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The dimensionality of the human visual evoked scalp potential
Electroencephalography and Clinical Neurophysiology, 40 (1976) 633--644
633
© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
THE DIMENSIONALITY OF THE H U M A N VISUAL EVOKED SCALP POTENTIAL * ROBERT N. KAVANAGH $, TERRANCE M. DARCEY and DEREK H. FENDER
California Institute of Technology, Pasadena, Calif. 91125 (U.S.A.) (Accepted for publication: December 10, 1975)
The bulk of data acquired from a multichannel evoked potential recording system presents the experimenter with a formidable analytic task before conclusions can be drawn from the data. Given an array of evoked potential recordings from N different channels, it is necessary to devise objective ways of studying the results. In this context, we wish to k n o w if all channels are independent of one another, or if the channels contain much redundant information. In this paper we will show, using the methods of principal components analysis and principal factor analysis, that N simultaneous visual evoked potential recordings from different electrode placements usually measure fewer than N independent variables and that processes chosen to model the underlying system should conform to this reduced dimensionality. The possibility of using a dipole to model the response is discussed and it is shown that its use is more plausible in light of the results of this study. It should be noted at the outset that, although the m e t h o d s described here produce a set of independent variables which retain the information contained within the total set of dependent variables measured, this representation is by no means unique and therefore
* This research was supported in part by a grant from the Alfred P. Sloan F o u n d a t i o n and by Grants NS 03627 and GM 01335 from the National Institutes of Health. $ Current address: Department of Computational Science, University of Saskatchewan, Saskatoon, Saskatchewan, Canada.
the specific variables found do not of necessity have any physiological significance in themselves. That is, the analysis may n o t lead to functions which describe physiological processes (Van R o t t e r d a m 1970). What is significant is that the analysis gives a good estimate of the dimensionality of the data space, and this is an important precursor to estimating the degree to which the data may retain the identities of specific underlying processes. This estimate tells us the number of underlying processes that are necessary to account for the data. The result narrows considerably the choice of models which may be used to explain the underlying processes. It is generally h o p e d a priori that all variables measured are n o t truly independent of the activity of all other variables, since if they were, there would be little hope of finding a tractable model to explain the data. If it appears that many variables measure activity that is also represented to a greater or lesser degree in other variables, then perhaps the data can be used to characterize a relatively simple equivalent generator. Previous studies have shown the power and limitations of the technique described here. Scher et al. (1960) and Cady (1969) have applied principal components analysis to multichannel recordings o f the human electrocardiogram and shown that there was redundancy in assuming more than a three-dimensional model. A number of workers have used factor analytic methods on spectral analyses of ongoing activity in humans (Walter and Adey 1965; Larsen 1969; Defayolle and Dindand 1974).
634 The results of Walter and A d e y (1965) show that the energy in certain spectral bands of the spontaneous activity is c o m m o n to several channels, hence they were able to show a factor structure of small dimensionality for this aspect of the data. Larsen (1969) tried to relate frequency c o m p o n e n t s to state of consciousness and concluded that although four factors account for nearly all the variability in each condition (waking and sleeping), only t w o factors per state of consciousness are important. Defayolle and Dindand (1974) have shown similar results to those o f Walter and A d e y {1965) with four factors accounting for 87% of the variance in the data and have shown that these factors correspond to the frequency bands described b y clinicians. Donchin (1966) has analyzed a single channel of human visual evoked potential over thirty different stimuli. His results indicate that ten factors summarize approximately 95% of the variance in the data from each of two subjects. Others have studied evoked potentials which were obtained using implanted electrodes located over the visual and auditory cortex of cats (Ruchkin et al. 1964; John et al. 1964, 1973; Suter 1970). John et al. (1964) applied principal factor analysis to visual evoked potentials during the establishment of conditioned avoidance and found that approximately 90% of the total variance is accounted for b y four factors. Suter (1970) applied principal c o m p o n e n t s analysis to auditory evoked potentials and claimed that the first four factors accounted for a b o u t 90% of the variance, while the fifth factor accounted for less than 3%. John et al. (1973) used factor analysis to study the effects of various drugs on visual evoked potentials during conditioned feeding. Ruchkin et al. (1964) have described one w a y in which the physiological significance of principal factor analyses can be enhanced. They defined their various evoked potential waveshapes observed in different anatomical regions of the cat so that the resulting principal factor description would take the form of a linear combination of the electrical wave-
R.N. KAVANAGH ET AL. forms arising in a small number of specified anatomical locations. It is difficult to compare the above results, b u t it is notable that all of the attempts to date to reduce the dimensionality of ongoing activity or the evoked potential concur in the result that such a reduction is eminently possible. The consensus of most studies is that it is possible to obtain an adequate description o f the evoked potential by using three to six dimensions (Donchin 1969). In our own work we are concerned with the localization within the brain of the sources of electrical activity that give rise to the scalp potentials. For this purpose it is essential to use a model with sufficient disposable parameters to match the dimensionality of the data. If the model has t o o few parameters, it cannot represent the data adequately - - i f t o o many parameters, the model will represent some spurious events as well. This paper treats the estimation of the dimensionality of the data in a specific set of experimental conditions.
Method and material
1. Theory The details of principal c o m p o n e n t s analysis and principal factor analysis can be found in several sources, such as Seal (1964) or Morrison {1967). The following brief overview o f the technique gives a description of its use in the analysis of the evoked potential data obtained in this experiment. Principal components analysis is a process in which N correlated variables are transformed in N uncorrelated variables. If these are arranged in descending order of variance, only the first M(
DIMENSIONALITY OF THE HUMAN VESP
635
values of the variance-covariance matrix of the observed waveforms is the best M-dimensional approximation to the observed waveforms in the least squares sense (Morrison 1967). Consider a matrix V of the data obtained by sampling N observed waveforms at K equally spaced time intervals. We wish to derive the N dependent rows from a mapping of N independent orthogonal rows of some matrix W, that is: V = BW
(1)
If W is to be a matrix whose rows are independent, then its variance-convariance matrix must be diagonal, thus: Cov[WW ~] = A = diag (hi,X2, ..., XN) If we let C = B -~
(2)
then we can substitute from equation (1) for W giving: Cov[WW ~]
= Cov[CV(CVy] = Cov[CVV~C ~] = C Cov[VW]C ~ =
CEC ~
where E = Cov[VV T] is the variance-covariance matrix of the original data V, i.e.,
Since E is a real symmetrix matrix we k n o w that some matrix C exists which will diagonalize it, t h a t is A = CEC ~ can be produced, and the desired matrix of independent variates is W = CV. The diagonal elements of A are h~, hs . . . . . kN, the N independent eigenvalues of E. Furthermore, since the trace of
A remains the same as the trace of E, t h a t is:
the total variance of the original data is preserved by the new orthogonal system W. Each hi corresponds to an axis in the system W, and the value of hi indicates the contribution to the total variance of the corresponding axis. When the hi are arranged in descending order, it is often found that a certain M(
636
R.N. KAVANAGH ET AL
2. Analysis I PHOTOSTIMULATOR J In the analysis of visual evoked potential data two approaches were used. The first approach involved taking the N channels of the evoked potential recordings as variables and the various potentials sampled in time as the observations of each variable. The principal components analysis was then used to ask h o w m a n y independent channels were present in the data. The second approach involved taking the various sample times as variables and the N potentials at each sample time as the observations. With this interpretation it is then possible to ask if the sample times are independent of one another, i.e., during successive intervals of the response are the data dependent u p o n the data in preceding or following intervals. It should also be noted that the analysis was done on correlation matrices rather than variance-covariance matrices. This is consistent with the principal components procedure, since the correlation matrix is the variancecovariance matrix expressed as standard scores. This has the effect of assigning equal weight to the original data in determining the orientation of the eigenvectors.
