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Spectral analysis of seizures in humans
COMPUTERS
AND
BIOMEDICAL
Spectral
RESEARCH
8,503-521
Analysis
(1975)
of Seizures
in Humans*
B. R. THARP AND W. GERSCH Department of Neurology, Stanford University Medical Center, Stanford, California 94305 Received March 12.1975 The electroencephalogram of a patient with intractable epilepsy was recorded with chronically implanted depth electrodes. The activity during a seizure was analyzed by an autoregressive spectral analysis technique that permits the calculation of the power spectra for all channels and the coherence between all possible channel pairings. The partial or residual coherence between a channel pair was calculated after a regression analysis had removed the influence of a third channel. If the coherence between two channels, in a frequency range of high spectral energy, fell to zero after partialing on a third channel, the latter was considered to drive the other two channels. A single site was sought which appeared to drive all other areas by analyzing all possible channel pairings and triplings. The source and spread of the seizure was determined by analyzing sequential data epochs. The analysis revealed the ictal activity arose in a single region, then spread over several pathways. The power spectra of the ictal activity revaled peaks in several frequency regions. Calculation of coherences and partial coherences in each of these frequency regions revealed different driving channels and, with analysis of sequential epochs, different pathways of spread of the ictal activity occurring at these frequencies. The spectral analysis technique provides, therefore, details of the seizure propagation and pacemaker activity not possible with conventional visual analysis of the analog data.
The analysis of the complex rhythms generated by the brain during a seizure has relied on the visual perusal of analog data. The attention of the electroencephalographer is directed to the period immediately preceding the seizure, the initial phases of the ictus and the postictal rhythms. The ictal rhythms of the generahzed seizures (“cortico-reticular epilepsies”) (I) and focal seizures which spread rapidly to contiguous cortex or to homologous areas of the contralateral hemisphere, are usually too complex for simple visual analysis. The electroencephalographer is frequently unable to locate the source of the abnormal activity or to accurately map its propagation through the brain. * This investigation was supported in part by Public Health Service Grant No. R.R.-70 from the General Clinical Research Centers, Division of Research Resources and an Epilepsy Foundation of America Research Grant. Copyright 0 1975 by Academic Press, Inc. 503 All rights of reproduction in any form reserved. Printed
in Great
Britain
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AND
GERSCH
The introduction of computers and the mathematical discipline of time series analysis into the field of electroencephalography has provided new tools with which to analyze these complex rhythms (2-5). In particular, spectral analysis of the electroencephalogram (EEG) and the linear system “black box” properties of spectral coherences and transfer functions derived from the spectral analysis (6) can reveal relationships between the activity from different brain regions. Spectral analysis of sequential EEG epochs can reveal changes in these linear relationships and shifts of energy to different frequency bands that are not evident in the more classical visual analysis of analog data. Major issues in the analysis of focal seizures recorded by surface or implanted macroelectrodes include the localization of the area of the brain from which the seizures arises, the pathways of propagation of epileptic activity and the site of the epileptic “pacemaker” (2) which may change as the seizure evolves. The results of the application of a spectral analysis technique to the study of these issues in the complex multichannel time series that appear in human epileptic activity are reported in this paper. Our notion of a seizure pacemaker is that it causes or drives the epileptic activity observed in other parts of the brain. Operationally, that pacemaker is identified by a spectral analysis technique that is equivalent to the identification of causality in time series. Experiments on simulated data corresponding to our causaIity model (unpublished data), and on cats made epileptic by electrical stimulation of a single brain site, reveal that causality can be unambiguously identified in time series by our computational method even when visual examination of the analog data offers no apparent clues to the generator site (7). The analysis of human alpha rhythms by our technique demonstrated that electrophysiological driving could be identified from the scalp EEG (8). The analysis of the EEG obtained simultaneously from scalp and epidural electrodes in a human epileptic patient (unpublished data) revealed that frontal lobe origin of a diffuse seizure discharge (proven by surgical resection and subsequent absence of seizures) could be identified using our computational technique on the scalp EEG data alone. The analysis of the scalp EEGs revealed, therefore, that causality or electrophysiological driving could be identified despite the fact that the electrodes were not placed precisely at the site of origin of the seizure. In the last two years the EEG data from scalp, epidural, and depth electrode recordings (the latter supplied by John E. Adams, M.D., Department of Neurosurgery, University of California at San Francisco) of 15 epilepticpatients have been analyzed. Our experience has been that the computational identification of the driving sites of epileptic seizures has almost invariably both corroborated what could be observed by careful visual examination of the EEG and also added valuable information concerning the initiation and spread of seizure activity that could not be discerned from the examination of the analog EEG. In this paper, the analysis of a single seizure in a human subject is examined in some detail to illustrate the method and to describe the logic and results obtained in using our technique in tracing the spread of a focal seizure from its site of origin.
