Title: Ensemble characteristics of visual evoked cortical potentials in noise
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Title: Ensemble characteristics of visual evoked cortical potentials in noise
Alternate Title: Visual evoked cortical potentials in noise
Physical Description: viii, 97 leaves. : illus. ; 28 cm.
Language: English
Creator: Doyle, Timothy C., 1943-
Publication Date: 1969
Copyright Date: 1969
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Subject: Brain   ( lcsh )
Electrophysiology   ( lcsh )
Noise   ( lcsh )
Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
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Thesis: Thesis--University of Florida, 1969.
Bibliography: Bibliography: leaves 92-96.
Additional Physical Form: Also available on World Wide Web
General Note: Manuscript copy.
General Note: Vita.
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Bibliographic ID: UF00097755
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000955719
oclc - 16969575
notis - AER8348

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ENSEMBLE CHARACTERISTICS OF VISUAL EVOKED

CORTICAL POTENTIALS IN NOISE













By
TIMOTHY C. DOYLE















A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT O'F THE REQUIREMENTS FOR THE
DEGRIF OF DOCTOR OF PHILOSOPHY












UNIVERSITY OF FLORIDA
1969











ACKNO1LE I [CIEiTS


The author wishes to express his sincere appreciation to

Dr. D.G. Childers, chairman of his supervisory committee, for his

counsel and encouragement. He also wishes to thank Dr. N.W. Pernr

and Dr. J.R. Smith, the other members of his supervisory ccmvittee,

and Dr. A.H. Nevis for their guidance. He wouldd like to acknowledge

the Visual Sciences Laboratory and staff for their facilities and

help.












TABLE OF CONTENTS
Page

ACKNOWLEDG NTS . . . . . . . . .. .. ii

LIST OF TABLES . . . . . . . . ... .. v.

LIST OF FIGURES . . . . . . . . . . . vi

ABSTRACT . . . . . . . . . . . . vi ii

CHAPTER

1 INTRODUCTION .............. ......... 1

2 SEQUENTIAL AND ENSEMBLE AVERAGING . . . . . 4
2.1 Sequential Averaging . . . . ... . 4
2.2 Variability of the VER . . . . . . . 7
2.3 Theoretical Models . . . . . . .. 10
2.4 Multi-Subject Ensemble Model . . . . . . 12
2.5 Ensemble Averaging . . . . . . .... .13
2.6 Methods . . . . . . . .... ..... .15
2.7 Objectives . . . . . . . . ... . 16

3 RESULTS OF ENSEMBLE AVERAGING . . . . .... .19
3.1 Introduction . . . . . . . . 19
3.2 Selection of Bandwidth . . . . . . .. 21
3.3 Ensemble Evoked Response . . . . . . .. 22
3.4 Noise in the Ensemble Evoked Response . . .. 28
3.5 Quality of the Ensemble Evoked Response Estimate 32
3.6 Variations in the Ensemble Evoked Response with
Stimulus Number . . . . . . . . .. 36
3.7 Average of Ensemble Evoked Responses . . . .. 39
3.8 Summary . . . . . . . .... ..... .41

4 THEORETICAL ASSUMPTIONS UNDERLYING RESPONSE AVERAGING .. 43
4.1 Stationarity of the Background Noise . . ... .44
4.2 Dependence of Signal and Noise . . . . .. 56
4.3 Additivity of Signal and Noise . . . . .. 59
4.4 Summary . . . . . . . .... ..... .67

5 SULMMARY . . . . . . . . . . . . 68





iv


Page
APPENDIX

A EXPERIM-ENTAL PROCEDURE ................. 75

B SLIPPLEMENTARY FIGURES -AND TABLES . . . . ... 78

REFERENCES. .. . . . . . . . . . . .. 92

BIOGRAPHICAL SKETCH ......... ............. 97












LIST OF TABLES
Page

3.1 Average Correlations for Figure 3.2 . . . ... 25

3.2 Average Correlations for Figure 3.4 . . . . .. 28

4.1 Maximum Number of Standard Deviations of Ensemble Control
Average about a Zero Mean . . . . . . .... 47

4.2 Maximum Number of Standard Deviations of + Estimate
about a Zero Mean . . . . . . . .... 48

4.3 Maximum Number of Standard Deviations (times 0.1) of
EER Estimate about Zero Mean . . . . . .... .49

4.4 RMS Values in uv of + Averages Under Stimulus and
Non-Stimulus Conditions . . . . . . .... .57

B.1
through
B.7 Response Correlations Coefficients . . . . .. 88

B.8
through
B.13 EER Correlation Coefficients for the 0.3 to 50 Hz
Bandwidth . . . . . . . . ... ...... 89

B.14
through
B.20 Control Correlation Coefficients . . . . .... 90

B.21 Values for Cf(t, t-T) in Figure 4.1 . . . . .. 91












LIST OF FIGURES


Page
2.1 Ensemble Versus Sequential Averaging . . . . .. 14

3.1 Bandwjidth Coinparison . . . . . . . .... 21

3.2 Ensemble Evoked Response . . . . . . .... 23

3.3 Superimposed Ensemble Evoked Response . . .... 27

3.4 Ensemble Control Averages . . . . . .... .29

3.5 Noise Amplitude in EER . . . . . . . . .. 31

3.6 Superimposed Ensemble Standard Deviations ........ 32

3.7 EER Confidence Intervals . . . . . . .... .. 34

3.8 \ER Confidence Intervals .. . . . . . ..... 35

3.9 Amplitude Fluctuations of the Primary Response ...... 38

3.10 Latency Fluctuations of the Three Primnary Components .. 38

3.11 Avcrage of Response and Control Ensemble Averages . . 10

4.1 Ensemble Autocovariance Function for First Control 51

4.2 Superimposed Ensemble Standard Deviations ....... 53

4.3 Histogram Estimates of Ensemble pdf for First Control .55

4.4 S/N Ratio Versus Number of SLurations . . . . .. 65

B.1 EER for Two Bandwidths . . . . . ..... .. 79

B.2 Ensemble Control Average for Two Bandwidths ...... SO

B.3 + Ensemble Responses for Tw'.o Bandwidths ........ 81

B.4 + Ensemble Control Averages for Two Bandwidths ..... 82

B.5 Ensemble Standard Deviations Under Stimulus and
Non-Stimulus Conditions. . . . . . ..... 83

B.6 Ensemble Autocovariance Function for First Recpoinse . 84




Vi1


Page
B.7 Ensenmble Autocovariance Function for Fourth Pesponse .

B.S Histogram Estimates of Ensemble pdf for First Response 86

B.9 S/N Ratio Versus Number of Summations . . . .... 87








Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy

E'SEMBLE CHARRACTEPJSTICS OF VISUAL
EVOKED CORTICAL POTENTIALS IN NOISE

By

Timothy C. Dole

August, 1969

Chairman: Dr. D.G. Childers

Major Department: Electrical Engineering


To date, most investigations of evoked potentials monitored at

the human scalp have utilized the average of a series of responses

recorded from a single subject to define components of the evoked

response, ER. This technique has provided a fairly consistent estimate

of an individual's average ER. However, with this method it is not

uncommon to obtain quite dissimilar averages from different subjects

under identical experimental conditions. This contrast in the ER

estimates across a population has hindered the interpretation of the

effects of stimulus parameters on the ER and frustrated the attempt

to use this average for clinical diagnosis.

The present study investigated one method of obtaining informa-

tion about the character of the response to visual stimuli common to

a considered normal subject population. The estimation technique

utilized was to average across an ensemble of fifty subject EEG's

containing evol.ed activity. This method permitted estimation of

common inter-subject response activity evoked by a single stimulus.


viii








By averaging the resultant waveforms of this ensemble average across

stimuli, a reliable, relatively noise-free inter-subject response

template was obtained.

The resemblance of the ensemble of cortical potentials to a

classical stochastic process enabled the author to investigate

three prevalent assumptions about the statistical character of the

evoked response and background noise activity. These three assumptions

are: (1) the response and background noise are independent; (2) the

background noise is stationary; and (3) the response and noise are

additive. The results presented in this dissertation lend some

measure of support to each of the three assumptions.












CHAPTER 1


I NTRODUCT ION


The time-varying electroencephalographic potential specifically

elicited by a single controlled sensory stimulus is referred to as

an evoked potential or evoked response (ER). An invariant evoked

response to controllable stimulus parameters, monitored from the

human scalp, would provide an excellent empirical diagnostic tool

for brain research. How..ever, this has eluded all investigators

because of tw:o inherent limitations. First, the parameters affecting

the ER, such as physiological variability of the subject during

stimulation, are usually unkrcno.n and uncontrollable. Second, the

observability of a single response is severely handicapped by the

interaction of the desired signal (ER) w..ith the background bioelectric

on-going activity.

The physiological variability, or state of the subject, intro-

duces a random variable into i..lat micht othenrise have been a

deterministic response. The on-going background activity masks and

distorts the ER such that it usually is undetectable at the scalp.

Thus, these tw.o sources of variability deny the investigator anything

but tenuous conclusions about the relationship of a single bioelectric

response to specific stimulus parameters. In order to determine the

general characteristics of an ER, several data processing techniques

have been utilized. The most co!r.on technique is the sequential







average (see (Capter 2) of a series of responses recorded from an

individual subject. This average is fairly consistent for a parti-

cular subject under identical stimulus conditions. However, the

averages obtained from several subjects often appear to be quite

dissimilar. Thus, it is difficult to establish a certain average

response w.aveform, elicited by specific stimulus parameters across a

subject population.

In the present study the average of a single response across a

subject population is investigated. This average provides an estimate

of the characteristics of the response to a particular stimulus which

are common to an ensemble of subjects. From these common character-

istics one can develop a model or template of the average response

activity across the ensemble. The changes in this response average

with repetitions of a specific stimulus can also be investigated. In

addition, the collection of subject EEG w.aveforms can be viewed as an

ensemble of independent sample functions from a stochastic process.

Thus, assumptions underlying the averaging process can be examined

by classical techniques. Tlese assumptions generally involve the

statistical nature of the evoked signal and background noise.

Therefore, the two major objectives of this investigation are:

(1) to develop a better understanding of the common characteristics

or norm of the evolved responses within a subject population; and

(2) to provide an empirical technique to evaluate the prevalent

statistical assumptions about the character of the evoked response

uand background potentials. Since at this time the response signal

cannot be separated from the background ncise, the results are







limited to some combination of the signal and noise rather than either

by itself. The characteristics of the noise under non-stimulus condi-

tions are observed, but these may provide little insight into the

background noise activity during stimulation. Therefore, the second

objective is theoretically unattainable at the outset. However, the

results obtained in this study should contribute to the mkowledge of

the statistical character of evoked on-going activity.

The results of this investigation are divided into tw.o chapters.

Chapter 3 explores the inter-subject norm or ensemble average and the

transient changes in this average. Also examined are the non-stimulus

control estimates of the average background activity. Chapter 4 con-

tains an examination of three assumptions about the signal and noise

which are necessary to theoretically justify results obtained by these

averaging techniques. In this examination the statistical character

of the ensemble of ER's with background noise and the ensemble of

resting bioelectrical potentials is investigated. To facilitate

understanding of the results presented in these chapters, Chapter 2

provides an overview of the background material and methodology for

this investigation.







CHAPTER 2


SEQUENTIAL AD!L) ENSEMBLE AVERAGING

The pea,-to-peak amplitude of the potentials recorded from the

scalp usually have a range of 9 to 50 v\' (Cobb, 1963). In most cases

the amplitude of the evoked activity in these potentials is below the

threshold of observability. It therefore becomes necessary for the

researcher to preprocess the scalp potentials to extract some charac-

teristics of the ER. Generally, these processing techniques require

reiterative measures and estimate the response characteristics in

terms of probability statements or statistical averages.

2.1 Sequential Averaging

To date the most powerful statistical descriptor of the ER is the

sequential average which was implemented photographically by Galanbos

and Davis (1943) and later electromechanically by Dau.son (1954) and

electronically by Barlow (1957). Today this average is usually cal-

culated by a general or special purpose digital computer. The sequen-

tial average is essentially the cumulative\ sum or average of successive

trains of the scalp potentials which attend a series of repeated

identical stimuli. The mathematical expression for this average is


N-l
(t) f.(t+jT) for O j=0

where fi(t) is the time varying EEG (electroencephalogram) recorded

from subject i and the stimulus cccurs periodically N times at t=0, T,

2T . .. The ~ indicates the above is an estimate of the true mean




5



m(t) for the range of t between 0 and T. If the stimuli are visual,

the result of this average is called a visual evoked response (VER).

Since the present study employs visual stimuli, most of the discussion

will focus on responses evoked in this modality.

