: ISO-RESPONSE CONTOUR ESTIMAilON
An Analysis of the Tuning Cmw Algorithm
INTRODUCTION
The iso-response contour (IRC) measurement method is-biquitous in he fid
of auditory physiology.
This ubiquity stems from the utility of the IRC measurement method in frequency response estimation for
simple systems which model properties of the response of codilear neurons to sound.
We illustrate as fol-
lows.
ISO-RESPONSE AND ISO-STIMULUS METHODS IN THE MEASUREMENT OF FREQUENCY RESPONSE
CHARACTERISTICS
The frequency response of a linear time invariant (LTI) system (figure Z La.) is commonly obtained by
applying a stimulus with constant amplitude, A, and measuring the respon. amplitude, B(o),
angle, 4(w), as a function of radian frequency,
. The transfer function,
and phase
(o), which has magnitude
H (w) I=B (w)/A , and angle c(w), completely characerizes the relation of the stimulus to the response of an
LTI system. Alternatively, H (w) could be measured by detennining the input amplitude A (w) necessary to
produce a constant response amplitude B for each w. In this case
H(w) =B/A (w). The angle of H(w)
would also need to be measured to completely characterize the system. Fmr an LTI system the former
iso -stimulus and the latter iso -response methods are equivalent.
Now consider a system consisting of the LTI system of fure
ity (figure 2.lb).
2.la fdloed
Suppose we wish to measure the transfer fmction H(w).
measurements of the peak-to-peak amplitude of the response Willnot measue
nown function. The alternate iso-response measurem
by a memoryless nonlinearith the iso-stimulus method,
H(w)1 since F () is an unk-
t, measurement cdf the stimulus amplitude A(w)
required to keep the peak-to-peak amplitude of the response onstant wil allow H ()
to be determined to
within a multiplicative constant since if the output of F () is constant then the input amplitude of F () is also
constant (F () is assumed invertible.) Therefore the iso-response measurementof the cascade system is essentially an iso-response measurement of the LTI system since any miaation in A(gw)will be due to the LTI subsystem alone.
Now suppose that F() saturates, i.e-, F(x)=fm,
for g>t and F
an F () arises naturally if the response has a limited dynamic m.
-16-
Altwmj
=fm-'-,,j
for x <a (b >a). Such
F is invertible on (a ,b) it is
not invertible outside of this range. Thus if the dynamic range of level for B (w) in figure 2.1b is greater than
the range (b -a),
then the output would be driven to saturation with an iso-stimulus measurement method.
Using an iso-response method the value of F may be kept within its invertible range by fixing F =C where
f min < C < fma thus avoidingsaturation.
This issue of saturation is important in the study of the auditory system since the dynamic ranges of the
neural response and of the sound stimulus in the physiological range of the auditory system differ markedly.
The neural response saturates over a sound pressure level range of approximately 20 dB whereas the dynamic
range of the frequency selective properties (the dynamic range in level of IH(w)1) may be as large as 80 dB
in the alligator lizard (unpublished observations). Similar results may be inferred from data in other preparations (Kiang et al, 1965). Therefore, since we may crudely model the neural response to sound as the cascade of an LTI filter and a saturating memoryless nonlinearity, the usefulness of the iso-response method in
measuring frequency selectivity of the neural response is readily apparent.
As a final note consider the system depicted in figure 2.lc.
It has been shown that both G(o)
I and
IH (X) can be estimated from iso-response measurements (Holton & Weiss 1983a). This particular system is
of special interest since the transformation of sound pressure at the tympanic membrane to the receptor
potential of hair cells in the alligator lizard can be approximated as such a cascade system. Specifically, middle and inner ear mechanics can be represented by H(w) and receptor potential generation from stereociliary
displacement can be approximated by the combination of F(.) and G(w) (Weiss, Peake, & Rosowski
(1985)). Since it is likely that a similar system model is applicable to higher vertebrate auditory systems as
well, we that the measurement of IRCs is a generally convenient method of assessing the frequency selectivity
of auditory response variables.
FORMALIZATION OF A BASIC IRC ESTMA 7ON PROCEDURE
We have defined an iso-response contour as the set of stimulus points (frequency, intensity) for which
some auditory response variable (such as neural average rate) is held constant. To estimate such an IRC, a
very simple procedure is used. For the purposes of illustration we will formalize this concept
one stimulus variable the role of
Let us assign
free-variable" (i.e., frequency) and the other the role of "dependent-
variable" (i.e., stimulus intensity). The free-variable is held constant and the dependent-variable is adjusted
-17-
until the desired response level (criterion) is attained. this procedure is then used with a number of different
free-variable values to obtain a set of (free-variable, dependent-variable) points for which the response is constant.
Although such an estimation method is straightforward and easily implemented, it has a number of
practical limitations. Tacit in our formulation were the assumptions that, 1) the value the response variable is
accurately and precisely observed and 2) unlimited time is allowed in which to produce an IRC estimate.
Both of these assumptions are inappropriate in most practical applications. For example, the response of an
auditory nerve fiber is a stochastic process. Thus, the neural response may not be estimated with arbitrary
precision in a limited observation interval. In addition, the amount of time for which the neural response
may be stably recorded is limited as well. For these reasons we are forced to make a compromise between
measurement precision and measurement duration in the estimation of IRC's.
PREVIOUS RIGOROUS STUDIES OF IRC ESTIMATION ALGORITHMS: Inapplicability to Measurements on
Cochlear Neurons
We may readily extend our definition of an iso-response contour to include any set of points (e.g.
(time, excitation energy) ) for which a response variable is constant (e.g. luminescence) in order to completely generalize the problem of IRC estimation. We do this with the thought of applying the extensive and
rigorous literature which treats IRC estimation to the problem of IRC estimation in cochlear neurons. In
each field, however, the formulation of the algorithm used to estimate IRCs is peculiar to the particular
application.
For example, with PEST (Parameter Estimation by Sequential Testing, Taylor & Creelman,
1967) an algorithm used in psychophysics, the response variable is the probability of the response exceeding a
criterion value. PEST measures such iso-probability contours by forming a probability estimate at each value
of the dependent-variable. To estimate a probability, a number of observations are necessary at each (freevariable, dependent-variable) point. This necessity greatly increases the amount of time required to make the
measurement. In psychophysics, however, the primary obstacle to obtaining an accurate IRC estimate is the
stochastic nature of the human observer's response and not a limitation upon the measurement duration.
Thus, PEST and similar algorithms for which a great deal of literature exists (Findlay, 1978; Pentland, 1980;
Watson et al, 1983) are not readily applicable to the problem of neural IRC estimation.
-18-
Threshold Hunter (Raymond, 1978; Carley, 1978) is an algorithm used with peripheral nerve fibers to
estimate the stimulus shock intensity level (as a function of time) required to hold the neural response constant. Periodic shocks are applied to a peripheral nerve. The shock level is adjusted in fixed size up/down
increments depending upon the neural response to the previous shock. The set of (time, shock level) points
is the IRC produced by this algorithm. In principle, this algorithm could be applied to measurement of (frequency, stimulus level) IRC's in cochlear neurons by varying frequency as a function of time. Unfortunately,
there is a practical drawback to such an application. In the formulation of Threshold Hunter, it is assumed
that the difference between the stimulus level required to maintain a constant response (threshold level difference) for sequential points in time is no greater than the stimulus level increment. The threshold level difference between sequential frequency points for auditory neurons is usually much greater than the precision with
which we wish to specify the IRC. Thus the constraint that the threshold level difference be less than or equal
to the stimulus level increment is violated.
Due to the limitations of these previous studies as applied to measurements of codilear neuron IRCs
we have chosen to perform a rigorous analysis on a modern version of an algorithm which has been in use
for some time in the field of auditory physiology. This is the Tuning Curve Algorithm (TCA).
TCA HISTORICAL PERSPECTIVE
The TCA has evolved from previously used manual methods (Tasaki, 1954). These early methods of
iso-response contour measurement used audiovisual tedmiques and required the experimenter to observe the
reponse variable magnitude and manually adjust the stimulus level until the desired response magnitude was
attained. With cochlear nerve fibers this measuremnt is relatively easy to make with a minimum of equipment required, and, though it requires a subjective judgment, IRCs can be measured reliably in this manner
(Kiang et al, 1965). The measurement, however, is time consing;
taking ten to twenty minutes to obtain a
set of fifteen (stimulus level, frequency) points (Kiang, unpublished observation).
In an effort to assess the frequency selectivity of cochlear neurons in a rapid and objective manner, an
automated algorithm was developed which yielded IRC's that closely approximated the those obtained with
the subjective procedure (Kiang, Moxon & Levine, 1970). This procedure is the TCA. The TCA is objective, reliable and greatly reduces the time required to measure an IRC (approximately sixty points in
-19-
frequency per minute). Similar automated algorithms have been developed in several laboratories (Klinke &
Pause, 1980; Schmiedt, 1982; Allen, 1982). Thus, the TCA and other TCA-like algorithms (Pickles, 1984;
Geisler et al, 1985) are coming increasingly into use. However, there has been no comprehensive study of
the accuracy of the TCA in measuring iso-response contours. In this chapter we present such an analysis of
the TCA.
-20-
TOOLS USED TO ASSESS TCA PERFORMANCE
To test the efficacy of the TCA, we performed computer simulations of a model of the TCA as well as
simulations on an electrical circuit which mimicked certain properties of the response of auditory nerve fibers
to sound. Tests were also performed on single cochlear nerve fibers in the alligator lizard. See Weiss et al
(1976) or Holton (1980) for a description of methods.
STIMULI
The tone-burst stimulus used in this study is defined as a sinusoid of amplitude, A, and frequency, f,
multiplied by a periodic window function, W(t), of period TTB, duty cycle, [5,(1) and unit amplitude as depicted in figure 2.2. W(t) has a trapezoidal shape to reduce spectral contamination of the stimulus so as to
allow accurate measurements of frequency selectivity. The time interval spanned by the rising or falling edge
of W(t) is called the rise time,
stimulus: 1) Frequency f,
Tr,
of the tone-burst. Five parameters are required to specify a tone-burst
2) Amplitude A, 3) Duty cycle,
Parameter values of 5 =0.5, Tra =100 ms and
, 4) Period Tm, and 5) Rise/fall time
T,.
ms were used in this study unless otherwise indicated.
T,=2.5
NEURAL RESPONSE DEFINITIONS
A neural impulse train will be represented as a regular point process with instantaneous firing rate, X(t)
(Johnson, (1974)). Average rate, X, is defined as,
l1-t
lf(u)dau
X=
for arbitrary time t and observation interval I (I >0).
(2.la)
For a tone-burst stimulus, let Xo. be the average rate
during the "on window" of the tone-burst and let Xo/! be the average rate during the "off window". We
define the average rate difference to be
AX = AX, - XAff
(1) The nazero
roT/f= (1-
porticm at W(t) is tamed
the "an winmdow"and the
)T.
-21-
(2.1b)
.
ero prtin
the 'f
window". T.
=
T,
and
TEST CIRCUIT
A test circuit was designed to approximate the frequency selective and stochastic properties of neural
discharges. The system shown in figure 2.3 is composed of two cascaded blocks. The first block is a linear
bandpass filter which has a frequency response grossly similar to that of a single cochlear nerve fiber (Holton
& Weiss, 1983a) in the aligator lizard. The bandpass filter frequency characteristic has a maximum near
1000 Hz and decreases for lower and higher frequencies with asymptotic slopes of-
18 dB/octave (figure
2.4). The second block generates an approximately Poisson(2 ) point process of rate Xy (t)] in response to
the filter output y(t).
The method used to generate a Poisson process with rate modulated by input voltage y(t) is based
upon the distribution of threshold crossings of a Gaussian random process as illustrated in figure 2.5 (Rice,
1944; Bendat, 1958; Papoulis, 1965). Each threshold crossing produces a brief voltage pulse (spike). The
spike rate, XIc , produced by the threshold crossings for a stationary Gaussian noise process, n (t) with correlation function R, (T) is,
(2.2)
X(c)= ·
where c is a constant threshold (Papoulis, 1965)(3 ) . We assume that eqn. 2.2 is approximately correct for
c=y(t)-yo
where y(t)
is a sinusoid of frequency, f,
and amplitude a with f much smaller than the
bandwidth of the noise process and a much smaller than y0 (both a and y > 0).
The Poisson criterion, which dictates that the probability distribution of spikes in non-overlapping intervals of arbitrary duration be independent, is approximately satisfied by setting the correlation time of the
noise process (inversely proportional to the bandwidth of the noise process,
R()
)
to be much
smaller than the average inter-zero-crossing interval 1/A. Thus, except for very small time intervals, interspike intervals will be independent.
(2) Spowaneos mchlear neuron actvity
(3) R(O) and R(O)
am moed
n be appromated
crudelyby a Polson
to ext.
-22-
roms (Fraz,
1976).
The average rate of discharge of a cochlear nerve fiber increases with increasing stimulus level. To
ensure that the test circuit also exhibits this property, the negative phase of y(t) must produce increases
above the quiescent rate (X0 =h (y)) that are larger than the decreases elicited by the positive phase of y(t).
This was achieved by setting threshold level to 2 times the RMS level of the noise, yo =2[R, (0)]'.
choice of yo we see from the rate function of eqn. 2.2 that I(yo+8)-X(Yo)I
<
For this
Iy 0-8)-X(y)I for posi-
tive 8 and this asymmetry of rate difference about Xo increases with increasing B. This assures that as the
peak-to-peak magnitude of y (t) increases, the average rate also increases.
To simulate the effects of different spontaneous discharge rates upon IRC estimation algorithms we
changed the quiescent spike rate of the test circuit. From eqn. 2.2 and figure 2.5 we see that the spontaneous rate, X(yo), may be changed by adjusting the value of yo or R, (O) or by adjusting the cutoff frequency,
fc, of the noise shaping filter S(f).
( )
e - y°
Since small changes in Yo and R, () produce relatively large changes in
, adjustment of Y0 or R(O)
bandwidth of the filtered noise signal,
(IR' ()
I
fe and R. (0) a ft).
was deemed inappropriate.
R ()
We therefore chose to adjust the
, which varies linearly with the filter cutoff frequency
We used an automatic gain control (AGC) circuit to maintain the RMS
level, (R, (O))I, at some preset value, thereby holding e - Y
"( )
constant. In this manner the average rate
could be repeatably and conveniently adjusted.
Shown in figure 2.6 are samples of the test circuit response to simulated tone-bursts. We display an
estimate of the instantaneous rate as a function of time, or a post-stimulus time (PST) histogram (Johnson,
1974) in response to a tone-burst stimulus. The average rate of discharge in the on-window increases with
increasing stimulus level at each frequency. The spontaneous discharge rate is seen in the off-window. The
frequency selective properties of the test circuit are apparent in that a fixed level stimulus elicits diminishing
average rate response as frequency is varied from 1 kHz as does the frequency response shown in figure 2.4.
Thus, the circuit crudely exhibits some of the frequency selective properties of cochlear neurons.
-23-
A MARKOV MODEL OF TCA BEHAVIOR: Theory and Experiment
It should be noted that although the neural average rate is the response variable in this analysis, these
results apply with appropriate modification to any quantifiable response variable. For example, in Chapter 3
we apply these results to the measurement of neural synchronization to a sinusoidal stimulus.
A DESCRIPTION OF THE TCA
The TCA presents a tone-burst to the ear of the experimental animal at a fixed frequency and level
(figure 2.7). The number of neural impulses that occur during the off interval, (Noff) is subtracted from the
number of spikes that occur during the on interval (N,).
If Ne,-No/>C,
This difference is compared to some criterion C.
then the stimulus level is lowered by one step (DOWN).
If N 0 1-No
C, then the
stimulus level is raised by two steps (UP). When either of the stimulus level sequences UP-DOWN-DOWN
or DOWN-UP-DOWN occurs, a success is declared and the final level (that level attained by the last down
step) is defined to be the threshold for that frequency. The frequency is then incremented. The level of the
last stimulus presentation at the previous frequency becomes the level of the first presentation at the new frequency and the hunting process begins anew.
It is tempting to assume that at frequency f, the TCA finds level P such that AX(P J)=XA
X, = CuT.o (T,
is the duration of the tone-burst on-window).
A(P
where
) is the average rate difference
between the spike rates in the on and off stimulus windows. It is also tempting to estimate the rate criterion
that the algorithm seeks. For example, for a criterion, C =0, the level sought should be that at which
O<(N, -Nof/)<1
on the average (berman,
1978). Thus, for a tone-burst stimulus with T,, =50 ms and
To = 50 ms, a rate criterion between zero and 20 spikes/sec would be expected. likewise we might expec
that for a criterion of C the algorithm seeks a difference N, -No// such that C <Nmo-Nof/<C +1 on the
average. These intuitions are, strictly speaking, false. The purpose of this section is to specify a model of
the algorithm and to analyze its performance so that the precise behavior of the TCA can be determined. In
the following section we will show that the TCA attempts to hold a probability (not an average rate of
discharge) constant.
Specifically, the 1CA seeks a stimulus level,
response exceeds criterion is equal to 2/3.
-24-
, such that the probability that the
MODEL FORMULATION
We assume that previous stimulus presentations have no effect upon the response during the present
stimulus presentation. More explicitly, for fixed frequency we let the stimulus level at time t (t an integer)
be represented by an integer l (t). Hence,
Prob( (No-Noff ) > C, 1(t)|I(t-1),
(t-2), ... ) = Prob( (No,,-Nff)
> C, (t)).
(2.3)
While this assumption is probably not strictly correct (e.g., Gray, (1966)), it is a reasonable starting point for
mathematical analysis.
Such an assumption suggests the Markov model represented in figure 2.8a for TCA stimulus level
presentations at fixed frequency. g (l) is the probability that (No. -Noff)>C
the algorithm increments the level by 2, whereas if (N., -Nff)>C
at level . If (N, -Noff )-C
the level is decremented by 1.
Defining p, (l) as the probability of being in state I at time t we have
ptl(1)=g(l+1)p,(l+1)
+ (1-g(l-2))p,(l-2)
A success level
(2.4)
is generated by either the sequence UP-DOWN-DOWN (UDD) or DOWN-UP-DOWN
(DUD) starting from state (4 ) I and ending in state 1. The probability of occurrence of each of these
sequences is
DUD,= g(l)(1-g(l-1))g(l + )
(2.5a)
UDDI = (1-g(l))g (+2)g ( +1)
(2.5b)
We define DUD and UDD 1 as the probabilities of occurrence of these two sequences from a starting state I.
Let us now examine the probability of success in a given state at time t.
The history of the process has an effect upon which state transitions result in successes. For example,
the sequence DDUDDUD would be grouped as D(DUD)(DUD) where the subsequences in parentheses are
successes. The grouping DD(UDD)UD would not occur since the sequence DUD occurred first and after a
sucess is declared, the algorithm disregards the prior sequence of state transitions. Thus the algorithm
operates on the sequence of state transitions generated by the Markov process in a selective manner and does
not identify all subsequences UDD or DUD as successful. With this property in mind we derive an expression for the probability of success in state I at time t, p, (,t).
(4) State and level will be used interdmngeably
In order to declare a successin state I at time t the process must be in state I at t -3. In addition, the
determine which success sequences generate a success in . For
transitions which lead to state I at t-3
example, if the process were in state +1 at t -4 and made the D transition to I at t -3, then the success
sequence UDD from I at t-3 would generate a success in + 1 at t -1 via a DUD success rather than a sucess in I att via the UDD sequence.
Since the success sequences are DUD and UDD, the prefixes (prior sequences) DU and UD cannot
occur with DUD otherwise the premature success (DUD)UD or (UDD)UD would result. Similarly, D cannot
occur as a prefix for UDD without the same effect. Table 2.1 lists the sequences which will produce a success in l at time t.