SPECTRAL
ANALYSIS
OF SEIZURES
505
METHODS
The depth electrodes used in this patient were stereotactically placed in both frontal and temporal lobes. Each electrode consisted of a pair of interwoven insulated stainless steel wires. Bipolar recordings were obtained at each site with an interelectrode distance of 2 mm. The EEG was passed through low level amplifiers (band pass essentially linear between 1 Hz and 500 Hz) and stored on magnetic tape (Ampex FR 1300). The EEG was simultaneously displayed on a cathode ray oscilloscope and written on paper by a Grass Model VI electroencephalograph for continuous visual monitoring. Selected portions of 10 channels of magnetically recorded data were simultaneously digitized at the rate of 200 samples per set, using an IBM 1800 computer. The digitized data was immediately transferred and stored in an IBM 360/50. The stored EEG was serially displayed on a television screen for verification and 5- or 9.6-set epochs were selected for spectral analysis. METHODS
OF ANALYSIS
Spectra, cross spectra, and spectral coherences were computed by the autoregressive (AR) representation method (9). Ten channels of data were analyzed simultaneously and a 10 x 10 matrix of cross correlation functions was calculated. In this method, a formula, best in the least-squares sense, is computed that expresses the observed time series in terms of its own past history (a regression of the data on itself) plus an additional unpredictable, uncorrelated random component. The formula is an explicit parametric model of the observed EEG data. The order of the AR model was automatically computed by a method due to Akaike (10-12). For each of the EEG epochs selected for analysis, spectral densities and coherences and partial coherences on all possible pairs and triples of data channels were computed from the AR model over a frequency range of O-50 cjsec using formulas appearing, for example, in Gersch (9). The statistical properties of spectra computed by this method have been discussed by Akaike (IO), Parzen (13) Kromer (24) and Gersch (9,lO The spectral density of the ictal data was characterized by a pronounced peak or peaks. The spectral coherences and partial coherences over the frequency intervals with high spectral intensity were utilized in the identification of potential or candidate EEG driving channels. Briefly, consider three simultaneous EEG time series that have high pairwise spectral coherences over the frequency range of interest. Spectral coherence at frequency f may be interpreted as the square of the correlation coefficient between the random variables that describe the spectral intensity in each of the time series at frequency f. The high pairwise spectral coherence may be due to the fact that the two brain areas are connected and their behavior is interdependent or, alternatively, they may be connected to another area which drives each of them. Our technique for the identification of a causal or driving site involves the calculation of partial spectral
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GERSCH
coherences. Partial coherence expresses the linear relationship (the square of the correlation coefficient) that remains between two time series after the influence of a third time series has been removed by a partial regression analysis. In our approach, we conclude that one time series drives or is causal relative to a pair of time series when (i) the partial spectral coherence between the data pair conditioned on this third time series is zero over the frequency range of high spectral density and high spectral coherence and (ii) only one time series of three considered has this property. If the partial coherence is zero, then formally, the third time series explains the linear relationship (coherence) between the other two. If time series are jointly gaussian, zero coherence between two time series implies that the two time series are not linearly related and more strongly, that they are statistically independent. From this perspective, we say that a channel which explains the linear relationship or interdependence between other time series drives or is causal to those time series. Conversely, if the partial coherence of the two time series conditioned on a third time series is substantially different than zero, then the third time series does not explain the linear relationship between the first two series. In the analysis of human seizure data, we ideally hope to locate a single recording site whose electrical activity will explain all of the spectral coherence that is observed between other data channels. For more details on this approach to causality and some applications to electrophysiological data, see Gersch and Goddard (7), Gersch, Midkiff, and Tharp (8) Gersch (16) and Tharp (17). The analysis generates voluminous data. Corresponding to d data channels, there are C(d, 2) spectral coherence versus frequency pairs and C(d, 3) partial spectral coherence triples where C(n, k) is the number of distinct combinations of n things taken k at a time, and C(n, k) = n!/(n - k)! (k!). The data, therefore, consists of(i) the spectral intensity of each of the 10 channels, (ii) C (10, 2) = 45 coherence pairs, and (iii) three graphs for each distinct partial coherence triple, 3 C (10, 3) = 360. The spectral data was computed from the AR model at integer values of frequency from O-50 Hz. A computer program scanned all of the coherence and partial coherence computations and graphs were printed of the spectral coherences and partial coherences in which the criterion that pairwise coherence was greater than or equal to 0.35 (2 0.35) and that partial coherence was equal to or less than 0.05 (SO.05) at the frequencies of high spectral intensities were satisfied. Using this technique, clear patterns in candidate driving channels emerged and appeared to justify the reasonableness of this approach and the arbitrary selection of the minimal coherence and maximal partial coherence values. For almost invariably, candidate driving channels did not appear to be isolated cases. For example, if Channel 3 was causal to Channels 1 and 2 by this criterion in one situation, we also observed that Channel 3 was causal to other channel pairs. Furthermore, the identification of the driving channels did not seem to be at all sensitive to slight variations in the lower limits of acceptable coherence or the upper limit of acceptable partial coherence. In this sense our driving channel criterion appears to be statistically robust.
SPECTRAL
ANALYSIS
OF SEIZURES
507
The application of spectral analyses methods to time series data is based upon the assumption that the time series is a finite duration sample from a stationary random process. An adhoc technique was used to partially substantiate that assumption for the EEG data analyzed. The order of the AR model computed for the spectral analysis is determined by the data and N, the number of data points available for analysis. In addition to computing the AR model determined by the data and N, we computed the model determined by the data and the number 1.15 N and the corresponding spectral analysis. The rationale for this procedure is that if the time series were stationary and the AR model of the data was stable, a nominal increase in N might slightly increase the order of the AR model but would not significantly change the outcome or the interpretation of the spectral analysis results. Conversely, iFthe time series were not stationary or if the observation interval was not sufficiently long, the AR model order and the spectral analysis results would change significantly with increasing N. This approach was justified by empirical experiments on simulated data and on EEG time series. The statistical reliability of the estimate of coherence in the neighborhood of zero coherence is particularly critical for our application. That estimate is in general less reliable than the estimates at greater values of coherence. The topic of the statistical reliability of AR-modeled spectral analysis has a research status. Parzen (13) conjectured that asymptotically the AR-modeled spectral matrix has a complex Wishart distribution with a = N/p degrees of freedom, where p is the order of the AR model fitted to the data. This assumption leads to the result that in the neighborhood of zero coherence the quantity ug2 is approximately chi-squared distributed, where g2 is the estimate of the coherence squared. In a simulation study of the statistical behavior of AR-modeled spectra and coherence estimates (personal observations), Parzen’s estimate of u in the neighborhood of zero coherence estimates has been found to be conservative. Typically, for d = 10 data channels and N = 500, we found 6
This 40-year-old male had a 30-year history of seizures characterized initially as generalized tonic clonic episodes. At age 13 they ceased, and two types of minor seizures appeared which had been relatively refractory to medication to the time of
SPECTRAL
E-5
ANALYSIS
OF SEIZURES
509
E-6
1(A-C) Electroencephalogram obtained during a clinical seizure characterized by tonic extension of upper extremities followed by automatic behavior (the three tracings are continuous). The depth electrodes are located in both anterior hippocampi (Hipp-La, RH), lateral amygdala (Amy-L*, R,,), mesial frontal basal area (Fron. Bas.-L FB,RFB), mesial frontalcortex 15 mm superior to frontal basal electrodes (Fron-Lr, RF), and anterior cingnlum (Cing-Lc,,, Rci.). The autoregressive spectral analysis was performed on the six segments of EEG underlined, E,, E,, . . ., Eb. Calibrations: Channels l-8,500 @; Channels 9,10,200 @; 2 sec. FIG.