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3. Equipment and experimental configuration The subject, stimulation apparatus and preamplification equipment were all situated in a shielded, darkened, sound-attenuating chamber. A modified troposcope (Fig. 1) delivered visual stimuli to the subject. Care was taken to ensure that the subject was comfortable in the apparatus and was well-adapted to the lowambient illumination. Dilation of the pupils was necessary to standardize retinal illumination by t h e stimuli. Recording time was short and constant, and a standard fixation procedure was used to insure stimulating identical retinal areas. Three stimuli were used: unstructured light flashes to the left eye, to the right eye, and to both eyes respectively. These stimuli were derived from a photostimulator with an intensity setting of 42000 horizontal candela.
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DIMENSIONALITY OF THE HUMAN VESP
637
Fig. 3. Top and right side views of electrode placement for subject KW (left) and subject DALO (right).
The EEG activity was monitored from goldplated electrodes held in place with electrode cream. The reference used was the so-called p h a n t o m electrode (Fig. 2) due to Offner which is created by using a summing n e t w o r k which weights each channel equally to form as average electrode. The signals were amplified and then analog samples of each channel were multiplexed onto FM tape. Also placed on the tape were signals denoting stimulus onset times and other timing information necessary to recover and sum the responses from individual channels. The FM tape was later demultiplexed and summated using a CAT signal processor. The visual evoked potentials, then in digital form, were punched onto paper tape for analysis. In the experiments being reported here, two subjects were used: an adult male (KW) and an adult female (DALO). Both subjects were normal. In the case of subject KW, 41 channels were recorded, while the DALO data were derived from 38 channels. The electrode placements are shown in Fig. 3. Each channel was sampled at 1 msec intervals for the 256 msec following each flash.
Each of the three stimulus conditions reported here was presented a total of three times for each subject, allowing extraction of a mean response for each condition. The data were then subjected to the above described analysis and also displayed as equipotential lines plotted on an animation o f the head.
Results
1. E v o k e d potentials There were six sets of data to be analyzed; three conditions for each of the two subjects. Fig. 4 gives a series of equipotential c o n t o u r maps for the two subjects which have been selected as a summary of the evoked potential data. Both subjects were normal and there is little difference between the maps for the three flash conditions within each subject.
2. Factor analysis using channels as variables Table I shows the cumulative percentage o f the variance summarized by the first six
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Fig. 4. Equipotential maps o f evoked potential activity at 85, 110, 120 and 185 msec after stimulus, shown in clockwise fashion. The upper three maps in each case are for subject KW while the lower three are for subject DALO.
TABLE I C u m u l a t i v e p e r c e n t a g e o f v a r i a n c e s u m m a r i z e d b y eigenvalues o f c o r r e l a t i o n m a t r i c e s w i t h r e c o r d i n g c h a n n e l s c o n s i d e r e d as variables. Eigenvalue
D a t a set a n a l y z e d
1 2 3 4 5 6
DALO Flash right
DALO Flash left
DALO Flash both
KW Flash right
KW Flash left
KW Flash both
47.5 65.7 79.7 93.3 97.5 98.4
56.6 79.0 90.3 95.9 97.7 98.3
52.1 69.9 84.3 91.4 94.9 96.7
56.8 83.0 95.3 96.4 97.3 97.9
56.2 84.9 94.6 96.6 97.6 98.5
51.3 90.6 96.1 97.2 98.0 98.5
T A B L E II F a c t o r l o a d i n g s f o r e a c h c h a n n e l o f D A L O d a t a f r o m flash left c o n d i t i o n . Channel
11 12 13 14 15 16 21 22 23 24 25 26 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 55 56 58 61 62 63 64 65 66 68 54
Principal factors 1
2
3
4
5
6
0.681 0.0744 0.626 0.279 --0.303 --0.842 0.842 0.916 0.862 0.252 --0.624 --0.730 0.855 0.954 0.878 0.141 --0.497 --0.576 0.781 0.894 0.890 --0.238 --0.648 --0.707 0.502 0.375 --0.287 --0.813 --0.846 --0.075 0.505 0.445 0.008 0.158 --0.349 --0.832 0.095 --0.620
--0.526 --0.410 --0.060 0.280 0.506 0.035 --0.404 --0.231 0.342 0.781 0.733 0.275 --0.392 --0.196 0.450 0.913 0.578 0.304 --0.527 --0.425 0.254 0.882 0.534 0.194 --0.814 --0.822 --0.039 0.222 0.050 --0.917 --0.662 --0.531 --0.445 --0.890 --0.722 --0.417 --0.891 0.521
--0.464 --0.474 --0.755 --0.900 --0.766 --0.419 --0.279 --0.204 --0.342 --0.409 0.037 0.570 --0.265 --0.068 --0.062 --0.019 0.611 0.680 --0.299 --0.025 --0.018 0.211 0.461 0.653 --0.179 --0.077 --0.061 0.335 0.470 0.095 --0.513 --0.684 --0.853 --0.303 --0.239 0.196 --0.035 0.272
0.145 0.190 0.149 0.057 --0.179 --0.107 0.178 0.203 0.097 0.071 --0.139 0.194 0.109 0.172 0.088 0.030 0.174 0.288 --0.043 --0.051 --0.259 --0.264 --0.215 0.111 --0.109 --0.376 --0.948 --0.372 --0.181 0.146 0.137 0.165 0.025 --0.237 --0.415 --0.236 0.311 --0.515
0.008 0.080 0.011 --0.104 --0.066 0.040 0.069 0.126 --0.073 --0.387 --0.161 0.060 0.059 0.098 --0.080 --0.369 0.038 0.128 --0.089 0.018 --0.130 --0.174 0.146 0.126 --0.068 --0.017 0.013 0.161 0.133 --0.299 0.067 0.095 0.179 --0.101 --0.244 --0.095 --0.089 0.013
0.052 0.012 --0.037 --0.086 --0.075 --0.034 --0.040 --0.066 --0.054 --0.025 --0.036 --0.023 --0.024 --0.036 0.012 0.036 --0.022 --0.069 --0.015 --0.058 0.123 0.095 0.066 0.043 --0.048 --0.133 0.000 --0.034 --0.015 0.124 0.082 --0.005 0.108 --0.072 --0.140 --0.114 0.283 0.025
R.N. K A V A N A G H ET AL.