The basic advantage of sequential averaging over other data pro-

cessing methods is that it extracts an estimate (e.g., the VER) of the

response waveshape from the random potential fluctuations monitored

at the scalp. Other data processing techniques such as autocorrelation

(Barlow, 1959), spectral analysis (Childers, et al., 1968), matched

filtering (Negin, 1968), and synchronous correlation (Regan, 1966a)

may detect the periodicity, frequency composition, presence, or phase

locked components of the ER, respectively, but they do not produce the

w:aveshape information provided by averaging. In addition, the apparent

underlying integration mechanisms noted in most complex neurological

systems tend to justify or lend a degree of rationale to the averaging

technique, as Rosenblith (1959, p. 540) reasons:

In the handling of sensory information, organisms
behave as if they were acting on the basis of
activity averaged over substantial regions of the
nervous system, such a weighted averaging process
could be carried out by comparing the outputs of
a large number of neural elements. Since there are
at present no devices of performing this task
(even if one knew exactly where and how to perform
it), one chooses to average over an ensemble of
responses to repeated identical stimuli in order
to bring out certain typical aspects of behavior
of the nervous system.

The major disadvantages of sequential averaging are: (1) only

those components of the ER which are time locked to the stimulus will

be extracted; (2) the responses to succeeding stimuli may differ and

distort the average response estimate because of the effects of those








preceding them; and (3) the resultant average of many responses is

not identical under invariant stimulus conditions. By extracting only

the time locked components, the averaging process obscures any phase

and frequency deviations of the individual responses. In addition,

the degree of amplitude variation of these time locked components is

hidden within the resultant waveform. However, it is impossible to

improve the averaging process by weighting and phase locking each res-

ponse without the a priori knowledge of the specific angle or amplitude

modulation which may affect a series of responses. Recent studies

by Harris and W'oody (1969) and Woody (196') have attempted to estimate

the degree of angle modulation of the ER by a correlation technique.

From this estimate, they' have suggested an adaptive filter to improve

signal extraction by averaging. However, to most investigators such

modulation is a perturbing factor distorting the estimate obtained by

averaging.

The second disadvantage is imposed by the practical limitation

that the large number of stimuli necessary to extract an average res-

ponse is generally presented over a relatively short period of time.

This limitation introduces twno disturbing factors. The first factor

can be termed overlap interference, i.e., the transient tail of one

response overlaps and distorts the initial components of the subse-

quent response. This usually occurs only with relatively high stimu-

lus rates. The critical stimulus frequency at which this interaction

occurs is not clearly established, but the consensus of evidence

indicates that it lies in the region of 0.7 Hz (Barlow, 1960) to

2.7 Hz (Kitzsato, 1966). There is some indication that higher rates








are permissible if tie stimulus is presented at random intervals

(Ruchkin, 1965).

The second disturbing factor is habituation, which can generally

be described as the decrease in the subject's attention to monotonous

repetitive stimulation, h.'hich changes the successive responses. Since

the extraction by averaging of a reliable ER requires many repetitions

of the stimulus (see Perry and Childers, 1969) most investigators

discuss habituation in terms of the changes in consecutive averages of

large groups of successive responses. Therefore short term habituation,

particularly transient changes of die ER's to the initial stimuli, is

difficult to record. However, "erlin and Davis (1967) utili-ing a

highly responsive subject, have noted a systematic doi-.. i.'ard drift in

the amplitude of individual responses to successive auditory stimuli.

This result agrees with the less specific gradual .wa-xing and waning

decrease in aiplitude of consecutive averages (VER's) found by many

authors (e.g., Perry and Copenhaver, 1965; Haider, et al., 1964; and

Garcia-Austt, et al., 1963). Bogac:, et al. (1960) have also noted

that habituation to visual stimuli is also reflected in increased

phase lag (latency) of certain averaged response components.


2.2 Variability of the VER

The third shortcoming of sequential averaging, the variability of

the resultant, poses the greatest dilemma to the researcher using this

tool. The VER's taken on different occasions from a single subject

are relatively stable over a period of time (Dustman and Beck, 1965);

how-.ever, when r:e compared s the V'ER's obtained under identical stimulus

conditions from a population of subjects, variability of the waveform








makes it difficult to interpret the effect of certain stimulus para-

meters on the human visual system. Some investigators have attempted

to reduce this inter-subject variability by expanding the resultant

waveform into orthogonal sets for comparison, with little success

(see Donchin, 1966 and John, et al., 1964). Others have organized a

system of classification of the certain maximna and minima of the \ER

waveform in terlms of amplitudes and time delays latenciess) from stim-

ulus onset (see Bergamini and Bergamasco, 1967 and Kooi and Bagchi,

1964). With this latter technique, investigators have been able to

occasionally detect trends of change common to a subject population

with gross variations in the stimulus parameters. Eut such findings

are again hampered severely by the variability of the VER. This

variability appears to be the product of changes in the psychological

state of the subject (e.g., attention) and the inherent statistical

character of the ER and background activity @.'erre and Smith, 1964).

In an attempt to understand the statistical properties of the ER

and background noise, a vast amount of effort has been concentrated

on the classification of the physiological origin of these scalp

potentials. However, to date there is only a rudimentary knowledge

of the mechanisms which underlie the generation of these EEG wave-

forms (Elul, 1967). The present evidence indicates that EEG poten-

tials arise from some complex integration of excitatory and inhibitory

post synaptic potentials, psp (Humphrey, 1968; Creutzfeldt and Kuhnt,

1967; Eccles, 1966 and Amassian, et al., 1964). The relationship

between these subcranial potentials and the bioelectric waveform

monitored at the scalp has been investigated in animals by De Lucchi,








et al. (1962). They have hypothesized that the scalp acts as a

further average of psp's. In similar investigations involving human

subjects, Cooper, et al. (1965) demonstrate that only an average of

those potentials which are widely synchronized can be observed in the

scalp. It is therefore interesting, as Rosenblith (1959) notes, that

even with this gross averaging of psp's by the scalp electrodes, we

still find variability of the ER to identical stimuli.

The amount of this variability produced by the interaction of the

response signal with the background activity (noise) is generally

unknown. To date, most investigations of this interaction have con-

centrated on only one component of the noise (resting EEG), the

alpha rhythm. This component is defined by Brazier, et al. (1961,

p. 647) as the "rhythm, usually with frequency of 8-33 c/sec., of

almost sinusoidal form, in the posterior areas, present during relax-

ation when the eyes are closed, attenuated during attention, particu-

larly visual." The physiological origin of this regular rhythmic

activity is yet undetermined. However, Andersen and Andersson (1968)

point to the thalamTus as the source of the rhythmic impulses which

trigger the psp's monitored at the scalp. This agrees with the

observations of Creutzfeldt, et al. (1966). The alpha rhythm is the

most striking compncent of the resting EEG of the majority of normal

adults. This compcnont is so prevalent that a resonant peak: around

10 H: can readily be seen in the EEG frequency spectrum 0 alter,

et al., 1967). In addition, the late rhythmic component (after

discharge) of the ER and \TR has the general form and dimensions of

Lhe 3pcrOtartnec.s alpha wave (Bishop and Clare, 1953 and Barlow, 1960).

It is tiherefcre not unexpected tha: studies involving the relationship








of this predominant EEG component to evoked responses, particularly

those elicited by visual stimuli, received precedence.

A review of the studies focUtsing on this interaction with the VER

is provided in the recent work by Childers and Perry (1969). They

note that many investigators suspect two fcrms of S to 13 Hz.

activity: the first, the predominant non-provoked EEG activity, and

the second, the components of the evoked response which are alpha-

like in character. The evidence presented indicates that if two dis-

tinct forms of activity exist, they probably share common neural

ensembles. However, this relationship is unclear and Childers and

Perry stress the need for further experimentation to secure the

additional information necessary to determine the interaction of

these components and other background activity with the ER.


2.3 Theoretical Models

Several investigators (e.g., Ruchkin, 1969; Kitasato and Hatsuda,

1965; Bendat, 1964; and Goldstein and Weiss, 1962) attempt to under-

stand the mechanisms of the VER and hence its variability by developing

theoretical models of the statistical character of scalp potentials.

These studies generally rely on the hypothesis that the signal (ER)

and noise (background activity) are additive. This assumption permits

a certain amount of mathematical tractability to these models but

contains little physiological basis. However, it is justified in

part because of the ability to extract a somewhat reproducible esti-

mate of the signal from the scalp potentials by sequential averaging.

For example, if the signal and noise were multiplicative, this








extraction technique would provide completely inconsistent results.

In order to facilitate statistical analysis of the scalp poten-

tials, these model developers generate an artificial stochastic

process. This process is constructed by segmenting (usually at the

points of stimulation) the provoked EEG record into an ensemble of

shorter records. The statistical moments of this ensemble can then

be calculated from the empirical data and compared to the theoretical

model (e.g., the first moment is the sequential average). Unfortu-

nately, few authors have progressed past the modeling stage to

compare the theory with practice. This gap exists, because, as

Barlow (1967) notes, the theoretical ability to separate the statis-

tical components (i.e., signal and noise) does not have an experimen-

tal counterpart. Such a separation would require a much greater

knowledge of the character of either the signal and/or noise and

their interdependence than is available at the present.

In addition, the assumptions made about the stochastic model are

generally violated in the empirical ensemble. For example, the

empirical statistics are derived from temporally related members,

which requires a stationary ergodic random process. This is highly

improbable if one notes the existence of a time varying mean (the

'ER signal). Also the members would probably have some statistical

dependence, reducing the quality of the empirical statistics.

Finally, each subject would have to be examined separately, .which

would eliminate the estimation of the inter-subject statistics.

lHo.wever, is there a better stochastic model and is such a model

feasible? This question initiated the present study and led to the








development of the following model of a multi-subject random process.


2.4 Milllti-Subiect Ensemble Model

It was decided that a more appropriate empirical stochastic

process might be generated from an ensemble of individual EEG's

(i.e., each EEG which contains evoked activity is recorded from a

different subject). Such an ensemble would eliminate the temporal

and probably the statistical dependence of its members. Thus, this

ensemble would more accurately comply with the classical definition

of a stochastic process as a process composed of an ensemble of inde-

pendent simultaneously monitored sample functions. The empirical

ensemble statistics would then provide a more precise evaluation of

the statistical assumptions necessary to theoretically justify

averaging to extract a consistent response estimate. In addition,

these ensemble statistics, particularly the first moment (average)

of the ensemble, would provide information about common inter-subject

characteristics of the ER.

The inherent limitations of this artificial process are two-fold.

First, the individual records are not simultaneously monitored, but

serially recorded over a several month period. Second, the evoked

activity in the member functions may result from several stochastic

processes instead of simply one. The first restriction becomes

significant if the nature of a normal adult EEG is dependent upon

the day of the year or the hour of the day. Fortunately, however,

the present evidence points to a relatively stable individual EEG,

particularly in frequency composition (Berkhout and Walter, 1968 and








Johnson and Ulett, 1959). The second limitation becomes important if

the population of ER's can be divided into discrete classes. To

date, there has been some indication (Perry, et al., 1968) that such

sub-divisions might exist: however, the method of classification and

the number of classes is yet undetermined. Therefore, the present

research effort will assume a continuum of EEG and response types

which can be modeled tender the same stochastic process.

2.5 Ensemble Averaging

The ensemble mean or average provides an estimate of the common

inter-subject response to a single stimulus. The mathematical

expression for this average is

1 N
m (t) Z f (t) (2.2)
Si= 1

where f.(t) is the EEC containing evoked activity recorded from

subject i out of a population of N subjects. This ensemble average,

VER, and unprocessed scalp potentials are schematically represented

in Figure 2.1. This figure demonstrates the basic difference in the

ensemble and sequential average (i.e., \TR). The ensemble average

provides an estimate of the inter-subject average response to a single

stimulus and the VER provides the inter-stimulus average response for

a single subject. The ensemble average eliminates the inter-subject

response variability while it reflects the changes in the average

response to a series of individual stimuli. The \VER average elimi-

nates the inter-stimulus response variability while it reflects the






















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individuality of a subject's average response.

The average of a set of \TER's from a subject ensemble equals the

average of the set of ensemble averages for a series of stimuli, pro-

vided both averages are obtained from the same inclusive collection

of data. This double average provides an overall estimate of the

response activity by sacrificing all variability information. This

overall average has been suggested by Dawson, et al. (1968) to reduce

all excess information about the response. Therefore, this single

estimate might provide a general inter-subject, inter-stimulus

response norm to specific stimulus parameters.