this study. The most common minor episode consisted of a feeling of tightness about the left chest followed by abrupt loss of consciousness. The patient slowly slumped to the ground, frequently elevating the flexed left arm and became flaccid. There were no automatisms or lateralizing postictal signs. The other seizure type consisted of a sudden absence and irrelevant verbalization without loss of tone or abnormal motor activity. Neurological examination was essentially normal. Pneumoencephalograms, cerebral arteriography, and cerebrospinal examinations were normal on several occasions. Following this electroencephalographic study, the patient’s seizures were controlled with the addition of carbamazepine to his other anticonvulsants.
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GERSCH
Electroencephalograms. The computer analysis was performed on a seizure recorded with depth electrodes implanted bilaterally into the frontal parasagittal region and temporal lobes (Fig. 1). Multifocal spikes and spike and slow waves were noted prior to the seizure, particularly in the right cingulate and frontal basal regions and the left frontal and cingulate leads. The onset of the seizure was characterized by an abrupt diminution of the amplitude of the EEG in all leads. High amplitude, rhythmical 9 cjsec spiking appeared in the right cingulate gyrus (Rc,,) and right frontal basal (a,,) region and spread rapidly to both amygdala (RA, LA). This activity persisted for 5 set, abruptly disappeared, and 30 c/set spiking appeared in the right frontal basal (RF& left frontal basal (LFB), and right frontal (RF) areas. Approximately 6 set later, low voltage 10-l 2 cjsec spikes again appeared at R cIN, gradually increased in amplitude (maximum 600 /lV), and spread to involve both right frontal leads and L,,, interrupting and then replacing the fast spiking in these areas. The high amplitude activity at Rc,, persisted throughout the remainder of the seizure, though the frequency gradually slowed. Approximately 25 set after the onset of the seizure, high amplitude 2-3 clsec sharp slow waves appeared at R, and R,, and spread rapidly to LA and LH. This slow activity rapidly spread to all the remaining recording sites, admixing and finally replacing the fast spiking at R,, and the rhythmical IO-12 cjsec sharp waves in the frontal areas. There was negligibIe spread to Rcr,. Approximately 40 set later, the seizure waned and high amplitude, multifocal amorphous delta activity appeared in all leads. Visual analysis of the analog data suggested that the seizure had its onset simultaneously at electrodes Rcr, and RFB. Morphologically distinct ictal rhythms (low voltage 30 c/set spiking) appeared several seconds later in both frontal (LF, RF) and frontal basal (L,,, RFB) leads. The seizure rapidly spread to involve both frontal, frontal basal, and cingulate electrodes. Approximately 30 set after its onset, the ictal activity appeared in the temporal leads with a concomitant decrease of activity in the frontal and cingulate cortex. Six epochs, respectively E,, E,, . . ., Es, were analyzed, the first five were 5 set in duration, the sixth 9.6 set (Fig. I). Epoch 1 ends 6 set before the onset of the seizure, Epoch 2 includes the initial 5 set of the seizure, Epochs 3,4, 5, and 6 begin approximately 15 set, 20 set, 30 set, and 35 set, respectively, after the onset of the seizure.
RESULTS
In view of the large number of graphs (415/epoch for 10 simultaneous EEG channels for each epoch of data analyzed), we present below an abstraction and interpretation of those results that distinguish left hemisphere, right hemisphere, and interhemispheric performance.
511
SPECTRAL ANALYSIS OF SEIZURES
1. Coherences
The coherences occurring in a frequency region of high spectral intensity are listed in Table 1. Although significant coherence was often noted in more than one frequency region in many channel pairings, the highest coherence only is listed in the table. Prior to the onset of the seizure (El), high coherence is present only between the two frontal areas. At the onset of the seizure (E2), high coherence at 7-8 c/set appears between the frontal areas, cingulate regions, and the amygdala. As the seizure continues (E3), the high coherence between the frontal and cingulate areas persists, particularly at 19-21 c/set, whereas there is negligible involvement of the temporal areas with the exception of R,. Approximately 20 set after the onset of the seizure (E4), the high coherences now appear at 11-15 c/set, and the right temporal lobe again becomes involved. Ten set later (E5), the left temporal lobe, particularly L,, becomes coherent with ipsi- and contralateral temporal and frontal areas at 3 c/set. Channel pairings involving RclN are coherent at 10 c/set and 3 c/set. In E6 high coherences occur between the majority of recording sites at 4 c/set as the seizure propagates to the temporal lobes (Fig. 1). 2. Partial Coherenceand Driving
Table 2 lists all the candidate driving channels in Epochs 2-6 that were identified by the coherence criteria discussed above. A. Epoch 1 (preictal). Only one channel tripling revealed significant driving; Lr drove RF-RCIN. B. Epoch 2 (initial 5 set of seizure). Though high coherences developed between many areas, there was relatively little significant driving. The coherence between L,, 0.63 .