PREFIX
Table 2.1: Success State History
START STATE (t6)
SUCCESS SEQUENCE
DDD
DDU
DUD
UDD
1+3
I
DUU
UDD or DUD
1-3
UDU
UUD
UUU
UDD
none
UDD or DUD
1-3
1-3
1-6
success in I
UDD or DUD
I
Therefore,
Ps(1,t ) = pt-6( +3)DDD _3DUD,+p,-t_
1 UDD,
6 (1)DDU
+P,-6(1 -3)DUU -_3(UDD, +DUD, )+p,(1 -
(2.6a)
3)UDUl _3UDD,
+pr-6(1-6)UUU- 6 (UDDi+DUD,)+p,( ,t -3)(UDDi+DUD)
Rewriting some of the prefixes in terms of the transition probabilities, g(1), and regrouping we have,
p, (l,t ) = p, (1,t -3)(UDD, +DUD,) + p.-6(1 +3 )g (1+3 )g (1+2)g (1+1)DUD,
(2.6b)
+ [p,-,(1-3)g(i-3)+p,-6(l-6)(1-g (1-6))](1-g (1-4))(1-g (l-2))(UDD,+DUD,)
+ [P -6(1)g(l)+p,-6(1-3)(1-g (-3))]g (1-1)(1-g (1-2))UDD,
Using equation (2.4) repeatedly we obtain,
-26-
(2.6c)
p, ( ,t) = p, (1,t - 3)(DUD,+ UDDi)+p - 3(1)(DDI +DUD,)
-p, _,( + 1)g (1+ 1)UDD,
-P-S(l -I1)[(1-8(l-1))8(l+1)+S(l-1)(1-g(l-2))
-pr-6(l)(1-
DUD
())g (1 +2) (1 + 1)DUDI.
Thus, given g(1), the probability of meeting criterion at stimulus level 1, and equations 2.4 and 2.6c we
have specified the behavior of the tuning curve algorithm if it is allowed to choose stimulus levels and sucess
levels (threshold estimates) at a constant stimulus frequency. Equation 2.4 determines the relative frequency
of stimulus levels, 1, which are presented by the algorithm and equation 2.6c governs the relative frequency
of success levels, ,.
Using these equations we will now examine the steady state distributions of stimulus
level and success level. Through these distributions we will ascertain what stimulus level the TCA would
choose as threshold on the average, were an infinite amount of time allowed for the TCA to choose threshold
points. We will show that the average threshold level is near the value of I for which g(l1)=2/3.
STEADY STATE BEHAVIOR OF THE MODEL
Theoretical Considerations
Derivationof Steady State Equations
The state occupancy probability as well as the probability of success would become time invariant as t
tends toward infinity if the Markov process were composed of a single ergodic drain (Drake, 1967; Papouis,
1965). However, note that if state
is occupied at t =0, three steps later at t =3 only states l-3,
1, +3 and
1+6 could be occupied (see figure 2.8a). Every third state belongs to the same subchain if time is measured
in three step intervals. Thus the Markov process which determines the behavior of the TCA does not consist
1
of a single ergodic chain and it is not clear that a steady state exists. For example, if po(l)=l then
P3-1(l)=Pi-2(1)=O
(i =1,2,...) and no steady state aocns. Nonetheless, a steady state does exist for the
initial conditions that are likely to be encountered. To ilustrate, we wil identify the set of initial conditiomns
for which a steady state exists and show that they are likely to occur.
process is derived from the process depicted in figConsider the transition diagram of figure 2.8b. ThIbis
ure 2.8a by using only 3-step observation intervals as discussed in the previous paragraph. This nw process
-27-
is a single ergodic chain and therefore is guaranteed to have a steady state. The original process states, , are
spanned
by
three
( ... l-2, +1,l+4..
such
consisting
subchains
states
( *
l-3,
),
, +3
H,and
). Wewillca thesesubchainsI,
...
-1,1+2,1+5
), and..
(
the
of
respectively. A one-step transiton from subchain I will leads to subchain II. ikewise, me-step transitions
from subchains I and HI leads to subchains III and I respectively.
Consider an initial state occupancy probability for subchain I,
l =3i i=..-1,O,1...
Po(l) =
otherwise
(2.7)
where a! is the initial state occupancy probability of state I on subchain I at t=O. We also assume that
C a'
= I so that the probability of being in subchains II or m at t =O is zero. Since a one-step transi-
j=-x
tion from subchain I leads to subchain fl, at t=1 only subchain II is occupied.
ikewise, at t =2 only sub-
chain m is occupied. Thus an initial condition on any one subchain will be transformed into a nonzero state
occupancy probability distribution on the other subchains. Since each of the subchains is ergodic, a nonzero
state occupancy probability at any time t leads to a steady state state occupancy distribution as t-..
There-
fore, for the initial conditions proposed on chain I in eqn. 2.7, each of the subchains will achieve a steady
state (for time in three step intervals). This may be written as,
1 l=3i, t=lim 3k
p,(l)
1,
l=3i+1, t=lm 3k+
=
f"
where pB, 13, and pf
(2.8)
l =3i+2, t =im 3k+2
are the steady state distributions on chains I, II and III respectively and i is an
irnteger. We mw modify the initial condition of eqn. 2.8 to obtain,
(ai
po(l)=
1=3i
pl"al4
Pdam
1=3i+1
l=3i+2
I
where
P'+ P' + p
=1
In this manner we can account for arbitrary initial conditions on the original chain of figure 2.8a.
-28-
(2.9)
Since the difference equation 2.4 is linear, scaling the initial conditions by p scales the steady state solution by p as well. Therefore,
I=3i, r=lim 3k
1 =3i +1, t =lim 3k +1
p'pi
p
p
1=3i +2, t =lim 3k +2
A--
I =3i, t =lim 3k +1
l =3i +1, r =lim 3k +2
p,(l) =
(2.10)
l =3i +2, t =lim 3k
I =3i, t =lim3k +2
&-x
l =3i +1, t =lim 3k
l =3i +2, t =lim 3k +1
From equation 2.10 we see that if p =
= 1/3 we obtain in compact form,
1=3i
3
P())
=3i+l
3
(2.11)
l=3i +2
Thus we conclude that if the initial state occupancy probability is equally distributed over the three subchains,
then a steady state for equation 2.4 exists.
Although the TCA starts from a particular state at t=0 for a given frequency, a priori, there is some
uncertainty about the identity of the starting state. This uncertainty will be large enough that we would be
unable to say which of any three adjacent states (therefore three different subchains) is a more likely starting
state. Therefore, to find the time invariant state occupancy probability p (1), we consider eqn 2.4 with an
initial condition which spans each of the three subdchainswith equal probability and thus assure a steady-state
state
occupancy probability,
p () = p (+l)g(l+
1) + p (-
In light of this result, equation 2.6c becomes,
-29-
2
)(1-g (-
2 ))
(2.12)
pI(l ,t ) = p,(l,t -3)(UDD,+DUDi)+ p(l)(UDDI+DUD,)
(2.13)
-p (I+1) ( +1)UDD,
(-1))g ( +1)+g(l-1)(1- (-2)) ]DUDI
-p (- 1l)[(10-g
-p (I)UDDtDUD
Equation 2.13 may be written as a simple difference equation for fixed 1,
p,(l ,t) - a(l)p,(lI,t-3)
= b (1)
(2.14)
where,
a () = (DUD,+UDD,)
and
b(l) = p ()(UDD +DUD,)- p(l +1)g(l+ 1)UDD,
p(l -
1)[(1-8 ( l -1))g (l + 1)+g (l -1)(1-
(1-2)) DUD,
-p (l)UDDiDUD,
Ignoring imaginary roots, the homogeneous solution to equation 2.14 is,
t
p,(l,t) = A[a(l)l3
(2.15)
Since a (l)< 1, the homogenous solution decays to zero as t approaches infinity, and we are left with the particular solution of,
P,( ,) = P (l) = I()
(2.16)
We now have equations for the steady-state state occupancy probability, p (1), and the steady-state probability of success at a given state, p, (1). Thus, we have shown that steady-state distributions of stimulus
level, , and success level, I,, exist. Using these distributions we will provide bounds for the average stimulus
level, E (), for arbitrary g (I).
The AverageLevels E () and E (,)
E(1) is the average stimulus level presented by the TCA. Knowledge of the properties of E(l) should
help us determine the expected value of the threshold obtained by the TCA, E (),
intimately related (eqn. 2.16). Thus we shall examine the properties of E(I).
since p () ad p(l ) are
We will show that E(I) is
dosely associated with the level 1:23where g (1z,3)=2/3. To assure that E(l) exists,(5 ) g(l) will be restricted
(5) The existence of E()
is an impoant
csideration
ince without a unique and finite E(I) no unique threshold detemi-
-30-
to a certain class. This dass is described in Appendix A. Here we assume that g(1) is in this class.
Before deriving the bounds on E (1) we derive two equations which will be of general use. The first of
these is obtained by taking the z-transform of the steady state probability equation 2.12. This results in
+ 2P(z) - z 2 G,(z)
P(z) = z-G(z)
(2.17)
where
G,(z) =
and
Zg(l)p (1),
P(z) = i z'p(l)
Rearranging terms in equation (2.17), we obtain
P( z=
1-)
G(z(z)
(2.18)
(l+)
Setting z =1 and noticing that P (1)=1 and Gp (1)=E (g(l)) we find
E (g(l)) =
2
(2.19)
The second of these useful equations is derived now for convenience. We will make use of it later. By
taking the inverse transform of eqn. 2.18 we obtain the difference equation,
p (l )8 () = p (l-1)(1-g (1-1)) + p (l-2)(1-g (I-2))
Equation 2.20 may also be obtained by drawing an imaginary vertical line between states
(2.20)
and !-1
of the
Markov chain in figure 2.8a. For the system to be in steady state the probability of a transition to the right
must equal the probability of a transition to the left. Equating these probabilities leads directly to equation
2.20.
We now determine upper and lower bounds for E ().
By differentiating equation 2.18 with respect to
z and using the moment generating properties of the z transform (Papoulis, 1965; Drake, 1967) we find that,
(2.21a)
3E(g(1)) -2 E () = 1
or equivalently,
l (3 ()-2)p(l)
Let g()
=
be a monotone increasing function for which E(I) eists(
nation an be made by the algaithm
(6) An exac desiption
of g(l) may be found in appendix A.
-31-
(2.21b)
6
). Consider a shifted version of g(l),
g
'(l)=g(l-+),
p(i),
p '(l)=p(l-).
2
where g '(-l)-,
2
g '(+ )>
This new function g '(l) gives rise to a shifted version of
If p '(I) is substituted into equation (2.21b) we obtain,
(2.22)
p'(1)(3g '(l)-2)=1
but since
(3g'(l)-2) > 0
for
1>0
0
for
l<O
(3'(l)-2)
(2.23)
we also have
Ilp'(1)(3g'(I)-2) - 1
(2.24)
1=--
Since all the terms in the summation of eqn. 2.24 are positive
lp'(l)(3g'()-2) < Ip'(1)(3g'(l)-2) < 1 ,
I=k
k1
(2.25)
1=0
Since p '(I) is a probability function,
A-1
0
(2.26)
< lp '(l )-<k - 1
1=0
And since g'(1) is monotone, eqn. (2.25) yields
lp '(l)
1
<-
3g'(k)-2
I=k
(2.27)
Combining eqn. 2.26 and eqn. 2.27 we have,
E(/')
=
'() -< p'()
k
38 '(k)-2
+ k-1
(2.28)
Tberefore,
1
(3g'(k)-2) +
(2.29)
0
A simil
derivation applied to
lp
E(l)
yields
"()(3g "()-2)
j+1 +
1
-
I
j< -1
(2.30)
To test the bounds we apply them to a 102 point sequence g (I)=0, 0.01, 0.02,... 1.0, 1.0 where the average
stimulus level is known (E(l)=66.66) with
14
= 67, _1= 66, j =-5,
56.12 s E(l) - 77.25
-32-
k =5 results in
(2.31)
We now turn to the study of E (,).
To compute E (4) we must first obtain p,(l |suacess). This is
p,(lIsuccess)=
P(()
(2.32)
Thus
E(,)
=
(2.33)
lp,(l Isuccess)
I
Unfortunately, p,(l)
is quite complex (eqn. 2.16), hence it is difficult to derive an analytical expression far
E (1,). Therefore, we compute p,(l) and E (l,) numerically.
NumericalDeterminationsof p (1) and p (l Isuccess)
In this section we compute p (), p,(l), E(l) and E(l,) for several spedfic g(l) functions. With these
E(t,)
g(l) we illustrate that E(l)
l2.,3where l2,3 is the leve such that g(l23) is doser to 2/3 than any
other level, I. These results are shown to hold for a larger dass of g(l) than is represented by the specific
g (l) functions we have chosen.
The technique used to compute p (l) consisted of constaining g (l)=O for I<O and g(l)=1
(L
is therefore the number of nonzero terms in the sequence g (l) for which I -L.,)
for l -Lm
thereby constaining
p (l) to be a finite sequence. Since p () was finite and equal to zero for l <0, the difference equation p (,)
(eqn. 2.20) was solved numerically by assuming initial conditions of x(-1)=0
and x(0)=1 with the recursive
equation,
x(l)
Ix(l-l)(l-g(1-1))+ x(1-2 )(1-g(t- 2 ))1
(2.34)
Equation 2.34 is an algebraic rearrangement of eqn. 2.20 with x substituted for p. The state occupancy probability function, P(l)=
L
p,(l success) was computed using equations 2.16 and 2.32.
Five
C
x(l)
i=0
specific g(l) were used. Of these five, two were linear functions of I (g(l)=I/L,)
spacings (/L.)
(figures 2.9a&b).
with different point
For contrast, we chose two other g (1) to be semicircular arcs (figures
2.9c&d); one concave, the other convex. In addition we also chose a discontinuous linma g (1) (figure 2.90).
The associated p(l)
(I/L.).
and p,(l success) are superimposed and plotted as a function of normalized level
We see that in every case except figure 2.9e both E()
-33-
and E(l, ) lie within on step in level of 123
(the l3 point is circled). For figure 2.9e, E(,)
lies within one step in level of lz3 while E(I) lies just out-
side a level step from 12,3. Thus, for the five different g () considered, the average stimulus presentation
level and the average success level are dosely associated with the level la3.
Some generalizations may be made on the basis of the numerical results obtained for linear g(l).
Notice in figures 2.9a&b that the state occupancy probability, p (1), is concentrated between two values of I.
In the case of figure 2.9b, Mp ()
.95 for .5_g()<.8.
Intuitively it would appear that any g(l) which
may be approximated by the linear g(l) of figure 2.9b in this interval will have E(l) and E(I,)=lZ.
intuition is verified analytically in Appendix A (eqns. A.18 through A.20).
briefly. Let p'(l)
.5-g(1).82.
'Ihis
We present the result here
be the steady state state occupancy probability for a sequence g'(l), similar to g(l) for
p'(1) and p(l),
the distribution associated with g (), will be approximately equal in this
range if p (ll- 1)+p (l) and p (12)+p ( 12+ 1) are much smaller than one. As a case in point, notice the similarity of the probability distributions of figures 2.9b and 2.9c. This similarity results from the fact that the
average slope of the circular g(l) for .5-g()
.8 at .023 per level-step (fig. 2.9c) is approximately equal to
the slope of the linear contour at .02 per level-step (fig. 2.9b).
In summary we see that for the g(l) we have examined, the average stimulus level E(l) and the average success level (threshold estimate), E(1,) is approximately 123 where g (1z3)=2/3. We have also seen that
the approximate congruence of the steady-state stimulus level distribution, p (1), for simila g (l) in the interval (0.5, 0.82) is a very useful property. Using linear g(l) of various slopes (various L.)
for which p(l)
outside of the prescribed interval is negligible, we may approximate p (l) produced by any other sequence,
g'(1), as long as the linear g () and g'() are similar in the interval (0.5, 0.82).
The TCA Does NOT Necessarily Satisfy a Rate Criterion
The Markov process model of the TCA depends only upon g(l), the probability of being above criterion. 'IThus,p () and p,(l) which characterize the steady state behavior of the TCA, depend explicitly on
the fumction g () and not Ak(/). Furthermore, in the absece of a known empirical relation between AX(l)
and g () or equivalently knowledge of the underlying spike generation process, g (1) provides no information
about A(l).
We illustrate as follows.
-34-
Let A = N, - N
.
We define a probability function, q (A ll), whose value is the probability of
obtaining a spike difference A at stimulus level 1, The probability of being above criterion, g (l), is then
g(l) = Prob(A>C)=
E q(All)
A=C+1
(2.35)
The average rate difference AX(l) is then,
IAX()= E(AII)
-= Aq(AI)
(2.36)
A=-z
where I is the duration of the tone-burst on and off windows (T, =Toff =I).
We will now show that given I
and a value of g ()= a ( < a < 1) that we may construct q(A i ) so as to leave AX(I) unconstrained.
For g (I)= a we define the probability functions,
h+(A) =
(
)
h_(A) =
(ac
A> C
(2.37a)
(2.37b)
hf(A) and h_(A) are arbitrary within their prescribed domains. Thus Eh+(A) and Eh_(A) may take on any
value within these domains.
-=<E,
C+1
()=
E+()
1
=
Aq(11l)
Al
A C+1
C
(2.38a)
q(A I) <
(2.38b)
Using equations 2.36 and 2.38a&b we obtain,
Y Aq(A1l) = (-a)E_(A)
From equation 2.39 and equations 2.38 we see that E_(A)
-x
< A(l)
<
(2.39)
+ aEh,(A) = IAX(l)
and E+(A)
may be chosen so that
for 0 < a < 1. Thus, fixing g(l)=cx does not necessarily limit AX(l) to any particular
range of values.
In conclusion, we see that AA(l) is not necessarily related to g(l).
Although an empirical
relation
between these two variables may exist, if the relation is unknown, then without spefication.of
AX(l) cannot be inferred from g ().
q (Al ),
In short, it should simply be noted that the TCA average stimulus level
and average success level are not determined by an average rate difference but by the function g ().
Any
relation between AX and g is due to the nature of the spike generation process and not the properties of the
-35-
TCA.
Experiments to Examine TCA behavior in the Steady State
In this section we will examine properties TCA steady state behavior implied by the Markov model.
For each property considered we provide a qualitative description and the results of experiments on the test
circuit and cohlear neurons which are designed to test the qualitative description.
PredictedValuesof E (1) and E (I,) UsingMeasurementsof g (1)
Measurement procedure
If the Markov model accurately predicts the behavior of the TCA then g(l) can be used to predict the
mean and standard deviation (s.d.) of 1,1, in the steady state. Thus, the experiment devised to test the Markov model of TCA behavior consisted of measuring the function g (1) as well as the sample mean, ,, and
s.d.,
,l, at fixed frequency for both the test circuit and codlear nerve fibers in the alligator lizard. Specifi-
cally, using tone-burst stimuli with 50 ms on and off windows and a criteria of C =0, the tuning curve algo(7 )
The
rithm was allowed to determine 80 successive threshold levels while the frequency was held constant.
sample mean, l,l, and variance, Fal were computed from the last 50 trials. This procedure was used to
reduce the effects of startup transients on the calculated average threshold level. g(l) was then estimated for
levels above and below I, by determining the relative frequency of No, -Nof >0 for many tone-burst presentations at each level.(8 )
A line of slope a (linear regression) was fit to the measured g () and from this slope the linear approximation to g(l), g, (l), was obtained. The mean and s.d. of l,l were then predicted using g,,(l)
dB level steps). Finally, the sample mean, l,,, and sample s.d.
E(l, llgu,), and predicted s.d., E(a,llg).
(using 1/4
,l were compared to the predicted mean,
Typical sets of measurements are presented for the test circuit
(7) Frequencies of 2KHz and 1KHz were used for the test writ
and nerve fibers revely.
The level step ize was 14
da
the tandhrd deviatim ot the measurement is r=((1-g())g(Q.Y¥)
(8) Under the aumpfion o independet tri,
We computed the san
where R is the number of stimuluspresentations. We assume the maximum value of acr=%(IK).
dard error to be .017 for the tee crcuit (800 trials) and .04 for the nerve fibers (150 trials).
-36-
(figure 2.10a) and an alligator lizard cochlear neuron (figure 2.10b).
Measurements on the test circuit
For the test circuit, the slope of g (I) is changed by varying the spontaneous rate X0 . Average threshold
determinations were done at two spontaneous rates, Xo= 9 spikes/second and 46 spikes/second to compare
results obtained with different g(l).
E (,
The difference between E(lQJ1glm) and 1,1 in dB is
Als,
aj1
and
gli,) are measured in dB. a, the slope of gjl, (), is in units of 1/dB.
Table 2.2: TCA steady state threshold determination (800 trials/point).