640
eigenvalues of the correlation matrices of the data when channels were considered to be variables. The factor coefficients (loadings)
for each of the six data sets were also computed; the factor structures for DALO left and KW left are shown in Tables II and III.
TABLE III Factor loadings for each channel o f KW data from flash left condition. Channel
1 2 3 4 5 6 7 9 10 11 12 13 14 15 17 18 19 21 22 23 25 26 27 28 29 30 31 33 34 35 36 37 38 39 41 42 43 44 45 46 47
Principal factors 1
2
3
4
5
6
0.013 0.181 0.063 --0.265 --0.886 --0.957 --0.568 --0.790 --0.627 --0.869 --0.930 --0.915 --0.889 --0.966 0.100 0.355 0.137 --0.872 --0.851 --0.962 0.276 0.539 0.582 0.205 --0.863 --0.843 --0.934 0.179 0.480 0.628 0.680 --0.828 --0.864 --0.827 0.203 0.392 0.356 --0.083 --0.916 --0.948 --0.614
0.955 0.959 0.969 0.895 0.239 --0.101 0.741 0.189 0.094 --0.040 --0.224 --0.341 --0.404 --0.010 0.426 0.382 0.167 --0.436 --0.453 0.065 0.760 0.554 0.468 0.287 --0.392 --0.435 0.169 0.878 0.771 0.626 0.474 --0.342 --0.344 0.408 0.927 0.878 0.893 0.936 0.226 0.153 0.715
0.195 0.146 0.141 0.212 --0.191 --0.183 0.018 0.514 0.749 0.362 0.139 --0.006 --0.122 --0.139 0.823 0.834 0.956 --0.015 --0.178 --0.141 0.494 0.612 0.639 0.850 --0.171 --0.234 --0.139 0.343 0.344 0.404 0.435 --0.371 --0.267 --0.160 0.248 0.199 0.201 0.121 --0.238 0.228 --0.039
--0.054 --0.082 --0.171 --0.231 --0.310 --0.128 --0.302 0.121 --0.028 0.022 0.040 0.053 0.086 --0.148 0.133 0.002 --0.067 0.041 0.103 --0.172 0.268 0.072 --0.011 --0.002 0.144 0.158 --0.234 0.252 0.142 0.039 0.001 0.101 0.166 --0.334 0.160 0.083 --0.041 --0.089 --0.150 0.030 --0.312
0.184 0.080 --0.015 --0.047 --0.008 0.001 0.162 0.075 0.041 --0.003 --0.037 --0.034 --0.001 --0.053 0.212 0.145 0.079 0.028 0.054 --0.059 0.104 --0.034 --0.142 --0.227 0.112 0.078 --0.076 0.051 --0.174 --0.199 --0.297 0.170 0.150 --0.052 0.018 --0.153 --0.161 --0.202 0.032 0.086 0.069
--0.091 --0.083 --0.030 0.138 0.110 0.138 --0.030 --0.080 --0.116 --0.314 --0.228 --0.192 --0.136 --0.121 0.103 0.082 --0.071 --0.186 --0.132 0.100 0.103 0.108 0.072 --0.083 --0.061 0.010 0.102 0.063 0.052 0.075 --0.002 0.017 0.075 0.062 --0.027 --0.007 0.036 0.142 0.051 0.090 --0.009
DIMENSIONALITY OF THE HUMAN VESP
641
T A B L E IV
Cumulative percentage of variance summarized by eigenvalues of correlation matrices with sample times considered as variables. Eigenvalue
D a t a set a n a l y z e d
1 2 3 4 5 6
u~
DALO Flash r i g h t
DALO F l a s h left
DALO Flash both
KW Flash right
KW F l a s h left
KW F l a s h both
46.0 71.8 86.5 95.5 97.7 98.3
51.9 76.9 90.3 95.3 97.6 98.3
61.5 77.1 86.2 92.3 95.8 97.6
62.5 79.6 91.7 94.7 96.1 97.1
56.2 72.8 85.2 90.7 95.3 97.2
68.1 83.1 91.7 94.3 96.2 97.7
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Fig. 5. Factor coefficients at successive sample times for six most significant factors in D A L O flash l e f t d a t a a n a l y z e d with sample times as variables.
Fig. 6. Factor coefficients at successive sample times for six most significant factors in KW flash left d a t a a n a l y z e d with sample times as variables.
3. Factor analysis using sample times as variables
Discussion
1. Evoked potentials Table IV shows the cumulative percentage of the variance summarized by the first six eigenvalues of the correlation matrices of the data when sample times were considered as the variables. Factor coefficients for each of the six data sets were computed; data for DALO left and KW left are shown graphically in Fig. 5 and 6.
Considering the equipotential maps in Fig. 4 it is apparent that there is considerable similarity in the responses of either subject to all three stimuli, but there is not much similarity between the responses of the two subjects, except in the range 175--195 msec. Certain simple patterns in the equipotential maps are
642
quite stable; in the case of the KW data, two well-defined potential troughs are prevalent from about 90 to 130 msec; and over this same range a stable but different pattern prevails in the DALO data. The simple structure of the equipotential maps indicates that neighboring channels are interdependent; also over several intervals in the responses, the maps suggest a simple, stable configuration of generators.
2. Factor analysis using channels as variables Table I shows that most of the experimental variance can be accounted for by only six eigenvalues -- in fact a great deal is summarized by only three eigenvalues, thus the channels are not independent of one another. The data are well accounted for by six factors and a model having six disposable parameters would have some justification in these results. Note that, if desired, one could select from Tables II and III six apparently independent recording channels by grouping channels whose coefficients for the first three factors are similar in magnitude and sign. This process leads to the classical gross classification of cortical areas, that is: left temporal, occipital, right temporal, right occipital, left ear group, and right ear group. The Tables clearly show that electrodes adjacent to one another record similar evoked potentials, hence the dependence from channel to channel. The important result is thus that a small number of independent processes is present, probably six or fewer. This suggests that it is appropriate to att e m p t models which are characterized by a small number of parameters.