I
I
I
1
1
I
I
I
---RH.R 8 0.36 5 & g0.250
F (R GIN’ E3
-
0
1
5
10
15
20 25 FREQUENCY
30 (C/SEC)
36
40
45
50
FIG. 2. Graph of coherence versus frequency of right hippocampus and right mesial frontal area during Epoch 3. Solid line, coherence; broken line, partial coherence (coherence remaining between RH-RF when influence of right cingulum (RCIN) removed by regression analysis). RcrN “drives” RW and RF at 11 c/set only. There is no driving at 21-22 c/set, the second frequency at which RH and Rr are highly coherent.
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and sites in the contralateral frontal and cingulate areas was explained by the influence of LF. The only driving of two ipsilateral hemispheric areas by a contralateral site was that of R,-RcrN by L,,,. C. Epoch 3 (15 set following onset). The spectra revealed dominant peaks at 19-21 c/set in Rn, LA, and both frontal, frontal basal, and left cingulate areas. A secondary peak at IO-12 cjsec was present in R, and R, and was the frequency of highest spectral density of RCIN. The partial coherence data reveals clear driving of both inter- and intrahemispheric channel pairings by L, at 20-21 cjsec. In the only evidence of right hemisphere driving, RCIN drives Ru and R, at 11 cjsec (Fig. 2). Rn and R,,, are coherent at 21 and 11 c/set and are driven by Lr and LFB at 21 cjsec. D. Epoch 4 (20 set foZZowing onset). The spectral curves are characterized by peaks at 12-14 cjsec. A secondary peak at higher frequencies is present at Ln and L, (38.40 c/set), and LFB, RFB, RF, and R,i, (30-33 cjsec). All areas, with the exception of R C,N,have significant activity below 4 c/set. TABLE EPOCH
EPOCH
1
2
I
SPECTRAL EPOCH
ANALYSIS
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OF SEIZURES
4
LA
.._
.--
.. .
---
.44
.--
36
EPOCH
%a
-42
.42
42
.60
In E4 there are four right hemisphere candidate driving sites: RA, RCIN, RF, and R,. R, and R,-,* are highly coherent, not influenced by other areas, and also responsible for the majority of the coherences that appear over the right hemisphere. RcrN
514
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TABLE
Channel
1
Channel
LB LFB
R
2
Channel
II
3
Frequency
Epoch
2
LF
7
RF
LF
7
LFB
&IN
J-F
7
R*
R GIN
LClN
R*
RF
RFB
R”
LFB
LF
RH
R FB
RH
RF
R”
LN
LF I-1 J-F
&I
RC,N
LF
LFB
RFB
L
LFB
LIN
LF
FB
7 7
0.03 0.03 0.00 0.05 0.00
21 21 21 21 21 21 21 20 21 21 21 21 11
0.48 0.42 0.50 0.48 0.38 0.52 0.60 0.52 0.50 0.56 0.38 0.38 0.46
0.00 0.05 0.03 0.03 0.00 0.05 0.05 0.05 0.03 0.05 0.00 0.03 0.00
RF
12 12 12 12 13 11 13 12 13 15 13 13 13
RF
10 12 15 14 13 11 12
0.48 0.48 0.75 0.60 0.35 0.60 0.48 0.48 0.35 0.54 0.46 0.53 0.48 0.38 0.67 0.61 0.36 0.48 0.35 0.67
0.04 0.00 0.03 0.00 0.04 0.03 0.02 0.04 0.04 0.00 0.00 0.03 0.04 0.04 0.00 0.00 0.00 0.00 0.04 0.04
3 3 3
0.65 0.46 0.46
0.03 0.00 0.03
RF&l
RF
LF
RF
LX
LCIN
R C1N R GIN RC,N RF
LF LF L FB L GIN &IN
RH RH
Epoch
RF
R FB R FL3
RH
LF
RFB
RH
RA R GIN &IN R GIN R GIN R CU.4 R GIN RF
RA
R l=B
RF
RCIN
RFB
RF
L
RF
L FB L Lax L
RF
RF RH
RH
L FB L* L
LF
LXN
LF
RC,,
LFB
RH
RF L FB RF
L LF L
RFB RFB RF RA RA
R FB R FB RH RClN
R cm R GIN &IN
RH RH RH RF
LA
3
4
RF
LFB
LB Epoch
LA
Partial Coherence
0.46 0.40 0.60 0.40 0.54
Epoch
RH
Coherence
5
SPECTRAL
TABLE
Channel
LFB L FB L FB RF LF RF
R LIB LA L.4 R.4 RH LB LF RF
ANALYSIS
II-continued
1
Channel
2
Channel
RF
LF
RF
LF
R GIN R GIN
LF
&IN R GIN
LFB
RF
LFB
RF
RH
R GIN R FB R FB R FB
RF &IN
R GIN
RA
L GIN R GIN LA LF LH R GIN R FB L GIN LH
LF
3 10 3 10 10 10 10 3 3 3 3 3 10 10 10
&IN
R UN Epoch
&IN
RFB R FB R* RA RA
RH RH RH RN
LH
RF
&IN
LH
RFB R FB
LH
LF
LH
LClN
L"
LA
LA
LF
L GIN
RH
RF
LF LF
RA
L GIN R FB
RA
LF
LFB
LH
RF
LH
LA LClN R” RH
L FB
R FB R GIN L GIN LF
Frequency
LFE
RF
LFB
3
LF
RH RH hi
RFB RA
515
OF SEIZURES
LFB
L FB LFB LFB
L
FB
LFB LB
L
LN
LA
LF
L FB
L”
L GIN
LFB
Coherence
Partial coherence
0.73 0.75 0.46 0.65 0.77 0.65 0.42 0.46 0.48 0.46 0.38 0.42 0.60 0.46 0.42
0.00 0.00 0.00 0.03 0.00 0.00 0.00 0.03 0.00 0.00 0.03 0.03 0.00 0.00 0.00
0.42 0.70 0.56 0.42 0.57 0.70 0.48 0.80 0.90 0.90 0.75 0.52 0.57 0.86 0.86 0.38 0.52 0.50 0.48 0.84 0.90 0.55 0.73 0.84 0.86 0.88 0.88 0.86
4 4 7 7 4 4 4 4 4 4 4 4 4 4 4 7 8 7 4 4 4 4 4 4 4 4 4 4
6
516
THARP AND GERSCH
TABLE
II-continued
Channel 1
LF L RA RH
R L,’ LFB
Channel 2
Channel 3
Frequency
Coherence
Partial coherence
0.86 0.70 0.48 0.73 0.53 0.55 0.52
drives many areas, particularly the temporal region, whereas RF influences particularly those channel pairings including the frontal areas. R, also drives some of the pairings including RA and RN; however, R cIN is clearly responsible for the high coherence between RF-R, and RF-R,. The high coherence of RCiN with R, and R, is not lowered significantly when partialed on any other channel, including RF. This suggests a relationship of R cIN with the right temporal lobe differing from that of RCrN with the frontal region or contralateral sites. Less ubiquitous driving is manifested by L,, and occurs only with channel pairings including LF. The left temporal lobe was not participating in the driving of the seizure at this time. E. Epoch 5 (30 set following seizure onset). During this epoch the spectral curves contain significant peaks at 3 c/set and 10 clsec. A single peak at 3 cjsec was present in the temporal lobe leads, and at 10 c/set in the cingulate leads. The coherence and partial coherence data was quite complex. A clear pattern of driving could be developed if(i) the results were separated into two groups on the basis of the frequency at which the maximum coherence occurred (3 c/set and 10 cjsec) and (ii) channel pairings with high coherences that are not influenced by conditioning on a third channel are identified. The frontal, frontal basal, and cingulate leads are coherent at 10 cjsec. LF-LFB, R FB are highly coherent and not “driven” by a third channel at LFB-&IN, and &INthis frequency. LFB, LF, and RCIN are the major driving channels at this frequency. R,,, drives several channel pairings but only those including RFB, suggesting that interhemispheric linear relationships including R FB are due to their connections to R CIN. L,, drives RF-R,,, and RF-RF,. Coherences between intrahemispheric pairs on the left are not affected by right hemisphere channels. The analysis of the coherence and partial coherence data occurring at 3 cjsec reveals that LF and R, are the major driving channels. The coherence between L, and R, is not altered significantly by conditioning on a third channel, but neither is a driving channel in any channel tripling. RH, however, drives many inter- and intrahemispheric channel pairings that include R, and LA.