_o
spikessec
9
46
In both cases E(lllgj)
Als
dB
0.0
0.0
1
is identical to 1,I and
a
1/dB
.07
.05
_
sI
dB
.95
1.22
E (,l
1gli,)
dB
1.08
1.28
sll is comparable to E(ao, lg,,).
Thus, it appears that the
Markov model of the TCA predicts the steady state behavior of the TCA for the test circuit.
Measurements on cochlear neurons
Nineteen threshold determinations were made on fourteen units in two alligator lizards. As with the
test circuit, the results are tabulated (table 2.3).
-37-
Table 2.3: TCA steady state threshold level determination on auditory nerve fibers (14 units in two animals) (150 trials/point). The standard error for rate measurements is .5
spikes/second unless otherwise noted. For five units g (I) was measured twice frcxn a single determination of threshold. The second measurement of g(l) is denoted by a " in the
,J1 column.
L_
.
a
At
E(arIlg1 )
i/dB
.08
.07
.06
.09
.09
.09
.12
.09
.05
dB
.78
.82
1.38
1.47
.9
.68
dB
1.02
1.08
1.17
.95
.95
.95
.83
.95
1.28
spikes/sec
4.1
1.8
2.0
5.6
11.7
8.5
9.7
5.8
10.0
dB
.21
.38
-.05
-.48
.629
-.48
.03
-.37
.33
8.8
.166
.1
7.3
7.6
9.6
.24
.07
.44
.07
.1
.06
15 +-1
-.03
.1
1.6
6.6
4.5
4.4
.38
.33
.08
-.58
.07
.08
.08
.04
1.03
1.32
.97
1.08
1.02
1.08
1.44
.82
1.08
.90
1.8
.38
.07
.09 +-.34
.08 +.02
In table 2.3 we see that in only two cases does E(l,IIg,,)
ured variance,
l (correlation
1.02
.86
1.09
.9
1.17
.90
mean +s.d.
The predicted variance, E (r
.83
1.66
1.04 +±.30
1.04 ±.14
lie more than 2 steps in level (.50 dB) from 1,.
lgj, ), however, on a fiber by fiber basis, does not compare well to the measoefficient = .22 ; correlation slope = .11). We feel this is due to the fact
that the measurement of g () is less precise for the nerve fibers (150 trials) than for the test circuit (800 trials) thereby causing a larger variation in the predicted slope, a. For all except one of the five units in which
g () was measured twice we see slope estimation differences of at least .03/dB. As can be seen by comparing the values of E(a,l Igtic) predicted from the two slopes, the predicted variance is sensitive to changes in
the slope of g,,(l).
Nonetheless, the sample variance values, cl, are comparable to the predicted variance
for the range of slopes measured (compare the mean variances in table 2.3). Thus, it appears that for auditory nerve fibers in the alligator lizard, the model of the TCA does predict the average sucxess level chosen
by the TCA in the steady state and roughly predicts the deviation of the success levels chosen about this aver-
-38-
age level.
Variation of AX with Spontaneous Rate, Xo
Prediction of AX vs. Xo assuming a homogeneous Poisson spike generation process
It is possible that the average rate difference associated with g =2/3 may depend upon the fiber's spontaneous rate. To explore this possibility we perform the following analysis. Let the previously introduced
random variables No,, and Nolf have Poisson distributions with average intensities X,,(l)T,,
and XoffToff
respectively. T, and Toff, as before, are the observation intervals in which we record No, and Noff respec
tively. By definition,
p(No,
I1)=
N,
eC
Tohatl
p(Noff) = (ohow
Nof !
O
ideal-lf tr(f)
We have shown that ideally the TCA chooses level l23 as threshold such that g (12)2/3.
g (1) = Prob (No,-Nif
=
C
(2.40a)
(2.40b)
We form g (l) as,
(2.41)
>C 11)
N, -(C +)
p(No. I)
N.=C +1
E
N
P(Nof)
=O
-~o~t)7t0",
^of..-C+')
(.(t)T,
)N""
(off
Trft
),'
N,=C1-INff
N. ! Noff
=O
We determined X,, (l23) by setting g(1)=2/3 and solving eqn. 2.41 numerically with C =0,1 and 2 and
with various values of Xolf Tol . We list the results in table 2.4 assuming that, T,. =To, =50ms; a standard
value used in measurements on cochlear neurons. The assumption of a Poisson spike distribution predicts
that AX at g =2/3 will show substantial increases with increasing Xoff.
-39-
>C)=2/3 for T, =Tolf =50 ms. Off-window rate is
Table 2.4: AX for Prob(N,.-Nf!
Xf!. All entries in spikes/second.
- XAf
AX =
1\°1f
0
10
20
30
40
50
60
70
80
90
C=0
C=1
C=2
22.4
24.0
26.4
28.5
30.5
32.5
34.2
35.7
37.2
38.6
46.1
47.6
49.3
51.2
53.0
54.6'
56.1
57.5
59.0
60.0
68.7
70.2
71.9
73.7
75.1
76.3
77.9
79.2
80.5
81.7
Measurementson the test circuit
To investigate whether AX varied with Xof
mined by the TCA. Tone-bursts with T,=Tff
AX was
nmeasured
at the average success level, 1,1, deter-
=50 ms were used with a criterion of C=O0. AX for three
different off-window rates are listed in table 2.5. As can be seen, the average rate difference increases with
increasing Xof in a manner similar to that predicted in table 2.4 for C=0.
Table 2.5: Average rate difference, AX, vs. off-window rate, ff , measured at stimulus level, 1,1 for the test circuit. T,. =T,l =50 ms and the
TCA criterion, C=0. All entries in spikes/second. The standard error
of all measurements listed is .5 spike/sec.
9
30
46
22
26
30
Measurements on cochlear neurons
To determine whether the spontaneous rate of a fiber has an effect upon the rate, A(12,),
associated
with threshold (g =2/3), we first determine whether the off-window rate, Aoff depends upon spontaneous
rate, Xo. We do this since the TCA does not directly use the spontaneous rate in the estimatiqn of l2 but
rather, uses the off-window rate Af I . We have compiled PS histograms of the response to tone-bursts
(T, =T.o=50
ms, T,=2.5 ms) from miscellaneous eents.
We have included only histograms for
which A > 20 spikes/sec in order to assure that the fiber response differed from the spontaneous.
The
spontaneous rate,
o0,for each fiber was also measured. As can be seen in figure 2.11 where we plot
off
versus the spontaneous rate, there is a strong correlation (corr: .88, slope: .67) between off-window rate Xof
and spontaneous rate X0 (43 units in 4 alligator lizards). Thus, in light of this result and the prediction of
table 2.4 for Poisson spike distributions we would expect that the average rate difference, A?, corresponding
to g =2/3 to increase with increasing spontaneous rate in alligator lizard cochlear neurons.
In order to test this hypothesis, AX(l)
gator lizards.
and Xqf (12,3)were measured for fourteen units ( 9 ) in two alli-
he results are listed in table 2.6.
dependence upon Xff (l2).
Specifically, A(12,)
AX(l 2 3) for the nerve fibers does not seem to show a
is not correlated with Aolf (correlation: .05, slope: .26).
This result is at odds with the prediction of increased Ai with increased Xo and suggests that the assumption
of a Poisson spike probability distribution is incorrect
However, the assumption of a Poisson distribution
does provide a rough estimate of the value of AX(12 3). With a criterion of C =0 and a spontaneous rate
range of 0 to 20 spikes/sec we predict using the Poisson spike distribution assumption that AX(1l2) is
between 22 and 26 spikes/sec. The average value of AX(1z23)falls roughly in this range (22 +3 spikes/sec).
(9) bse are the same mits as In table 23.
age values are given (italics)
For the five uits t.r whidM~(v,)
41-
and
ff/(t:)
were meaured twice ar.
Table 2.6: 19 average rate differences, AX, vs. off-window rate, Aof,
measured at stimulus level, 2I for 14 units in two alligator lizads All
entries in spikes/second. The standard error is .5 for Xof and 1 for
AX(12,3) unless otherwise noted. For units in which duplicate measrements were taken we give the average resoonse values (italics).
4.1
1.8
2.0
5.6
11.7
23
19
24
28
22
9.1
24
5.8
28
9.4
7.5
22
20
12.3 - 1
19
4.1
20.5
4.5
4.4
1.8
18
20
19
Variation of AA(l2)
with Frequency
Analytic description
It is possible that the average rate difference Ah measured at the TCA chosen threshold, 123, varies
with stimulus frequency. To see why this is so let us first consider the variation of g(l) with frequency. The
simplest possibility is that the shape of g () is independent of frequency, f, except for a tanslation along the
l-axis, i.e.,
g(lf)
= 8 (t-z(f))
(2.42)
where z (f) defines the shape of the IRC. However, since the TCA holds probability but not rate constant,
different frequencies, f, may have l/3 that correspond to different average rates, AX(12/3). If this is the case
the TCA would seek a different average rate for each frequency (for convnience
'probability/rate frequency shift").
we will call this a
he presence of large probability/rate frequency shifts would imply that
the TCA iso-probability contour estimate is be different from the iso-average rate contour which it was originally thought to estimate (Liberman, 1978; Holton, 1980; Holton & Weiss, 1983a). Thus, any previous conclusions which explitly depended upon the average rate difference being constant along a TCA-estimated
-42-
IRC may be in error.
Measurements on cochlear neurons
To investigate whether large probability/rate frequency shifts exist in the alligator lizard, g(l) and AA(I)
were measured for three different sets of 1; each set in the vidnity of threshold for frequencies one octave
below CF, at CF and one octave above CF. If Ati(l3) is a strong function of frequency, then the value of
AX(l) in the neighborhood of g (1)=2/3 should differ appreciably for different frequencies.
In figure 2.12a we plot for one representative unit, g(Q) vs. A(l)
to generate the curves g(AX). As
can be seen the curves (A)) overlap over a large range of g. Also, it is apparent when all data are superimposed (figure 2.12b, 18 units in four animals) that the rate associated with the lz point is = 20 spikes/sec
for many different fibers. In figure 2.12c&d we see that for C =1 and C =2, AX(12)
40 spikes/sec and
60 spikes/sec respectively.
These data suggest that at the frequencies studied, the average rate difference associated with the 23
level is not strongly variant with frequency.
Furthermore, due to the observed large region of overlap of the
g (AT) at various frequencies and criterion levels, and its relative invariance across units and animals, in the
alligator lizard there probably exists an empirical relationship between g (1) and A(l).
Summary of Steady State Properties
We have shown analytically and experimentally that the TCA seeks as threshold the stimulus level
2
where the probability of exceeding criterion is equal to 2/3. In addition we have shown analytically that the
TCA uses only the function g(l) and not the average rate difference, AX(I), to determine threshold. We
have also shown that this property suggests that the average rate difference empirically associated with g =2/3
may vary with both fiber spontaneous rate and stimulus frequency. We have found, however, that there does
appear to be an empirical relationship between g (l) and A(l)
which is common to many fibers in many
animals. Thus the TCA iso-probability contour estimate is also an iso-rate contour estimate.
-43-
TRANSIENT BEHAVIOR OF TE MODEL AND TCA IRC ESTMATION ERROR
In this section we will show that the TCA produces a biased estimate of the 12,3level and that the magnitude of this bias may be affected by the frequency characteristic of the acoustic system as well as the frequency response of the system under study. Errors in the estimation of Q (the sharpness of tuning), P
(minimum sound level required to elicit a response), and CF (frequency at which Pi,
exists) will be dis-
cussed.
Theoretical Considerations
We have previously studied the steady state behavior of the Markov model at fixed frequency. During
the estimation of an IRC, however, the TCA increments frequency immediately after a success is declared
and begins to search for the first success level at the new frequency. In terms of our model, we specify this
process as follows. Consider the TCA just after it has declared a success at level k and frequency fo.
frequency is incremented to f
and the first stimulus level presented at f
1
The
is the previous success level (thres-
hold estimate), k. The stimulus level is then varied until a success, ,1, is dedared at frequency f 1. This
process continues until threshold estimates have been made at each frequency. In choosing the first success
at a given frequency, however, the TCA does not allow a steady state to be attained. Thus, E(1, ) will not in
general equal 12, and the TCA estimate of threshold will be in error. We now quantify this error for the
Markov model of TCA behavior.
The probability that 1,1 is the first successful state given the starting state is k may be written as
p5 l(lli,k) =
where p (,l,k)
i
(2.43)
p,(1,l,k )sj(l 1)
is the probability of taking path i from k to ,1 with no intervening successes and the si(l,l)
are the allowable success sequence probabilities which terminate path i in a sucess at 1,1.(1 0 ) The expected
value and variance of the first success from state k may be computed from p, 1(l,l,k).
p,l(l,,k)
for a given 8(1) by simulating the Markov chain. We chose
gga.,()=/L.
We have estimated
(1) to be the linear function
(see g(l) in figures 2.9a&b). We will later show that this choice af g(l) will provide a lower
(10) Recall the premature sumess argument used in fomulatig
the state traitin
-44-
equation for , (eq. 2.6).
bound on the expected error made by the TCA.
as a normalized measure of the difference between
We define the quantity Al=[E(Ilk)-1,3/L.
the expected first success level starting from k and the threshold 12,3. Plotted in igures 2.13ab&c is Al
versus k/L.,
50 and 100, (solid line) where L,,.
for L==25,
defines the point density of gj.(l).
The
dotted lines represent + one standard deviation about the mean.
If AI were zero then the success level would equal l23 and be independent of the starting state. This
occurs only for k = 1X3. Al is nonzero for k different from l2,3 Thus, the algorithm chooses as threshold a
level that depends strongly on the starting level. For starting levels below l1a, Al is negative (threshold
underestimation) whereas Al is positive (threshold overestimation) for starting levels above 123. As L,
increases, AlI versus k/Lx
approaches the (dashed) line AI =(k -12)IL=
more dosely, i.e., the expected
value of the threshold estimate approaches the starting state. This results from the fact that the algorithm
is increased, thereby raising the probability of an intervening
must traverse more states to reach 12,3as L.
success. If we equate the range of k/L,
(0,1) to any fixed range in dB sound pressure level then we see
that as sound pressure level resolution is increased (by increasing L..),
the TCA will produce an increasingly
Therefore the TCA estimate of threshold is biased.
poor estimate of 13 for starting levels away from 13.
This bias depends upon the previous value of threshold, and this bias increases with increasing level resolu-
tion.
To show that gu (1) provides a lower bound on TCA estimation error, consider the following class of
8(l)
fL'..
ga() = g
where gm<0.5
and
,.>0.82.
(1 1 )
m <L/LL < g
/L,.l > 98
(2.44)
Sudl g(l) that neither not monotonically approach zero with decreasing
I nor monotonically approach one with increasing I are likely to exist in cochlear nerve fibers. The probabilistic nature of spontaneous activity imposes a lower limit on g (l) since even in the absence of aural stimuli
(11) 7Tee boxum were determined in the previous ectia to atwe E(,)
-45-
= J".
there is a nonzero probability of exceeding criterion(
2)
and for high stimulus levels the fiber response may
saturate such that an above-criterion response does not occur with probability one.
Intuitively we see that the TCA estimation error associated with gt (l1) provides a lower bound for the
TCA estimation error caused by 8,(l)
as follows. For l <O, g,,(l)=O
cess for stimulus levels 1<0 is zero. Likewise for l -L,,
which I >Lu
g,i,(l)=l,
so that the probability of a false sucthereby making a success at a level for
impossible as well. Thus, no matter how distant the starting state, k, is from 123, the thres-
hold level, with gu.(l), the TCA estimation error, e, must be such that -l3<e<L,-l
for g,r(l),
2
3. Conversely,
there is always a nonzero probability of success at all levels. If a starting level k, is chosen distant
from 12,l, there may be a sucess declared erroneously well outside the range allowable for gi, (l). Thus the
magnitude of the expected error for g, (l ) can only be greater than that produced by gui ().
In figures 2.14ab&c we present Al vs. kL,,
three values of g
and g
(quantities as previously defined for figure 2.13) for
(see caption for details). For reference, Al vs k/L.
included in each figure (dotted line). For all values of L,.
and all values of g,
g8,h(l) provides a lower bound for the expected error magnitude
IAl I.
for gi.(l)
is also
and gm we see that
Experiments to Examine TCA Transient Behavior
Topical Outline
In this section we examine properties of the TCA suggested by the transient behavior of the Markov
model. We will cover four topics which conaern TCA IRC estimation error. These are:
71CAErrors in Estimating Neural IRCs
TCA Errors Caused by Correctionfor the Acoustic System Frequency Characteristic
(12) In tbe abene of aural timuli, the average rates in the -window and df-window c the tmeburst pcse wlMbe
idenical. Amning 50 ms on and off windows, a TCA criterion of C =0, and an approdmately Poisson probabiity disibutio of spike mccurnnces (Frem, 1976; Litlefield, 1973), for spcntanemorates cf 10 qpid/asecand 40 spikedwc the robbilities of exceeding criterion, Pob(N.,-Nf
>0), we .27 and .4 pectively. This probability approaches 12 as sqontanus
Rae
-x.
Miscellaneous Observations
TCA Errors Caused by Non-monotonicg (l)
TCA Errors Caused by Neural Adaptive Effects
In addressing these issues we will use Monte Carlo simulations of the Markov model. In most cases the
linear, probability of exceeding criterion function, 8u, (1) will be used to estimate the g (1) of cochlear neurons. This function supplies a lower bound for the average TCA error.
TCA Errors in Estimating Neural IRCs
Qualitative descriptions and simulations
tracking of a frequency-invariant IRC
As an introduction to TCA estimation error we consider the TCA as it estimates threshold for a very
simple IRC; one that is invariant with frequency. We introduce the constraint, however, that the first level
presented by the TCA is substantially different than threshold. In Part I we showed that the expected value
of the TCA threshold estimate is not necessarily the threshold level, l.
Instead, if the starting level, k, is
different from 123, then the expected value ot the threshold estimate will lie between k and 12,3. In figure
2.15a we show schematically that starting at a level (X) above threshold (solid line) the TCA estimates threshold (0) and then uses this threshold estimate as the starting level for the next frequency. Since the initial
starting level is above 12,3, however, the expected TCA threshold estimate will also be above 123. Thus
several frequency steps will be required for the TCA threshold estimate to converge to l.2/3 This property is
illustrated for an initial starting level below iz/3 in figure 2.15b. Notice that the estimation error magnitude
will be larger in this ase since, as shown theoretically, for starting levels lower than threshold the expected
value of the threshold estimate is larger than for comparable starting levels above threshold. Since the IRC
depicted does not vary with frequency, we see that the TCA has produced an artifactual component in its
estimation of the IRC.
Simulated TCA estimates in which the initial level was above and below threshold (dotted line) are
shown in figure 2.15c, thus validating our qualitative argument. The TCA will produce an artifactual cornm-
-47-
ponent in its IRC estimate if an initial starting level is chosen much different than threshold.
This com-
ponent is larger for initial starting levels below threshold than for initial starting levels above threshold. We
extend this result in the next two sections and show how it will cause predictable errors in the estimation of
V-shaped neural IRCs.
trackingof a linearIRC
Iso-response contours of cochlear neurons are roughly V-shaped. They may be crudely approximately
by two line segments in the frequency-level plane: a low-frequency segment where threshold decreases with
increasing frequency and a high-frequency segment where threshold increases with increasing frequency. To
assess the accuracy of the TCA in measuring such contours we shall first consider the performance of the
TCA in tracking a linear contour which is an approximation to one segment of a neural IRC. figure 2.16a
indicates schematically the types of errors expected of the TCA given the transient response properties of the
Markov model. Consider first a segment with negative slope (figure 2.16a). Suppose that the first threshold
point chosen is exactly l2/3. Then the starting level at the next (higher) frequency will be above threshold.
Thus, the threshold estimate at this frequency will be above threshold as discussed in the previous section.
As is depicted, we would expect this process to continue and the average threshold levels chosen by the TCA
to be above the IRC. We would expect this pattern to be reversed were the IRC of positive slope as depicted in figure 2.16b. On the average, threshold levels would be chosen by the TCA which were below threshold. In addition, the expected error would be larger as indicated (also as discussed in the previous section).
