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Mapping the functional proximity of cortical regions by multidimensional scaling
Neuroscience Lettere, 8 (1978) 99--104 © Elsevier/North-Holland Scientific Publishers Ltd.
99
MAPPING THE FUNCTIONAL PROXIMITY OF CORTICAL REGIONS BY MULTIDIMENSIONAL SCALING JACKSON BEATTY
Department of Psychology, University of California, Los Angeles, Calif. 90024 (U.S.A.) (Received January 18th, 1978) (Revised, version received February 13th, 1978) (Accepted February 14th, 1°o78)
SUMMARY
Multidimensional scaling procedures provide a new means of studying the spatial organization of brain electrical activity. A three-staged analysis process is proposed and tested on humvn electroencephalographic data. Simple, high accuracy solutions with anatomically meanin~ul dimensions were achieved. The procedure is not limited to the application presented, but is of genera] utility.
In the electrophysiological analysis of brain ftmction, elegant and power.?ul of activity at a techniques have been developed to study the temporal pa~em "* single recording site, but rather less progress has been made in solving the more difficult problems of analyzing the spatial patterning of activity at a numbe:r of simultaneously recorded sites (see refs 7, 8, 9 and 10). Ultimately much of our understanding of the organization of brain functions must depend upon such configurational analyses [12]. In the present paper, I report that recently developed techniques of multidimensional scaling provide useful analytic ~o~ls for the extraction of ~nforrnation about the functiional relations between the sources of simultaneously recorded multichannel electrophysiological data, using the multichannel electroencephalogram (EEG) as an example. The procedure involved consists of three logically distinct steps. In the first, multi-channel electropnymologmal data is acquired from the preparation, inspected for artifacts, preprocessed if necessary, land then stored for analysis. In the second step, some aspect of that data is selected by appropriate featureselection and proxixmty-estimation algorithms which are applied to all channels of data in a pairwise fashion. The product of this step is a numerical estimate of the similarity or dissimilarity of each pair of channels with respect to the features of interest. In the third step, the resulting proximity matrix is ana.~yzed by a multidimensional scaling procedure to determine both the number of
100
dimensions m.~essary to reproduce the measured similarity and the corJi~m~ration of the channels in that multidimensional space. Nonmetric multidimensional scaling analysis attempts to derive the configuration of points in an N
dimensiOnality "~,~inga general pro~am for 2-way multidimensional scaling, KYST [ 6]. The results of this analysis for each of the I0 subjects appears in Fig. 1. In
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Fig. 1. Function configurations of EEG activity at the eight recording sites with respect to ~hared synchrony as determined by a 2-way mul~dhnensional scaling procedure, KYST. Note the orderly arrangement of voints about an anterior-posterior and left-right axis. F u n c ~ o n ~ l y more similar points are closer together in this representation. The solutions were reflected on one or both axes for some subjects before plotting to achieve uniformity in the figure.
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these solutions, points that cluster together are functionally similar, whereas points separated by greater distances in the solutions are functionally more dissimilar. From these configurations it appears that orderly information concerning functional proximity of brain regions is contained within the m ultichannel electroencephalogram. Although the configurational solution may be freely rotated in 2-way Euclidean scaling procedures such as KYST, in the present example the rotation to principal components achieved automatically by the program was sufficient. The two dimensions that appe~r~ J e anatomically meaningful and may be labelled anterior-posterior and left-right respectively. O n these dimensions, with the exception of a revers~ in the occip~*~Llchannels of Subject 6, every ~:hannel is approprhtely placed with respect to the other channels given the anatomic~ arrangement~ from which the electroencephalogram wa~ ~e,-orded. But the distance~ in these configurations do not represent anaton~ical distances which may be measured in millimeters, but rather reflect functional distances, which ~ c measured in the units required by the algorithm generating the original proximiW matrices, hi this case the unit of distance reflects the degree of synchronous EEG activity at the various sites measured. These 2-dimensional configurations represent quite accurate fits to the data. In the KYST procedure for multidimensional scaling, the configurations are altered to minimize "stress", a measure of the badness of ~he fit. Stress as used here is defined as the ratio of the root mean squared error in estimating the original proxim:'ty data from the computed dimensional configuration of distances and the root mean square of the distances in tl e configuration. By ~his measure, the fit of the configurations shown in Fii~. 