SPECTRALANALYSISOFSEIZURES
517
F. Epoch 6 (35 set following seizure onset). The analysis of data occurring for durations of 5 set and 9.6 set after E5 yielded similar results. The data in Table 1 and Table 2, E6, corresponds to the analysis of the 9.6~set epoch. The analysis revealed that high coherence (greater than 0.60) developed between all channel pairings except for L,-RF and pairings including R,, (Table 1). The spectral curves revealed a single peak at 2-4 c/set in all channels, with a lesser peak at 6-8 c/see+ particularly in the cingulate and frontal regions. The temporal leads were all highly coherent with each other and not significantly influenced by conditioning on nontemporal lobe channels. L,, LFB, and L, were responsible for much of the significant driving that occurred at 4 c/set (Table 2). L,, in particular drove several channel pairings that included ipsi- and contralateral frontal and cingulate channels. SUMMARYOFRESULTS
The spectral density curves and coherence data suggest the major areas of brain involvement during the seizure, and the partial coherence analysis with the identification of driving channels provides more specific data to dissect the details of the evolution of the seizure. If high coherence between two areas is considered as participation of those areas in the seizure, the evolution of the ictal activity can be traced from the frontal and cingulate regions to the right temporal lobe and finally to both temporal lobes. Both amygdalae are involved during the initial ictal epoch (E2); their linear relationship to other brain areas decreases during the next 15 set (E3) and increases again, initially involving R, (E4), as the seizure spreads into the temporal lobes. Visual analysis of the analog data is consistent with the computer analysis. The partial coherence analysis of the initial 5 set of the seizure (E2) fails to reveal a single dominant driving site. The driving of R,-R oIN by LCIN and the driving of several interhemispheric pairings by LF were suggestive evidence for lateralization of the seizure pacemaker to the left hemisphere. The high coherences between other brain sites without evidence of driving, however, raises the possibility of multiple, relatively independent, or rapidly changing pacemakers or a more complex, diffuse interrelationship between the frontal and cingulate regions. Rapidly changing pacemakers or multiple independent ones would militate against the emergence of definitive driving channels. Fifteen seconds after the onset of the seizure (E3), two distinct seizure processes can be identified. Faster ictal activity at 21 c/set is present in the frontal and cingulate areas and the right hippocampus, with clear driving by L,. The spectra reveal peaks at IO-12 c/set in R,, R,, and R,,,, and the partial coherence analysis shows driving of Rn-RF at 11 c/set by RCIN. Five seconds later (E4), the faster activity has waned and the ictal discharge at 12-l 5 c/set has now become the dominant frequency. The major driving sites during
518
THARPANDGERSCH
this period are the right mesial frontal area and, as in E3, the right cingulate area. The apparent linearity between sites in the frontal lobes can be explained by their connection to the right frontal lobe and coherence between the right temporal lobe and frontal areas by their connection, particularly, to the right cingulate. Two distinct ictal processes are again apparent 10 set later (E5). Ten cjsec activity is present in the frontal and cingulate areas but the major driving site has shifted to the left frontal and frontal basal regions. The slower activity (3 cjsec) is driven by R, and LF. This is the first indication of driving by a temporal lobe channel. Analysis of the final epoch (E6) reveals that the seizure has spread into the temporal lobes with a concomitant change in the morphology and frequency of the analog activity. During this epoch it is not possible to define a single source or to lateralize the seizure to a hemisphere by visual analysis of the EEG. The computer analysis reveals that the ictal activity at 4 c/set, driven by L, and R, in the previous epoch, is now driven by L,, L,, and L,,. The analysis reveals three distinct seizure patterns: (i) a 21 cjsec rhythm that appears transiently in the early phases of the seizure (E3), with driving by Lr, (ii) an intermediate frequency rhythm (10-l 5 c/set) that is noted initially in E3 with driving by RCIN, becomes the dominant activity in E4, and is again driven predominantly by RF and &IN and wanes in E5, with the pacemaker now located in L,, LFB, and RCIN, and (iii) a slow ictal rhythm at 3-4 cjsec that first appears in E5, with the pacemakers located predominantly at R,, and L,, and becomes the dominant activity in E6 with the pacemaker shifting to the left hemisphere, i.e., L,. LH, and LFB. DISCUSSION
The present study describes the application of a spectral coherence analysis method for the identification of causality in human ictal EEG data. Several earlier attempts to apply spectral coherence measurements to cerebral electrical activity have been reported. Brazier (2,3) described changes in the percentage coherence between areas in the temporal lobes and between activity from these depth sites and the scalp during changing levels of consciousness, seizures, and following intravenous barbiturates. In a later study, coherences and phase information were employed to trace the propagation of seizures through the brain of humans (4). Larsen (29) demonstrated differences in the coherence between the occipital regions prior to photic stimulation, which could be correlated with the type of photic response obtained. Walter et al. (20) employed graphic displays to illustrate the evolution of the spectral intensity and coherence of the EEG recorded from the occipital areas in normal humans, but did not attempt to explain the coherence that developed between two areas of the brain, i.e., were the two sites directly related or was there a “shared-trigger or pacemaker?’ (2). Walter and Adey (21) used multivariate spectral analysis and a principal component analysis to investigate the concept of biophysical generators, but their approach does not appear to have been applied to practical EEG analysis or to