2.6 Methods

In the initial stages of this investigation it w.as decided that

an ensemble size (subject population) of fifty would provide

reasonable statistical accuracy and remain i-ithin the realm of experi-

mental practicability. Both stimulus (signal) and non-stimiulus

(noise) trials were recorded from each of the subjects. The stimulus

was a monocular periodic flash of light 'which had a repetition rate

of about one flash per second. The stimulus was visually occluded

during noise (control) trials. The potentials recorded in these

trials were monitored from a pair of scalp electrodes vertically

oriented on the occiput midline. These potentials were amplified

by a factor of 100 K; separated into t.'o data channels (i.e., a

wide band, 0.3 to 50 H:, channel and a narrow band, 0.3 to 15 Hz,

channel); and recorded on magnetic tape.

These recorded potentials were then digitized and stored in an








IBM 1S00 computer in order to facilitate necessary mathematical

computation. Computer memory limitations restricted the amount of

data analyzed. Therefore, most of the results presented in this

study are derived from activity following ten stimulus triggers (both

signal and noise trials) selected from the periodic (approximately

1 Hz) stimulus train. The triggers chosen are the first through

fifth, tenth, twentieth, thirtieth, fortieth and fiftieth in the

repetitive stimulus train. All of the computation was performed on

the first 950 milliseconds of data following each trigger. A more

detailed discussion of the data collection and analysis procedure

is provided in Appendix A.


2.7 Objectives

The two objectives of this study, as stated in the introductory

chapter, are: (1) to develop a better understanding of the common

characteristics or norm of the evoked responses within a subject

population and (2) to provide mi empirical technique to evaluate

the prevalent statistical assumptions about the character of the ER

and background potentials. The inter-subject ensembles recorded under

stimulus and non-stimulus conditions provide data from which both

these objectives can be pursued. The average across the ensemble

under stimulus conditions furnishes an estimate of the common inter-

subject evoked activity. The data provided by these ensembles is

appropriate for the investigation of theoretical assumptions about

the EEG stochastic process since these potentials are a reasonable

facsimile of such a process.








In pursuit of the first objective the ensemble average across

fifty subjects under stimulus conditions is examined. It is antici-

pated that the results from this average will reveal the components of

the evoked activity which are the most consistent across the subject

population. These components should reappear after each stimulus with

some degree of variability. The variability would probably be the

result of transient changes (e.g., habituation) in the inter-subject

response with the number of stimuli, variations introduced by the

inherent error of the estimation process, or background noise poten-

tials which may remain in the average. A transient variation should

result in some overall trend of change in the response average while

noise interference and estimation error should affect the average in

a more random manner. The transient trends of response estimate and

confidence in this estimate across a series of stimuli will be inves-

tigated. The ensemble average under ron-stimulus conditions is

expected to provide an approximation of the amount of noise contained

in the response estimate. The average of a collection of ensemble

averages should eliminate transient variability, reduce the components

of the background noise and provide a fairly stable inter-subject

response template.

In order to theoretically justify the above response estimation

by averaging,it is necessary that (1) the background noise come from

a relatively wide sense stationary random process; (2) the signal and

noise processes be statistically independent; and (3) the signal and

noise be additive. Since the signal and noise potentials cannot be








separated at the present time, the above properties are generally

assumed to be true. The second objective of this study, evaluation

of these assumptions, is hindered by the inability to segregate the

signal and noise. However, by hypothesizing one or tw.o of these

assumptions an attempt can be made to empirically assess the third.

For example, by hypothesizing the signal and noise to be statistically

independent, the inter-subject ensemble tnder non-stimulus1 conditions

should approximate the noise activity under stimulus conditions. The

stationarity (i.e., time dependence of the ensemble statistics) of

the background noise can then be examined using the resting EEG as

noise approximation. Evaluation of the second assumption can be

obtained in a similar manner by hypothesizing the third assumption

and examining the + ensemble average and ensemble variance under

stimulus and non-stimulus conditions. The assessment of the third

assumption is more difficult. However, by hypothesizing the first

t-.o assumptions w.e can examine the growth of the signal to noise

(S/N) ratio, speculating the signal to be the ensemble average and

the noise the ensemble average under stimulus conditions. If the

signal and noise are additive, the S/N;. ratio should grow linearly

with the number of averages.

The results presented in the t1w.o succeeding chapters attempt to

realize the above objectives. The response activity common to the

subject ensemble is first examined, followed by the evaluation of the

statistical assumptions about the character of the ER and background

noise.












aIAPTER 3


RESULTS OF ENSEMBLE AVERAGING


3.1 Introduction

In this chapter the inter-subject average response to a monocular

visual stimulus is examined. The average across the fifty subject

ensemble record (see Figure 2.1) is utilized to extract an estimate

of the response activity from the random scalp potentials. This

ensemble average should provide an inter-subject response norm if:

(1) the assumptions mentioned in the previous chapter hold; (2) there

exist stimulus locked response components which are common to the

subject population; and (3) the amnplitude of the average background

activity in this estimate is much less than the amplitude of these

response components. The first condition, tie validity of assiriptions

about the response signal and background noise necessary to justify

averaging, is the subject of Chapter .1. The present chapter will

h.pothesi:e these assumptions to be true and exam-ine the ensemble

average for any inter-subject stimulus locked response components. In

addition, an estimate of the amount of background noise contained in

this average is investigated.

In, order to siLplify discussion of the ensemble average evoked

response estimate the term ens-emble evoked response (EER) will be

utilized. This te:i is analogous to the term visual evoked response







(VER) in that it is not intended to iiply that the average response

estimate contains only evoked activin'. Theoretically the \ER and EER

contain noise components those variance decreases directly with the

number of sums in the average (see Perrn and Childers, 1969). An

estimate of the amount of noise in the EER is obtained from the average

across the ensemble muder non-stimulus conditions. This average pro-

vides a series of control waveforms, each attending a particular

occluded stimuli, whose amplitudes should provide an estimate of the

background noise in the EER. This background noise activity which is

not related to the stimulus will introduce variations in the EER from

one stimulus to the next.

As noted in the previous chapter, other parameters ..hich could

affect the EER across stimuli are deviations of the estimate about

the true mean and variations in this response mean with stimulus

niuTiber. The deviation of the EER from the true mean is the result of

the variance of the potentials across the ensemble (see Bendat and

Piersol, 1966). Both the response signal and background noise contri-

bute to this .variance since it is unlikely that the response activity

from every subject is identical and the noise is random. However,

since the signal activity generally has less amplitude (Perrn and

Childers, 1969) and greater sirmilarity across the ensemble than the

noise potential, the latter should account for most of the ensemble

variance. Therefore the estimation deviations of the EER should be

primarily the result of the background noise. The range of these

deviations can be predicted utilizing the Tchebycheff inequality and









ensemble variance. This prediction takes the form of confidence

intervals about the EER.

In order to examine the second parameter, the variations in the

EER caused by changes in the actual inter-subject response, it is

necessary to separate response signal from the noise activity in the

ensemble average. Since this is not possible at this time, the

present investigation will generally be restricted to variations in

the more prominent components of the EER's which are fairly consistent

across stimuli. Since the stimuli are identical, these variations

should result in some trend of change rather dtan random fluctuations.


3.2 Selection of Bandwidth

The EER and other results presented in this chapter and the

succeeding chapter are obtained from the stimulus provoked and resting

EEG ensembles mentioned in Chapter 2. In addition, most of the dis-

cussion of these results is limited to narrow band filtered (0.3 to

15 H:) potentials. Although the fidelity of this more restrictive







2 1
S4 Standard Deviation








o 3
0I -

Eandiuidth Comparison
Th-i ensemble mean and standard deviation
for nro bandJwidths are SLpecr imposed.








bandwidth is not as good as the wider band potentials (0.3 to 50 Hz),

it still retains tie significant EEG potential fluctuations, as

Figure 3.1 demonstrates. In this figure the first and second ensemble

moments for both bandwidths are superimposed and reflect only minor

variations, including a slight phase shift introduced by the filter.

In addition, the upper limit of 15 Hz exceeds the minimum range of

ten times the frequency of stimulation suggested by Ciilders, et al.

(1968) to recover a VER average with good fidelity. Therefore, only

the narrow band averages which provide an accurate description of the

EEG activity will usually be discussed. However, many of the results

presented here are duplicated for the wider band potentials in

Appendix B. This appendix also contains other results which are not

pertinent to the discussion. Any reference to these results will

have a prefix B.


3.3 Ensemble Evoked Response

The ensemble evoked response, EER, to the selected photic stimuli

is provided in Figure 3.2. Each average is designated by the number

signifying which stimulus within the train evoked it. In each

waveform the flash stimulus occurs at the extreme left of each 950

millisecond response average. The first four ensemble response

averages can also be found in Figure 2.1, which depicts the method by

which the averages were obtained. The waveforms are segmented into

seven 135 millisecond intervals to facilitate comparison and dis-

cussion. Again, it should be pointed out that each EER is only an

estimate of the actual evoked potential and contains both evoked and

background activity.








Interval


4
C-

z 5
c-,

' 1'


FIGURE 3.2
Ensemble Evoked Response
Ensemble averages for ten
selected stii:muli are
divided into 7 intervals
of 135 ;rlliseconds.


A cursory glance at the EER's in

Figure 3.2 provides the reader with little

indication that the fluctuating waveforms

contain any response activity. However,

if one utilizes the inverted sinusoid in

the second interval of the fourth res-

ponse as an ER template, the most common

attribute of the w1aveform set becomes

more apparent. This inverted sinusoid,

although sometimes partially distorted,

appears in the second interval of each

response in the figure. The potential

fluctuations across the ten EER's also

share a degree of likeness in the latter

portion of the first and initial part of

the third intervals. However, neither

approaches the conformity found in the

second interval.

In order to better establish the

bounds of the response activity it is

necessary to find some quantitative

measure of this similarity across the

EER's. The degree of similarity between

two responses for a specific interval

can be quantified by a correlation








coefficient which would d provide a total of 45 correlations per

interval for the ten EER's of Figure 3.2. This coefficient, p, can

be expressed as:


1 x'
1= r (3.1)


where P is the time covariance between EER's x and v in interval
xy
i and Y~ and Y are the time standard deviations of these w.aveforms
x y
in this interval. How to reduce these paired correlations to a single

measure of the similarity across the entire set of EER's becomes

the major concern. The method chosen in this study a.'s to average

the correlation coefficient for one interval. This average provides

a conservative measure of the conformity across the ten responses

since most of the correlation coefficients between w\aveform pairs

must be significant to obtain a significant average correlation. The

average correlation reduces the effects of sporadic deviations in

these correlation coefficients, which randomly appear between

paired waveform segments. The significance of these correlations is

questionable, since they do not have any consistent relation to the

stimulus.

The correlation coefficients for each paired EER (Figure 3.2) at

a specific interval are provided in Tables B.1 through B.7 in

Appendix B. The averages and maxima and minima of these correlations

are provided in Table 3.1. The average correlations reinforce the

previous observation that the activity in the second interval across

the ten EER's is the most consistent. In fact, even the minimum








TABLE 3.1

Average Correlations for Figure 3.2

INTERVAL 1 2 3 4 5 6 7

Pave .30 .88 .12 .05 .06 .04 -.07

Pmax .89 .99 .88 .88 .94 .87 .98

min -.45 .58 -.71 -.92 -.87 -.92 -.87


correlation of .58 is above the 1 per cent significance level, i.e.,

assuming samples are normally distributed and independent (Undenrood,

et al., 1954). The average of the correlation coefficients across

the segmented EER's obtained from the wider band potentials were

generally less than those in Table 3.1 (see Tables B.8 through B.13).

Thus filtering the data appears to eliminate some of the variability

across the response estimates.

The first interval of the EER's is the only other 135 millisecond

time period where the waveforms demonstrate any degree of consistent

correlation. This consistency appears to be the result of the more

common components in the latter half of this interval rather than

the initial deflections after stimulus onset. In intervals 3 through

7 (i.e., 270 to 950 milliseconds after the stimulus) the EER's

demonstrate negligible correlation. This variability of these late

components of the EER seems to indicate that: (1) the average back-

ground activity is much greater than the average response activity

during this period; (2) the individuality of the response for different

subjects is greater, which increases the deviations of the estimate








in this region; or (3) the variations with stimulus number are greater

in the late response. In similar correlation tests, Perry and

Childers (1969) have demonstrated a significant decrease in the

average correlation coefficients of inter-subject VER's following

the first 300 milliseconds of the response, and a decrease in intra-

subject correlations after 500 Inilliseconds. They also note that

the highest correlations are generally found in the second and third

100 millisecond intervals, which is in close agreement with the

previous results. Therefore, the EER and \ER results seem to reject

the reasoning that it is the individuality of the evoked response

which introduces the variability in the late components of the

ensemble average.