3. Factor analysis using sample times as variables Table IV shows that most of the experimental variance is accounted for by three eigenvalues, and over 97% is summarized by six. Note that the mean of each sample time in this analysis is a measure of the average potential at all electrodes at that sample time,
R.N. KAVANAGH ET AL.
and the correlations between variables are the correlations of the average activity at one sample time with that at another. Thus Fig. 5 and 6 are plots of the coefficients of this average activity at each sample time with each of the six underlying factors. These factor coefficient curves show that for each subject there are roughly three intervals of importance during the responses. If we consider factors one and two for each subject, we see that they generally have the same sign and about the same magnitude over specific intervals, namely between 50 and 90 msec, 100 and 140 msec, and between 150 and 180 msec. Remembering that the factors are arranged in decreasing order of summarization of the data variability, it is seen that these first two factors, which account for about 75% of the variance, are the most important to consider. In each of the remaining factors there are various peaks or troughs which generally occur in one or more of the three intervals singled out. This is in good accord with the appearance of the equipotential maps, where for each of these intervals there is a stable pattern in the c o n t o u r lines. The differences between the responses of the two subjects do not manifest themselves generally in the factor coefficients until the last three factors. The results suggest the temporal sequence of generators of the evoked potentials -- over quite long segments of the responses the activity at some sample time is strongly related to the activity at times preceding and following, indicating that the activity over that period arises from a stable set of generators.
4.
Modeling
the
visual evoked
potential
The primary motivation for discussing the h u m a n visual evoked potential in terms of dimensionality of the data lies in considering the implications this has for modeling. One hypothesis is that the potentials on the surface of the head result from equivalent dipoles located somewhere in the brain. If this is so, then at any point on the surface of the head Vi(t i) = aixPx (tj) + aiyPy (tj) + aizPz(tj)
(3)
DIMENSIONALITY OF THE HUMAN VESP where Vi(tj) is the voltage which appears at electrode i at time tj. The three components of the dipole, Px(tj), Py(tj) and Pz(tj) are combined in a linear manner at any electrode site, and the constants aix , aiy and aiz are dependent upon only the dipole location, the reference potential location and the location of the recording electrode. Note t h a t the dipole components may be a function of time but the dipole location is assumed constant; the a i's are not functions of time. A single dipole is characterized by six parameters: three position coordinates and three orthogonal components of the dipole strength. Equation (3) is apparently a function of only three parameters because the dipole is assumed to be fixed in space and hence the dependence upon the actual position of the dipole is included in the coefficients aix, aiy and aiz. Thus, if such a relationship were to exist in the evoked potential data, the model would be of the form V = AP, where V is the N × K matrix of experimental potential values from N electrodes sampled at K instants of time, P is the 3 × K matrix of model dipole components, and A is the N × 3 matrix of model dipole coefficients. In practice it is improbable that the matrix of coefficients A is time-invariant, since in the case of evoked potentials one would expect to have either different sources active at different times during the response or moving sources.. This suggestion is borne out in the results obtained using channels as variables, in which it would appear t h a t three factors are not sufficient to summarize the data. However, if one uses six factors, it is clear that the data are well accounted for, and one could use this result to argue that over the duration o f the response one or perhaps two dipoles would suffice: if necessary these one or two dipoles could be allowed to be relatively fixed for some interval and then move to another relatively fixed site over another interval of the response; i.e., Equation (3) applies piecewise over the complete response. Again, it should be emphasized t h a t m a n y other model formulations involving six parameters could
643 account for the same data. Thus the choice of a particular model cannot be justified on the basis of these results alone. The important conclusion is that a small number of independent processes are apparently present, probably six or fewer. Likewise, as previously discussed, the results obtained using sample times as variables suggest a small number, possibly three, of stable generators in temporal sequence.
Summary A small number of processes can account for most of the evoked potential activity in the two subjects studied. Principal components analysis indicates that six independent processes can account for approximately 97% of the variability in the data. Moreover, the factor analysis and plots of the factor coefficients yield indications that the times during which these principal factors are active agree quite well with the times at which the equipotential maps show some organized activity. The question of dipoles being the underlying cause of the observed activity is not answered by the factor analysis. The principal factors are not unique, but models which have a small number of parameters are more justifiable in light of the results of this study.
R6sumd La dimensionalitg du potentiel @icrdnien dvoqug visuel de l'homme Un petit nombre des processus peut expliquer la plupart de l'activit~ potentielle 6voqu~e chez les deux sujets 6tudi~s. L'analyse en composantes principales indique que six processus ind~pendants peuvent expliquer approximativement 97% de la variabilit6 dans les donn~es. De plus, l'analyse factorielle et les trac6es des coefficients factoriels indiquent que les temps durant lesquels ces factuers prin-
644
cipaux sont actifs s'accordent assez bien avec les temps pendant lesquels les cartes 6quipotentielles montrent une certaine activit6 organis6e. La question des dipoles 6tant la cause fondamentale de l'activit6 observ6e n'est pas r6solue par l'analyse factorielle. Les facteurs principaux ne sont pas uniques, mais des modules qui ont un petit nombre des param~tres sont plus justifiables par suite des r6sultats de cette ~tude.
References Cady Jr., L.D. Computed relationship of standard electrocardiographic leads. Med. Res. Engng, 1969, 8: 37--42. Defayolle, M. et Dindand, J.B. Application de l'analyse factorielle ~ l'6tude de la structure de l'EEG. Electroenceph. clin. Neurophysiol., 1974, 36: 319--322. Donchin, E. A multivariate approach to the analysis of averaged evoked potentials. IEEE Trans. biomed. Engng, 1966, BME-13: 131--139. Donchin, E. Data analysis techniques in averaged evoked potential research. In E. Donchin and D.B. Lindsley (Eds.), Average evoked potentials. NASA, 1969, SP-191 : 199--217. John, E.R., Ruchkin, D.S. and Villegas, J. Experimental background : signal analysis and behavioural correlates of evoked potential configurations in
R.N. KAVANAGH ET AL. cats. Ann. N.Y. Acad. Sci., 1964, 112: 362--420. John, E.R., Walker, P., Cawood, D., Rush, M. and Gehrmann, J. Factor analysis of evoked potentials. Electroenceph. clin. Neurophysiol., 1973, 34: 33--43. Larsen, L.E. An analysis of the intercorrelations among spectral amplitudes of the EEG: a generator study. IEEE Trans. bio-med. Engng, 1969, BME-16 : 23--26. Morrison, D.F. Multivariate statistical methods. McGraw-Hill, New York, 1967, 338 p. Offner, F.F. The EEG as potential mapping: the value of the average monopolar reference. Electroenceph. clin. Neurophysiol., 1950, 2: 213--214. Ruchkin, D.S., Villegas, J. and John, E.R. An analysis of averaged evoked potentials making use of least mean squares techniques. Ann. N.Y. Acad. Sci., 1964, 115: 799--826. Scher, A.M., Young, A.C. and Meredith, W.M. Factor analysis of the electrocardiogram. Circular. Res., 1960, 8: 519--526. Seal, H.L. Multivariate statistical analysis for biologists. Methuen and Co., London, 1964, 209 p. Suter, C.M. Principle components analysis of average evoked potentials. Exp. Neurol., 1970, 29: 317-327. Van Rotterdam, A. Limitations and difficulties in signal processing by means of the principlecomponents analysis. IEEE Trans. bio-med. Engng, 1970, BME-17: 268--269. Walter, D.O. and Adey, W.R. Analysis of brain-wave generators as multiple statistical time series. IEEE Trans. bio-med. Engng, 1965, BME-12: 8--13.