SPECTRAL
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have been further pursued. A technique for identifying causality, purportedly useful in the analysis of economic time series, was developed by Granger (22). Granger’s definition of causality is based entirely on the predictability of time series and not on phase lead or lag. A critical comparison of our technique with Granger’s appears in Gersch (16). Applications of our technique are discussed in Gersch and Goddard (7) and Gersch, Midkiff, and Tharp (8). Our approach to the concept of causality in time series is developed from considerations of extensions of concepts of regression and partial regression in the multivariate analysis of random variables. A single time series is identified to be causal relative to other time series if it uniquely explains the pair-wise linear relationships between other time series over a relevant interval in the frequency domain. That is, if the influence of a driving or causal time series is removed from the other time series by a regression analysis, the residual or partialed time series are pairwise uncorrelated, and hence, incoherent over that frequency interval. This definition lends itself to explicit statistical tests, to multivariate extensions of the definitions of causality, to the case in which a linear combination of time series may be causal relative to other time series, and to the physical plausability arguments that we think are essential in order to conclude that causality is present in time series. The results of the spectral analysis of a sequence of epochs during the propagation of a seizure are reported in this paper and reveal the migration of the driving sites, the appearance of new pacemakers concomitant with the cessation of driving in others, and the separate pacemakers for concomitant ictal activity at different frequencies. Though a seizure may be initiated in a relatively small neuronal aggregate, it will drive other areas to which it is anatomically related, either through long association pathways or short intracortical connections. Such pacemaker activity has been demonstrated by Petsche et al. (23) and Gogolak et al. (24) in the analysis of septal theta activity, and the concept was applied by Brazier (4) to the analysis of seizures. These secondary sites may then develop their own inherent rhythmicity and, in turn, will drive other populations of neurons within their sphere of influence even when the site of the seizure initiation has ceased discharging. This pattern of selfpropagation has been called the “will-o’-the-wisp” response (25). The almost simultaneous recruitment of many widespread areas, however, often leads to difficulties in localizing these pacemakers in analog data even with electrodes implanted near the participating neurons. The computer analysis was unable to locate a single site of seizure initiation in this patient, but it did delineate more clearly the propagation of the seizure and also revealed complexities not apparent in the analog data, such as the separate pathways of spread of ictal activity of different frequencies. Similarly, this technique appears to offer potential for inferring mechanisms for the initiation and evolution of the less understood generalized or “cortico-reticular” epilepsies (1). Current research techniques for locating the site of seizure generation in humans include the indirect method of intracarotid drug infusion (14) or depth electrode implantation with subsequent visual analysis. The limitations of these
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latter experimental approaches and the inadequacies of a simple morphological analysis have been reviewed by Ajmone Marsan (26). Conjectures about the participation of thalamic nuclei in generalized seizure activity may be tested using the spectral analysis methods of identifying causality in time series. Our current approach has been to emphasize the computer analysis of epileptic data that could be understood by more conventional EEG techniques. The objective is to gain confidence and experience in the spectral analysis method and to clarify the new information obtained by this approach, which was not available in the visual examination of the EEG. The patient analyzed in this study was treated medically; therefore, confirmation of the site of seizure initiation by surgical removal of the involved tissue could not be obtained. We do not feel that this technique has been perfected to a degree such that the site of surgical intervention can be determined by the computer analysis. Considerably more experience is required before the technique can reliably supplement existing analysis techniques used for the identification of seizure pacemakers or foci in patients being evaluated for seizure surgery. REFERENCES 1. GLOOR, P. Generalized cortico-reticularepilepsies. Someconsiderations on the pathophysiologq of generalized bilaterally synchronous spike and wave discharge. Epilepsia 9, 249-263 (1968). 2. BRAZIER, M. Studies of the EEG activity of limbic structures in men. Electroencephalo~r. C/in. Neurophysiol. 25309-318 (1968). 3. BRAZIER, M. Electrical activity recorded simultaneously from the scalp and deep structures in the human brain. J. Nerv. Merit. Dis. 147, 31-39 (1968). 4. BRAZIER, M. Spread of seizure discharges in epilepsy: Anatomical and electrophysiological considerations. Exp. Neuvol. 36,263-212 (1972). 5. WALTER, D. AND BRAZIER, M. Advances in EEG analysis. Eiectroencephalogr. C/in. Ncurophysiul.