It is interesting to note the occurrence of alpha-like activity

in the latter portion of response 40. This singular occurrence may

be the result of a particular combination of the random EEG potentials.

However, these potential fluctuations have a strong resemblance to

the after-discharge activity noted in the studies mentioned in

Chapter 2.

In Figure 3.3 the ten EER waveforms are superimposed with the

time mean removed from each average. In this figure the consistency

of the responses in the region from about 60 to 300 milliseconds

after the stimulus can easily be seen. Because of the similarity of

the EER in this region (R2) it will be designated the primary

response region. In this region there are three prominent peaks.

These peaks are identified in Figure 3.3 as components one, two









uad three, and occur approximately 90 Tsec, 150 msec, and 250 msec

after stimulus onset, respectively.

The average correlation coefficient in the primary response

region is 0.75, which is slightly less than the average correlation

in the second interval of Figure 3.2. This decrease is probably the

result of the increase in interval width from 135 to 240 rrec. Thus,

the second region of Figure 3.3 has more components and greater

variability than the second interval of Figure 3.2. The average

correlation of the third region in Figure 3.3 is 0.23, which indi-

cates a minor degree of similarity not previously found in the corres-

ponding correlations in Table 3.1. This similarity is possibly the

result of some low frequency activity comrimon to the tails of the EER's

which would not be detected by the correlations of the 135 millisecond



3









/"



2
i








R., R R,
FIGURE 3.3
Superimposed Ensemble Evoked Response
Overlapping EER's are divided into three regions for comparison.
R, = 0 to 60 rrsec; P2 = 60 to 300 rrsec; R- = 300 to 950 msec.




28



intervals. However such a low frequency component is not visually

detectable, except for a very slight negative drift in the late

response components.


3.4 Noise in the Enserble Evoked Response

The EER's in the previous figures contain an unl no.,n amount of

background noise. One estimate of the amplitude of the background

activity which remains in the EER is the ensemble average under

non-stimulus (control) conditions. The accuracy of this estimate is

dependent on the relationship between the signal and the noise. For

example, if the signal and noise are independent and the noise is

stationary, this average provides a good measure of the amount of

background activity in the EER. The waveshape of this average is

important only if a particular component appears time locked to the

occluded stimulus, indicating the existence of some ex-perimental

artifact.

In Figure 3.4, this average is subdivided into ten control ..,ave-

forms attending the ten occluded stimuli. The format of this figure

is identical to that of Figure 3.2. The average correlations of the


TABLE 3.2

Average Correlations for Figure 3.4

INTERVAL 1 2 3 4 5 6 7

ave .07 -.02 -.03 .01 -.07 -.00 -.07

Pmax .81 .92 .95 .99 .87 .92 .81

Pmin -.79 -.99 -.93 -.92 -.98 -.93 -.93








Interval


FIGURE 3.4
Enserble Control Averages

Ensei a'le averages for ten
occluded stimuli are
dividc-d into 7 intervals
of 155 milliseconds.


intervals of Figure 3.4 given in Table 3.2

demonstrate the lack of correlation between

the control w.aveforms. These correlations

are similar to the average EER correlations

in the last four intervals of Table 3.1.

This supports the previous observation

that there may be little response activity

in this region.

In every interval the root mean square

of the control w.av-eforms, Cr is less

than the rrrs value of the responses, R rMs

in the second interval of Figures 3.2

(where these rms values have the dc removed).

The iiniinum RP is only 1.03 times the

maximum Cs and the magnitude of the

largest R nis 9.7 times the smallest

Crm. The average of all ratios of R

to C is 2.6. Therefore, assuming that

the ensemble control average provides a

good estimate of the amount of background

activity remaining in the averaged response

and that the signal and noise are additive,

what is the signal to noise ratio (S/N) in

the EEF? To date, the answer to this

question cannot be resolved, since it is








necessary to also know the phase of the average background activity.

How,,ever, with only a magnitude estLmate, the upper and lower bounds of

this signal to noise ratio can be found by adding and subtracting the

magnitude of the control from the EER and dividing by the control

amplitude, i.e.,

R -C R + C
TES TmtS Tllms ITS
Ms < S/N < lrs ms (3.2)
C C
riS lIms

It should be noted that in the case when the lower bound is negative

(which is a theoretically impossible event) the lower bound is set

to zero.

Rather than establish these bounds, a more common practice is

to estimate the signal to noise ratio in the average response by

dividing the magnitude of the average response, Rs, by the average

noise, Cms (Perry, 1966), i.e.,

S R
S = rms (3.3)
N C
rms

Using this estimate, the relative signal to noise measures on the

previous page provide the maximum, minimum and average S/N. Thus, one

would predict that the EER's in the second interval of Figure 3.2

contain approximately 10 to 50 per cent noise with an expected value

of 28 per cent. In Chapter 4 it is shown that under certain assump-

tions these predicted S/N values are reinforced by the degree of

similarity (i.e., correlation) previously found between the EER's.

Therefore, the average amplitude under control conditions probably

provides a reasonable estimate of the amount of noise distorting the




31


EEP. waveform.

If the overall rms value of the control w aveforms approximatelyy

0.5 pv) represents the expected noise distortion in the response

average, then Figure 3.5 provides an upper and lower bound of the

noise distortion exTected in the EER. The actual magnitude of the

background noise potentials at any instant could be several times

their rms value. Hown.ever, the bounds in Figure 3.5 provide an

average approximation of the amplitude of the background noise which

could be contained in the EEP estimate. That is, the average

response without noise should be expected to be within these bounds

if the control w.aveforms provide a good estimate of background

noise and if tde EER is a good estimate of the average signal plus

noise activity. The latter stipulation, the quality of the EER

estimate, provides further confidence limits in the average evoked

activity. Thus a more accurate bound on the evoked activity is a

combination of the noise bounds and the confidence limits on the

EER.


1 Ayv


1 2 3 4 5
Response Number
10 20 30 40 50



FIGURE 3.5
Noise Amplitude in EER
Overall pr s control value is added and subtracted from the center
EER waveform.


24~


:2
'U




32



3.5 Quality of the Enserable Evoked Pesponse Estimate

The EER is a good estimator of the signal plus noise mean if the

deviations of this estimate about the mean are relatively minor. A

measure of this estimation deviation is provided by the standard

deviation, of(t), of the underlying process. The actual value of

Of(t) is unknown. However, an unbiased estimate of of(t) is the

ensemble standard deviation, rf(t), which is calculated from the

equation

1 N 1/2
Jf(t) = { N i-1 [fi(t) ff(t)]-} (3.4)


.where N is the nmabuer of fi(t)'s, subject EEG's, and mf(t) is the

ensemble average given by equation 2.2.

A plot of 3f(t) for the potentials attending e3ch of the ten

is provided in Figure B.5. These ten waveforms are superimposed in

Figure 3.6 in order to observe any stimulus effects on 8f(t). As

S

0 4
> I HI-2




S *-i '- /iV /

'- FIGURE 3.6
1 Superimposed Ensemble Standard Deviations

Each standard deviation attends one of the ten
selected stimuli which occur at the extreme
left of the figure.








can be seen in this figure, the fluctuations of the af(t) waveforms

do not appear to have any consistent relationship to the stimulus,

other than a slight average increase around the primary response

region.

If the ensemble standard deviations in Figure 3.6 provide a good

estimate of the standard deviations of the process, then the standard

deviation of the mean estimate, mf(t), is


o (t)
sd[lf(t)] ~ (3.5)


This standard deviation can be incorporated into the Tchebycheff

inequality to provide reliability or confidence bounds on the EER

estimate. This inequality provides a lower bound on the probability

that the true mean and estimated mean will differ by less than n

standard deviations of the estimate. For example, we can be at least

89 per cent confident that the true fiean and estimated mean (EER)

will differ by no more than three standard deviations (calculated

by Equation 3.5) of the estimate, regardless of the underlying

probability distribution. If the underlying probability distribution

is Gaussian, the confidence increases to 99 per cent.

The S9 per cent confidence intervals for ten EER's are plotted

in Figure 3.7. The bounds in this figure are constructed from ten

EER waveforms plus or minus three-sevenths of the ensemble standard

deviations superimposed in Figure 3.6. As can be seen in Figure

3.7, or calculated from Figure 3.6, the overall width of the intervals

varies from about 2 to 4 Dv. Although this appears to be a relatively








large amount of variability, the shape of the bounding waveforms

generally conforms to the EER waveshape, with a few exceptions. The

most notable exception occurs in the first response, around 300

milliseconds after stimulus onset. This variation could have been

anticipated from Figure 3.6, since the standard deviation of the

potentials in this region of the first response partially stands

above the remaining response deviations.

If the confidence intervals in Figure 3.7 are compared with the

VER confidence intervals in Figure 3.8, the reliability of the VER

is greater than the EER in two out of the three cases shown (J.M.

and B.A.). In the third case (J.F.) the reliabilities are approxi-

mately equal, if the greater amplitude of the VER is taken into

consideration. In all three cases the VER's were obtained by

averaging the subject's response to fifty stimuli. Therefore, for




V I 1 grv



1 2 3 4 5
Response Number
10 20 30 40 50




FIGURE 3.7

EER Confidence Intervals
The above intervals are plus and minus three standard deviations of
the EER estimate. This is about 3/7 of the deviations in Figure 3.6.








the same number of averages, the EER appears to provide as good an

estimate of the response activity as some VER's.

The VER amplitude is about one and one-half times that of the

EER's in all of the cases show.n. This larger amplitude and greater

reliability of the VER estimate implies that the inter-subject

response variability is greater than the variability of responses

obtained from a single subject. This is i.hat one would predict from

the comparison of \VR's obtained from a single subject and those

obtained from several subjects. In addition, the background noise

variability would influence tie reliability of the estimate. Thus

inter-subject EEG variability (e.g., amplitude and frequency deviation

in the background activity) probably reduces the confidence in the

EER estimate. One variable parameter which might reduce confidence

in the VER and not in the EER would be a general change in the evoked

activity with the rnmber of stimuli. In order for this variation not

to affect the FER it is necessary for this change to be fairly con-

sistent across the stimuli for the subject population.




J.F. J.M. B.A.


I\ ;' l: llJ Y



FIGURE 3.8
VER Confidence intervals
Scale and procedure is the same as in Figure 3.7. Each of the above
suhiects are members of the ensemble.









3.6 Variations in the Enseirble Evoked Response .withi Stimulus Number

As noted previously, the EEP provides an estimate of the changes

in tl.- evoked rc-sonse (cofuion to the subject population) with repe-

titions of the stimulus. That is, each EER in Figure 3.2 is an

estimate of the activity which is time locked to a specific stimulus

within the periodic train. In order to obtain an accurate estimate

of the response changes related to the stimulus number, it is necessary

to remove the EER variations resulting from the backCground noise.

Since this cannot be accomplished at this time, the EER's across

stimuli will be examined for trends of change, noting that variations

caused by the background noise should be random in nature.

In a preliminary investigation of the EER's in Figure 3.2 as

a function of stimulus number, little transient change is discernible.

The only noticeable change is a general decrease in amplitude between

the first and remaining responses. The first EER appears to contain

a low frequency component, particularly evident in the latter portion

of this response, which is not as predominant in the EER's that

follow. In the superimposed EER's in Figure 3.3, this low frequency

activity causes the initial components of the first response to rise

above the other responses and the later components to descend below

the remaining responses. In addition, the EER to the first stimulus

contains a sustained.positive component about 300 msec, which does

not have a counterpart in any of the following EER's. However, the

reliability of this component is questionable because of the increase

in the ensemble standard deviation in this region. It therefore








appears that not only does the average response to the first stimuli

differ from those that follow, but that the predominant component

(around the third peak: of Figure 3.3) of this response is more

variant than any other.

If the activity outside of the primary response region (60 to

300 msec) in Figure 3.3 is ignored and the means within the correlated

section are removed from each response, the amplitude of the first

response in this region does not stand out. For example, if the

magnitude of the waveforms in the primary response region are plotted

as a function of stimulus number, the amplitude of the first response

is no longer so predominant. A plot of both the nns and peak to

peal:, ptp, amplitude variations for the individual response in the

interval is provided in Figure 3.9. For this figure the nms value is

about a :ero mean and the ptp value was measured from the second and

third primary deflections indicated in Figure 3.3.