633
© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
THE DIMENSIONALITY OF THE H U M A N VISUAL EVOKED SCALP POTENTIAL * ROBERT N. KAVANAGH $, TERRANCE M. DARCEY and DEREK H. FENDER
California Institute of Technology, Pasadena, Calif. 91125 (U.S.A.) (Accepted for publication: December 10, 1975)
The bulk of data acquired from a multichannel evoked potential recording system presents the experimenter with a formidable analytic task before conclusions can be drawn from the data. Given an array of evoked potential recordings from N different channels, it is necessary to devise objective ways of studying the results. In this context, we wish to k n o w if all channels are independent of one another, or if the channels contain much redundant information. In this paper we will show, using the methods of principal components analysis and principal factor analysis, that N simultaneous visual evoked potential recordings from different electrode placements usually measure fewer than N independent variables and that processes chosen to model the underlying system should conform to this reduced dimensionality. The possibility of using a dipole to model the response is discussed and it is shown that its use is more plausible in light of the results of this study. It should be noted at the outset that, although the m e t h o d s described here produce a set of independent variables which retain the information contained within the total set of dependent variables measured, this representation is by no means unique and therefore
* This research was supported in part by a grant from the Alfred P. Sloan F o u n d a t i o n and by Grants NS 03627 and GM 01335 from the National Institutes of Health. $ Current address: Department of Computational Science, University of Saskatchewan, Saskatoon, Saskatchewan, Canada.
the specific variables found do not of necessity have any physiological significance in themselves. That is, the analysis may n o t lead to functions which describe physiological processes (Van R o t t e r d a m 1970). What is significant is that the analysis gives a good estimate of the dimensionality of the data space, and this is an important precursor to estimating the degree to which the data may retain the identities of specific underlying processes. This estimate tells us the number of underlying processes that are necessary to account for the data. The result narrows considerably the choice of models which may be used to explain the underlying processes. It is generally h o p e d a priori that all variables measured are n o t truly independent of the activity of all other variables, since if they were, there would be little hope of finding a tractable model to explain the data. If it appears that many variables measure activity that is also represented to a greater or lesser degree in other variables, then perhaps the data can be used to characterize a relatively simple equivalent generator. Previous studies have shown the power and limitations of the technique described here. Scher et al. (1960) and Cady (1969) have applied principal components analysis to multichannel recordings o f the human electrocardiogram and shown that there was redundancy in assuming more than a three-dimensional model. A number of workers have used factor analytic methods on spectral analyses of ongoing activity in humans (Walter and Adey 1965; Larsen 1969; Defayolle and Dindand 1974).
634 The results of Walter and A d e y (1965) show that the energy in certain spectral bands of the spontaneous activity is c o m m o n to several channels, hence they were able to show a factor structure of small dimensionality for this aspect of the data. Larsen (1969) tried to relate frequency c o m p o n e n t s to state of consciousness and concluded that although four factors account for nearly all the variability in each condition (waking and sleeping), only t w o factors per state of consciousness are important. Defayolle and Dindand (1974) have shown similar results to those o f Walter and A d e y {1965) with four factors accounting for 87% of the variance in the data and have shown that these factors correspond to the frequency bands described b y clinicians. Donchin (1966) has analyzed a single channel of human visual evoked potential over thirty different stimuli. His results indicate that ten factors summarize approximately 95% of the variance in the data from each of two subjects. Others have studied evoked potentials which were obtained using implanted electrodes located over the visual and auditory cortex of cats (Ruchkin et al. 1964; John et al. 1964, 1973; Suter 1970). John et al. (1964) applied principal factor analysis to visual evoked potentials during the establishment of conditioned avoidance and found that approximately 90% of the total variance is accounted for b y four factors. Suter (1970) applied principal c o m p o n e n t s analysis to auditory evoked potentials and claimed that the first four factors accounted for a b o u t 90% of the variance, while the fifth factor accounted for less than 3%. John et al. (1973) used factor analysis to study the effects of various drugs on visual evoked potentials during conditioned feeding. Ruchkin et al. (1964) have described one w a y in which the physiological significance of principal factor analyses can be enhanced. They defined their various evoked potential waveshapes observed in different anatomical regions of the cat so that the resulting principal factor description would take the form of a linear combination of the electrical wave-
R.N. KAVANAGH ET AL. forms arising in a small number of specified anatomical locations. It is difficult to compare the above results, b u t it is notable that all of the attempts to date to reduce the dimensionality of ongoing activity or the evoked potential concur in the result that such a reduction is eminently possible. The consensus of most studies is that it is possible to obtain an adequate description o f the evoked potential by using three to six dimensions (Donchin 1969). In our own work we are concerned with the localization within the brain of the sources of electrical activity that give rise to the scalp potentials. For this purpose it is essential to use a model with sufficient disposable parameters to match the dimensionality of the data. If the model has t o o few parameters, it cannot represent the data adequately - - i f t o o many parameters, the model will represent some spurious events as well. This paper treats the estimation of the dimensionality of the data in a specific set of experimental conditions.