In order to systematically study the effect of IRC slope upon TCA estimation error, we studied the
behavior of the Markov model with a Monte Carlo simulation (hereinafter called simulation). IRCs ,z(f),
of slope a were obtained from the equation
z(f) = alogldJ
(2.45a)
where the set fi is composed of consecutive frequencies that are fixed fractions of a decade apart (i.e., 1/40
decade). Thus a determines the difference in level between adjacent threshold points. The function g ( Jf)
was of the form g (l ,f )=g (I -z(f )) with
-48-
1max
+L7m
2a
2
Lm a
g(l)= 1
3 ma
l2-L ,,
0
(2.45b)
otherwise
This particular form of g(l) was chosen so that threshold at any given frequency corresponded 12/3=z(f)
(threshold is 12/3and g (0)=2/3).
For a given set of (Lm x, a), using the TCA simulation twenty five estimates were made of an IRC of
slope a. The first point on the IRC was chosen as the TCA starting point. This set of measurements is
described graphically for a =5 and Lm=100
in figure 2.16c (only four estimates are shown for clarity). To
quantify the error made by the TCA, for each a an average IRC estimate was computed. The average
difference between this estimate and the known IRC divided by Lmax was defined as the average error.
In figure 2.17 we plot average error ±s.d. as a function of normalized slope for L,=25,
(1 3
)
50, 75 and 100.
Immediately apparent is the fact that error is greater for positive slopes than for negative slopes and that this
asymmetry increases with increasing L
(increased level resolution). In addition we see that for positive
slopes the error grows more quickly with slope magnitude than fr
negative slopes. Thus, for greater IRC
slope magnitudes we would expect larger differences in estimation error between TCA estimates using positive and negative frequency increnents.
tracking of a V-shaped IRC
To illustrate the types of errors which may our
in the TCA-estimation of neural IRCs we construct an
IRC grossly similar to a neural IRC by joining an IRC of negative slope to one of positive slope as depicted
in figure 2.18a. For such a V-shaped threshold vs. frequency contour we would expect the TCA when starting at low frequency (low-high) to underestimate the true threshold on the high frequency side and overestimate an the low frequency side in a manner consistent with the previous discussion of TCA error vs. IRC
slope. If the algorithm were to start at high frequency (high-low) on this same contour, we would expect this
(13) The last 80% of the estimate was used to avoid the transient effects asociated with stating the TCA at a level much
different than 2l as discussed in the previous ection.
-49-
error pattern to be reversed. The general shape (high and low frequency slopes) of the true threshold contour is preserved in the estimate although it is slightly distorted near the low threshold tip. Such distortion
may lead to a systematic error in estimating the characteristic frequency and Pro,
the sound level at the
characteristic frequency.
The overestimation of Pa,
may lead to an underestimation of QOs(14) which we will call tip-
blunting". This effect may be intensified for IRCs resembling high-sensitivity neural IRCs where the Q near
CF is high (Holton & Weiss, 1983b). Such an IRC is approximated in figure 2.18b. We see a substantial
decrease in Q near CF in the qualitatively predicted TCA IRC estimate.
We used a neural IRC archetype with a
Q1 /s near CF of 2.5 and high and low frequency slopes of 80
dBldecade and -40 dB/decade respectively (Holton & Weiss, 1983b). For the simulation we used standard
level and frequency steps of 1/4 dB and 1/40 decade respectively. Using the average value of g (l) slope,
a =.08/dB (table 2.3), L,=50.
figure 2.19 shows a low-high, and high-low average IRC estimate produced
by the tuning aurve algorithm on this archetypical IRC.
As predicted the general shape of the IRC is
preserved in the TCA estimate (the high and low frequency slopes of the IRC and the average estimate are
similar), and when starting at high frequences the TCA will overestimate on the high frequency side of the
V-shaped contour, underestimate on the low frequency side, and underestimate CF (1/40 decade). This error
pattern is reversed when the TCA starts at low frequencies. In addition, Prm/ is overestimated (
an appreciable decrease in Qioa for both estimates (
5dB) and
30% from 2.5 to 1.8 in the high-low estimate and
1.7 in the low-high estimate).
Measurement of TCA errors for cochlear neurons
ffects of initialstartinglevel
Starting at high frequency and decrementing to low frequency, we estimated the IRC of a cochlear
nerve fiber using a set of initial starting levels which spanned a 40dB SPL range. The results of .these meas-
(14) Q.,, is defined as the ratio of the chraeistic
frequency to the x dB bandwidth, where the x dB bandwidth
as the difference in fquency between the fist poins to either ide of CF which are x dB above Pi,.
-50-
defined
urements are shown in figure 2.20.
The high frequency portion of the IRC estimate shows a component
which strongly depends upon starting level in the manner qualitatively predicted and shown in simulation. If
we assume that the IRC started at 70 dB SPL is most nearly correc then we see transient overestimation for
the IRC estimate started at 90 dB SPL and transient underestimation for the IRC estimate started at 50 dB
SPL. In addition the underestimation transient is larger than the overestimation transient. Regardless of the
accuracy of the IRC estimate which was started at 70 dB SPL, however, we see that the TCA will produce
IRC estimates with grossly different characteristics if the initial starting level is either much different than
threshold.
TCA error and IRC slope
For each fiber, an estimate of the IRC was measured using the TCA with positive frequency increments
(i.e., from low frequency to high frequency, low-high). For comparison, we used a more accurate IRC estimation algorithm CCA3 )( 1 5 ) to estimate the neural IRC as well. Another TCA IRC estimate was taken
with negative frequency increments (high-low). The results of these measurements are displayed in figures
2.21a&b where we show the low-high and high-low TCA IRC estimates compared to the same TCA 3 estimate.
Also shown is the difference between the TCA 3 estimate and the TCA estimates.
We see the
predicted error pattern of overestimation to one side of CF and underestimation to the other side.
Results from 9 units with different CF are compared in figure 2.21c. Hre
we plot the difference
between the TCA 3 and TCA estimates on a frequency scale normalized to the TCA 3 CF. We see average
overestimation errors of 3.5 dB and 2.3 dB on the low and high frequency slopes respectively. We see average underestimation errors of -4.5 dB and -9.6 dB on the low and high-frequency slopes respectively. To
determine whether the Markov model roughly predicts the magnitude of these errors we recall that the average sope of g(l) was determined to be .08/dB (from table 2.3). We assume that this result holds for the
units used here. Using 1/4 dB level increments as was done here this correspds
igure 2.17b where L,=50
to L.
=50. Looking to
we see that using the linear approximationto g(l), gA(l), the maimum TCA
(rther
(15) The more aiute
estimate used the .ame TCA algorithm except that 3 ucces
each frequency. The tird success was declared threshold. We dtherere cell tis algarithm the
in TCA ACCURACY IMPROVEMENT the CA 3 is a muc more anrate IRC etiatorn.
-_51-
dun me) we required at
CA 3 . As we denribe later
overestimation error is 2.5 dB (1/4 dB level increments). The maximum underestimation error is -4.0 dB.
Thus, except for the underestimation error on the high frequency portion of the IRCs the Markov model
predicts the approximate magnitude of the expected TCA estimation error. However, it should be recalled
that the linear approximation to g (), g, (l), only provides a lower bound on the TCA average error. Thus,
we find that the experimental results are consistent with the Markov model predictions of TCA behavior.
TCA IRC parameterestimationerror
The TCA and TCA 3 were used to obtain IRC estimates in the same unit. Parameters derived from
these two estimates are compared in the scatter plots of figure 2.22a-e (28 units in 5 animals). figures
2.22a&b show that the low and high frequency slope estimates are, on the average, similar for TCA 3 and the
TCA (Low-frequency: corr=0.76,
slope=1.06
.17; High-frequency: corr=0.69, slope=.97
.19).
This
corroborates the result of the previous section where we saw that the TCA estimation error was constant at
high and low frequencies (see figure 2.21c). Overestimation of P,
the fact that in only six of 28 cases does the value of Pr
is seen in figure 2.22c as evidenced by
taken from the TCA 3 estimate exceed that taken
from the TCA estimate. The average Pm, overestimation is 2.8 ± 3.5 dB. figure 2.22d shows the TCA
estimate of CF plotted as a function of the TCA 3 CF estimate. The low-high (ircle) estimate slightly overestimates CF whereas the high-low (triangle) underestimates CF. Far the most part the difference is on the
order of 10%. The Qiod derived from the TCA estimate is plotted against the
Q10dB
of the TCA 3 estimate
in figure 2.22e. The TCA estimate consistently has a Qloaw smaller than that of the TCA 3 estimate. Quantitatively, the average Q0lo
1.79±.4.
of the TCA estimates is 1.19±.35
whereas that of the TCA 3 estimates is
This is difference of approximately 30%.
Thus the TCA performs as predicted for alligator lizard cochlear neurons. High and low frequency
slopes are estimated correctly on the average whereas Pi
is overestimated, CF is slightly over or underes-
timated depending upon whether the algorithm uses positive or negative frequency increments respectively,
and Qe0, is underestimated.
-52-
TCA Error Caused by Correction for the Acoustic System Frequency Characteristic
Qualitative description
The spectral properties of the acoustic system may cause artifactual components to be generated in TCA
IRC estimate. Consider the cascade of the acoustic system with iso-response contour X(f),
re 2.23a, and the ear with iso-response contour, H(f).
obtain H(f)
as shown in fig-
Assume that X (f) is known but H(f)
we would measure the response of the cascade logR (f)=logH(f
is not. To
)+logX (f) and then subtract
logX(f ).
For log/F(f) a line of constant slope (figure 2.23b), the overall transfer function and the expected value
of the tuning curve would be as shown in figure 2.23c. However, if we subtract the known iso-response contour of logX (f) from this estimate we obtain the contour of figure 2.23d as our estimate of the threshold
contour of logH(f ). This estimate is different from the real threshold contour of logH (f) which should be a
line of constant slope. The acoustic system, X (f) has caused an artifact in the measurement of H (f ).
We demonstrate this property of the TCA in a straightforward manner using a Monte Carlo simulation
of the Markov model with L,=50
(corresponding to the average slope of g ()= .08/dB for cochlear neurons
and level increments of 1/4 dB). Using the acoustic system and ear iso-response contours of figure 2.24a&b
respectively, we show the cascade characteristic (logX(f)+log/f(f)),
the average IRC cascade estimate
(ow-high) and the average corrected estimate in figure 2.24c&d (25 trials). The corrected estimate obviously
differs from the linear contour of figure 2.24b thus showing that the simulated acoustic system may induce a
prominent artifact in the TCA iso-response contour estimate.
Measurements on cochlear neurons
The TCA 3 was used to estimate the IRC for a cochlear nerve fiber Ihis IRC and the iso-response contour of the acoustic system are shown in figure 2.25. Upon examining the TCA 3 estimate of the neural IRC
we see decrease in sensitivity of the fiber with increasing frequency in the region between 4 kHz and 6 kHz.
This is similar to the iso-response contour used in the simulation (figure 2.24b). The acoustic system isoresponse contour in this frequency range is also similar to the acoustic system iso-response contour of figure
2.24a (between 2.5 and 4 kHz) in that it shows, in sequence, a decrease and increase that are comparable in
-53-
magnitude to the slope of the neural IRC. Such a combination of slopes caused error in the simulated estimate of figure 2.24d. We see a similar type of error in the TCA IRC estimate of a cochlear neuron in figure
2.25. This prominent error is marked with an arrow where we see underestimation of the IRC by approximately 15 dB. This error corresponds to the error seen in the simulation above 3 kHz (figure 2.24d).
Miscellaneous Observations
TCA errors caused by nonmonotonic g (l)
qualitativedescription
If g (I) is not a monotonically increasing function of , then the TCA may not find a unique criterion
level. To illustrate, consider the g (l) shown in figure 2.26. If the TA
chooses a starting level greater than
1., it is unlikely that criterion will be met. Therefore, the level will be raised toward lm.
At this level it is
even less likely that criterion will be met so the level wil be raised once more. It can be seen that the TCA
in this regime will fruitlessly seek the desired criterion level of 12,3 by incrementing until the maximum level is
reachded.
measurementson a cochlcarneuron
Evidence of nonmonotonic g (l) was found in one unit.( 16 ) The comparison of two IRC estimates (one
incremented from low to high frequency (low-high) and the other from high to low (high-low)), revealed
gross differences on the high frequency portions (figure 2.27a). The curves were measured using the TCA 3 .
The large descrepancy in levels chosen by the algorithm in the two cases at high frequency prompted the
measurement of g ()(17) at a point in this frequency range (figure 2.27b). g(l), after reaching a maximum
of .8 at
73 dB SPL began to decrease.
This form of g(l) can account for the deviation in the high-low and low-high estimates at high frequency as follows. The first level used by the algorithm will be the threshold chosen for the previous fre(16) It should be noted that although this pbenomena was found in only one unit, no ehaustive search for uch unlts was
instituted. Thus the behavior of this unit may not be unique mung cochlear nerve fibers in the alligatr lizard
(17) See Prected Valuesof E() andE(I,) UsingMmaemem
of g(l).
-54-
quency. On the high frequency side of the contour since we increment from low to high frequency, this first
level will be substantially below the true threshold since the fiber sensitivity decreases with increasing fre
quency. Thus the algorithm will be operating on the lower level portion of g (1) and will likely choose a level
close to the lower 12,3.
The reverse is true when decrementing from high to low frequency. At high frequency the algorithm
uses the high level portion of g (1) because even were the threshold chosen accurately at one frequency point
the starting level at the next frequency point would be substantially above threshold since the sensitivity of the
fiber increases with decreasing frequency on this portion of the IRC. Thus the level likely to be chosen in
closer to ,,.(18)
When the algorithm reaches the frequency region where g (1) is again monotonic (below 450 Hz in this
case), there may be a 60 dB level difference to traverse between l,
and threshold. We have previously
shown that if the starting level used by the TCA is above threshold then the expected threshold estimate will
lie between the starting level and threshold.
This explains the continuous rather than abrupt transition
between the high and low frequency portions of the high-low estimate.
Thus, g(l) functions which are nonmonotonic can induce large errors in the IRC estimate produced by
the TCA. It should be noted, however, that most sequential estimation algorithms which depend upon iteratively comparing a response measurement to a criterion level are subject to similar problems. Therefore, the
possibility of a nonmonotonic 8 (1) should be considered before application of such an estimation procedure
to the measurement of iso-response contours.
TCA errors caused by neural adaptation
qualitative description
Tacit in the use of a Markov model for TCA behavior is the assumption of a stimulus-historyindependent neural response. It was assumed that a given tone burst had no effect upon the response to subsequent tone bursts. Here we present some evidence that at higher response criterion levels this assumption
may be invalid and that the effects of neural adaptation may cause smal but noticeable TCA threshold
(18) A maximum sound level of 90 dB SPL is imposed by the algithm
-55-
to avid damaging the ear.
measurement error.
As seen in figures 2.21a&b the CA produces a distinctive error pattern when applied to a V-shaped
IRC: overestimation of threshold on one side and underestimation on the other. Consider the side on which
the TCA overestimates threshold. The levels presented to the ear are substantially above threshold and it is
possible that the nerve fiber shows adaptation with repeated suprathreshold tone burst presentations. Thus, a
still higher level would be required to elicit the criterion response than if the fiber were unadapted. On the
side where the TCA underestimates threshold, since the levels presented are substantially below threshold, we
presume that our original assumption of stimulus-history-independence is valid. Thus, the TCA may elevate
threshold by overstimulation.
measurements on cochlear neurons
To test the hypothesis that TCA overstimulation may cause a shift in the sensitivity of a neuron near
the TCA-chosen threshold, we investigated the effects of suprathreshold and subthreshold stimulation on
threshold in cochlear nerve fibers. We used the TCA to estimate a neural IRC. We then chose a frequency
where the TCA was likely to have overestimated threshold (in this case on the high frequency IRC slope).
Starting at a level below the TCA-chosen threshold, tone bursts were presented consecutively with 2 or 3 dB
level increments. When threshold was reached the stimulus sequence was repeated until ten seconds of measurement time was spent at each level. Likewise, starting above threshold, tone bursts were presented consecutively with 2 or 3 dB level decrements until a point 2 or 3dB below threshold was attained. In both cases
the average rate in the on and off windows was obtained at each level. Shown in figure 2.28a are plots of
rate difference A vs. level for subthreshold (ex) and suprathreshold (triangle) stimuli in two units. Threshold was determined using the TCA with 50ms on/off tone bursts and a criterion of 0. There is virtually no
difference between the two level curves for each unit.
For other units the same experiment was performed with one modification; the measurement was made
10 dB above the TCA-chosen threshold. Shown in figure 2.28b are sub-"threshold" and supra-'threshold"
rate vs. level curves for two units. In both cases the two rate vs. level curves are shifted relative each other
by a noticeable amount. The cases shown here showed the largest shifts (~ 6 dB). For 19 units the average
shift was 2.5 dB +1.5.
This result suggests that as aiterion level is raised, the TCA may make errors in
-56-
threshold determination by artificially raising threshold through overstimulation. This effect does not appear
large in the alligator lizard, however.
SUMMARY
(1)
In the steady state the TCA estimates the level,
N,
and Nff
such that g(12)
= Prob(N,,(l)-Nof
(l)>C).
are the number of spikes recorded during the on and off windows of the tone burst
stimulus and C is a constant. In the alligator lizard there is a relationship between the probability of
exceeding criterion, g(1) and the average rate difference AX(l).
Using tone burst with 50 ms on and
off windows with criteria of C = 0, 1 and 2 we have found that A(Iz
) is approximately 20 spikes/sec,
40 spikes/sec and 60 spikeslsec respectively. In general, however, since the TCA holds probability constant, the rate at the level, 12,, is not necessarily as simply related in other preparations. Nonetheless,
if similar spike generation processes are assumed for different preparations, then the TCA indirectly
estimates an iso-rate contour by estimating a uniquely related iso-probability contour.
(2)
The transient response of the TCA causes the inaccurate estimation of an iso-probability contour. The
magnitude of the inamcuracy depends upon the function g(l), the IRC being measured, and the frequency characteristic of the acoustic system which delivers sound to the ear. The primary source of
these errors is due to the TCA choosing the first success level, 1,1, at each frequency thereby not allowing a steady state to be attained. Thus, the threshold estimate, 1,I at any given frequency depends upon
the level of the first stimulus presentation at that frequency.
underestimation of Q (
The inaccuracy of the TCA causes
309%),the tuning sharpness and overestimation of P,.
(= 3 dB). Depend-
ing upon whether positive or negative frequency increments are used, the TCA will either overestimate
or underestimate CF respectively (
10%). TICA IRC estimation error, however, does not (on the
average) seem to affect the accurate estimation of low and high frequency IRC slopes.
-S7-
A CASE HISTORY OF TCA IRC ESTIMATION ERROR
INTRODUCTION
The physiological precursor of the neural action potential is the hair cell receptor potential. Thus, a
comparison of the tuning properties of the receptor potential to those of the neural response could shed light
on the trarnsduction process which converts receptor potential waveforms into action potential trains. To this
end, Holton and Weiss (1983b) compared iso-response contours of the D.C. component of the hair-cell
receptor potential with those of the average spike rate (1 9 ) of cochlear nerve discharges. Both were measured
with TCA. The hair cell and neuron IRCs were similar in many respects. The similarities induded comparable low/high frequency slopes and comparable sensitivities. The principal difference cited was that the hair
cell IRCs were less sharply tuned than the neural IRCs. Several possible sources of this difference were proposed by Holton & Weiss.
We will demonstrate an additional factor that is likely to have contributed to the difference in sharpness. This factor is a result of the IRC parameter estimation error discussed in the section entitled 7CA
Errors in Estimating Neural IRCs where we showed that the TCA underestimates Q, the quality of tuning.
We will show that TCA error alone could account for the reported difference. In order to compare the two
sets of measurmets,
however, we will first examine the conditions under which the algorithm was used in
each case.
The neural iso-AX response contours presented by Holton & Weiss (1983b) were measured using a frequency point density of 100 points/decade and a level resolution of 1/3 dB SPL whereas the iso-D.C.
con-
tours were measured with a density of 10 points/decade and a level resolution of 1/4 dB SPL The level resolution difference is insignificant. The frequency density difference, however, is ten-fold. We therefore
aspect that the accuracy of the 'ICA-estimated iso-D.C. contours is not as high as that of the iso-AX contours since high density in frequency results in greater accuracy. In order to quantitatively assess the performance of the TCA for both the receptor potential and the nerve discharge data we will need to determine
(19) Akhougb thee cmtoum were measured with the TCA and ae thefore,
was shown previousy that hee omtours are iso-ate ontours as well.