Since the responses are very similar in this region, the rms and

ptp measures provide essentially the same information. Both demon-

strate a slight decreasing trend in amplitude with the number of

stimulus presentations. However, the magnitude of the amplitude

fluctuations is not mUuch smaller than the overall decrease; therefore

the significance of the latter appears questionable. In a plot

(Figure 3.10) of the changes in the time delay or latency of the

three peaks in the 60 to 600 millisecond interval no significant

trend is discernible. Therefore, the above results imply that little,

if any, transient change exists in the first fifty responses for








0
rO
> in





4-- -r4'
ii .,-
G


1.1-

1.0


0.9

0.S


0.7

0.6-


10 20 30 40 50


FIGURE 3.9

Amplitude Fluctuations of the Primary Response


260 T


240 4


Component 3


Component 2


Component 1


Response
!_L __-J__--I.-I- I .C


1 2 3 4 5
1 2 3 4 5


10 20 30 40 50


FIGURE 3.10

Latency Fluctuations of the Three Primary Components


--- ITS

Sptp
A


Response


2 3


220

200


180


160

140


120


*U
r4


orH
,-



0-


100 -








responses averaged across a subject population. An exception to this

observation is the difference in the overall character between the

first and remaining responses. This may be evidence of some initiali-

zation of the visual system either at the receptor or higher neural

processing level. At the receptor level, Brazier (1967) has noted

that there is a large difference in the amplitudes of the electro-

retinogram attending the first and second stimuli in a repetitive

train. However, she also notes that the amplitude of the second ERG

is smaller than any that follow, indicating that the transient change

at the receptor input exists beyond the first flash. The persistent

transient variations (11 to 20 seconds) noted by Regan (1966b) in

responses recorded at the scalp also imply a more sustained initiali-

zation period than is found in the above EER results. However, Regan

utilized a continuous sinusoidal stimulus which restricts extra-

polation of his results to the present study. If initialization is

not a factor then a high degree of variability in the first response

might account for the relative uniqueness of the EER to the first

flash. The ensemble standard deviation following the first stimulus

partially supports this reasoning but it does not explain all of the

component differences between the first and succeeding EER's. There-

fore, system initialization and inter-subject response deviation may

both contribute to this difference.


3.7 Average of Ensemble Evoked Responses

The results of the previous sections indicate that the EER provides

a relatively stable response estimate in die region from 60 to 300





40












a,






a






o O






:), r-


o 0 c



CD 0 v
.. ,-- O O
oL .







S- 0 r-
.- [" O i ,

C) LUC







S 0
0 *r-
C ) L C }





I 0I
/ ro
























o
1 ^ Ci 'f








msec. If this region is stipulated to contain the primary activity

of the evoked response, then very little response infonnation should

be lost by averaging EER estimates across stimuli. This average

should further reduce the average background noise activity and

increase the quality of the response estimate. That is, the variance

of the background noise and the variance of the EER estimate should

both decrease with the increase in the total nLumber of averages

from fifty to five hundred.

Figure 3.11 provides the average of the ten EER's and the ten

control waveforms shown previously. As can be seen in this figure,

this average seems to extract the near sinusoidal complex in the

primary region and reduce the remaining components. The averaged

control is a general overall reduction of the individual control wave-

forms presented earlier. The S/N ratio in the primary response

region of the averaged EER is estimated from the averaged control to

be about 3.6. This is a two-fold improvement over the average S/N

ratio estimated for the EER to a single response from the individual

control waveforms. If confidence intervals were provided for the

averaged FER,each interval would be less than 0.5 iv from this

average. Therefore the average of the ten EER's provides a reliable,

relatively noise-free inter-subject response norm.


3.8 Summnar'

The average over an ensemble of subject EEG's (EER) extracts

a relatively consistent estimate of the response to visual stimuli.

1Te ccnsistenc- cf ten EER's is found to be greatest for components








in the 60 to 300 msec region following stimulus onset. The variations

of these components with stimulus nurmber are generally insignificant,

indicating little change, either habitual or otherwise, in the activity

evoked by the first fifty stimuli. However, a general modification

in the character of the entire average waveform is noted following the

EER to the first stimulus. It is suggested that this change is the

result of initialization of the visual system and a greater variability

of the first response across the subject population. This variability

of the first response is demonstrated in confidence intervals about

the ten EER estimates.

The amount of noise activity in the more consistent region of the

EER's is estimated by ensemble control averages. From these estimates

the average S/i' ratio in the EER is found to be 2.6. In order to

further extract the response signal from the noise, the average of

the ten EER's is calculated. This averaged EER provides a more

reliable response estimate than single EER's and improves the S/N

ratio. The response estimate extracted by tiis average is almost an

inverted sinusoid, beginning approximately 100 msec after stimulus

onset.












CHAPTER 4

THEORETICAL ASSUMFTIONiS UNDERLYING RESPONSE AVERAGING


In order to theoretically justify the results obtained from many

EEG data processing techniques, it is necessary to assume certain

hypotheses about the statistical character of the fluctuating poten-

tials. Investigators who employ the averaging techniques to extract

time locked signals from the incoherent background noise generally

assume that: (1) the noise is a wide sense stationary random process;

(2) the signal and noise are independent; and (3) the signal and

noise are additive. The latter assumption is probably the most

important since non-additive components cannot be separated by the

linear process of averaging. However, a signal dependent on non-

stationary noise can be extracted if the degree of dependency and

non-stationarity is minor.

In this chapter an attempt will be made to gain positive or

negative evidence about these assumptions by examining the data

obtained from the subject ensemble. The results of these examinations

will not, however, enable the experimentor to negate or to confirm

these assumptions because of the lack of knowledge about the poten-

tials being tested. Thus, to examine one of the above assumptions,

the test utilized must hypothesize the nature of the signal and

noise parameters. In many cases the t-st hypotheses depend on one

or n.to of the above assumpTticons, conditioning the conclusLiveness of









the result. The present study is therefore intended to provide

insight into the possible workability, rather than ultimate veracity

of these assumptions. Sudc an examination may then provide a foun-

dation for further studies into the statistical properties of evoked

signals in background noise.


4.1 Stationaritv of the Background Noise

A random process is said to be stationary if the statistics of

the process are time invariant (see Davenport and Root, 1958). For

example, a large nwrmber of statistically independent sample functions

which are simultaneously recorded constitute a wide sense stationary

process if the mean and autocorrelation function at one instant of

time equals those at all other times. In the present study the

sample functions are the EEG records of the fifty subjects. The

time varying function of particular interest is the ongoing activity

under stimulus conditions. However, these potentials cannot be

separated from the evoked response. Therefore, this investigation

must rely on the resting potentials under non-stimulus conditions to

estimate the background activity. It then becomes necessary to

work under the constraint of the second assumption (i.e., that the

signal and noise are independent) to examine the first.

Since an average is an estimate of the first moment of the

ensemble, it is the stationarity of background noise mean which is

the most critical to the averaging technique. If the noise mean is

highly non-stationary (e.g., the averaged potential fluctuations

exceed those of the response average) then the averaging process








cannot extract a detectable signal from the background noise.

Utilizing the resting EEG potentials to approximate the background

activity, the noise mean estimate (ensemble control average) seen in

Figure 3.4, is not time invariant. IHow.ever, are these fluctuations

the result of an underlying non-stationary process or caused by the

variations inherent in the estimation process? In order to answer

this question it is necessary to determine the inherent variance of

the mean estimate of N sample functions. If the N sample functions

are independent, which is plausible since individual subject EEG's

should not depend on one another, then this variance of the estimate

is (see Bendat and Piersol, 1966)


o}(t)
VAR\[if(t)]= N (4.1)

where o'(t) is the variance of the process which is wuknown and

must be estimated. This latter estimate also has variance associated

with it :.-hich again must be estimated, and so fcrth. In order to

prevent endless estimation of these statistics, the estimate of the

variance, .2-(t), will be assumed equal to the actual variance,

oa-(t), i.e.,

6-2(t) = 2(t) (4.2)

N;ow. utilizing the Tchebycheff inequality, the range of the mean

estimate can be predicted without knowing its exact probability

distribution. This irne,:uality states that the probability that the

difference between the true m'an and estimate mean will be greater








than C standard deviations of the estimate will be less than 1/C2,

i.e.,


PROB [Imf(t)-mf(t)l C{sd[r f(t)]}] > 1 (4.3)
C2

In Chapter 3 this inequality was used to provide S9 per cent confi-

dence limits (C=3) about the EEP and \EP estimate, noting that


OfCt)
sd[mf(t)] -= 3 (4.4)


Similar confidence limits can be obtained by hypothesizing a

mean and using 3f(t) to obtain confidence limits. For example, if

the noise mean, mf(t), is hypothesized to be stationary and equal to

zero, then Equation 4.3 becomes



PPOB[Jlf(t)|> C{ f(t)/N] < 1 (4.5)
C- 2

Again setting C=3, the 89 per cent confidence intervals about the

hypothesized stationary mean can be established. In addition, from

Equation 4.5 another value of C can be calculated from the ensemble

value of mf(t the ensemble control average, and 3f(t), the control

standard deviation, i.e.,

mf(t)
C < (4.6)
,(t)

The maximum value- of C found in Equation 4.6 can then be compared

to the threshold value of 3 -- rejecting the stationary hypothesis

for C>3 and accepting the hypothesis for C<3. This is equivalent








TABLE 4.1

Maximum Number of Standard Deviations of

Ensemble Control Average About a Zero Mean

Control 1 2 3 4 5 10 20 30 40 50

Cmax 2.56 1.62 2.53 2.43 2.24 3.00 2.37 3.07 3.69 2.37
max

to testing the h),yothesis at a significance level of greater than

5.5 per cent since the probability of acceptance of this two-tailed

test is greater than 1-1/C2, if the hypothesis is true. If the under-

lying distribution is Gaussian, this significance becomes greater

than 1 per cent. The maximum value of C for each of the control

trials, whose waveform is provided in Figure 3.4 is given in

Table 4.1. From this table the stationary mean hypothesis is

accepted seven times, rejected twice, with one indecision. These

results would therefore iLply that the ensemble mean of the resting

EEG, and perhaps the background activity, is nearly time invariant

with an occasional non-stationary trend.

Another estimate of the degree of fluctuation of the first

moment of the background activity is the plus-minus average suggested

by Schimmel (19671. This estimate is similar to the EER except that

alternate sample function are subtrtacted in the averaging process

rather than added. This estimate cam be expressed mathemratically by


odd even
ft) = p f (t 7- f ( t)] (4.7)


If the evoked response is assL-'ed to be nearly the same for each








subject, i.e., near deterministic, then the resultant average should

contain mainly background activity. The reliability of this estimate

is, ho\.ever, contingent upon the third assumption (i.e., that the

signal and noise are additive) in addition to the second.

With this restriction in mind, the preceding test for station-

arity can also be applied to the plus-minus average. The standard

deviation of this estimate about the true + average is the same as

the standard deviation of mf(t) provided in Equation 4.4 since +o (t)

is the same as the variance, a o(t), of the process (see SchiTmmel)

under stirmTlus conditions. By assuming the response signal to be

near deterministic, this variance is mainly the result of the back-

ground activity. If i.we again assume that the estimate of the variance

is very near the actual variance, the values for C can be calculated.

The values for Cma provided in Table 4.2 again indicate that the
max r
first moment of the background activity is stationary, i.e., the

stationary hypothesis is accepted for every + average.

If the signal is assumed to be non-deterministic, it contributes

to the estimated variance of the process and the below Cmax values




TABLE 4.2

Maximum Number of Standard Deviations

of + Estimate about Zero Mean


Response 1 2 3 4 5 10 20 30 40 50


1.93 2.34 2.69 2.00 2.21 2.14 2.30 2.49 2.27 2.82


C
max








increase. That is, the noise variance is now the estimated ensemble

variance, 8.(t), minus the signal variance. In addition, the average
a ef ra.e
will contain some signal activity. In order to accept the stationary

mean hypothesis for the ten estimates the standard deviation of the

signal cannot be greater than 10 per cent of the total, Of(t). Such a

limitation is reasonable since present magnitude estimates of the

evoked response indicate that the background EEG activity has an

amplitude about five to ten times as great as the response (Kitasato

and Hatsuda, 1965; Perry and Childers, 1969). Therefore, with the

incoherent nature of the noise and time locked character of the

signal, the ensemble variability of the former should be at least

ten times greater than the variations of the latter.

In order to check the validity of the previous test, it is

interesting to examine the stationarity of the response average, i.e.,

EER. Again, assuming that the signal and noise are additive and

independent, and that the signal standard deviation is at most 10 per

cent of the total ensemble variance, then the values of Cm for the
max\
ten EER's in Figure 3.2 are given in Table 4.3.



TABLE 4.3

lMa-imumT Number of Standard Deviations

(times 0.1) of EER Estimate About Zero Mean


Response 1 2 3 4 5 10 20 30 40 50

C 42.5 36.1 28.1 30.1 :2.6 38.3 33.0 35.8 28.6 19.0
m, x








These results clearly demonstrate that the mean of the signal with

noise appears to be non-stationary, which is imperative for the signal

to be extracted by averaging. Even if the signal variance accounts for

the entire ensemble variance, the Ca values calculated (i.e., 1/10
max

of the above results) would d indicate the EER to be non-stationary

sLx out of ten times. Therefore, the preceding stationary mean test

can detect non-stationary trends under the worst case conditions,

which indicates that it is a valid test.