Method and material
1. Theory The details of principal c o m p o n e n t s analysis and principal factor analysis can be found in several sources, such as Seal (1964) or Morrison {1967). The following brief overview o f the technique gives a description of its use in the analysis of the evoked potential data obtained in this experiment. Principal components analysis is a process in which N correlated variables are transformed in N uncorrelated variables. If these are arranged in descending order of variance, only the first M(
DIMENSIONALITY OF THE HUMAN VESP
635
values of the variance-covariance matrix of the observed waveforms is the best M-dimensional approximation to the observed waveforms in the least squares sense (Morrison 1967). Consider a matrix V of the data obtained by sampling N observed waveforms at K equally spaced time intervals. We wish to derive the N dependent rows from a mapping of N independent orthogonal rows of some matrix W, that is: V = BW
(1)
If W is to be a matrix whose rows are independent, then its variance-convariance matrix must be diagonal, thus: Cov[WW ~] = A = diag (hi,X2, ..., XN) If we let C = B -~
(2)
then we can substitute from equation (1) for W giving: Cov[WW ~]
= Cov[CV(CVy] = Cov[CVV~C ~] = C Cov[VW]C ~ =
CEC ~
where E = Cov[VV T] is the variance-covariance matrix of the original data V, i.e.,
Since E is a real symmetrix matrix we k n o w that some matrix C exists which will diagonalize it, t h a t is A = CEC ~ can be produced, and the desired matrix of independent variates is W = CV. The diagonal elements of A are h~, hs . . . . . kN, the N independent eigenvalues of E. Furthermore, since the trace of
A remains the same as the trace of E, t h a t is:
the total variance of the original data is preserved by the new orthogonal system W. Each hi corresponds to an axis in the system W, and the value of hi indicates the contribution to the total variance of the corresponding axis. When the hi are arranged in descending order, it is often found that a certain M(
636
R.N. KAVANAGH ET AL
2. Analysis I PHOTOSTIMULATOR J In the analysis of visual evoked potential data two approaches were used. The first approach involved taking the N channels of the evoked potential recordings as variables and the various potentials sampled in time as the observations of each variable. The principal components analysis was then used to ask h o w m a n y independent channels were present in the data. The second approach involved taking the various sample times as variables and the N potentials at each sample time as the observations. With this interpretation it is then possible to ask if the sample times are independent of one another, i.e., during successive intervals of the response are the data dependent u p o n the data in preceding or following intervals. It should also be noted that the analysis was done on correlation matrices rather than variance-covariance matrices. This is consistent with the principal components procedure, since the correlation matrix is the variancecovariance matrix expressed as standard scores. This has the effect of assigning equal weight to the original data in determining the orientation of the eigenvectors.
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3. Equipment and experimental configuration The subject, stimulation apparatus and preamplification equipment were all situated in a shielded, darkened, sound-attenuating chamber. A modified troposcope (Fig. 1) delivered visual stimuli to the subject. Care was taken to ensure that the subject was comfortable in the apparatus and was well-adapted to the lowambient illumination. Dilation of the pupils was necessary to standardize retinal illumination by t h e stimuli. Recording time was short and constant, and a standard fixation procedure was used to insure stimulating identical retinal areas. Three stimuli were used: unstructured light flashes to the left eye, to the right eye, and to both eyes respectively. These stimuli were derived from a photostimulator with an intensity setting of 42000 horizontal candela.
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DIMENSIONALITY OF THE HUMAN VESP
637
Fig. 3. Top and right side views of electrode placement for subject KW (left) and subject DALO (right).
The EEG activity was monitored from goldplated electrodes held in place with electrode cream. The reference used was the so-called p h a n t o m electrode (Fig. 2) due to Offner which is created by using a summing n e t w o r k which weights each channel equally to form as average electrode. The signals were amplified and then analog samples of each channel were multiplexed onto FM tape. Also placed on the tape were signals denoting stimulus onset times and other timing information necessary to recover and sum the responses from individual channels. The FM tape was later demultiplexed and summated using a CAT signal processor. The visual evoked potentials, then in digital form, were punched onto paper tape for analysis. In the experiments being reported here, two subjects were used: an adult male (KW) and an adult female (DALO). Both subjects were normal. In the case of subject KW, 41 channels were recorded, while the DALO data were derived from 38 channels. The electrode placements are shown in Fig. 3. Each channel was sampled at 1 msec intervals for the 256 msec following each flash.
Each of the three stimulus conditions reported here was presented a total of three times for each subject, allowing extraction of a mean response for each condition. The data were then subjected to the above described analysis and also displayed as equipotential lines plotted on an animation o f the head.
Results
1. E v o k e d potentials There were six sets of data to be analyzed; three conditions for each of the two subjects. Fig. 4 gives a series of equipotential c o n t o u r maps for the two subjects which have been selected as a summary of the evoked potential data. Both subjects were normal and there is little difference between the maps for the three flash conditions within each subject.
2. Factor analysis using channels as variables Table I shows the cumulative percentage o f the variance summarized by the first six
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Fig. 4. Equipotential maps o f evoked potential activity at 85, 110, 120 and 185 msec after stimulus, shown in clockwise fashion. The upper three maps in each case are for subject KW while the lower three are for subject DALO.
TABLE I C u m u l a t i v e p e r c e n t a g e o f v a r i a n c e s u m m a r i z e d b y eigenvalues o f c o r r e l a t i o n m a t r i c e s w i t h r e c o r d i n g c h a n n e l s c o n s i d e r e d as variables. Eigenvalue
D a t a set a n a l y z e d
1 2 3 4 5 6
DALO Flash right
DALO Flash left
DALO Flash both
KW Flash right
KW Flash left
KW Flash both
47.5 65.7 79.7 93.3 97.5 98.4
56.6 79.0 90.3 95.9 97.7 98.3
52.1 69.9 84.3 91.4 94.9 96.7
56.8 83.0 95.3 96.4 97.3 97.9
56.2 84.9 94.6 96.6 97.6 98.5
51.3 90.6 96.1 97.2 98.0 98.5
T A B L E II F a c t o r l o a d i n g s f o r e a c h c h a n n e l o f D A L O d a t a f r o m flash left c o n d i t i o n . Channel
11 12 13 14 15 16 21 22 23 24 25 26 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 55 56 58 61 62 63 64 65 66 68 54
Principal factors 1
2
3
4
5
6
0.681 0.0744 0.626 0.279 --0.303 --0.842 0.842 0.916 0.862 0.252 --0.624 --0.730 0.855 0.954 0.878 0.141 --0.497 --0.576 0.781 0.894 0.890 --0.238 --0.648 --0.707 0.502 0.375 --0.287 --0.813 --0.846 --0.075 0.505 0.445 0.008 0.158 --0.349 --0.832 0.095 --0.620
--0.526 --0.410 --0.060 0.280 0.506 0.035 --0.404 --0.231 0.342 0.781 0.733 0.275 --0.392 --0.196 0.450 0.913 0.578 0.304 --0.527 --0.425 0.254 0.882 0.534 0.194 --0.814 --0.822 --0.039 0.222 0.050 --0.917 --0.662 --0.531 --0.445 --0.890 --0.722 --0.417 --0.891 0.521
--0.464 --0.474 --0.755 --0.900 --0.766 --0.419 --0.279 --0.204 --0.342 --0.409 0.037 0.570 --0.265 --0.068 --0.062 --0.019 0.611 0.680 --0.299 --0.025 --0.018 0.211 0.461 0.653 --0.179 --0.077 --0.061 0.335 0.470 0.095 --0.513 --0.684 --0.853 --0.303 --0.239 0.196 --0.035 0.272
0.145 0.190 0.149 0.057 --0.179 --0.107 0.178 0.203 0.097 0.071 --0.139 0.194 0.109 0.172 0.088 0.030 0.174 0.288 --0.043 --0.051 --0.259 --0.264 --0.215 0.111 --0.109 --0.376 --0.948 --0.372 --0.181 0.146 0.137 0.165 0.025 --0.237 --0.415 --0.236 0.311 --0.515
0.008 0.080 0.011 --0.104 --0.066 0.040 0.069 0.126 --0.073 --0.387 --0.161 0.060 0.059 0.098 --0.080 --0.369 0.038 0.128 --0.089 0.018 --0.130 --0.174 0.146 0.126 --0.068 --0.017 0.013 0.161 0.133 --0.299 0.067 0.095 0.179 --0.101 --0.244 --0.095 --0.089 0.013
0.052 0.012 --0.037 --0.086 --0.075 --0.034 --0.040 --0.066 --0.054 --0.025 --0.036 --0.023 --0.024 --0.036 0.012 0.036 --0.022 --0.069 --0.015 --0.058 0.123 0.095 0.066 0.043 --0.048 --0.133 0.000 --0.034 --0.015 0.124 0.082 --0.005 0.108 --0.072 --0.140 --0.114 0.283 0.025
R.N. K A V A N A G H ET AL.