Supp.
27 (1968).
6. WALTER, D. Spectral analysis for electroencephalograms: Mathematical determination of neurophysiological relationships from records of limited duration. Exp. Neural. 8, 155-181 (1963). 7. GERSCH,W. AND GODDARD, G. Locating the site of epileptic focus by spectral analysis methods. Science
169,701-702(1970).
8. GERXH, W., MIDKIFF, A., AND THARP, B. Alpha rhythm generators. In “Computers in Biomedicine,” pp. 26-29. Suppl. to the Proceedings of the Fifth Hawaii International Conference on System Sciences, Honolulu, Hawaii, 1972. 9. GERSCH, W. Spectral analysis of EEGs by autoregressive decomposition of time series. Math. Biosci. 7,205-222 (1970). IO. AKAIKE, H. Statistical predictor identification. Ann. Inst. Statist. Math. 21,243-247 (1969). II. AKAIICE, H. On a semi-automatic power spectrum estimation procedure, In “Proceedings ofthe Third Hawaii International Conference on System Sciences,” pp. 974-977. Honolulu, Hawaii, 1970.
AKAIKE, H. “Information Theory and an Extension of the Maximum Likelihood Principle.” Research Memo 46, Inst. of Stat. Math., Tokyo (1971). To appear in Probl. ControEZnf: Theory. Akademiai Kiado, Budapest. 13. PARZEN, E. Multiple time series modeling. In “Multivariate Analysis” (P. Krishnaish, Ed.) Vol. 2, pp. 389-406. Academic Press, New York, 1969. 12.
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14. KROMER, R. “Asymptotic Properties of the Autoregressive Spectral Estimator.” Tech. Rept. No. 13, Dept. of Statistics, Ph.D. dissertation, Stanford University, 1969. 15. GERSCH, W. “Statistical Performance of the Autoregressive Model Method for Multidimensional Spectral Anaiysis.” Paper presented at NSF Regional Conference on Time Series Analysis, Chapel Hill, North Carolina, 1973. 16. GERSCH,W. Causality or driving in electrophysiological signal analysis. Math. Biosc. 14, 177196 (1972). 17. THARP, B. Autoregressive spectral analysis-a unique technique for the study of human seizure activity. In “Computers in Biomedicine,” pp. 2629. Suppl. to the Proceedings of the Fifth Hawaii International Conference on System Sciences, Honolulu, Hawaii, 1972. 18. GERSCH, W. A note on trapezoidal digital filter design. Comput. Biomed. Res. 6, 281-285 (1973)
19. LARSEN, L. Mechanisms ofepiiepticsusceptibility to photic activation. Case study of the photoconvulsive response. Epilepsia 10,473-480 (1969). 20. WALTER, D., RHODES,J., BROWN, D., AND ADEY, W. Comprehensive spectral analysis of human EEG generators in posterior cerebral regions. Electroencephalogr. Clin. Neurophysiol. 20,224237 (1966).
21. WALTER, D. AND ADEY, W. Analysis of brain-wave generators as multiple statistical time series. IEEE Trans. Biomed. Eng. 12,8-13 (1965). 22. GRANGER, C. Investigating causal relations by economic models and cross-spectral methods. Econometrics 3,424-438 (1969). 23. PETSCHE, H., GOGOLAK, G., AND VAN ZW~ETEN, P. Rhythmicity of septal cell discharges at various levels of reticular excitation. Electroencephalogr. Clin. Neurophysiol. 19,25-33 (1965). 24. GOGOLAK, G., STUMPF, C., PETSCHE,H., AND STERC,J. The firing pattern of septal neurons and the form of the hippocampal theta wave. Bruin Res. 7,201-207 (1968). 25. UDVARHELYI, G. AND WALKER, A. Dissemination of acute focal seizures to the monkey. I. From cortical foci. Arch. Neural. Chicago 12,333-356 (1965). 26. AJMONE MARSAN, C. Pathophysiology of the EEG pattern characteristic of petit ma1 epilepsy, a critical review of some of the experimental data. In “The Physiopathogenesis of the Epilepsies” (H. Gastaut, H. Jasper, J. Bancaud, and A. Waltregny, Eds.), pp. 237-248. Charles C Thomas, Springfield, Ill., 1969.
AND
BIOMEDICAL
Spectral
RESEARCH
8,503-521
Analysis
(1975)
of Seizures
in Humans*
B. R. THARP AND W. GERSCH Department of Neurology, Stanford University Medical Center, Stanford, California 94305 Received March 12.1975 The electroencephalogram of a patient with intractable epilepsy was recorded with chronically implanted depth electrodes. The activity during a seizure was analyzed by an autoregressive spectral analysis technique that permits the calculation of the power spectra for all channels and the coherence between all possible channel pairings. The partial or residual coherence between a channel pair was calculated after a regression analysis had removed the influence of a third channel. If the coherence between two channels, in a frequency range of high spectral energy, fell to zero after partialing on a third channel, the latter was considered to drive the other two channels. A single site was sought which appeared to drive all other areas by analyzing all possible channel pairings and triplings. The source and spread of the seizure was determined by analyzing sequential data epochs. The analysis revealed the ictal activity arose in a single region, then spread over several pathways. The power spectra of the ictal activity revaled peaks in several frequency regions. Calculation of coherences and partial coherences in each of these frequency regions revealed different driving channels and, with analysis of sequential epochs, different pathways of spread of the ictal activity occurring at these frequencies. The spectral analysis technique provides, therefore, details of the seizure propagation and pacemaker activity not possible with conventional visual analysis of the analog data.