In order for the background activity to be wide sense stationary

the autocorrelation function must also be time invariant. The auto-

correlation function, R(t, t-T), is a function of both time, t,

and the time shift of the correlation, T. For R(t, t-r) to be time

invariant the autocorrelation function at time t1 must be the same at

all other time t, for a given T, i.e.,


Rf(tl, tl-r) = Rf(t2, t2-r) (4.8)

The estimate of this function is obtained utilizing Equation 4.9.



Rf(t, t-T) = 1 f.(t)f.(t-T) (4.9)
1 1
i=l

Using Equation 4.9, a three-dimensional plot of the ensemble auto-

correlation function estimate versus t and t-T for the resting EEG

ensemble (Control 1) is provided in Figure 4.1. Since the previous

test indicated that the mean is stationary, this average is sub-

tracted from the autocorrelation to eliminate any sampling


















0


(U







4-J r
4-







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cm
O E-


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fluctuations of tiis first moment. Therefore the autocorrelation

function in Figure 4.1 is more correctly an autocovariance function

which has the form

N
Cf(t, t-T) = N [fi(t) mi(t)][f(t-T) mi(t-T)] (4.10)


The magnitude of this function, like the autocorrelation function, is

a maximum for T=0, which occurs along the main diagonal of Figure 4.1.

It also should be noted that the covariance function is symmetrical

about T=0; therefore the super and subdiagonal values are equivalent.

In order to examine the stationarity of this function one muLst

examine the fluctuations of Cf(t, t-T) for a fixed T and varying t,

which is analogous to viewing the variation C along the main and off

diagonal terms in Figure 4.1. Fluctuations of Cf(t, t-T) w itJ time

are noticeable in this figure; however, their significance is

questionable. From the actual values of Cf(t, t-T) provided in

Table B.21 of Appendix B, one does notice that the magnitude of these

fluctuations is generally independent of the diagonal used, i.e.,

the time shift, T. Therefore fluctuations of the main diagonal

for T=0, which is equal to the ensemble variance, should provide as

good a measure as any other Cf(t, t-T1) of the time fluctuation of

the correlation function.

The ensemble standard deviations of the ten control trials are

superimposed in Figure 4.2. These deviations are overlapped to pro-

vide a simultaneous estimate of the degree of fluctuation and to

provide a comparison with the standard deviations presented in the








sane maimer in Figure 3.6. The reader interested in the individual

ensemble standard deviations of each control is referred to Figure B.5.

The maximum, minimum, and average values of the waveforms in

Figure 4.2 are 4.5 pv, 1.S pv, and 2.S uv, respectively. If the

standard deviation is hypothesized to be stationary and equal to the

average of these w\aveforms, then the fluctuations of the estimated

values exceed a range of 61 to 160 per cent of this average.

This variation is probably greater than one would expect from the

estimation error; however, without knowledge of the higher moments

one cannot further quantify this statement.

These higher moments are essentially descriptors of the finer

detail of the uLnderlying probability structure of the process.

Therefore the profile variations of the ensemble probability density

function, pdf, should provide a rough evaluation of the degree of


5











FIGURE 4.2
1 I */- Iz

Superimposed Ensemble Standard Eeviations

Each standard deviation attends one of the ten
occluded stimuli %which occur at the extreme
left of the figure.








time dependence of these morrents. Previous investigations of the

amplitude probability density function of individual EEG records

ha'e provided conflicting results. Elul (196S) and Saunders (1963)

note that EEC amplitude histograms appear to approximate the normal

(GaLussian) distribution. However, the results of a study by Campbell,

et al. (1967) generally reject the hypothesis that the EEC amplitude

is no-rmlly distributed. A comparison of the ensemble histograms to

the normal distribution may provide additional information about the

underlying EEC pdf.

Histogram estimates of the ensemble pdf at every 10 millisecond

interval after the first occluded stimulus are shown in Figure 4.3.

Each of the seven row.'s in this figure approximately correspond to

the seven 135 millisecond intervals used previously to segment

the ensemble control averages (Figure 3.4). As can be seen in

Figure 4.3, there appears to be little consistency between the

ensemble histograms at different time instances. The significance

of these pdf deviations with time is conditioned by the variance of

the estimation process (i.e., the histogram approximation). However,

the w.ide contrast in contour between individual pdf estimates appears

to exceed the limitation of sample variability. For example, the

histograms in Figure 4.3 vary from unimodal (t=60 msec) to bimodal

(t=90 msec), -asymmetric (t=710 msec) to near s\-rmmtric (t=100 msec),

and leptokurtic (t=490 msec) to platykurtic (t=330 msec).

In addition to this variability, few of the histograms reserrble

the normal distribution. An attempt to quantify the degree of















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normality utilizing the Kolmogorov-Smirnov test proved unsuccessful.

Although the }olmogorov-Smirnov test is generally more efficient than

the chi-square goodness of fit test (f1iller and Freund, 1965) for the

number of available samples (50), the necessity of estimating the

mean and variance reduced the sensitivity of the test (see Lilliefors,

1967). That is, there was a very high probability of accepting

the hypothesis of normality when it w.as false.


4.2 Dependence of Signal and Noise

In the previous section .'e noted that if the signal and noise

are additive and the signal near deterministic, then the plus-minus

average should provide an average estimate of background activity.

If the same averaging process is performed on the resting EEG,

then a similar estimate of the noise under non-stimulus conditions

is obtained. A comparison of the general character of these

processed potentials should provide an estimate of variations in

the average noise under stimulus and non-stimulus conditions. How-

ever, hot. does one make such a comparison? The w.aveshape of these

potentials for ten signal and ten control averages (see Figures B.3

and B.4) has little meaning, since they were recorded at different

times. But the degree of fluctuation in the -ms measure of the

average should provide an estimate of average energy under both

conditions. Table -4. provides the rms values of each plus minus

control and response w.aveform. The range of rms values is nearly

equal under both conditions. Thus, the time fluctuating averages








TABLE 4.4

rZ'S Values in j' of Averages

Under Stimulus and Non-Stimulus Conditions


Waveform 1 2 3 4 5 10 20 30 40 50

Response 0.46 0.37 0.42 0.28 0.39 0.3S 0.32 0.36 0.27 0.46

Control 0.26 0.36 0.40 0.38 0.32 0.28 0.49 0.58 0.45 0.67


of the background activity and resting activity are of the same

magnitude. Since the signal potentials are present in one case and

not the other, the general implication is that the magnitude of the

average noise is independent of t]e signal.

Another property of the background and resting EEG which can

be investigated under the assumptions of a near deterministic signal

added to noise (s+n), is the ensemble standard deviation. If the

signal and noise are independent, the standard deviation of the

resting EEG ensemble (Figure 4.2) should have a character similar

to the standard deviation of the stimulus provoked activity

(Figure 3.6). That is, the ensemble standard deviation of signal

plus noise, as+n(t), is given by



c. (t) = [ (t) + o2(t) + 20 (t)]l/2 (4.11)
+n n sn

If the signal is near detenninistic and independent from the noise,

then the signal variance, a (t), should be much less than the noise

variance, oa(t), and the covariance between the signal and noise,

o2 (t), equals zero, which reduces Equation 4.11 to the following:
sn1








S ( (t) = n(t) (4.12)
n

Thus, the standard deviation of the signal plus noise ensemble

is primarily the result of the background noise and should be similar

to the standard deviation of the resting EEG, if the stimulus does not

affect the noise (i.e., signal and noise are independent). As noted

previously, the tine fluctuations of the standard deviations should

not be the same for both cases, since these deviations were recorded

at different times. If w.e compare the standard deviations, 3 s+(t)

and n(t), provided in Figures 3.6 and 4.2 respectively, the overall

average and peak to peak values are very similar, i.e.,

ave[ s+n(t)] = 2.82 ou ave[6n(t)] = 2.86 p\

ptp[8 (t)] = 3.1 uv ptp[n (t)] = 2.7 u\

From these measurements one could predict that the variance of the

noise is independent of the signal. However, in Figure 3.6 a slight

increase in the average standard deviation, 8 (t) w.as noted around
s+n I

the primary response region. Since similar trends do not appear in

the standard deviations, 8 (t), in Figure 4.2, this increase could

imply some signal and noise dependence.

If the near deterministic signal and noise are dependent and the

average value of the noise is small, then the ensemble standard

deviation of signal plus noise becomes


O (t) = {o2(t) + E[s(t)n(t)]} (4.13)
s+n n

where E[s(t)n(t)] is the mathematical expectation or stochastic

average of the product s(t) n(t).








The ensemble standard deviation for noise dependent on a deter-

ministic signal should then increase when the amplitude of the signal

increases. This is the case in Figure 3.6, particularly for the

standard deviation of the first response (see Figure 3.1). However,

if the signal is relatively non-deterministic, then the standard

deviation of independent signal and noise would also tend to increase

in the region of greatest signal variability. Since it is possible

that the standard deviation of a random signal could be proportional

to the magnitude of the signal, the existence of a random signal would

also explain the slight increase in Figure 3.6. In either case thiis

correspondence between signal magnitude and the ensemble standard

deviation is minor, except for the first response; this would imply

that the standard deviation of the noise is relatively independent

of the signal.

A comparison of autocovariance functions under stimulus and

non-stimulus conditions did not reveal any additional information

about the dependence of the signal and noise. The variations of this

function appeared to depend primarily on the variance, o2(t), as noted

previously (see Figures 4.1, B.6 and B.Y). The histogram estimates

of amplitude probability density functions under stimulus and non-

stimulus conditions were too variable to attempt a comparison (see

Figures 4.3 and B.8).


4.3 Additivity of Si-nal and Noise

In the previous chapter the maximLun (9.68), minimum (1.03) and

average (2.6) nms signal to noise, S/N, ratios were computed. These








ratios were calculated from the ratio of the rms value of the EER's

in the second interval to the rIms value of the ensemble control

average in any interval. In addition, the correlation coefficients

1.ere found between EER w.aveforms. Since the S/N ratio estimates

the ajrount of noise remaining in the EER w.aveforms, a decrease in

the S/N' ratio should be reflected by a decrease in the correlation

coefficients between waveforms. 1T1at is, if tie signal and noise

are additive, then the correlation between EER's should depend on

the relative amount of uncorrelated noise and correlated signal

contained in these averages.

In Chapter 3 the correlation coefficient between EER wiaveforms

x and y was calculated from the time standard deviations, T' and 4 ,

and the time covariance 'f' i.e.,



P~' = f (4.14)
x V

The S/l! ratios were found using the equation


R
S/N = (.15)
uns 4
n

where Ys is an estimate of the average signal time standard deviation

and 7' is an estimate of the average noise deviation under stimulus

conditions (assuming the signal and noise are independent). Both of

these functions are dependent on the time variance,, a, nd a mathe-

matical relationship between the two can be established with certain

constraints.









If the EER is composed of signal plus noise, i.e.,



x = s5 + n1 (4.16)

y = 52 + n2 (.17)

then the time covariance is



= T 'T + T (4.18)
x-v sis: Sn- s2nI nln2

If the signal and noise are statistically independent, which insures

linear independence, then the t-.o covariances between the signal and

noise in Equation 4.18 are zero and Equation 4.14 becomes

'" + 'T
SiS2 nin2
p = 1/ (4.19)
[() + (E+ )]1
5s n1 -2 n2

If the noise is stationary then the time standard deviation, ',' of

its estimated average over equal periods of time should be the same,

providing the period is sufficiently long. If we issuTme that the

period between stimuli is sufficiently long and that the ensemble

average signal is near deterministic for each stimulus, then



'll n = m'
sI 2 s (4.20)

and

T T = T (4. 21)
ni n2 n

The correlation coefficient between EFR w.aveforms can then be written

as in Equation 4.22.








'l' + 4'
SIS2 n1n2
S= ----(4. 2)
y'2 + 112
s n

which reduces to


PS(S/N)2 + p
p = (4.23)
1 + (S/N)2

where ps and pn are the correlation coefficients between the signal

and noise and S/N is defined by Equation 4.15 with the ~ deleted.

Since we assumed previously that the average signal was near deter-

ministic, ps is almost unity. The noise should be incoherent with

the stimulus; thus p is near zero and Equation 4.23 reduces to


(S/N)2
P = -(4.24)
1 + (S/N)2

Therefore, if a pair of identical signals are added to stationary

noise, which is independent from the signal, then the correlation

coefficient between the resultant waveforms can be calculated from

the S/N ratio.