1 is exceptionally ~ ~. . . . configurations rang,~,d between 0.04 and accurate; the stress values of 0.01~ for individual subjects. Thus the configum~ons re ~resent the data to an accuracy of ] percent or better. Wit~l respect to the d'~mensionality of the solution, the 2~iimensional solutions presented appear to be most adequate. The stress values associated witch 1-dimensional solutions were about twenty times as great as those for the 2~limensiona~ solution, ranging from 0.080 and 0o156 for individual subjects. With only 8 point.s in a configuration, a 3-dimensional solution is not warranted ~'
An e x ~ n a t i o u of the plot of computer distances in the configuration against the corresponding proximities in the PCC matrix (Shepard diagr~n) reveals a linear or near-linear relation for every subject. This indicates that a m~t~c analysis o~ the~e data could be employed. The ~mportance of these findings is as follows: First, multidimensional scaling procedures provide a method for clearly visualizing and comprehending the functional organization of electrophysiological activity as it simultaneously appears st s number of recording sites. This provides a solution to one of the most difficult problems in multichan~el electrophysiological analysis, representation of the data in a concvptuaUy meaningful form. Second, this method is not limi~d to the correlational analysis of the human
103
EEG. An~ source of multichannel electrophysiological data may be employed, providing chat a substantively meaningful procedure for estimating functional proximity may be found. One obvious application wouTId be in the study of multlple~hannel averaged evoked potentials~where a correlation statisticcould be used to assess waveform similarity of the potentials and multidimensicbna! scaling to provide a map of functionally related rc
I wish to thank N. Gaynor and R. Peeler, who assisted in various aspects of the experimental proced~tre and E. Ho!n~n, for his helpful comments. This research was sponsored by the Office of Naval Research under Contract N00014-76-C-0015. REFERENCES
1 Callaway, E. and Harris, P.R., Coupling between cortical potentials from different a~as, Science, 183 (1974) 873--875. 2 Carroll, J.D. and Chang, J.-J., Analysis of individual differences in multidimensional scaling via an N-way generalization of "Eckart-Young" decomposition, Psychometrika, 35 (1970) 283--319. 3 Harmony, T., Otero, G., Ricardo, J. and Fernandez, G., Polarity coincidence correlation coefficient and signal energy ratio of the ongoing EEG activity. I. ]Normative data, Br~in Res., 61 (1973) 133--140. 4 Jasper, H.H., The ten twenty electrode sy~'~temof the International Federation, Electroenceph, din. Neurophysiol., 10 {1958) 371--375.
104 5 Kruskal, J.B. and Wish, M., Basic concepts of multidimensional scaling (Sage University Papers on Quantitative Application in the Social Sciences, Berkeley, in preparation). 6 Kruskal, J.B., Young, F.W. and Seery, J.B., How to use KYST, a very flexible program to do multidimensional scaling and unfolding, Bell Telephone Labs, Murray Hill, New Jersey. 7 Lehmann, D., EEG phase differences and their physiological significance in scalp field studies. In G. Dolce and H. Kltnkel (Eds.),CEAN: Computerized EEG analysis, Gustav Fischer Verlag, Stutt~gart, 1975, pp. 102--110. 8 Livanov, M.N., Spatial organization of cerebral processes, Wiley, New York, 1977, 181 pp. 9 Pet~che, H., N~gypal, T., Prohaska, O., Rappelsberger, P. and Volimer, R., Approaches to ~he spatio-temporal analysis of seizure patterns. In G. Dolce and H, Kiinkel (Eds.), CEAN: Computerized EEG analysis, Gustav Fischer Verlag, Stuttgart, 1975, pp. 111-127. 10 Talbot, S.A. and Gessner, U., Systems physiology, Wiley, New York, 1973, 511 pp. 11 Walter~D.O. and Adey, W.R., Analysis of brain-wave generations as multiple statistical time series, IEEE trans. Biomed. Eng., BME-12 (1965) 8--13. 12 Zulch, J.K., Creutzfeldt, O. and Galbraith, G.C. (Eds.), Cerebral localization: an Otfrid Fo,.~rster symposium, Springer-Verlag, Berlin, 1975, 339 pp.; Luria, A.R., Higher coztical functions in man, Basic Books, New York, 1966, 513 pp.