-58
tricdty qealrlng, io.proluility
emoun,
it
g (l), the probability of exceeding criterion at level I for both measurements.
RECONSTRUCTION
OF RECEPTORPOTENTIALg (l)
We made direct measurements of g'(l) for 15 cochlear neurons in the alligator lizard. The dope of
g(l) was 0.08/dB SPL +-.02 in three alligator lizards. The g(l) for the receptor potential D.C. component
if we assume that the source of uncertainty
has not been measured directly. However, we can estimate ()
in the measurement is a zero mean additive white Gaussian noise process, w(t),
E (w (t )w (T))=Wuo(t
-T)).
of intensity W (i.e.,
Let us define the average value of the signal plus noise in the on-response win-
dow as s,,, and sof as the response in the off-window. Letting Vo(l) be the receptor potential in response to
a tone burst stimulus at level I we have,
,
- (vo( )+W ())d
.
(2.46a)
A1+A2
..
sff
~
f
f w(t)dt
=
2
(2.46b)
A
where (0,Al) and (Al,l 1 +A 2) are the on-response and off-response measurement intervals respectively.
and sof are derived from non-overlapping intervals and the process w(t) is white and Gaussian,
Since s
s, and
So
Gaussian random variable of variances W/A 1 and W/A 2 respectively.
are statisticaly indepent
Specifically,
p (so~,
1 ) =
1rW
· 2WAI
(2.47a)
P(sf)e=
6w
2WAI
2
(2.47b)
Letting s, -ssff =x we may obtain g(1) as,
8g() = Prob(x>V) = (
W( L +)
f,
2 Wo(1
A
))
d
2
= (2ir)f
-_
,2-dr,
a
where
=
V
-V(l)
and V is the criterion votage used by the TCA.
W( 1 +-)
Al
A2
-59-
(2.48)
The calculation of g(l) requires the specification of W, the noise intensity in the receptor potential
. P cc.
recording system bandwidth, Vo(l), the time constants A1, A2, and criterion voltage V¢. Holton and Weiss
measured the dependence of the D.C. component on sound pressure upon level. Using results given in table
2 of Holton & Weiss (1983a) we represent Vo(l) for low level stimuli as( 2 0)
Vlo."
necarCF1
lo10v
0 ' PICm
Vo(l)=
where I is the sound pressure level in dB (=0
j
(2.49)
is defifiMas the sound level necessary to produce the cri-
terion response voltage. W is obtained from the broadbantRMS
noise level, VRJS, and the doublesided fre-
-T r(
quency bandwidth bw using the relation,
W = V Mw
(2.50)
we may calculate W where VjRS is the brdadband, RMS&dise level and bw is the doublesided bandwidth of
the receptor potential recording system. VS
is
.im..1~ be the "ad hoc" noise floor spedfied by Holton
& Weiss (figure 9 (p221), 1983a) of .3 my and the bandddth,
the measurement system used in the hair-cell a
Weiss (1983a)). Using these patarmters and V, = (.1il,
bw, 8KHz.
1=47 ms and A2=2 ms as with
Methods:Iso-voltage contors (p210) Holton &
.3 my, 1 my), g(l) at CF and away from CF for
i in dB are shown in figure 2.29. In all cases the gQ>yirresponding
to V, =.1 my shows a much more gra-
dual dependence upon stimulus level the for other criterion voltages.
EFFECTS UPON REPORTED RESULS
Faor V, = .1 mv the slopes of the D.C. receptor potential g(l) are .07/dB SMP away, from CF and
.04/dB SPL near CF. The off-CF g(l) slope is similar to the average neural g(1) slope and the lope of g (l)
near CF is much shallower. Thus, were RP and neural IRCs to be estimated by the TCA with identical frequency point spacing and level increment size we would expect similar performance away from CF and much
poorer performance near CF since shallow g(l) implies larger L
and therefore greater eror magitude
(see figures 2.13 and 2.17). As seen previously, error is substantial for the ICA estimate of neural IRCs at
40 points/decade frequency spacing and 1/4dB level increments and this error causes undescmation
(20) The measurements presented in Holton & Weiss (1983a) do not extend to values of VO in the neighlbofamd of .mv due
to meann
m
i
aonideratins. Nonetblem we extrapola these resluts into
s vola
rane fr analyticsmpicity.
of Q.
The density of the D.C. RP contour measurement was 10 points/decade thereby increasing the difference in
threshold level between each point. As shown previously, increasing the slope of the IRC under measure
increases error (figure 2.17). We therefore expect poor fidelity of the TCA iso-D.C. contour estimate to the
IRC and we would expect the poorest fidelity in the neighborhood of CF. The implication is that a sharply
tuned D.C. IRC would be poorly estimated by the TCA algorithm at a V¢ =.lmv criterion level and the estimate would show decreased tuning sharpness, Q.
A simulation of the receptor potential IRC measurement made by Holton & Weiss (1983a) was done
on the archetypical neural IRC defined in TCA Errors in Estimating Neural RCs to quantitatively test whether
the sparse frequency point spacing used could account for the difference in Q between RP and neural IRCs.
We used the g(l) obtained in figure 2.29a&b with 10 point/decade frequency increments and 1/4 dB level
increments. The result is shown in figure 2.30. An obvious decrease in Q is apparent.
average
Qm
of the TCA IRC estimate(
1
) is 1.52 ± .18 versus Q
Quantitatively, the
=2.5 for the archetypical IRC. The
decrease in QOdB seen between neural iso-Ax contours and receptor potential iso-D.C. contours is from 2.34
to 1.2. Thus we have shown that it is possible for TCA estimation error to account for most of the
Qa10,
difference seen between neural IRCs and receptor potential D.C. IRCs in the alligator lizard.
IMPROVING TCA ACCURACY
MODIFICAI7ONOF TA PARAMEERS
In order to minimize inaccuracy in the TCA iso-probability contour estimate we must consider the
source of the inaccuracy. We have seen that as the number of level steps to reach threshold from a given
level increased, the expected error also increased. Conversely, taking larger steps in level, though it decreases
inacraacy, obviously will decrease level resolution and thereby decrease precision.
There are at least two ways to minimize inaccuracy in the TCA iso-probability contour estimate without
sacrifiDig precision. The first consists of a modification of a parameter in the TCA If we assume that any
.
TAh
iso-probability contour is continuous in frequency, then by choosing frequency increments snal enough we
tlur it
(21) The Q was mesured for twenty TCA muated estimtes.
rbt
Evmge
Q was the mean
t4AU- faE
-61-
te
meaemme.
are assured that successive threshold points will lie dose to one another in level. Therefore, once the algorithm finds a good threshold estimate at one frequency the expected values of the subsequent estimates are
assured to be dose to the 123 point. Choosing smaller frequency increments diminishes the difference in
level between adjacent frequency points.
In terms of the model we have proposed for the ICA, this is
equivalent to decreasing the apparent slope of the IRC being measured.
We have previously shown that
TCA inaccuracy decreases with decreasing IRC slope (figure 2.17).
Another approach would be to modify the TCA so that instead of choosing the first success as threshold, the mA success is chosen. In examining figure 2.13 we suspect that
IE((._l)1I8)-23JI > IE(
I)-l2,31
(2.51)
or in words, the expected value of each successive threshold estimate at a given frequency is closer to the true
threshold of 12,3. m could be chosen to achieve any desired accwracy. This suspicion is borne out in simulation. Shown in figures 2.31 and 2.32 are E (,
lo) for m =2,3.
As in figure 2.13 the standard deviation
about each point is shown. As m is increased further, (figure 2.33, m =5) E (,
lo) approaches l,
much
more closely. This property was exploited in the development of the TCA 3 IRC estimator used previously in
this chapter (m =3). We have verified the superior performance expected of the TCA 3 for ten cochlear neurons in the alligator lizard by measuring g and AX at several points on the TCA IRC estimate and on the
TCA 3 IRC estimate.
The results are illustrated in figure 2.34 (see caption for details). The TCA 3 is more
effective than the TCA at holding g and AA constant as a function of frequency. This increase in accuracy,
however, is achieved at the expense of an increase in the required measurement time. Using the test circuit,
we found that the
ICA estimates sixty points of an IRC in an average of approximately 60 seconds. The
TCA 3 requires approximately 150 seconds to estimate sixty points along the same contour.
COMBA7TING TCA ERRORS CAUSED BY CORRECTION FOR THE ACOUSTIC SYTEM FREQUENCY
CHARACTERISTIC
We have seen that the tuning curve algorithm is sensitive to the characteristic of the amustic system
which delivers sound to the ear. Either of the aforementioned techniques used to minimim TCA error
would effect a reduction of the IRC estimation error caused by the acoustic system. A more efficient means
-62-
to combat this effect, however, would be to nullify the effects of the acoustic system characteristic, H(w) by
preceeding it with a filter whose transfer function is H-l(w)=1/H(o)
thereby making the cascade indepen-
dent of frequency. This is easily accomplished under computer control if the acoustic system characteristic,
H(w), is known. Assume that the TCA normally controls Vi,, the magnitude of the sinusoidal input voltage
to H(t).
If at angular frequency w, the TCA premultipliesV, by I/IH(w) to produce Vj=Vj,/nH(o),
then for fixed V, the output magnitude of the acoustic system depends only upon the input, Vi,, and not
upon frequency. Thus, the premultiplication of VL, by 1/]H(w)J effectively removes the acoustic system
from the cascade. Although this technique has yet to be implemented and tested in this laboratory, it is felt
to be the most efficient and effective means to eliminate acoustic system induced artifact in the TCA IRC
estimate and it is certainly simpler than physically constructing an acoustic system which has a frequency
invariant characteristic.
-63-
SUGGESTION FOR FURTHER STUDY
In the case of both the improvements, to the TCA mentioned in P7CA Error Minimization", accuracy
is increased at the expense of increased measurement duration. In the case of increased frequency resolution,
more points must be taken to cover the same frequency range. For m >1, we must wait for m successes to
occur. Thus the question arises; how can we achieve the greatest amount of accuracy in the least amount of
time? One method is to implement an algorithm which uses all available "information" to estimate the set of
points (I J) which describe an iso-response contour.
Information" in this sense includes all a priori
knowledge of neural response properties such as the slope of g (l) and the general shape of neural IRCs as
well as the additional information supplied by the response to a probe toneburst. Synthesis of such an optimal
IRC estimation algorithm, even if the implementation were not technically feasible would provide a bound on
how well any algorithm could estimate an IRC with given accuracy.
-64-
v
C)
0-
a
,
a
m
2w
aD
t
. -S
I
'9
la8
I:
-65-
.
m
-
R§
a'
T--
-
~~~1.
-i
I
\
_~~~~V
I
Ii
-
lFz'
c
c _L
c'--'---
I -
-66-
a
M
c
4)
$XkO
L" CO CZ
U) 3 I
(1)
0
.a
Ad
f
'6
o~~~~~~c
g
b
L
r
C X
9*8C t if t
'L
LX
eh
8
fq 0t~-C 'tx
-4-,
-4-,
u)
-67-
1
1
.4
w
m
ci
w
0
0
100
10000
1000
FREQUENCY CHZ)
Figure 2.4: Frequency response caracteistic
-68
of LTI filter in figure 2.3.
0
Co
L.
g
ok
\
em~~~~~~~~~~~~~~~~~
N
%i
II~~~~
I.-
.5~~
.~~~~~~~~~~t
t=
Q
II2Zr
E
3
_
H
CA
e
L"
o
'~.
I
oC
~
~'c
v
-69-
IC
a
I
*
*
N
I
ii
a
.
a:
n
o(IS
za:
z
a:
NPI
V)
w
:3
U1
W
Lli
Ij
w
LL
1
z
0-4
R
i
j:
I
I
1,
I:~~~~~~~~~~~I-L
1
A
I
"i
C33S/S3MIdS)
>
31UU
m
-J
w
-70-
-J
=
ICA
A
STI
---RESPONSE
_·
I ·I
T
1
I
I
NOFF = 2
NON=S
Figure 2.7: Schematic representation of the neural response to a tone-burst (see text).
-71-
, '-
*
-
-
(-)
-3)
(.+)
a
WA)
Ok tat )
I.
I. .
.
a() = DD i
b() = uo O + 0U D + 0ou
oc~)-
V CuO
+ U OUj
puuA
dOC)= UUUl
b
Figure 2.8: Discrete-time discrete-state Markov models of the tuning curve algorithm at fixed frequency, a) Markov chain modeling TCA behavior at fixed frequency. g(l) is the probability of
exceeding criterion a stimulus level I; b) Ergodic Markov subchain derived from the Markov of chain
of part a) (see text).
-72-
+ p(l)
g(I)
° P (ll success)
2.I
.I
.
a
.S
*
I
*I
e
I
___
a
*
a
I-1
I.
.S
-
I
·
2.3
-
.3
I
I
eI
b
.
9*^
.
**
a
8 , '~-
09
~
.e.
-.~
S
,I~,,
I
w
.
.
.
U
1.I
.
.
w
w
-
.
'. . . :.
.S
·
.I
].W
.1
*
0.e-..
*
P
.5
C
. .
* .9*
e
a
-- A
.5
. . I . e
I .e
! I
I
S
I
.$
I
I
.9
I/L max
Figure 2.9b&c: Demonstation that E(I) and E(I,) lie near 12. Numerical dterminations of p(I)
and p,( Isuccess) for five model g(l) plotted as a funtion of /Lm,. p(l) - (plus), p,(l) - (diamond). Along the abscissae we have put diamonds and plus signs at the points E(l)/L,n
and
E(, )/L
respectively. These quantities were derived fromnthe probability distibutions shown. With
the exception of part e) the points are sufficiently dose so as to be indistinguishable. Perpendicular
lines are drawn from the points E(l )/La,
and E ()IL,.m to (). 12/3is cired.
(L,=20); b) g(l) = g(!)/L.m (L=50); c) g() = [/Lm
-73-
- (/Ln)
a) (l) = /L
2 ]~ (L=50);
+
p
)
SS)
l I succe
o p
.!
.t
I
g(l)
_
4
_ ·
.5S
d
1.0
.2A
]e3'
F
e
0
4
.!
i
.
I
L
.
-- - - - ,
,
I
I.B
W
--
I/L max
Figure 2.9d&e: d) g(l) = -1-[1-(/Lx)
(Ln=20).
2
(Lax=50);
-74-
e) g(l) = I/L,m,
l-14,
0 othrwise
0
m
0
l- (n
4
I-
a)
II
4
0
LJ
I
00
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00
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0
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start
q)
ii__
_ · ___
_ i __
(a)
Q)
frequency
·
(b)
-
)
start
start
- - rg
_
frequency
Figure 2.15a&b: Schematic and simulation of TCA error dependence upon IRC initial starting level for
a frequency invariant IRC. Circles- points chosen by the algorithm as threshold, X- starting level at a
given frequency. a) Initial starting level above threshold; b) Initial starting level below thresabold.
-88-
I
II
I I II
II
II
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(BP)
leAlI
(4
t
B
rt 10
m
.4
stort
a)
(a)
Q)
I
frequency
.4
(b)
Q)
Q)
tart
frequency
Figure 2.16a&b: Schematic and simulation of TCA error dependence upon IRC slope. Cirdes- points
chosen by the algorithm as threshold, X- starting level at a given frequency. a) negative slope; b) positive slope.
_-n-
c0
r&
rn
Go 11
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Figure 2.18: Schematic of TCA estimation error for V-shaped IRCs. Solid line: IRC. Dashed line:
TCA estimate. Xs: starting level at indicated equency; 0: TCA threshold estimate at indicated frequency a) moderately sharp frequency selectivity. b) sharp frequency selctivity.
-96-
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0
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-20-
.
..
100
. ..
.
1000
...
10000
frequency (Hz)
Figure 2.21a: TCA estimate comparison to the TCA estimate in alligator lizard nerve fibers; Upper
panel: TCA (dotted) TCA 3 (solid line), lower panel: difference between TCA and TCA3 estimate;
Positive frequency incremets.
-99-
I
100
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80
03
()
12,
60-
. . . .
. I
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m
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0-
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V1
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100
1000
10000
frequency (Hz)
Figure 2.21b: TCA estimate comparison to the TC
estimate in alligator lizard nerve fibers; Upper
panel: TCA (dotted) TCA 3 (solid line), lower pane: difference between TCA and TCA
estimate;
3
Negative frequency increments.
-100-
I
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1.0
frequency
(re TCA3 CF)
I
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mean error
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--if%
- -I
10.0
0.1
1.0
frequency
(re TCA3 CF)
Figure 2.21c: Difference between TCA and TCA 3 for nine fibers and average diffcrenc as a funcion
of frequency referenced to CF of TCA 3 IRC estimate (see text for explanation); Upper panel: positive
frequency inrements; Lower panel: negative frequency inacreents.
-101-
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levas
level
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Figure 2.23: Schematic of error caused by correction for the acoustic system frequency characteristic.
a) Acoustic system iso-response contour. b) IRC to be measured c) Iso-response contour of IRC and
acoustic system in cascade (solid line). TCA cascade IRC estimate (dashed line). d) TCA cascade IRC
estimate corrected for acoustic system frequency characteristic (dashed line in c) minus contour in a)).
The arrow points to the most prominent error caused by the acoustic oorrection.
-107-
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Figure 2.26: Schematic of a pathological nonmonotonic g (I) (see text).
-110-
80
v)
60
_V
40
20
_
1
· ____
1000
100
frequency (Hz)
Figure 2.27a: Effects of nonmonotonic g(l) on neural IRC estimation. High-low and low-high TCA2
IRC estimates as shown for a ochlear nerve fiber in the alligator lizard. Frequency inacrement: 1/40
decade, level incrcnant: 1/4 dB.
1.0
0.8
X
X
x
0.6
X
g(1)
X
x
0.4
x
x
X
X
X
0.2
x
0.0
50
60
70
80
90
100
level (dB SPL)
Figure 2.27b: Effects of nonmnotonic g(l) on neural IRC estimation; Noamonotonic g()
f =530Hz for unit whose IRC( are shown in part a) of this figure.
-111-
at
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dL
^^
IUU
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a)
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40e<
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100
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120:
0
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suprathreshold
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TCA threshold
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70
I I I I I I
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80
lIl
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90
level (dB SPL)
Figre 228a A vs. level for suprathreshold and subthreshdd stimuli (see text for description).
Threshold (AX(l23) 20 piessc)
is marked by an arrow.
-112-
1111111111111111111111111111111
100 -
A
x
0
80 -
C,
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C,
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80
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90
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level (dB SPL)
!
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n~ I
=
!
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6040
a
suprathreshold
x
subthreshold
I
i
20
80r
80
I90 100
90
100
)dB
110
110
I
120
level (dB SPL)
Figure 2.28b: AX vs. level for prae hold ad subthiresholdstimuli (see text for description). Measurament of figure 2.28a repeated at = 10dB SPL above the AX(12) = 20 spikes/sec theshold level.
-113-
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frequency (Hz)
100
1.0-
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0.5
0.69
0.4
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nn
V.
0.5
1.0
2.0
frequency re CF
. . . . . . . . . ·. . I.
60
0
0
a]
I
20-
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05
0.5
n.
.
L
.
......
1.0
2.0
frequency re CF
Figure 2.34.: Depiction of measuremnts of g and Ai at points on an iso-responsecontour estimate.
The frequencies used were .5CF, .707CF, 1.OC, 1.41CF, 2.OCF where CF is the characteristic frequency of the fiber obtained from the appropriate IRC estimate. Tone burst stimuli with To. = Tol =
50 as, Tr = 2.5 ms were used. In all cases both the TCA and the 'ICA 3 used a criterion of C=0, 1/4
dB level increments, -1/40 decade frequency increments and the starting frequency was 6310 Hz. Tbe
initial starting level was 90 dB SPL; Upper panel: TCA
IRC estimate (solid line). The triangles are
3
the points at which g and AX were measured. middle panel: g measured at points depicted in upper
pane; Lower panel: AX asured at points depicted in pper panel
-126-
I
.
1.U
i.
..
..
..
..
..
..·
.
I
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. ·
.