Using the three S/N ratios given on page 59, the following

values of p are found from Equation 4.24:

Pmax = 0.989

Pmn= 0.515
Pmin
Pave = 0.871

Therefore, subject to the above stipulations, one would predict the

above values for correlation coefficients between the EER responses









in the second interval. A comparison of these predicted correlations

with those calculated in the previous chapter, i.e.,

ma = 0.99

Pmin = 0.53
P min O.S
Pave = 0.88

discloses that the former and latter are nearly equivalent.

The implications of this strong similarity between the predicted

and measured coefficients are two-fold. First, the R to C
11RS ImS
signal to noise ratio provides a good estimate of the background

activity remaining in the EER. Second, the assumption of an additive

signal to noise is further supported. One argument which might be

presented against the validity of the predicted coefficients is that

the ;assumptions of pn=0 and p.=1.0 are not realistic. How;.ever, the

average correlation coefficients calculated across segments of the

control w aveforms were very near :ero. This w-ould partially reject

the argument, particularly in the case of the average predicted

correlation coefficient.

Another method to investigate the additivity of the signal and

noise is to observe the growth of the signal to noise ratio with the

number of averages. For example, if a near deterministic signal and

random background noise are additive and independent, then the

waveform of the average signal will remain essentially constant and

the variance of the average noise will decrease directly with the

nulnber of stumiations in the average. If we assume that the noise is

stationary, then the true noise mean is a constant, which in the








present study is preset to zero. The nIs value of the average noise

\.ill approach this zero value as the number of sLmmmiations, N, becomes

very large (noting that the variance of this average about the zero

mean decreases by a factor of N). The rms value of the average signal

should be relatively invariant with the number N. Therefore the

signal to noise ratio, S/N, should be expected to increase as the

number of summations in the average increases.

The minimum rate of the S/N ratio increase can be specified in

terms of the true ensemble variance, oa(t), and a probability deter-

mined from the Tchebycheff inequality (see Equation 4.3). Thus, we

should expect the estimated S/N ratio to reflect a random increasing

trend with the number of summations in the average. However, how

can this increase be examined? The technique chosen to observe this

growth was to calculate the rms value (in the 60 to 300 millisecond

region) of the EER at every even number of summations and to compare

this value to the rms value of the average of the same data. After

a number of summations, the average of the data should provide an

amplitude estimate for the background noise and the EER a signal

amplitude estimate. The S/N ratio can then be formed by dividing the

EER by the average at every even number of summations, N, within

the average. This procedure was repeated for each set of data

following the ten selected stimuli.

A plot of the S/N ratio versus the number of summations N for

the EER's to the first five stimuli is provided in Figure 4.4. Although

the S/N ratio demonstrates an overall slight increase with N in each







6 t S/N


N

24 50


Response 1


Response 2


3 t S/N


N
- --- --j 1 i
24 50
Response 3


S/N


Response 5




"x/

N

24 50


S/N









N

-------esponse 4
24
Response 4


FIGURE 4.4
S/IN Ratios Versus ILmuber of
Sumabnat ions

S/iN Ratios for the EER are
calculated at every even
nLmTber of summations, N.


-----+------r








of the plots in this figure, the random fluctuations of S/N over-

shadow this gro-.th. In addition, it should be noted that the S/N

ratio fluctuations of the first response are nearly tw-ice those of the

remaining responses. In a similar plot of the S/N ratio for EER's to

later stimuli (see Figure B.9), an overall decrease in this ratio with

N can be noted. These plots appear to discredit the signal plus

noise assumption. However, this conclusion contradicts the previous

results in this chapter and questions the results achieved by the

established technique of response averaging. Therefore, the adequacy

of the above test becomes highly suspect.

The inadequacy of this test appears to be the result of: (1) the

large variations in amplitude of individual EEG's which appear to

introduce abrupt changes in the noise estimates; and (2) the S/N

ratio estimate being probably inaccurate except for a large number

of sums (i.e., EER is more noise than signal). The first short-

coming might be eliminated by performing the same test on a single

subject (i.e., using sequential as opposed to ensemble averaging);

the latter shortcoming could be reduced by beginning with forty to

fifty sums and increasing the number of summations. Still another

technique would be to utilize Equation 4.24 and discuss the improve-

ment in S/N by the degree of correlation of the average after N

summations with a preset response template or norm. This correlation

technique would be more sensitive to the waveshape of the response,

which would help to separate signal and noise energy in the case

of fewer summations.








4.4 SLummair

The separation of the foregoing examinations into three groups

is somew-hat artificial because of the inter-dependence of the

results obtained. Therefore, many of the results obtained actually

provide tests for more than a single .assumption. The above examina-

tions are grouped into three areas to help organize the test results

and produce a more readable text.

The results of these tests strongly imply that the mean of the

background noise is time invariant and that its amplitude is rela-

tively independent of the signal activity. The higher order statistics

(i.e., standard deviation, autocovariance and pdf) of the noise appear

to be non-stationary. How-ever, the time fluctuations of the estimates

of these higher order statistics are dependent on the variance of the

estimation process, which was not investigated. The average and

peak-to-peak fluctuations of the ensemble standard deviations under

stimulus and non-stimulus conditions were very similar, which further

indicates that the signal is independent of the noise. The average

correlation coefficients of a theoretical signal plus noise were

nearly identical to those calculated from the actual averaged res-

ponses. Thus, the assumption of signal plus noise was partially

supported. However, a subsequent test of the theoretical increase

in the S/N ratio by averaging signal plus noise data provided incon-

clusive results.












CHAPTER 5

SU.N-A4RY

To date, most investigations of evoked potentials monitored at

the htuian scalp have utilized the average of a series of responses

recorded from a single subject to define components of the evoked

response. This technique has provided a fairly consistent estimate

of an individual's average ER. However, with this method it is not

uncommon tc obtain quite dissimilar averages from different subjects

under identical experimental conditions. This contrast in the ER

estimates across a population has hindered the interpretation of the

effects of stimulus parameters on the ER and frustrated the attempt

to use this average for clinical diagnosis. It may be possible that

the character of an ER is, like a signature, unique for each subject.

If this analogy is valid, utilization of the ER as a research or

diagnostic tool might be limited. However, if one could establish a

particular feature of ER estimates common to a considered normal

subject population, then this template or norm might identify that

quality associated with a population of responses evoked by a specific

stimulus. This would be analogous to identifying the characteristics

attributed to a specific letter of the alphabet in a set of signatures.

The influence of variations in stimulus parameters and physiological

abnormalities might then be recognizable in the degree of deviation

of a subject's ER estimate about this norm.









The present, study investigated one method of estimating such

a response norm. The estimation technique utilized was an average

across an ensemble of subject EEG's containing evoked activity. The

advantages of this method were two-fold. First, it permitted estima-

tion of common inter-subject response activity evoked by a single

stimulus. Second, the collection of subject EEG's strongly resembled

an ensemble of independent sample functions from a stochastic process.

This second advantage permitted an investigation of several prevalent

assumptions about the statistical nature of the evoked signal and

background noise.

The ensemble average revealed a fairly consistent, recurring

component complex following each of a series of photic stimuli, i.e.,

ten stimi.uli chosen between the first and fiftieth stimuli in a

periodic stimulus train. This averaged complex or primary response

mainly consisted of a positive to negative to positive potential,

near sinusoidal in form, which began approximately 60 milliseconds

after the onset of each stimulus. Significant correlation coefficients

w-ere found between all pairings of the ten ensemble averages in this

primary region. Outside of this region most of the paired correlations

between wavefonrms were insignificant. The average of these paired

correlations provided a reliable measure of the conformity across

semnents of the ten responses. Control estimates of the amount of

background noise contained in the primary response region predicted

that this complex could contain from 10 to 50 per cent noise, with

an expected amount of 2S per cent.








Under the assumption that die evoked signal and background noise

are additive, this prediction was further substantiated by the range

of the correlation coefficients between paired response estimates.

Therefore, these results implied that the control estimate provided

a good evaluation of the amount of distorting noise remaining in

the ensemble evoked response, EER. Since the primary response was

detectable in each EER, these results also imply that even with

a signal to noise ratio of unity the average response is observable.

The ensemble average technique also provided an estimate of

the changes in the evoked response 'ith the nLunber of stimulus

repetitions. That is, each EER w'as the average of the activity

attending a single stimulus. An examination of the changes in the

response estimates following each stimuli disclosed moderate com-

ponent fluctuations in amplitude and latency of the primary response.

Since these fluctuations did net exhibit any specific trends or

transient variations wiith the number of stimulus repetitions, they

appeared to be the result of background noise distortion rather than

response changes. The one noted exception, the uniqueness of the

EER tc the first flash, is hypodthesized to be the result of inter-

subject response variability or some initialization of the visual

system. This first response estimate appeared to contain a low

frequency ccimpnent which was not evident in the remaining estimates.

In addition, the time fluctuations of the ensemble standard deviation

abo-it the ensc:i.ee average resembled the waveshape of this average

for the first 15 :.,illisecor.s foll'...'ling ite first stimulus. This









was a characteristic relatively unique to the standard deviation

attending the first stimulus and resulted in a decrease in the con-

fidence of the first EER. Another singular property of the potentials

attending the first stimulus was demonstrated in plots of the EER

signal to noise ratio as a function of the number of subjects in the

average. The magnitude of the variations of the S/N ratio in the

plot for the first response was nearly twice that found in the

remaining graphs. Thus, these graphs reflected the greater varia-

bility of the initial potentials. Because of these unique properties

of the first response estimate it may be advisable to delete this res-

ponse when comparing \TER's across subjects.

The average of the ten EER's waveform appeared to provide a

reliable, relatively noise-free response template. In this template

the primary sinusoidal complex was enhanced and the amplitudes of

the remaining components were reduced. The character of these reduced

components strongly resembled that of a similar single control

template. Thus, this response template further supported the hypo-

thesis that the common inter-subject response activity generally

is restricted to the primary response region. This average response

template also implied that investigations which compare the sequential

average obtained from different subjects should possibly restrict

this comparison to the 60 to 300 milliseo.nd region after stimulus

onset. However, since the above template would not extract inter-

subject signal components whose latency varies from one subject to

the next, this restriction might be inappropriate for some studies.








Three common assumptions about the statistical character of the

signal and noise necessary to theoretically justify response averaging

were examined using the ensemble of subject EEG's. The results of

these examinations generally supported the assumptions that: (1) the

noise mean is stationary: (_) the signal and noise are independent;

and (3) scalp potentials consist of signal plus noise. The first

assumption was investigated by testing tw:o amplitude estimates of

the average background activity against a hypothesized stationary

mean. Of the twenty times tested, the stationary hypothesis was

rejected only twice. The conclusiveness of these tests was dependent

on the second assumption, i.e., the independence of signal and noise.

The combination of signal and noise activity is demonstrated to be

generally non-stationary. If the signal and noise are dependent, the

hypothesis tests on the + average should demonstrate this dependence

by rejecting the stationary hypothesis. Since this was not the case,

the results of the hypothesis tests appear to definitely support the

assumption of a stationary noise mean. An investigation of the

stationarity of the higher noise moments was generally unsuccessful.

Since the variance of the estimate of these statistics was unknown,

the time dependence of these functions could not be evaluated. However,

the relative degree of the time fluctuations of the variance, auto-

covariance, and pdf functions implied that the higher moments of the

noise might be non-stationary.

An investigation of the second assumption, the independence of

signal and noise, was accomplished by comparing the ensemble + average









and standard deviation under stimulus and non-stimulus conditions.

The average provided an estimate of the noise amplitude with and

without the evoked activity. The range of amplitude fluctuations

of this average was nearly equal under both conditions. The average

and peak-to-peak fluctuations of the ensemble standard deviation

under stimulus and non-stimulus conditions were very similar.

Therefore, both the noise average and the deviation about this

average appear to be independent of the signal activity.

An examination of the third assumption, additivity of signal

and noise, was generally hindered by the inability to separate these

tv.'o components. This shortcorrng thwarted an attempt to empirically

demonstrate the theoretical improvement of the signal to noise ratio

by averaging the assuTmed signal plus noise potentials. However,

the relative stability of the average estimates and the previous

prediction of the signal-noise ratio by the correlation coefficient

between w.aveforms supported this signal plus noise assui ption.

The results of this investigation of an ensemble of subject

EEC's which contained evoked responses have provided an estimate

of the common inter-subject response. The estimated response template

found in this study was limited to responses evoked by visual stimuli,

specifically a monocular, periodic flash of light. The applicability

of this teniplate to other studies of responses in this modality

is at present uncertain. However, the method utilized in this

investigation has provided at least one technique for obtaining an

inter-subject response norm to a specific stimulus. It is impractical








to utilize this ensemble technique in most experimental or clinical

investigations because of the large subject population required.