99
MAPPING THE FUNCTIONAL PROXIMITY OF CORTICAL REGIONS BY MULTIDIMENSIONAL SCALING JACKSON BEATTY
Department of Psychology, University of California, Los Angeles, Calif. 90024 (U.S.A.) (Received January 18th, 1978) (Revised, version received February 13th, 1978) (Accepted February 14th, 1°o78)
SUMMARY
Multidimensional scaling procedures provide a new means of studying the spatial organization of brain electrical activity. A three-staged analysis process is proposed and tested on humvn electroencephalographic data. Simple, high accuracy solutions with anatomically meanin~ul dimensions were achieved. The procedure is not limited to the application presented, but is of genera] utility.
In the electrophysiological analysis of brain ftmction, elegant and power.?ul of activity at a techniques have been developed to study the temporal pa~em "* single recording site, but rather less progress has been made in solving the more difficult problems of analyzing the spatial patterning of activity at a numbe:r of simultaneously recorded sites (see refs 7, 8, 9 and 10). Ultimately much of our understanding of the organization of brain functions must depend upon such configurational analyses [12]. In the present paper, I report that recently developed techniques of multidimensional scaling provide useful analytic ~o~ls for the extraction of ~nforrnation about the functiional relations between the sources of simultaneously recorded multichannel electrophysiological data, using the multichannel electroencephalogram (EEG) as an example. The procedure involved consists of three logically distinct steps. In the first, multi-channel electropnymologmal data is acquired from the preparation, inspected for artifacts, preprocessed if necessary, land then stored for analysis. In the second step, some aspect of that data is selected by appropriate featureselection and proxixmty-estimation algorithms which are applied to all channels of data in a pairwise fashion. The product of this step is a numerical estimate of the similarity or dissimilarity of each pair of channels with respect to the features of interest. In the third step, the resulting proximity matrix is ana.~yzed by a multidimensional scaling procedure to determine both the number of
100
dimensions m.~essary to reproduce the measured similarity and the corJi~m~ration of the channels in that multidimensional space. Nonmetric multidimensional scaling analysis attempts to derive the configuration of points in an N
dimensiOnality "~,~inga general pro~am for 2-way multidimensional scaling, KYST [ 6]. The results of this analysis for each of the I0 subjects appears in Fig. 1. In
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Fig. 1. Function configurations of EEG activity at the eight recording sites with respect to ~hared synchrony as determined by a 2-way mul~dhnensional scaling procedure, KYST. Note the orderly arrangement of voints about an anterior-posterior and left-right axis. F u n c ~ o n ~ l y more similar points are closer together in this representation. The solutions were reflected on one or both axes for some subjects before plotting to achieve uniformity in the figure.
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102
these solutions, points that cluster together are functionally similar, whereas points separated by greater distances in the solutions are functionally more dissimilar. From these configurations it appears that orderly information concerning functional proximity of brain regions is contained within the m ultichannel electroencephalogram. Although the configurational solution may be freely rotated in 2-way Euclidean scaling procedures such as KYST, in the present example the rotation to principal components achieved automatically by the program was sufficient. The two dimensions that appe~r~ J e anatomically meaningful and may be labelled anterior-posterior and left-right respectively. O n these dimensions, with the exception of a revers~ in the occip~*~Llchannels of Subject 6, every ~:hannel is approprhtely placed with respect to the other channels given the anatomic~ arrangement~ from which the electroencephalogram wa~ ~e,-orded. But the distance~ in these configurations do not represent anaton~ical distances which may be measured in millimeters, but rather reflect functional distances, which ~ c measured in the units required by the algorithm generating the original proximiW matrices, hi this case the unit of distance reflects the degree of synchronous EEG activity at the various sites measured. These 2-dimensional configurations represent quite accurate fits to the data. In the KYST procedure for multidimensional scaling, the configurations are altered to minimize "stress", a measure of the badness of ~he fit. Stress as used here is defined as the ratio of the root mean squared error in estimating the original proxim:'ty data from the computed dimensional configuration of distances and the root mean square of the distances in tl e configuration. By ~his measure, the fit of the configurations shown in Fii~. 