.·
0.8
0.6
g
0.4
0.2
n n
0.5
I.U
I
I
I
a
I
a
I
I
a
I
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I
I
I
I
2.0
1.0
frequency
4
I
re CF
I
I
,I
I
I
I
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I
0.8
0.6
9
0.4
0.2
nn
%v.u
0.5
I
I
1.0
frequency
I
I-
2.0
re CF
Figure 2.34b: Comparison of TCA and TCA
perfom ane. Meauts
of g at points
iso3
response ontour estimates. The frequencies used were always .5CF, .707CF, 1.0CF, 1.41CF, 2.0CF
where CF is the charaderistic frequency of the fiber obtained from the appropriate IRC estimate.
Tone burst stimuli with T = Tf
= 50 ms, Tr = 2.5 ms were used. In a cases both the TCA and
the TCA used a criterion of C =0, 1/4 dB level increments, -1/40 decade frequency increments and the
starting fiequency was 6310 Hz. The initial starting level was 90 dB SPL; Upper panel: Measurements
of g at points on TCA estimated iso-respose contours for 10 units. The dashed line is the threshold,
8 =2/3. Lower panel: same as upper panel only TCA3 was used.
-127-
I
· ,.,
bU
O0
C)
(D
I
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I
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I
I
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I
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40
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20
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0
i
I
I
I
I
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0.5
I
2.0
1.0
frequency
I
60
I
t
I
I
i
I
re CF
I
i
I
I
I
\Q) 40
n0
"/ 20
I
TCA 3
0
I
0.5
I
I
I
I
I
I
I
I
I
I
I
2.0
1.0
frequency
I
re CF
Figure 2.34c: Comparison of TCA and TCA 3 performance. Measurements of AX at points on isoresponse contour estimates. The frequencies sed were always .5CF, .707CF, 1.CF, 1.41CF, 2.OCF
where CF is the characteristic frequency of the fiber obtained from the ppropriate IRC estimate.
Tone burst stimuli with To= T
= 50
50 ms, Tr= 2.5 ms were used. In all cases both the TCA and
the TCA 3 used a criterion of C =0, 1/4 dB level increments, -1/40 decade frequency increments and the
starting f'equency was 6310 Hz. The initial starting level was 90 dB SPL; Upper panel: measurments of
AX at points on TCA estimated iso-response contours for 10 units. Dashed lines denote the expected
range of threshold rate values. A range rather than a specific value is supplied since the TCA and
TCA 3 are iso-probability contonurestimators and a unique value of threshold for rate may not be specified. We approximate the range of average rate difference threshold by examining the spread of average rate difference about g =2/3 in figure 2.12b and find that 18 spikes/sec < AX < 30 spikes/sec.
Lower panel: same as upper panel only TCA3 was used.
CHAPTER 3: THE DEPENDENCE OF NEURAL SYNCHRONIZATION UPON FREQUENCY
INTRODUCTION
Cochlear nerve fibers respond to acoustic stimuli by a change in their instantaneous firing rate. When
this rate has a waveform whose structure is similar to that of the acoustic stimulus, the response is said to be
synchronized to the stimulus. Synchronization is a basic property of the firing patterns of cochlear neurons in
response to sound. (Kiang et al, 1965; Rose et al, 1967; Johnson, 1974; Littlefield, 1973; Kim & Molnar,
1979; Anderson, 1973). Synchronization, however, is limited. If the rate of variation of the stimulus with
time is large, then the neural response will not contain the fine structure of the stimulus waveform. Specifically, in mammals, the synchronized response is limited to stimulus frequencies below a few kilohertz (Gray,
1966; Rose et al, 1967; Anderson, 1973; Littlefield, 1973; Johnson, 1980). We therefore ask: What coichlear
mechanisms limit neural synchronization?
Studies of hair cell receptor potential, a physiological precursor to the nerve fiber response, have shown
the magnitude of the AC. component of receptor potential decreases with increasing frequency (Holton &
Weiss, 1983a; Russell & Sellick, 1978). his attenuation of the A.C. componenthas been attributed to the
low-pass filtering properties of the hair cell membrane (Weiss, Mulroy & Altmann, 1974; Russell & Sellick,
1978). However, it is not known whether the decrease of synchronization with frequency in the nerve can be
fully accounted for by attenuation of the A.C.
receptor potential. In order to address this issue, we have
studied neural synchronization as a function of frequency for single cochlear nerve fibers in the alligator lizard
using techniques sinilar to those used to measure the hair cell receptor potential A.C. component in this
animal (Holton & Weiss, 1983a).
7he most systematic and quantitative previous study of synchronization was performed by Johnson
(1980). Johnson measured the maximum level of synchronization as a function of frequency in codilear
nerve fibers of the cat. This measurement technique is limited in that: 1) synchronization may not be studied
as a function of stimulus level, 2) synchronization may not be measured at low levels and 3) the measurement
is time consuming since first synchronization must be measured as a function of stimulus intensity in order to
determine the saturated (and therefore maximum) synchronization level. In this study we have developed
-129-
two new methods of studying neural synchronization analogous to the methods used by Holton & Weiss
(1983a) for hair cell studies. The first method allows the study of synchronization variation with frequency
with out the complicating effect of average rate variation with frequency; we measure synchronization at constant average rate. This technique also allows the measurement of synchronization at low levels. The second
method allows the measurement of contours in the (frequency, stimulus level) plane along which synchronization is constant (Littlefield, 1973). These contours are measured rapidly using a new synchronization estimation technique and an automated measurement algorithm. We have found that synchronized responses of low
characteristic frequency (low-C)
(1
) fibers (CF < 900 Hz) and high-CF fibers (CF > 1 kHz) differ
markedly; synchronization decreases with frequency much more rapidly for high-CF fibers than for low-CF
fibers. In addition, we have found that synchronization in high-CF nerve fibers decreases more rapidly with
frequency than does the AC.
component of receptor potential (Holton & Weiss, 1983a) between receptor
potential and the neural response for high-CF fibers. Thus, it appears that the receptor-potential/nervefiber-discharge transduction process behaves as a low pass filter.
(1) The dcaratestic fequency (Co) is deined as the frequncy at which a minimmw stimulus level, Pm wi produce a
threshold response. This definitim applies to any responsevariable. By high-CF and low-CF fibers, howeva, we specficlly
mean fibers whoseaverage rate respcmseis most sensitiveat high and low frequendes,respectively.
-130-
METHODS
In this section we describe the stimulus paradigms and methods used to measure the average rate and
synchronization of cochlear nerve fibers. Detailed descriptions of the dosed acoustic system used to deliver
sound to the ar of the animal, surgical preparation and neural recording in the anesthetized alligator lizard
are described elsewhere (Weiss et al, 1976; Holton, 1980). Unlike Weiss et al (1976) and Holton (1980),
however, the anesthetic used was de-ionized H 2 0 and Urethane (4:1 by weight, respectively). The dosage
was .01cc per gram body weight. Immediately after surgery and periodically throughout an expiment,
the
health of the preparation was assessed by measuring the electrical response (at the round window) to acoustic
clicks. The sound level necessary to produce a just discernable response is called the Visual Detection Level.
The details of this preparation assessment procedure are described elsewhere (Holton & Weiss, 1983a).
Animals with an initial VDL of greater than -55 dB were not used in this study and if the VDL during an
experiment rose above -55 dB, the experiment was terminated. We maintained the body temperature of the
animal between 200 and 21°C.
S7IMULI
The tone-burst stimulus is a sinusoid of amplitude, A, and frequency, f, multiplied by a periodic window function, W(t), of period T,
duty cycle, 3, and unity amplitude as depicted in figure 3.1. The
nonzero portion of W(r) is termed the on-window" and the zero portion the off-window". T, =
Tolf = (1-P)Te.
T, and
The trapezoidal shape of W(t) helps minimize spectral contamination of the stimulus and
thereby allows accurate measurements of frequency selectivity.
he time interval spanned by the rising or fal-
ling edge of W(t) is called the rise time, ,, of the tone-burst. Five parameters are required to specify a
tone-burst stimulus: 1) Frequency f,
,.
2) Amplitude A, 3) Duty cycle, , 4) Period T,, and 5) Rise/fall time
Parameter values of T, = 2.5 ms(2 ) J3= .5 and To = 100 ms are used in this study unless otherwise
noted.
(2) A rise time of 2.5 ms produces little pel
100 Hz in this study.
contamination
eragy above 100 Hz. We wi cly use fequeies
-131-
above
NEURAL RESPONSE COMPONENTS
A spike train will be represented as a regular point process with the instantaneous firing rate function,
A(t). We will be concerned with two properties derived from this rate function; average rate and synironization.
We now provide definitions of these measures. Derivations of estimates for these quantities are
presented in Appendix B.
Average Rate (3 )
In response to a tone-burst input we define average rate difference as the difference between the rates
in the on and the off windows. Specifically, let A0 be the average rate during the on-window and let Xf
be
the average rate during the off window. Then we define the average rate difference to be
Ak = A 0- Xo
(3.1)
.
An estimate of average rate, i, is
I, I
._
(3.2)
where N is the number of events recorded in an observation interval, 1.
Synchronization
Synchronization is defined in the folowing manner.
We assume the instantaneous rate,
(t), in
response to a tone is periodic with period T equal to the tone period. ThIus X(t) can be expanded in a
Fourier series (en,
1970, Liu & Liu, 1975),
A(t) =
2
Ae
.
(3.3)
where k is an integer. We define synchronization, S, as the ratio of the complex amplitude, AI( 4 ) of the
fundamental component of x(t) in the tone-burst ant-window to the average on-window rate, A,
(3) Avee
rate =
f(t)dar
for arblitrry t md obeatin
interval,.
(4) In this dvelpnat
we treat the esimation of the compex quantity A&. This is impler dan e
arg (Ai) a has been done previously (Johnm, 1974; Littlefield, 1973; Kim & Moiar, 1979).
-132-
_sitig A I ad
S = A
(3.4)
The magnitude of this measure is identical to that used by Johnson (1974). It should be noted, however,
that we estimate synchronization during the tone-burst on-window rather than for a continuous tone as has
been done in previous studies (Kiang et al, 1965; Rose et al, 1967; Anderson, 1973; Littlefield, 1973; Johnson, 1974; Kim & Molnar, 1979). We discuss this difference in Appendix D. Briefly, we do not feel this
difference should pose a major obstade in the comparison of our results with previous work.
If we assume a bandlimited rate function, X(t), as described by
h,
X(,)
where K
=
(3.5)
.IT is the highest frequency component of X(t), then we may estimate Al by quantizing the tone-
burst on-window into M consecutive time intervals of duration ?.( 5 ) For each of Q tone-bursts presented the
number of neural spikes which occur in time interval (bin) m, n,,
occuring in bin m over all tone-burst presentations, i.e.,
, =
q1
is recorded. N. is the sum of spikes
(n,,)q. Thus, the set N. comprises a PST
histogram (Johnson, 1974) of spike occurrences within the tone-burst on-window. Letting MQ T=I, the total
observation interval, we obtain
M-1
A
1
-iIf
-j2wm
N. e
e
(3.6)
0
Isinc- T
A measure of synchronization as defined in equation 3.3 is,
s =A/(N
+ 1)
(3.7)
where N is the total number of spikes occurring in the on-window observation interval (N = C N,).
(5) In this study
was set to
T/10. We assume that the mte _fulin. o(),
harmonic(.=5).
-133-
ontins no harmonics ighMr thn the 5th
Measurement of Iso-fundamental Contours
To measure iso-fundamental contours, an automated algorithm similar to the tuning urve algorithm
(see chapter 2) is used. P=
-
IA 12, the power in the fundamental component of the rate function, is used
as the response estimate since it is unbiased for small observation intervals (please see Appendix B). If we
that assume
X(t)has no frequency components above the second harmonic (i.e., (t) well described by eqn.
3.5 with K.2),
then the estimate of P is
=
(
[(No-Ne + (N-N 3)- N]
(3.8)
The N are obtained by subdividing the tone-burst on-window into intervals of length T/4 as indicated in fig
ure 3.2. Thus, each tone cycle is divided into four equal intervals which we number 0, 1, 2, and 3. We
record the number of nerve impulses which occur in each interval over the tone-burst and the sum of the
spikes from intervals numbered i is Ni.
N =No+N,+N2+N3
(6) Often methan
one
N is the total number of spikes which occur in the on-window,
(6 )
nburst
is usd in older to obtain a more acrte
estimate of
. in mch a ae the mpones
to the Q tone-bursts,(N,),, are added togetherto fom Ni', and the obevation interval,T,, becomesQ,,.
-134-
RESULTS
The response of a fiber to a low and high frequency tone burst is shown in figure 3.3. Both responses
have similar average rates (AA=94 spikes/sec at 500 Hz, AX=88 spikes/sec at 1000 Hz). Howeve,
the
response to the low frequency tone-burst shows a large component synchronized to the individual cycles of
the tone, whereas no such synchronized component is apparent in the response to the higher frequency toneburst. To examine this synchronization decrease in cochlear nerve fibers, two types of measurements were
taken: measurement of iso-fundamental contours and measurement of synchronization at constant average
rate difference.
ISO-FUNDAMENTAL CONTOURS: Dependence Upon Fiber CF
Iso-funmdamentalcontours for a given fiber were produced by estimating the power in the fundamental
component of the firing rate,
A, 12, during one or two tone-bursts and using an iso-response contour (IRC)
estimation algorithm to determine the sound pressure level which produced a criterion response. Thus these
contours are iso-A 1 12contours.7) Iso-fundamental (iso- A2) and iso-rate (isoAX) contours of representative low and high-CF units are shown in figure 3.4. For the low-CF units (fig3.4b), the shapes of the isofundamental and iso-rate contours differ only slightly whereas there are differences between the iso-rate and
iso-fundamental contours for the high-CF units. For high-CF units (fig 3.4a), the synchronized component
falls off rapidly at a frequency much lower than the iso-rate CF. Thus, the CF det
ined from the iso-
fundamental contour is much lower than that determined from the iso-rate contour. For low-CF units the
iso-rate and iso-fundamental contour CFs are similar. These comparisons are smmarized in figures 3.5a&b
where the iso-fundamental CF is plotted against the iso-rate CF for both low and high-CF units, respectively.
As can be seen, the iso-fundamental CF for high-CF units is different from the iso-rate CF. In addition, the
value of the iso-fundamental CF does not seem to depend upon the iso-rate CF (corr: 0.07, slope: 0.02)
whereas the iso-rate and iso-fundamental CFs differ only slightly for the low frequency units (corr: 0.93;
(7) A tuig crve algorithm(deumbed in cpter 2 as TCA3 ) was ed to estime io. lA I rtorin.
Tirefore, to ioIA 12 contour i actually an iotpmrobaility ,i,
ur with the probbiity of IA,12C equal to 2/3 where C
the atedon.
We ame
that the probability 2/3 level arsp
to a uique value d IA, 12 o al values of
(see appedix B for
autha detail).
-135-
slope: 0.9). The average CF of high-CF iso-fundamental contours was 346 ±90Hz (27 units). Thus, for
high-CF units, the synchronized response sensitivity begins to decrease in the vicinity of 350 Hz and this frequency is independent of the fiber iso-rate CF.
As a final note, for high -CF units the average low and high frequency slopes( 8) were -33.6 -10.8 dB
SPIJdecade (min: 49.2, max: -18.6) and 74.0
23.2 dB SPUdecade (min: 40.0, max: 114.0) respectively.
This result will be used later when we consider previous measurements of receptor potential iso-A.C. component contours.
SYNCHRONZATlON AT CONSTANT AVERAGE RATE DIFFERENCE:
DependenceUpon CFfor Low and High-CFFibers
The fundamental component of the instantaneous rate of firing, A1, was measured at constant average
rate, (9 ) AK, for low and high-CF units. The synchronized response,
nitude taken. The resulting quantity,
, (eqn.3.5) was estimated and its mag-
A1 /A0, is the synchronization index. In figure 3.6a we plot syn-
chronization index at constant rate versus frequency for representative low and high-CF units. Immediately
apparent is the rapid decrease of synchronization for the high-CF unit at frequencies above 250 Hz vs. the
more gradual decrease of synchronization with frequency of the low-CF unit. Since these data were taken in
the same animal, this difference is not attributable to response variation between preparations.
To determine whether the decrease of synchronization is dependent on frequency selectivity, we measured the frequency at which the synchronized response is 3 dB below the average maximum synchrony,
.
3
,(10) and compared f.3
to the iso-rate CF. Figure 3.6b shows f
3
vsus
the iso-rate CF for 21
(8) To estimate the high. and low-frequency slopes, a linear regression was performed on 10 points above and below CF
respectively. Ts mll number of points was used far the following reaon
Iso-fundamental contours wre measured at
high timulus levels (average Pin=73.7
6.8 dB SPL). To avoid damage to the ear, a maximum stimulus level of 90 dB
SPL was impued. Only 15 of the 27 contours from which CF could be estimated bad the requisite number of points below
90 dB SPL to either side of (F. The large scatter observed in these meurm
ts may have been prtially due to the man
number of points used to estimate the slopes.
(9) ao-X contomurwere measured using the aurate
CA3 algorithm presented in Part111 of chapter 2.
dhw that this algorithm does indeed hold A cunstant for cear
nerve fiben in the lligator lizard.
in appadix B we
(10) Thef ,a point ws estimated by dwing straight l
segments through the data (as in figure 3.6e) and noting the equency t hich the synchronized respome was 3 dB below the average low frequency maximum. The average low frequeney
maximum was computed by aveingg the value of the sadunirtixim
index for dt below 300 Hz. Thus, f
wa aily
estimated for hig;C
units for which at least two synchronization data points below 300 Hz existed. This ad oe method for
determining a low frequency reference syncronization level was based on the qualitative observtion that the ydmraimtion
index approaches a low frequency plateau below 300 Hz (see figs
3.6d&e).
-136-
high-CF fibers. The mean f
is 369 + 37 as compared to the mean iso-rate CF of 1674 ± 268. A slight
negative correlation is seen between f3,
slope is only -.1 indicating that f3,
f3s
and the iso-rate CF (correlation= -0.18), however, the regression
and the iso-rate CF are approximately independent. Figure 3.6c shows
versus the iso-rate CF for 16 low-CF units. The mean f 3a is 498 + 38 Hz as compared to the mean
iso-rate CF of 285 - 108 Hz. The correlation between the iso-rate CF and f3ad is -0.32 but, as for the
high-CF units, the regression slope is small (-0.08). These results indicate that for high and low-CF units the
decrease of synchronization with frequency is not dependent upon the fiber CF.
The pooled data from 39 high-CF units (7 animals) and 19 low-CF units (4 animals) are shown in figure 3.6d&e Synchronization decreases steeply above 350 Hz for high-CF units whereas it decreases much less
steeply for low-CF units. A regression line, computed from the synchronization vs. frequency data above 350
Hz for high-CF units, had a slope of -41.8 ± 2.1 dB/decade. A similar line for the low-CF data (f >300
Hz) had a slope of -13.0 - 1.2 dB/decade.
SYNCHRONIZATIONAT CONSTANT AVERAGE RATE DIFFERENCE:
Dependence Upon AX for High-CF Fibers
The measurements of synchronization at constant rate presented thus far were all made with AX
20
spikes/sec. For two units we were able to measure synchronization at constant rate for AAX 20 and 40
spikes/sec. Figure 3.7a shows synchronization as a function of frequency for these two units at Ai
20
spikes/sec and at at AK= 40 sp/sec. The general dependence of synchronization upon frequency is siilar
for
these units. Figure 3.7b shows pooled data from 9 high-CF units at the AX= 40 spikes/sec. The regression
line fit to these data points above 350 Hz had a slope of -60.4 +4.4 dBdecade which is steeper than the
slope of 41.8 dBldecade obtained for A=20
depended upon the choice of frequency, f,
spikes/sec. To determine whether this apparent difference
above which the regression line was computed, we varied f
between 350 Hz (the qualitative "knee" for both sets of data in figures 3.6b and 3.7b) and 500 Hz (the frequency above which data for AX=40 spikestsec are scarce; <20 points) and found that the regression slope of
the data in figure 3.6b did not vary by more than 3 dl/decade and the regression slope of the data in figure
3.7b decreased monotonically as a function of frequency to -81.2
8.8 dBldecade at 500 Hz. Thus it
appears that, on the average, the slope of synchronization versus frequency at constant average rate for
AX=40 spikey/sec is significantly (at least 14 dB/decade) steeper than the average slope of the lower criterion
-137-
measurement at AXh 20 spikes/sec. This result, however, differs from the finding that in single units (figure
3.7a) the synchronization vs. frequency slope does not vary greatly between the two average-rate criterion levels. To resolve this issue, further experiments must be performed in which syndlronization variation with
frequency is measured as a function of criterion level in single units.