Therefore, the applicability of the technique is restricted to

isolated studies. The results achieved in such studies, however,

should be useful in many other investigations which compare the intra-

subject response average across a subject population.

The ensemble of subject EEG's appears to provide a worthwhile

tool by which to evaluate theoretical assumptions about the statistics

of the EEG. However, the degree of sample variation noted in this

study seems to indicate that a greater number of sample functions

would provide a more reliable evaluation. This is particularly true

in the case of the estimation of the higher ensemble moments

and the ensemble probability density function. In order to obtain a

significant improvement, a fair amount of increase in the number of

sample functions would be necessary. The advantages of such an

increase would probably be overshadowed by the difficulties presented

in acquisition of this larger ensemble.












APPENDIX A

EXPERIMJf.TAL PROCEDURE


A.1 Data Acquisition

The data used in this experiment was collected from fifty subjects

(32 males, IS females). The right eye acuities of all subjects were

normal or correctable to normal as measured on a Snellen chart. Sub-

jects were asked their hand preference and given a far vision sighting

test for eye dominance. One pair of bipolar electrodes (Lexington

C-100) were placed S cm apart on the scalp, located vertically on the

occiput midline with the lower electrode over the inion. This con-

figuration was chosen becuLse it has been noted to produce more

reliable \ER's across subject populations. Each subject \was seated

upright in an electrically shielded Faraday cage and asked to concen-

trate on the light seen through the right eye-piece of a modified

troposcope. During a stimulus (signal) trial the light consisted

of a xenon flash of intensity 3.6 log ft-lamberts subtending 215' in

an invariant incandescent surround of intensity 1.3 log ft-lamberts

subtending 18024'. The flash was presented periodically (approximately

0.9 H:) in t.o trials and periodically (average rate of approximately

0.4 H:) in a third. Each signal trial was followed by a control

(noise) trial identical to the preceding stimulus trial with the

xenon flash occluded. After each trial the subject was permitted

a rest period of two to three minutes: total time required of each









subject was usually less than forty-five minutes.

The potential difference monitored bet\.een the scalp electrodes

w.as fed simultaneously through two Grass P5111 differential amplifiers

(channels 1 and 2) with a gain of 100K and bandpass of 0.3 to 50 Hz.

The bandpass of the signal in channel 2 was further restricted by a

Khron-Hite 330B electronic filter from 0.3 to 15 H=. These amplified

signals were then fed into a Sanborn 2000 F1 magnetic tape recorder

and a Mnemotron 400C computer of average transients (CAT). Also

recorded on magnetic tape was the output of a Crass S-S stimulator

which provided an external trigger for the CAT and a Grass PS-2

photostimulator which powered the xenon flash. In order to detect

changes in equipment gain from one subject to the next a calibrate

signal from a Medistor C-lA Calibrator was processed in a manner

similar to EEG data prior to the experirrnntal run for each subject.

A.2 Data Processing

In each trial the CAT sequentially summed each one second epoch

of the bioelectric potentials following each stimulus trigger. In

this manner VER and control wvaveforms were provided for each subject.

These waveforms were permanently registered on a Sanborn 701A Strip

Chart Recorder. From these records the "best" signal and noise trials

(under periodic stimulus conditions) were chosen. Judgment of "best"

was obtained strictly on an amplitude basis, disregarding the v.aveshape

of the summed response to avoid biased results. That is, the larger

sequential averaged signal and smaller sequential averaged noise

amplitudes were judged to be the best.








The unprocessed analog signals from these better trials were

then digitized and stored on a memory disc of an IBM 1SOO computer.

Since the disc could only accommodate a total of approximately 4.4

(105) data words, it became necessary to limit the amount of analog

data stored. Therefore, the storage of analog potentials (digitized

at a rate of 200 samples per second) was restricted to ten near one

second epochs of data for each subject and each channel. The ten

epochs chosen followed the first through fifth, tenth, twentieth,

thirtieth, fortieth and fiftieth stimuli in the periodic stimulus

train. These ten epochs were chosen to provide a measure of both

the initial (first few seconds) and prolonged (fort)' to fifty seconds)

changes in the evoked activity with repetitions of the stimulus. Each

epoch consisted of the first 190 digitized points following each of

the ten stimulus triggers; thus a single epoch is 950 milliseconds

in duration.

In order to eliminate any near DC potentials introduced by equip-

ment base line shifts, the mean was removed from each 190 point epoch.

Equipment gain fluctuations across the subject population were also

eliminated by multiplying the digitized potentials by a constant

inversely proportional to the magnitude of the recorded calibrate

signal. Computaticns on this adjusted array of data provided most

of the results contained in this study. The programs utilized to

obtain the data were fairly simple in nature and are therefore omitted

from the text. Tne numerical results of these computations were

converted to graphical displays by a Computer Industries 135 On-Line

Incremental Plotter.












APPENDIX B



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FIGURE B.9
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1 1 00

u aI 1 0J

I -u I5 0 00 1 00

S-0 .. u l. 0 M. 1.00

u 2i u :L -0.,0 -0 57

l2 -0 0? *u 2J 0.a: 0 33

:0 0 n. 0 I 0 -0 &: -u 7

10 -0 I. -u -t 0 -0 .:

j- -0 -0 7B u.ll 0 0e

0u -0 01 *0.O *0 1. -0.1;


].i21,tI. 0

**.., I


1 00

0 )u I O0

0.15 0 e.


I or

-0 Il 1 00

0 aO. 0 O I.CO


10 :o0 0 0O 0O


Ia[ .i. 7

-,., -.06


I 00

-0 5. 1.00

0.1. -0 I

-0 at 0.0

0 GE 0 S.


I .00

-0 06 1 00

-O.I7 0 '3 I O0

0. 1 -0 1: -0 0' 1 03


TABLES B.S through B.13


EER Correlation Coefficients for the 0.3 to 50 Hz Bancdwidth


The above tables provide the correlation coefficients betw-een the

ten wide band response waveforms in Figure B.1. These response wave-

forms were shifted to compensate for phase differences introduced by

the filtering process. The above intervals therefore correspond to

those segmrnted to narrow-band EEP's.

















C- l I 5 1

1 I.O 2

C. 1 I C'.'

S0 8I -0.1] 00O

0 0 0 -0 0.0: ] '00


0. 0' 0 !,

10 -0 01 -0 '.

"O 0. t 0.: :

C0 0 ' -0 L.O

.0 0 00C -0 :L

0C. -0 6i -0 .


5 10 :0 510 0.O 0




ri eril IE

1cPt


0 E1 .O00

j- ;0 -CO. L


10 C0 0 C

O.'1 0 S%

0 1" *0 (12


I.00

-0.IL I 00

-0. 1 I' I. 1
- j,.J -) ,I C,. I C-0


1 i 00

C 0 :' 1.000

7 .0 L7 .0 1 2.00

.O I6 -0. .7 0 ) 00O

5 -0 8 0 0 01 -0 88

10 -0 'i -0 "I .0 10 '

C:0 -0 OC. -'0 1) -1'0 [1 C-.8

7510 -,.6 0.(l 0 'I5 *.0''

'*0 0 I] -0 8: -0 17 0 ,

,0C 0 11 -0 aJ -0 I01 ,. :


10 10 '0 C0 0





*0


I 00

-C0 hO .00

-0. (' C.81 1.00

0 O1' -0..' 1.00

-0. 0 0 ': -0.6 1].00

-0. .8 0. .88 O'." -0.uL C."0 1 00


(CNI ] : 5

1 C00

1 O. I1.00

I O. C 0 I I.C.00

r .07 0.. -0 (iC I

S0 21 -0 P' -0 0o -0.'

20 -0 !: -0 .5 0 C0 -0.


:0 -0 '.

10 -0 1

OC 0C.:
,C C,. C,


10 :0 5 0 0o 0





-dC e o0


00

I 0 .00

> O. i. 1.00


C.,6 -0. :: -0. :. 0.65

-0 '1J 0 0O -0 "I -0 i.

0 10 ', '( -0 -0 'I.

*O.]? 0 L -0 r "(' 1)


J. 100



0 '] 0 IO


hNT| I



: -0 1-

1 "0. :



S 0

C '10 0.
CC. .


60 -0 0


0 *0 !1 '

*C' -0. 1.


1.00


I 10 :0 7o0 L0 0


1 00

-0. 1.00

-0. 0. 1' 1.00

-0 :5 -0 1: -0. 1 1 00

0. !8 -0 -C0.8. 0 (: 1.00

,* 0" o.7 o 0 7 o -1 0 7.

0 0 0 ! -0 l* -0 18 0 1:7

0 :: 0 7' 0.70 -0 1' 8 0.7 1


I C :10 1'0 O, 0


2 2.001

: 0. 1.00

5 0.0. -0.'" 1 00


0 87 0 : -0 1'

0 1. -0.78 -0 :'



:0 0.69 -0 5, O.li.

0 "0 :' -0.'.O -0 '7

-0 -0 *., 0 1: -0 I

0 0.i C. -0 C L


c.e .BL *
r. 0'


J.CO

*O. : 1.00

0 5 -0 1: 1.00

o0.: 0o i. o0 6

-0. 0 71 -0.'


-C0. : -0. -0 C.


I '00'

0. :1 1.00

0 0' '. ].0o'


C TL ]

I I 00

: -0.88 I 00

7 0 '' -0 '1'

i. -0 28 0 '.7

0.0'' -0.1:

10 0 7: 0 7:

0 ''. -I t


70 0 '1 -0 '1
0 -0 02 0 .'
:O -C. C.


10 7 0 0 '.0 0


d e


1.00

0.0' 1 00




0. 0 -0 :*
*O. i' -0 1


( '" -0 1 .


1 00

0 :0 2.00

-, 8 I 0 80 1.00

O. B 0 0C 0 1i 1 ..O

0.,: -0 5. -1.1' -0 ,8 I 00


CnlL : I

J 1 OC,



-0 : ..0 I 0 I

.0 .5 0. 0 0 81

S0 10 -0 ., CL




CO C.'. : -( -C. :1

O 0.' -0 1 -C0 C.

'0 -0. 0 0.0 '.


1 10 :0 1' S .0 G 0


Il -i.4L '


1. 00


-0 !o 0 0 100

0 : : 0 '2 -0 10 1.00

0 1" -0 1a (.'0 0.7O 12 00O


TABLES B.14 through B.20


Control Correlation Coefficients


The above tables provide the cor-

relation coefficients between the

ten control i.avefonns in Figure

3.4 for each 135 msec interval.


I,7i2.al a

c e 0'








.:a 00


0.81 0I ,
0 80 0 4I. J 00


$ I.







T

.00 .05 .10 .15 .20


.25 .30 .35 .40 .45


.00

.05

.10

.15

.20

.25

.30

t .353
.40

.45

.50

.55

.60

.65

.70

.75

.SO

.S5

.90


TABLE B.21

Values for Cf(t, t-T) in Figure 4.1

The following values are measured relative to Cf(0,0) and the
values of t and T are given in seconds.


1.00

0.90

0.SS

0.79

0.71

0.86

1.07

0.58

0.63

0.57

0.51

0.62

0.S9

0.53

0.73

0.77

0.59

0.57

0.79


0.41

0.53

0.30

0.10

-0.03

0.04

0.10

0.07

0.20

0.0S

0.13

0.20

0.27

0.25

0.30

0.26

0.20

0.36


0.52

0.2S 0.22

0.35 0.27

0.17 0.03

0.38 -0.07

0.34 -0.06

0.21 -0.17

0.06 0.00

-0.01 0.01

0.22 -0.02

0.32 -0.09

0.12 -0.04

0.18 0.05

0.13 -0.01

0.2S 0.12

0.1S 0.18

0.14 -0.01


0.28

0.09 0.11

-0.06 -0.09 0.00

-0.07 -0.06 -0.09 0.00

0.10 -0.12 -0.04 -0.0S 0.16

0.06 -0.01 -0.OS 0.01 -0.13 0.15

-0.21 0.12 -0.10 0.00 -0.19 -0.05

0.06 -0.16 -0.00 -0.24 -0.27 -0.23

0.05 -0.02 -0.23 -0.2S -0.2S -0.35

-0.12 -0.OS 0.05 -0.14 -0.17 -0.24

-0.02 -0.16 -0.09 -0.09 -0.29 -0.32

0.02 -0.13 -0.16 -0.14 -0.15 -0.30

0.03 -0.02 -0.05 -0.21 -0.14 -0.16

0.02 -0.07 -0.10 -0.19 -0.12 -0.12

0.OS -0.01 -0.01 -0.09 -0.03 -0.13




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