640
eigenvalues of the correlation matrices of the data when channels were considered to be variables. The factor coefficients (loadings)
for each of the six data sets were also computed; the factor structures for DALO left and KW left are shown in Tables II and III.
TABLE III Factor loadings for each channel o f KW data from flash left condition. Channel
1 2 3 4 5 6 7 9 10 11 12 13 14 15 17 18 19 21 22 23 25 26 27 28 29 30 31 33 34 35 36 37 38 39 41 42 43 44 45 46 47
Principal factors 1
2
3
4
5
6
0.013 0.181 0.063 --0.265 --0.886 --0.957 --0.568 --0.790 --0.627 --0.869 --0.930 --0.915 --0.889 --0.966 0.100 0.355 0.137 --0.872 --0.851 --0.962 0.276 0.539 0.582 0.205 --0.863 --0.843 --0.934 0.179 0.480 0.628 0.680 --0.828 --0.864 --0.827 0.203 0.392 0.356 --0.083 --0.916 --0.948 --0.614
0.955 0.959 0.969 0.895 0.239 --0.101 0.741 0.189 0.094 --0.040 --0.224 --0.341 --0.404 --0.010 0.426 0.382 0.167 --0.436 --0.453 0.065 0.760 0.554 0.468 0.287 --0.392 --0.435 0.169 0.878 0.771 0.626 0.474 --0.342 --0.344 0.408 0.927 0.878 0.893 0.936 0.226 0.153 0.715
0.195 0.146 0.141 0.212 --0.191 --0.183 0.018 0.514 0.749 0.362 0.139 --0.006 --0.122 --0.139 0.823 0.834 0.956 --0.015 --0.178 --0.141 0.494 0.612 0.639 0.850 --0.171 --0.234 --0.139 0.343 0.344 0.404 0.435 --0.371 --0.267 --0.160 0.248 0.199 0.201 0.121 --0.238 0.228 --0.039
--0.054 --0.082 --0.171 --0.231 --0.310 --0.128 --0.302 0.121 --0.028 0.022 0.040 0.053 0.086 --0.148 0.133 0.002 --0.067 0.041 0.103 --0.172 0.268 0.072 --0.011 --0.002 0.144 0.158 --0.234 0.252 0.142 0.039 0.001 0.101 0.166 --0.334 0.160 0.083 --0.041 --0.089 --0.150 0.030 --0.312
0.184 0.080 --0.015 --0.047 --0.008 0.001 0.162 0.075 0.041 --0.003 --0.037 --0.034 --0.001 --0.053 0.212 0.145 0.079 0.028 0.054 --0.059 0.104 --0.034 --0.142 --0.227 0.112 0.078 --0.076 0.051 --0.174 --0.199 --0.297 0.170 0.150 --0.052 0.018 --0.153 --0.161 --0.202 0.032 0.086 0.069
--0.091 --0.083 --0.030 0.138 0.110 0.138 --0.030 --0.080 --0.116 --0.314 --0.228 --0.192 --0.136 --0.121 0.103 0.082 --0.071 --0.186 --0.132 0.100 0.103 0.108 0.072 --0.083 --0.061 0.010 0.102 0.063 0.052 0.075 --0.002 0.017 0.075 0.062 --0.027 --0.007 0.036 0.142 0.051 0.090 --0.009
DIMENSIONALITY OF THE HUMAN VESP
641
T A B L E IV
Cumulative percentage of variance summarized by eigenvalues of correlation matrices with sample times considered as variables. Eigenvalue
D a t a set a n a l y z e d
1 2 3 4 5 6
u~
DALO Flash r i g h t
DALO F l a s h left
DALO Flash both
KW Flash right
KW F l a s h left
KW F l a s h both
46.0 71.8 86.5 95.5 97.7 98.3
51.9 76.9 90.3 95.3 97.6 98.3
61.5 77.1 86.2 92.3 95.8 97.6
62.5 79.6 91.7 94.7 96.1 97.1
56.2 72.8 85.2 90.7 95.3 97.2
68.1 83.1 91.7 94.3 96.2 97.7
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Fig. 6. Factor coefficients at successive sample times for six most significant factors in KW flash left d a t a a n a l y z e d with sample times as variables.
3. Factor analysis using sample times as variables
Discussion
1. Evoked potentials Table IV shows the cumulative percentage of the variance summarized by the first six eigenvalues of the correlation matrices of the data when sample times were considered as the variables. Factor coefficients for each of the six data sets were computed; data for DALO left and KW left are shown graphically in Fig. 5 and 6.
Considering the equipotential maps in Fig. 4 it is apparent that there is considerable similarity in the responses of either subject to all three stimuli, but there is not much similarity between the responses of the two subjects, except in the range 175--195 msec. Certain simple patterns in the equipotential maps are
642
quite stable; in the case of the KW data, two well-defined potential troughs are prevalent from about 90 to 130 msec; and over this same range a stable but different pattern prevails in the DALO data. The simple structure of the equipotential maps indicates that neighboring channels are interdependent; also over several intervals in the responses, the maps suggest a simple, stable configuration of generators.
2. Factor analysis using channels as variables Table I shows that most of the experimental variance can be accounted for by only six eigenvalues -- in fact a great deal is summarized by only three eigenvalues, thus the channels are not independent of one another. The data are well accounted for by six factors and a model having six disposable parameters would have some justification in these results. Note that, if desired, one could select from Tables II and III six apparently independent recording channels by grouping channels whose coefficients for the first three factors are similar in magnitude and sign. This process leads to the classical gross classification of cortical areas, that is: left temporal, occipital, right temporal, right occipital, left ear group, and right ear group. The Tables clearly show that electrodes adjacent to one another record similar evoked potentials, hence the dependence from channel to channel. The important result is thus that a small number of independent processes is present, probably six or fewer. This suggests that it is appropriate to att e m p t models which are characterized by a small number of parameters.