-138-
DISCUSSION
NOVEL SYNCHRONIZATION MEASUREMENT SCHEMES
We have developed two new methods of studying synchronization in single auditory nerve fibers: the
measurement of synchrony at constant rate and the rapid measurement of iso-synchronization contours.
Using these methods we have uncovered a phenomenon which appears independent of the average rate CF: a
rapid decrease in the synchronized component above 350Hz. Both methods can be used to study the dependence of synchronization upon response level by adjusting the rate criterion for synchrony at constant rate,
and by adjusting the criterion used by the iso-fundamental algorithm. Previous methods (Anderson, 1973;
Johnson, 1980) did not permit such a study. Furthermore, the rapidity with which these measurements can
be made permits the detailed study of synchronization in individual neurons. Previous methods due to their
time-consumin nature (Johnson, 1980) did not permit as detailed study of individual units.
RELATION OF NEURAL SYNCHRONIZATION TO RECEPTOR POTENTIAL A.C. COMPONENT:
Free-Standing Region
According to previous studies (Weiss et al, 1976) high and low-CF fibers project to two morphologically distinct types of hair cell. Low-CF fibers project to "tectorial" hair cells (in a region of the papilla
covered by the tectorial membrane) and high-CF fibers project to "free-standing" hair cells (in a region
without a tectorial membrane) Thus we can compare the synchrony of high-CF fibers with the A.C. receptor
potential of free-standing hair cells studied by Holton & Weiss (1983a). Figure 3.8a shows hair cell and
neural responses to different frequency tone-bursts for a hair cell and nerve fiber of similar CF (2.0 kHz and
1.6 kHz respectively). The A.C. component of the receptor potential declines gradually with frequency. At
1 kHz a robust A.C. response is seen. At 2 kHz the magnitude of the A.C. component has declined by a
factor of two. The decline in the magnitude of the synchronized component for the neural response is much
more precipitous. At 500 Hz a robust neural synchronized reponse component is present, but it becomes
undetectable at 1 kHz and above.
In figure 3.8b iso-response contours of tectorial hair cell and high-CF cochlear neuroms are compared.
Comparisons between iso-D.C. and iso-average-rate-difference have been described previously (Holton &
Weiss, 1983b) and indicate that these contours are similar. The iso-A.C.
and iso-fundamental contours of
figure 3.8b, however, are markedly different. The neural synchronized response declines rapidly the range of
-139-
frequencies (350 Hz <f < 1kHz) for which A.C. receptor potential becomes more sensitive. For 15 highCF units the average iso-fundamental high frequency slope was 74 - 23 dB/decade whereas for 28 freestanding hair cells the average iso-A.C. contour high frequency slope was -33.6
+
2.1 dBldecade. This indi-
cates a large reduction in synchronized component between receptor potential and neural impulse train.
Measurements of AC. component of receptor potential at constant D.C.
receptor potential for hair
cells in the free-standing region revealed that the A.C. component decreases with frequency at a rate of
approximately 20 dBldecade (Holton & Weiss, 1983a). We compare these data with our measurements of
syruonization at constant rate in figure 3.8c. Consistent with examination of tone burst responses and isofundamental contours, the neural response declines much more rapidly with frequency than the receptor
potential A.C. component. To make a more quantitative comparison we again note that free-standing hair
cell iso-D.C. contours and high-CF neural iso-rate contours are very similar (Holton & Weiss, 1983b). Let
us assume that these iso-rate and iso-D.C. contours are identical.
Specifical
let us assume that
receptorpotential D.C. is constant along a neural iso-rate contour. Thus during measremnt
of syn-
chronization at constant rate, the D.C. receptor potential is also constant as well. Hence, we can compare
the measurements of A.C. at constant D.C. and synchronization at constant rate directly. Figure 3.8c indicates that neural synchronization declines at a rate of 42 dR/decade above 350 Hz as compared to the dedine
in A.C. receptor potential of 20 dB/decade (Holton & Weiss, 1983a). Under our assumption that neural
average rate difference and receptor potential are equivalent for free-standing hair cells, these results suggest
the existence of a low-pass process relating the receptor potential to the neural response which causes further
synchronized response attenuation at a rate of about 20 dB/decade.
COMPARISON OF SYNCHRONIZATION FOR FREE-STANDING AND TECTORIAL FIBERS
We have shown that the decrease of synchronization with frequency (at constant rate) is much steeper
for free-standing fibers than for tectorial fibers. Since no receptor potential data exist for tectorial hair cells
we propose two hypothetical explanations. First let us we assume that the transduction of receptor potential
into neural impulses is similar for the two types of hair cells. In that case a possible explanation for this
difference in synchronized response is a difference in receptor potential behavior in the free-standing and tectorial hair cells. Receptor potential may increase with increasing frequency in tectorial hair cells thereby
-140-
countering the low-pass process relating receptor potential to the neural response. Alternatively, the receptor
potential to neural transduction process may be much different in free-standing and tecorial hair cells. Previous work has shown that there are many differences in structure between these two dasses of hair cell (Mulroy, 1974) and recently, morphological differences between the synapses of tectorial and free-standing hair
cells have been discovered. Specifically, tectorial hair-cells are more richly endowed with afferent synaptic
bodies than free-standing hair-cells (Mulroy, (personal communication)). This morphological evidence suggests the possibility that synaptic transmission at the receptor cell/neuron junction differ for these two populations.
COMPARISON TO THE CAT
As with any comparison, differences in preparations and recording techniques must be explicitly stated.
These are:
(1)
The temperature of the alligator lizard was kept between 200 and 210 C. The body temperature of the
cat is between 370 and 380 C. Such a difference in temperature, even were the sound transduction
mechanisms identical in both animals, would probably cause differences in the response to sound of the
two systems; synaptic transduction consists of a series of electrochemical events (Aidley, 1978) and most
chemical reaction rates depend upon temperature.
(2)
In previous studies (Rose, 1967; Littlefield, 1973; Anderson, 1973; Johnson, 1974; Kim & Molnar,
1979) a continuous tone stimulus used. In this study we have used the tone-burst. The effects of this
difference are considered minimal, however (please see appendix D).
(3)
We have measured the frequency dependence of synchronization at much lower levels than either Johnson (1980) or Anderson (1973) and the synchronized response at high and low levels may differ due to
neural adaptation or fatigue.
Johnson (1980) reported a decrease in the maximum synchronization index for frequencies above 1kHz
in the cat. As in our study of synchronization in the alligator lizard, this phenomenon appeared independent
of the fiber's iso-rate CF. In figure 3.9 we show synchronization data for free-standing and tedorial nerve
fibers in the alligator lizard along with Johnson's data (1980) for nerve fibers in the cat replotted on a log-log
-141-
scale. The magnitude and frequency dependence for the tectorial fiber synchronization and the cat cochlear
neurons do not differ appreciably for frequencies below 300 Hz but the tectorial fiber synchronization begins
to decrease above 350 Hz.
he magnitude of the free-standing fiber synroniztd
response is at all times less
than that of the cat fibers and also begins to decrease near 350 Hz. These two set of data, however, bear a
stiking resemblance; both exhibit a rapid decline of synchronized component. The slopes of these declines
are virtually identical: -41.8
2.1 dBldecade in the alligator lizard (above 350 Hz) and 43.6
2.8
dBdecade in the cat (above 2.5 kHz).
To attempt more than a simple phenomenological comparison we would need to know the behavior of
receptor potential in the cat. Unfortunately, receptor potential data do not exist for this animal. However,
there are some hair cell data pertinent to synchronization loss in the guinea pig. Russell and Sellick (1978)
found that A.C.
component of receptor potential in guinea pig inner hair cells declines relative the D.C.
component at a rate of approximately 20 d(/decade.
This is similar to the findings for receptor potential in
the lizard. Based on this similarity, let us then assume that A.C. receptor potential behavior in mammals is
similar to A.C. receptor behavior in the alligator lizard. This would imply the maximum A.C.
receptor
potential decreases at a rate of 20 dB/decade in mammnal as well. Let us now assume that in Johnson's
measurements of maximum synchronization index as a function of frequency that the A.C. receptor potential
was saturated as well.
In conjunction with Johnson's data which indicate that neural synchronization
decreases at approximately 40 dBrdecade, our assumptions imply that the receptor potential/neural transduction process in the cat produces an additional 20 dB/decade attenuation of synchronized component. This is
the same result as obtained for alligator lizard free-standing nerve fibers.
Thus the primary difference between the synchronized response of alligator lizard free-standing neurons
and cochlear nerve fibers in the cat may be the frequency at which synchronization begins to decline. It will
be of interest to study the dependence of synchronization attenuation in the alligator lizard as a function of
tamatn ture. Specifically, the investigation of whether syn
Monization decrease
with increasing temperature would be the next logical course of study.
-142-
begins at higher frequencies
AI
I
T
I
r
rl
I
m-
(1-P8)TTB
PTTB
Figure 3.1: The tone burst stimulus (see text for details).
stimulus
02
intervals
spikes
1
llllllllllllll11111
1111Iif]II ll
13
I
I
20
I
I
3
12
I
IL
Figure 3.2: Tone hburststimulus divided into M equal time intervals. We divide the toneburst onwindow into M time intervals of duration T/4 where T is the period of the tone. Spikes occuring during time interval i are snmed to obtain N i . In this schematic, N 0 =, N 1=], N 2=2, and N 3 =1.
-143-
U)
E
a,
E
-. _
o
0
o
0
o
0
00
.i1i
N
I
/
I
O
o
j
8D
(oas/sa>!ds) aGDJ
.9i in
4
.
iiu
E
N
U.e
E
0
I e5 B
°~
S
o0
(o
o0
o
0
0
B
jq~iI
X e1
0
C
(oas/sa8!ds)
ii%-I
91DJ
-144-
I v1 '4
irt
I
I
I
-J
0Ln
cn
-I
80 -
I
I
r
I
'
II
I
0
a:
40
-
w
a
w
:D
20
ISO-FUNDRMENTAL
a:
ISO-RATE
LI
344.341
1
0
_ I
I
I I
TI
I
I
I
I
I
I
10000
FREQUENCY
-
IIJ M
1
1000
100
CHZ]
I
!
Cf
I
I
\I,
m
0
60 Li
a
40 -
- - - - ISO-FUNDAMENTAL
(n
w
a 20 U)
a:
ISO-RATE
-
0~
4n. 4m
0 !-
100
_-I
I
I
.I
I)
I
I
FREQUENCY
I I i
l li
10000
1000
CHZI
Figure 3.4a: Iso-rate and Iso-synchrony contours for two high-CF fibers. The response criterion was
approximately AI1 = 30 spikes/sec.
-145-
100
II
-J
0cn
0m
:I
II
60
-
40
w
a:
Ln
RL
20
w
-__
O.
I SO-RRTE
)
0
I
!
___
I
I'
II Ij
'~~4 lr2tlf
I
I vilaK
1000
100
(HZ)
FREQUENCY
100
80
-
0
m
'
I-
a:
40
ar
LU
IENTRL
20
a:
0n
J3
,.
0II
14M2
...
100
I
T
1 I
I
10000
1000
FREQUENCY
1
(HZ)
Figure 3.4b: Iso-rate and Iso-synchrony contours for two low-CF fibers. The response criterion was
approximately A1i= 30 spikes/sec.
-146-
N
I
0.6 0.5
X
x
X
XX
XX
xx
0.4
LL
I
x
X
X
0.3
C)
X
x
X
(a)
x
0.2
I
0
*-_
0.1 -
I
1.0
w
w
r
I
2.0
3.0
iso-rate
wl
-
4.0 5.0 6.0
CF (KHz)
I
0.6 -
I
N
x
0.5
x
0.4
X
0.3
XX
C)
x
(b)
XX
0.2
x
ni - Ir
0.1
0.2
iso-rate
0.3
0.4 0.50.6
CF (KHz)
Figure 3.5: Scatter pd of iso-fundamental contur CF versus iso-rate cmntour CF for low and highCF
units in 12 alligatr
lizrds; a) 27 high-CF units; b) 19 low-CF units.
-147-
I
I
100
.!.
i
80O
c,
'0 60), 40
20
_
________
100
_
_
_ _
1000
I
10000
Frequency (Hz)
x
4
^
.
!
.
.
.
.....
I
I
I.U
4-
0.1
N
C-
A
L
high CF unit
-
X -
>n
cn
v.v
1nn
I
I
100
low CF unit
.
. ..
1000
10000
frequency (Hz)
Figure 3.6.: Synchronization at comtant rate (A =20 sp/sec) for low- and high-CF units; Upper
paneld:
bw- and high-CF iso-rate cmontours
for two fibers in a single alligator lizard. Synchronization was measured at marked (frequency,level) points along each contour, Lower panel: Syndronization
masuremnts at constant rate for the two fibers (Xs, low-CF; triangles, hiSh-Q);
-148-
I
I- r^,
-
.
500
la
A
,A
A
400
A
f3dB
a
aA'k
Aa
A
300
fln
_
_
A
-
1 000
iso-rate
Cnn
(b)
41~
A
I
3000
2000
CF (Hz)
.
.
I
x
x
xx
xX
,
X
x
x
400
f3dB
(C)
Ann
I,U
400
200
iso-rate
600
CF (Hz)
Figure 3.6b&c: Synironization at constant rate (AX=20 sp/sec) for low- and high-C units; b) scatter
plot of f 3
(see text) versus iso-rate CF for high-CF units (21 units in 6 alligator lizards). c) scatter
plot of f 3 (see text) versus iso-rate CF for low-CF units (16 units in 4 alligator lizard).
-149-
v
n
I.U
IL
-'1
Q)
I
[
....
-n
I
I
C:
C
00
N
(d)
0.1
A
c-
o'
O
A tiLA
A
cn
A
la
0.01
A
A
100
W
w
I
w
1000
10000
frequency (Hz)
x
()
1
I.U n
.
'
*
Z
C:
|
*
* -
.
x
x
0
0N
*
(e)
0.1
LO
c)
0.01
1
100
·
·-
1-·--
_
1000
10000
frequency (Hz)
Figure 3.6d&e: Synchronization at constant rate (AX 20 sp/sec) for low- and high-CF units; d) Pooled
data for synchronization versus frequency at constant rate for high-CF units. The straight line is a
linear regression for points above 350 Hz; slope= -41.8 -2.1 dB/decade, (39 units in 7 alligator
lizards); e) Pooled data for synchronization versus frequency at constant rate in low-CF units. The
straight line is a linear regression for points above 250 Hz; slope= -13.0 + 1.2 dB/decade (19 units in 4
alligator lizards);
-150-
x
1.0 -
0o
0.5
l
I
I
I
·
·
I
'1
·
Q)
o
0
N
0
A-
c-'
>,,
X-A=20
0.1
spikes/sec
_
_
__
_~~~~~~~~~~~~~~~~~~~~~~~~~~~
1000
200
frequency (Hz)
x
-o
n
4
I.U
I
.
I
0.5
0
C
N
*0
Co
l
nu
ri
I
I
200
1000
frequency (Hz)
Flgure 3.7.: Synchronization versus freqeincy for different constant AA criterion: high-F units; Comparison of synchronization at constant AX for two units. AX=20 spikes/sec (Xs), AX.40 spikes/sec
(trianIs).
II-/
~
'~
I.U
l
~
f"Nk
.
.
.
^
I
I
^
.
I
I
()
F
U
N
C
c
cc-
nf
w
,(
0. x
r),
I
U.
I
I
100
.
..
1000
I
3000
frequency (Hz)
Figure 3.7b: Synchronization versus frequency for different constant AX criterion: high-CF units; Syndchronization at AX=40 spikes/sec linear regression line for data above 350 Hz; slope= -60.4 + 4.4
dB/decade, (9 units in 4 alligator lizards).
1 re
jfL
CU
o
CJ
c'J
I
-O
;u
0 9c
I
dL
LLi
.L
In
0_
I
I
It.
i o 1
X1X'E
q 8i
-;
CJ
In
Oc
I
0
0
I1
rI
0
0
0to
LJ > O
>rJI
0a~)
0
co
a~"
a
I
I
II
I
I
t
11111
I
11111t
100-
w
00
S
80-
0
0
S
\\ 0%
60cr n
U3
U)
.
0
-
-
T 40-
w
0W
20-
0.1 mV
*,VO=
_
I
rI
I
VI =0.1 mV
---
22-5-1.03
0-
_ _ ______
_I__
10.0
1.0
0.1
I
I 11
I I
I
11111
I
(kHz)
FREQUENCY
100 -
0
0
80 o
0
00
0
a
60 -
Io
a
0
%0
.-
0o
tc
rr U) 40-
0
soee
031
Lo
0
a
I SO-RRTE
0
0
o
o
I SO-FUNDRMENTRL
j
4 1-23
*
100
I
..
..
. . I,,.
I I I 0I 0 0
1000
I
II
.
i .i .
I
101300
FREQUENCY (HZ]
Figure 3.8b: Comparison, for the free-standing region, of hair cell receptor potential and neural instantaneous rate. Upper panel: iso-D.C. (filled cirdes) and iso-A.C. (solid line) contours for hair cell
receptor potential. Lower panel: iso-rate (open cirdes) and iso-fundamental (solid line) contours for
single cocdlear neuron.
-154-
I
.
p
I a I,
fih
I
I
,,,
CU
w
m
1 0-
F
C
0
a
P
U
C
S
S
0
V3w
0-1-
E
0
z
0.01-
L
01
I I·1111 i[Iw
I
I
p
l[
,I
UTTTUTT
1
. ,,,1
1.0
100
Frequency (kHz)
x
1.n
'I.-
L
.....
-
*
E E
.
_., e !
J
Q)
_C
N
0.1
A
0
A
L_
I
O
a
>,
o3
A
0.01
a
-
100
.
.
1000
10000
frequency (Hz)
Figure 3.8c: Comparison, for the free-standing region, of hair cell receptor potential and neural instantaneous rate. Upper panel: Frequency dependence of A.C. receptor potential at cnstant D.C. (iso-Vo
frequency response) compared to frequency dependence of synchronization at constant rate (lower
panel). Neural data: 39 units in 7 alligator lizards, Receptor potential data reproduced from Hdton &
Weiss (1983a): five hair cells.
-155-
X
-1 r %
I.U
--
1. A , , I
1I
I
11
1
CAT
-_0
/
C
o
0
N
o
C
0
U
0.1
o
LIZARD
high-cf
,')
*C
0.05
.
A
d
w~~~
100
-
w
ww
a
o
a
..
..
1000
10000
frequency (Hz)
Figure 3.9: Comparison of the dependence of neural synaonizati
upon frequency for cochlar
nerve fibers in the alligator lizard and the cat. Triangles: high-CF units in the alligator lizard; Xs: lowCF units in the alligator lizard; boxes: cochlear neurns in tlhe cat. Tbe cat data are taken from Johnson (1980) and replotted on a log-log scale.
-156-
APPENDIX A: Existence of E (1)
Equation 2.20 facilitates the existence proof for E (I). Suppose g (l) is of the following dass.
g ()
(A.1)
g ( -1) forall I
there exists L 1 , L 2 s.t.
g (L2-2) >2
g(L 1 ) -<,
Since g (l) is an increasing function we may rewrite 2.20 as
(1-g ) )
,(l) _(p(l-1)+p(l-2))
For l cL
1
(A.2)
where g (L 1)-<2,
P(1)>-p(-1) + p(l-2).
(A.3)
p()
(A4)
Since p (l)>O for all l,
p( -1)
I sL
Therefore from equation A. 3,
p(l) > 2p(-2)
which implies using the geometric series
a
a with a =112,
-a
=
I sLI
p(l-2i)
p(l)
(A.5)
l<L1
i=l
(A.6)
We now determine a lower bound for E ().
E(l)
L' I
L1
I lp(l)_
I=-x
lp(l)
(A.7)
for
LI
L'
and
1
(A.8)
L'-c0
The right side of equation A. 7 must be less than zero since all the p (l) are positive and all I are negative.