1 is exceptionally ~ ~. . . . configurations rang,~,d between 0.04 and accurate; the stress values of 0.01~ for individual subjects. Thus the configum~ons re ~resent the data to an accuracy of ] percent or better. Wit~l respect to the d'~mensionality of the solution, the 2~iimensional solutions presented appear to be most adequate. The stress values associated witch 1-dimensional solutions were about twenty times as great as those for the 2~limensiona~ solution, ranging from 0.080 and 0o156 for individual subjects. With only 8 point.s in a configuration, a 3-dimensional solution is not warranted ~'
An e x ~ n a t i o u of the plot of computer distances in the configuration against the corresponding proximities in the PCC matrix (Shepard diagr~n) reveals a linear or near-linear relation for every subject. This indicates that a m~t~c analysis o~ the~e data could be employed. The ~mportance of these findings is as follows: First, multidimensional scaling procedures provide a method for clearly visualizing and comprehending the functional organization of electrophysiological activity as it simultaneously appears st s number of recording sites. This provides a solution to one of the most difficult problems in multichan~el electrophysiological analysis, representation of the data in a concvptuaUy meaningful form. Second, this method is not limi~d to the correlational analysis of the human
103
EEG. An~ source of multichannel electrophysiological data may be employed, providing chat a substantively meaningful procedure for estimating functional proximity may be found. One obvious application wouTId be in the study of multlple~hannel averaged evoked potentials~where a correlation statisticcould be used to assess waveform similarity of the potentials and multidimensicbna! scaling to provide a map of functionally related rc
I wish to thank N. Gaynor and R. Peeler, who assisted in various aspects of the experimental proced~tre and E. Ho!n~n, for his helpful comments. This research was sponsored by the Office of Naval Research under Contract N00014-76-C-0015. REFERENCES
1 Callaway, E. and Harris, P.R., Coupling between cortical potentials from different a~as, Science, 183 (1974) 873--875. 2 Carroll, J.D. and Chang, J.-J., Analysis of individual differences in multidimensional scaling via an N-way generalization of "Eckart-Young" decomposition, Psychometrika, 35 (1970) 283--319. 3 Harmony, T., Otero, G., Ricardo, J. and Fernandez, G., Polarity coincidence correlation coefficient and signal energy ratio of the ongoing EEG activity. I. ]Normative data, Br~in Res., 61 (1973) 133--140. 4 Jasper, H.H., The ten twenty electrode sy~'~temof the International Federation, Electroenceph, din. Neurophysiol., 10 {1958) 371--375.
104 5 Kruskal, J.B. and Wish, M., Basic concepts of multidimensional scaling (Sage University Papers on Quantitative Application in the Social Sciences, Berkeley, in preparation). 6 Kruskal, J.B., Young, F.W. and Seery, J.B., How to use KYST, a very flexible program to do multidimensional scaling and unfolding, Bell Telephone Labs, Murray Hill, New Jersey. 7 Lehmann, D., EEG phase differences and their physiological significance in scalp field studies. In G. Dolce and H. Kltnkel (Eds.),CEAN: Computerized EEG analysis, Gustav Fischer Verlag, Stutt~gart, 1975, pp. 102--110. 8 Livanov, M.N., Spatial organization of cerebral processes, Wiley, New York, 1977, 181 pp. 9 Pet~che, H., N~gypal, T., Prohaska, O., Rappelsberger, P. and Volimer, R., Approaches to ~he spatio-temporal analysis of seizure patterns. In G. Dolce and H, Kiinkel (Eds.), CEAN: Computerized EEG analysis, Gustav Fischer Verlag, Stuttgart, 1975, pp. 111-127. 10 Talbot, S.A. and Gessner, U., Systems physiology, Wiley, New York, 1973, 511 pp. 11 Walter~D.O. and Adey, W.R., Analysis of brain-wave generations as multiple statistical time series, IEEE trans. Biomed. Eng., BME-12 (1965) 8--13. 12 Zulch, J.K., Creutzfeldt, O. and Galbraith, G.C. (Eds.), Cerebral localization: an Otfrid Fo,.~rster symposium, Springer-Verlag, Berlin, 1975, 339 pp.; Luria, A.R., Higher coztical functions in man, Basic Books, New York, 1966, 513 pp.