3. Factor analysis using sample times as variables Table IV shows that most of the experimental variance is accounted for by three eigenvalues, and over 97% is summarized by six. Note that the mean of each sample time in this analysis is a measure of the average potential at all electrodes at that sample time,
R.N. KAVANAGH ET AL.
and the correlations between variables are the correlations of the average activity at one sample time with that at another. Thus Fig. 5 and 6 are plots of the coefficients of this average activity at each sample time with each of the six underlying factors. These factor coefficient curves show that for each subject there are roughly three intervals of importance during the responses. If we consider factors one and two for each subject, we see that they generally have the same sign and about the same magnitude over specific intervals, namely between 50 and 90 msec, 100 and 140 msec, and between 150 and 180 msec. Remembering that the factors are arranged in decreasing order of summarization of the data variability, it is seen that these first two factors, which account for about 75% of the variance, are the most important to consider. In each of the remaining factors there are various peaks or troughs which generally occur in one or more of the three intervals singled out. This is in good accord with the appearance of the equipotential maps, where for each of these intervals there is a stable pattern in the c o n t o u r lines. The differences between the responses of the two subjects do not manifest themselves generally in the factor coefficients until the last three factors. The results suggest the temporal sequence of generators of the evoked potentials -- over quite long segments of the responses the activity at some sample time is strongly related to the activity at times preceding and following, indicating that the activity over that period arises from a stable set of generators.
4.
Modeling
the
visual evoked
potential
The primary motivation for discussing the h u m a n visual evoked potential in terms of dimensionality of the data lies in considering the implications this has for modeling. One hypothesis is that the potentials on the surface of the head result from equivalent dipoles located somewhere in the brain. If this is so, then at any point on the surface of the head Vi(t i) = aixPx (tj) + aiyPy (tj) + aizPz(tj)
(3)
DIMENSIONALITY OF THE HUMAN VESP where Vi(tj) is the voltage which appears at electrode i at time tj. The three components of the dipole, Px(tj), Py(tj) and Pz(tj) are combined in a linear manner at any electrode site, and the constants aix , aiy and aiz are dependent upon only the dipole location, the reference potential location and the location of the recording electrode. Note t h a t the dipole components may be a function of time but the dipole location is assumed constant; the a i's are not functions of time. A single dipole is characterized by six parameters: three position coordinates and three orthogonal components of the dipole strength. Equation (3) is apparently a function of only three parameters because the dipole is assumed to be fixed in space and hence the dependence upon the actual position of the dipole is included in the coefficients aix, aiy and aiz. Thus, if such a relationship were to exist in the evoked potential data, the model would be of the form V = AP, where V is the N × K matrix of experimental potential values from N electrodes sampled at K instants of time, P is the 3 × K matrix of model dipole components, and A is the N × 3 matrix of model dipole coefficients. In practice it is improbable that the matrix of coefficients A is time-invariant, since in the case of evoked potentials one would expect to have either different sources active at different times during the response or moving sources.. This suggestion is borne out in the results obtained using channels as variables, in which it would appear t h a t three factors are not sufficient to summarize the data. However, if one uses six factors, it is clear that the data are well accounted for, and one could use this result to argue that over the duration o f the response one or perhaps two dipoles would suffice: if necessary these one or two dipoles could be allowed to be relatively fixed for some interval and then move to another relatively fixed site over another interval of the response; i.e., Equation (3) applies piecewise over the complete response. Again, it should be emphasized t h a t m a n y other model formulations involving six parameters could
643 account for the same data. Thus the choice of a particular model cannot be justified on the basis of these results alone. The important conclusion is that a small number of independent processes are apparently present, probably six or fewer. Likewise, as previously discussed, the results obtained using sample times as variables suggest a small number, possibly three, of stable generators in temporal sequence.
Summary A small number of processes can account for most of the evoked potential activity in the two subjects studied. Principal components analysis indicates that six independent processes can account for approximately 97% of the variability in the data. Moreover, the factor analysis and plots of the factor coefficients yield indications that the times during which these principal factors are active agree quite well with the times at which the equipotential maps show some organized activity. The question of dipoles being the underlying cause of the observed activity is not answered by the factor analysis. The principal factors are not unique, but models which have a small number of parameters are more justifiable in light of the results of this study.
R6sumd La dimensionalitg du potentiel @icrdnien dvoqug visuel de l'homme Un petit nombre des processus peut expliquer la plupart de l'activit~ potentielle 6voqu~e chez les deux sujets 6tudi~s. L'analyse en composantes principales indique que six processus ind~pendants peuvent expliquer approximativement 97% de la variabilit6 dans les donn~es. De plus, l'analyse factorielle et les trac6es des coefficients factoriels indiquent que les temps durant lesquels ces factuers prin-
644
cipaux sont actifs s'accordent assez bien avec les temps pendant lesquels les cartes 6quipotentielles montrent une certaine activit6 organis6e. La question des dipoles 6tant la cause fondamentale de l'activit6 observ6e n'est pas r6solue par l'analyse factorielle. Les facteurs principaux ne sont pas uniques, mais des modules qui ont un petit nombre des param~tres sont plus justifiables par suite des r6sultats de cette ~tude.
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R.N. KAVANAGH ET AL. cats. Ann. N.Y. Acad. Sci., 1964, 112: 362--420. John, E.R., Walker, P., Cawood, D., Rush, M. and Gehrmann, J. Factor analysis of evoked potentials. Electroenceph. clin. Neurophysiol., 1973, 34: 33--43. Larsen, L.E. An analysis of the intercorrelations among spectral amplitudes of the EEG: a generator study. IEEE Trans. bio-med. Engng, 1969, BME-16 : 23--26. Morrison, D.F. Multivariate statistical methods. McGraw-Hill, New York, 1967, 338 p. Offner, F.F. The EEG as potential mapping: the value of the average monopolar reference. Electroenceph. clin. Neurophysiol., 1950, 2: 213--214. Ruchkin, D.S., Villegas, J. and John, E.R. An analysis of averaged evoked potentials making use of least mean squares techniques. Ann. N.Y. Acad. Sci., 1964, 115: 799--826. Scher, A.M., Young, A.C. and Meredith, W.M. Factor analysis of the electrocardiogram. Circular. Res., 1960, 8: 519--526. Seal, H.L. Multivariate statistical analysis for biologists. Methuen and Co., London, 1964, 209 p. Suter, C.M. Principle components analysis of average evoked potentials. Exp. Neurol., 1970, 29: 317-327. Van Rotterdam, A. Limitations and difficulties in signal processing by means of the principlecomponents analysis. IEEE Trans. bio-med. Engng, 1970, BME-17: 268--269. Walter, D.O. and Adey, W.R. Analysis of brain-wave generators as multiple statistical time series. IEEE Trans. bio-med. Engng, 1965, BME-12: 8--13.