-
Using eqn. A. 5 we have
t 'I
L lp(l)-
(2)p(L
=O2
2(L' 1-2)p(L'
)(L1-22)+
1)
2
+ 2(L' 1 -3)p(L'1
p(L'-1)(L'-2
-
1)
(A.9)
1)
Therefore, by eqns. A. 7 and A. 8,
LI
Y lp(l) - 2(L '1-2)p(L '1) + 2(L'1-3)p(L '1-1 )
l.-S
-157-
(A.10)
Thus E(I) has a lower bound. To find an upper bound for E()
we use equation 2.20 and the mono-
tone property of g () again to obtain
p(l)
1 (l-2) (-1)+p(I-
2 ))
(A.11)
K O
(A.12)
First consider the sequence
x(I+2) = K(x ( +l)+x ())
If we let x(l)=a
and x(1 +1)=b,
then the ratio (x(l +2)+x ( +3))/(x(l)+x(l
(x(1+2)+x(1+3))_
=K 2+K 2
(x(l)+x(l+l)) =
Since K=(1-g(1-2))/g(l-2)
a+bj
+1)) is
(A.13)
<K2+2K
is a monotonically decreasing function of , so is K2 + 2K. Therefore, if
K =(l-g (L2-2))/g (L2 -2) then for l -L 2 ,
p (+2) + p(l+3))
(A.14)
Lp(I)+ (+l)p(l+1) < (I+)(p(l) + p(1+l))
(A.15)
p(l) +p(l+l) >
2
And since,
i
I=L2
If we consain K 2 +2K<l
tp(l)
(L2+21+l)(p(L2)+p(L 2+1))(K2+2K) t
(A.16)
I=0
then g (L2 -2)>
2 . Using the geometric series once more we obtain,
i Lp() <-(L2+1)(P(L
2K)
2)+P(L
2+1)) _(K2+
(A.1
+(p(L2)+P(L2+1)) ((1K2_2K9
Since both tails of the summation for E (l) converge, the summaion E (1) must itself converge. Therefore, for g (l) as described by eqns A. 1(1), E (1) exists and is finite.
We now derive a result which will be important for the comparison of different g(1). Equation A.6
implies that,
Lt-2
p(L 1 -1) + p(L1)
>p(t)
(1) In addition, E(I) must mst for ay finite sequence gt(1), but the proof
-158-
1
g(L) s
amitted for this ninftreing
(A.18)
case.
Thus, the total probability in the lower tail of the distribution, p (1), for arbitrary g(l) may be no greater
than the sum p(L 1) + p(L 1-1).
If p(L 1 ) + p(L 1 -1) is small then the probability associated with this lower
tail is negligible.
Similarly, if we let
we see that fo g(l) - .8165 we have K 2 + 2K
(A.19)
1-9g(f)
8(l)
K
<
1/2. Therefore, using equation A.14 and the geometric
series (see derivation of eqn. A.6) we obtain
p(L 2) + p(L 2 +1) >
x
p(l)
/=L 2-2
g(L 2) > .8165
(A.20)
for arbitrary 8 (l). Thus if p (L2) + p (L 2 +1) is small, the probability in the upper tail of p (1) is negligible.
-159-
APPENDIX B: Estimates for Average Rate and Synchronization
POISSON MODEL
We assume that the spike sequence can be represented as a Poisson process with rate function, X(t). The probability that N spikes occur in an interval I is,
p(N I) = ( J)N/
(B.la)
N!
where
t+l
=1 /IA(f)da
(B.lb)
AVERAGE RATE ESTIMATE
We obtain the maximum likelihood estimate, X,,, by determining the maximum of p (n IA) in
eqn. B.la with respect to X. For convenience we maximize the natural logarithm of p (n I).
n(p (N Ix)) = Nln(KI) -
- In(N!)
(B.2)
Differentiating eqn B.2 with respect to X and setting the result equal to zero yields,
fAi
(B.3)
= N
The expected value and variance of this estimate are,
E(.,,,~)
: =A
(B.4a)
and,
var(k.{)=
IA
(B.4b)
respectively. The Cramer-Rao lower bound is (Van Trees, 1968),
var () >
-
[E-
np (N X)
(B.5)
Using eqn. B. 2 to evaluate this expression yields,
vw(a)
A
(B.6)
Thus the estimate of eqn. B. 3 is efficient (has the minimum variance of any unbiased estimator).
An unbiased estimate of var (i,,,), which we use as an approximate error bound for i,, is
-160-
a.2 = N2
(B.7)
SYNCHRONIZATION ESTIMATE
To form the synchronization estimate of eqn. 3.4 we estimate A1 and l/oA and form the product,
S = ia
(B.8)
where & is an estimate of 1/Ao. ( 1 ) We now derive expressions for A1 and &. In addition, a simple
method used to estimate IA1 12is presented for use in the measurement of iso- A1 12contours.
Estimation
of A 1
Let X(t) be well described by eqn 3.5. We observe the average rate in the m6' time interval,
X(m) where
(m + 1)
X(m) = 1f
We wish to estimate
X(t)dt
m=0,1,...,M-1
(MT>>T)
(B.9)
from X(m). This may be done as follows. Combining eqns. 3.5 and B. 9
Ak
yields,
(m +1)
(
g
2
m)=
1
T mt
Ake
dt =
Rm
.2__
Ake T e
kj
Tsinc r kT
(B.10)
T
k =-K..
k=-KIow
We then find the Fourier coefficients of X(m),
_
Al =
1-1
M
A
_jlmt
X(m)e
-Kmax
<
(B.1)
-< Kmax
Substituting eqn. B. 10 into eqn. B. 11 yields,
K .
Al,:
(1) Later we will show tat
dent.
ht I
A= -K.
Ak
'T
e
.
-Kin,smc 'rr
=
T0
1M-I j
2 -Ia
(k -)m]'
a
when the number ci observaticns is lage then the quantities il and
-161-
(B. 12)
"
Ao0 amemtistilly
indepen.
We notice that for Km=T/T< 2) and k l, the term in parentheses approaches zero for large
M and is identically zero for M T an integer multiple of T. For k = l, this term is unity. Therefore,
kr
Ak
AT
sincerT
k c K,,,
and since sinc7r - O0 for -Km
T
r<
2Km
(B. 13)
e
we have using eqn. B. 11,
(m)e
e
Ak =
(B.14)
rT
T
We now form the estimate Ak by replacing X(m) with the efficient estimate,
est((m))
=
(B.15)
T
where n, is the number of spikes which occur in the interval (mT, (m +1)T).
Letting MT=To,
the duration of the tone burst on-window we form the efficient( 3 ) estimate,
AAi=
c
M1
:I,
T
03.16a)
J
=O
T,.sincrr
E(Ak)=Ak since E(nm)=h(m)T.
nBe
For a Poisson process, since the nm are obtained from nonover-
lapping intervals they are independent.
The variance of Ak, E[(Ak-Ak)(Ak-Ak)']
where
*
denotes complex conjugation is,
1 -1l_
(m
-0
var (Akk) =
)0
(B.16b)
=
C2 T
T, silnt
T sinc2ukT
We now modify equations B.16a&b for the case where Ajt is estimated from the response to Q
tone bursts. A simple average of the At obtained for each of the Q tone bursts and letting
(2) Th is exactlythe Nyqui citerion for sampled pnals at ample rate 1/r and maximum frequencyKU./T.
(3) Since the estimate of eqn. B. 15 is efficient, thi estimate is fficient as well due to the fact that the tranmatica
K(m) to A is linear (Van rees, 19fi8).
-162-
from
Q
Nm
q-1
(
)
(B.17a)
and
(B.17b)
I = QTf
produces as the estimate of At for Q tone bursts,
Ak =
N
·
Isinc7
(B. 18a)
T
T Jm=
and
M-L
(=18b)
var(.k) =
Ismnc2W
Isinc 2 ,7r
T
-
T
Estimation of
Ao
Let a
1
The distribution of N, the number of spikes occuring in the observation inter-
Ao
val, I is
N!
The maximum likelihood estimate of a, found by maximizing eqn. B.19 with respect to a is,
(B.20)
1 =v N
The expected value of this estimate does not exist due to the singularity at N = 0. Hence, we use
the asymptotically unbiased estimate of,
(B.21)
&L
= N+
N+1
whose expected value is,
I
E(-)
N+1
)=
(-e--)
-163-
d21
I
AO
(22a)
. The variance of this estimate is,
For large values of I, A0 > >1 thus E ()
(A0 ?A
1 )2(e-A12
2=E(a2)_(
(.22b)
(AOl)"
12e-A0
(
(A
N2O(N+1)!(N+1)
=
(A /)
)eAo
k=1
kk!
( 1 (1-e-d
A
=
(-)e -A[Ei (AO)-n (ol
Ao
)2(l-eA,)2
)2
)]-( )2(1-e-')
2
Ao
where y= .5772156... (Euler's constant) and Ei (x) is the exponential integral function( 4 )
The consistency of this estimate is shown as follows. For large x, Ei(x)=eX/x
(Abramowitz
& Stegun, table 5.2 (1972)). Thus for large values of Aod substitution of the limiting value ofxD
Ei(AOd) into eqn B.22.b yields,
2 ( ed(_l)-(A )2 0
(B.23)
Synchrony Estimate Synthesis
Before combining the estimates A1 and & as in eqn. B.8 we must first show they are independent. From eqn. B.17a and B.18a we see that the Ak, are the weighted sums of the independent
random variables n,.
As MQ tends toward infinity the Ak become Gaussian random variables
according to the Central Limit Theorem (Davenport & Root (1958), Drake (1967), Papoulis
(1965)). Thus the A,kare marginally Gaussian for large MQ.
To show they are jointly Gaussian we note that if z =ax +by is Gaussian for all nonzero a
and b then x and y are jointly Gaussian (Papoulis pp. 224 (1965)). This is true for any two Ak,
since they are themselves weighted sums of independent random variables. This makes z a
weighted sum of independent random variable as well and therefore as the number of terms in the
sum gets large z will also be Gaussian. Thus the Ak are jointly Gaussian. Fminally,to prove
(4) See Abramowitz and Stegun (1972) for identities (eqns. 5.1.2 and 5.1.10) and tabulatiom. Also West (1984) and
Gradshteyn & Ryzhik (1980).
-164-
independence,
we show that the Ak, are uncorrelated.
a vector of independent random variables, on the set of orthogonal vectors
[No,N 1 · ·- ,N(m-1)],
[1,e
-J2lrj-
,e
are the projections of
The A
-J2i-r(M - 1)
] and are therefore uncorrelated by definition (Van Trees (1968)).
Since the Ak are jointly Gaussian and uncorrelated they are also independent (Davenport & Root
(1958),
Papoulis
T[Ao]=&=l/(o+
(1965)).
l/I)=I/(N+l)
Thus
any
transformation
T[-]
of
Ao
(i.e.,
is independent of A1 (Papouis (1965)).
We may now form the synchrony estimate of eqn. B.8 using eqns. B.18a & B.21,
|
-J2=-
||lM-1
eicT 1 Ne
(B.2a)
T
rith variance,
cr = var(A 1 )
(B.24b)
ia
SYNCHRONIZATION MEASURES FOR IRC ESTIMATION
An iso-synchronization contour can be defined as the set of (tone frequency, intensity) points
for which IS I is constant. We may define an iso- IA1 contour similarly. To measure such an isosynchronization contour a suitable estimate of synchronization is needed. The estimates of eqns.
B. 18a & B. 24, however, are not immediately useful since they are estimates of a complex quantity rather than its magnitude. Furthermore, the derivation of unbiased estimates of IA1 I or IS I
for small observation intervals I is difficult.
11 2, a
We now show that if instead of choosing A 1 1 as the response variable we choose 1A
simple
unbiased
(though
var(x) = E(xx')-E(x)E(x)
inefficient) ( 5 )
and that E(,)
IAk12
estimate
is
obtained.
Recall
that
= Ak. If we letx=A* we obtain,
= E(IAk
12 )
-
(B.25)
ar(Ak)
Using eqn. B. 18b we see that,
exist using N, (a definedinn eqution B. 15) as the basic efficient stastic since I A I and
(5) No efiient eimat
(Van Trees, 1968).
cmmute over nlinear tran(cmatis
nlinear functicns of ?)(m) and efiiency does mnot
-165-
ISwe
E
(B.26)
= ,ar(,(k)
Therefore,
EI&k12
= IAkl2
N
Ix
(B.27)
Thus, letting Pk = Ak 12and substituting the expression for IAk I from eqn B.18a we can form the
estimate
I
Nco
2
N.cos
Isinc
T
2Mrm
2 T+
N-1sin21rmT
Nsin 2'm
+
=0
=0
-
- N,
(B.28)
m 0
The form of this estimate for k=1 is simple when X(t) is well described by eqn 3.5 with
Kmax 5 2. Setting T=-,
b
4
eqn. B.28 becomes,
+ (N-N
P 1 = 2I((2)2[(No-N2)2
21
M/4
whereNi =
N, +iandN =
i=0
3
3)
2)
- N]
(B.29)
M-1
Ni = Y N,.
i=O
m =O
Due to its computational simplicity, eqn. B.29 is useful in the rapid calculation of a A112
estimate immediately after the presentation of a single toneburst. This estimate can then be used
with an iterative iso-response algorithm in much the same way as an average rate estimate is used
(see tuning curve algorithm description; Chapter 2).
-166-
APPENDIX C: Tests of Iso-fundamental Contour Estimation Algorithm Accuracy Using Single Cochlear
Neurons in the Alligator Lizard
Here we show the accuracy of the new iso-fundamental contour estimation algorithm we have
developed. In six cochlear nerve fibers (in the alligator lizard) we measured iso-fundamental contours. We
used the TCA 3 algorithm (CHAYER
estimate
1
2: IMPROVING TCA ACCURACY) with the synchronized response
described in Chapter 3 (eqn. 3.8) and Appendix B (eqn. B.29). Thus, similar to the TCA when
used with the average rate response, P 1 was compared to the criterion, C. The tone burst stimulus had
T
= 100 ms, 01= 0.5 and
,=
2.5 ms. The TCA3 was used with a criterion of C =0, -1/40 decade fre-
quency increments and level increments of 1/4 dB. In order to obtain a better estimate of P 1, two tonebursts were used (see Chapter 3: METHODS).
At several (frequency, level) points along each iso-fundamental contour we measured,
IA1 ,
the magni-
tude of the fundamental component of the neural firing rate function. These measurements are shown in figure C.1. Solid lines connect the measurements of All along each contour. These data illustrate that the
iso-fundamental algorithm does indeed hold
IA1 approximately constant For the set of measurements
showing the most variation with frequency (as marked by the arrow) in figure C.1 the deviation from the
mean value of
IA!1
26 spikes/sec is within + 5 spikes/sec.
-167-
ii
-
I
I
I
I
0O
I
1 I-%
.
N
I s
8 '-
a)
O8
at
/I/
I
D
C
I
I
I
i
0
0o
I
I
0
0
(3@s/s>!ds)
-168-
ILVi
v
'X
ala
.
I
APPENDIX D: Justification of Comparisons between Synchronization Measured with Tone Burst Stimuli
and Synchronization Measured with Continuous Tone Stimuli
We have consistently used a single stimulus paradigm throughout this study. Thus we can directly comrnpare measurements of iso-fundamental contours and synchronization at constant average rate difference. The
choice of the tone burst as the stimulus was primarily motivated by its utility in the rapid estimation of isofundamental contours. However, in previous studies of neural synchronization (Rose, 1967; Littlefield, 1973;
Anderson, 1973; Johnson, 1974; Kim & Molnar, 1979) and hair cell receptor potential A.C. component
(Russell & Sellick, 1978), continuous tone stimuli were used. Thus, in order to justify a comparison of our
results with these previous studies we need to determine whether tone-burst stimuli produce roughly the same
response as do continuous tone stimuli.
Let us first examine the neural tone burst response (figure 3.3). Both histograms of spike activity exhibit two common features: a rapid transient response and a steady state response. At stimulus onset, the
neural response shows a large transient which decays to a steady state within 10 ms of stimulus onset. The
tone bursts of figure 3.3 have on-windows 12.5 ms in duration. Our standard stimulus has an on-window
duration of 50 ms. The response to such a stimulus is shown in figure D.1. The transient portion of the
tone burst response will form only 20% of the total on-window response to a 50 ms tone burst. In addition,
we notice that the detailed shape of the transient response is such that the average rate over the first 10 ms of
the response does not seem (qualitatively) to be too different from the steady state average rate. We would
therefore expect that the transient response will not cause the value of response averaged over the full duration of the tone burst to differ substantially from the tone burst steady state response. To test this notion, we
measured response variables Ao , IA11 and IS I over both the steady state portion of the tone burst response
(last 80%) and over the full tone burst response for our standard tone burst with a 50 ms on-window. Variables measured for the steady state portion of the response will be superscripted with ss.
AVERAGE RATE MEASUREMENTS
In figure D.2 we plot A0 vs. A.
The data span a tone burst average rate difference response range of
0 spikes/sec to 136 spikes/sec and come from fibers with a range in CF of 170 Hz to 3980 Hz (111 units in 9
animals). It can be readily seen that the steady state and averaged tone burst responses are very similar
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(correlation: .99, regression slope: .85 -. 008).
SYNCHRONIZATION MEASUREMENTS
In figures D.3ab&c we plot Ao versus A',
IA11 versus 1 1 1" and IS I versus IS S, respectively (10
units in one alligator lizard). The average rate difference, AX, for all these measurements was between 20 and
30 spikes/sec. The relations between the steady state responses and average responses are (correlation: .96,
regression slope: .80
.03) for A and AS, (correlation: .99, regression slope: .88
IA1 155,and (correlation: .99, regression slope: .98 +.008) for IS I and
.02) for IA1
I and
Is! .
In light of these results which relate the average rate and synchronized components in the steady state
portion of the toneburst response to the average tone burst response we conclude that measurement of the
average tone burst response is comparable to measurement of the tone-burst steady state response. In the
case of synchronizaton these two response measures are virtually identical. If we assume that the steady state
response to a 50 ms tone-burst is proportional to the steady state response produced by a long tone-burst
(Westerman, 1985) then it is likely that measuing synchronization over a tone-burst is comparable to
measuring steady state synchronization in response to a continuous tone.
-170-
C)
Co
w
0
(I)
100
0
TIME
Figure D.:
CMS)
Neural instantanous rate respome to a tme-burst (T,=Tff
= 50 ms).
Rate in
10 ms af tone-burst
spikes/scc. Notice that the transient portion of the response has disappeard by
Znpoe.
toneabumt
the
of
80%
mprises
tbIrcore
cuset. The Beady state portion of the response
-171-
I
11111,,,,,,,,,,,
liii, lilt l
IX
-N
I
C
I
r
I
r
- Lr)
I
U)o
E-0
(1
Ii~
- t
0
rr
r -t
II
<
iv4i
-1
00
N
I II 111111 II I I I II I
5
0iL
t
00
°
-172-
0L
0 "]
Ei
L
mA=
I
L
I
1
·
Du.
N=70
4*
40.
a
Ao
20
JP6P
0
0
40
20
40
60
N=70
30
AI,
b
20
10
1,
AAll
0
1.0
N=70
0.8
0.6
ISi
0.4
0
C
0.2
0.0
0.0 0.2 0.4 0.6 0.8
1.0
ISI
Figure D.3: Neural rate response averaged over tone-burst on-window pltted against the response
averaged over the final 80% of the tone-burst on-window (see text for details). a) Average rate, A0
(spikcessec); b) Fundamental rate aomponent, a, (spikes/sec); c) Syndronization index, IS I (nmit-
ks).
-173-
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Journal
BIOGRAPHY
Cristopher
Rose was born in New York City on January 9, 1957. In June of 1979 he received the
Bachelor of Science degree from the Massachusetts Institute of Technology in the Department of Electrical
Engineering and Computer Science. In June of 1981 he was awarded the degree of Master of Science in
Electrical Engineering and Computer Science also at M.I.T.
Dr.
Rose's
research
interests
include
auditory
physiology,
chronic
and
recording/stimulation, prosthetic device control, stochastic processes and telecommunications.
Awards
1979- Bell Labs Cooperative Research Fellowip
1979- NSF Fellowship (declined)
1979- GEM Fellowship (dedined)
1977- General Motors Scholarhip (declined)
1975- National Achievement 4-year Corporate Sholarip
-177-
acute
neural