The analysis of the complex rhythms generated by the brain during a seizure has relied on the visual perusal of analog data. The attention of the electroencephalographer is directed to the period immediately preceding the seizure, the initial phases of the ictus and the postictal rhythms. The ictal rhythms of the generahzed seizures (“cortico-reticular epilepsies”) (I) and focal seizures which spread rapidly to contiguous cortex or to homologous areas of the contralateral hemisphere, are usually too complex for simple visual analysis. The electroencephalographer is frequently unable to locate the source of the abnormal activity or to accurately map its propagation through the brain. * This investigation was supported in part by Public Health Service Grant No. R.R.-70 from the General Clinical Research Centers, Division of Research Resources and an Epilepsy Foundation of America Research Grant. Copyright 0 1975 by Academic Press, Inc. 503 All rights of reproduction in any form reserved. Printed
in Great
Britain
504
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GERSCH
The introduction of computers and the mathematical discipline of time series analysis into the field of electroencephalography has provided new tools with which to analyze these complex rhythms (2-5). In particular, spectral analysis of the electroencephalogram (EEG) and the linear system “black box” properties of spectral coherences and transfer functions derived from the spectral analysis (6) can reveal relationships between the activity from different brain regions. Spectral analysis of sequential EEG epochs can reveal changes in these linear relationships and shifts of energy to different frequency bands that are not evident in the more classical visual analysis of analog data. Major issues in the analysis of focal seizures recorded by surface or implanted macroelectrodes include the localization of the area of the brain from which the seizures arises, the pathways of propagation of epileptic activity and the site of the epileptic “pacemaker” (2) which may change as the seizure evolves. The results of the application of a spectral analysis technique to the study of these issues in the complex multichannel time series that appear in human epileptic activity are reported in this paper. Our notion of a seizure pacemaker is that it causes or drives the epileptic activity observed in other parts of the brain. Operationally, that pacemaker is identified by a spectral analysis technique that is equivalent to the identification of causality in time series. Experiments on simulated data corresponding to our causaIity model (unpublished data), and on cats made epileptic by electrical stimulation of a single brain site, reveal that causality can be unambiguously identified in time series by our computational method even when visual examination of the analog data offers no apparent clues to the generator site (7). The analysis of human alpha rhythms by our technique demonstrated that electrophysiological driving could be identified from the scalp EEG (8). The analysis of the EEG obtained simultaneously from scalp and epidural electrodes in a human epileptic patient (unpublished data) revealed that frontal lobe origin of a diffuse seizure discharge (proven by surgical resection and subsequent absence of seizures) could be identified using our computational technique on the scalp EEG data alone. The analysis of the scalp EEGs revealed, therefore, that causality or electrophysiological driving could be identified despite the fact that the electrodes were not placed precisely at the site of origin of the seizure. In the last two years the EEG data from scalp, epidural, and depth electrode recordings (the latter supplied by John E. Adams, M.D., Department of Neurosurgery, University of California at San Francisco) of 15 epilepticpatients have been analyzed. Our experience has been that the computational identification of the driving sites of epileptic seizures has almost invariably both corroborated what could be observed by careful visual examination of the EEG and also added valuable information concerning the initiation and spread of seizure activity that could not be discerned from the examination of the analog EEG. In this paper, the analysis of a single seizure in a human subject is examined in some detail to illustrate the method and to describe the logic and results obtained in using our technique in tracing the spread of a focal seizure from its site of origin.
SPECTRAL
ANALYSIS
OF SEIZURES
505
METHODS
The depth electrodes used in this patient were stereotactically placed in both frontal and temporal lobes. Each electrode consisted of a pair of interwoven insulated stainless steel wires. Bipolar recordings were obtained at each site with an interelectrode distance of 2 mm. The EEG was passed through low level amplifiers (band pass essentially linear between 1 Hz and 500 Hz) and stored on magnetic tape (Ampex FR 1300). The EEG was simultaneously displayed on a cathode ray oscilloscope and written on paper by a Grass Model VI electroencephalograph for continuous visual monitoring. Selected portions of 10 channels of magnetically recorded data were simultaneously digitized at the rate of 200 samples per set, using an IBM 1800 computer. The digitized data was immediately transferred and stored in an IBM 360/50. The stored EEG was serially displayed on a television screen for verification and 5- or 9.6-set epochs were selected for spectral analysis. METHODS
OF ANALYSIS
Spectra, cross spectra, and spectral coherences were computed by the autoregressive (AR) representation method (9). Ten channels of data were analyzed simultaneously and a 10 x 10 matrix of cross correlation functions was calculated. In this method, a formula, best in the least-squares sense, is computed that expresses the observed time series in terms of its own past history (a regression of the data on itself) plus an additional unpredictable, uncorrelated random component. The formula is an explicit parametric model of the observed EEG data. The order of the AR model was automatically computed by a method due to Akaike (10-12). For each of the EEG epochs selected for analysis, spectral densities and coherences and partial coherences on all possible pairs and triples of data channels were computed from the AR model over a frequency range of O-50 cjsec using formulas appearing, for example, in Gersch (9). The statistical properties of spectra computed by this method have been discussed by Akaike (IO), Parzen (13) Kromer (24) and Gersch (9,lO The spectral density of the ictal data was characterized by a pronounced peak or peaks. The spectral coherences and partial coherences over the frequency intervals with high spectral intensity were utilized in the identification of potential or candidate EEG driving channels. Briefly, consider three simultaneous EEG time series that have high pairwise spectral coherences over the frequency range of interest. Spectral coherence at frequency f may be interpreted as the square of the correlation coefficient between the random variables that describe the spectral intensity in each of the time series at frequency f. The high pairwise spectral coherence may be due to the fact that the two brain areas are connected and their behavior is interdependent or, alternatively, they may be connected to another area which drives each of them. Our technique for the identification of a causal or driving site involves the calculation of partial spectral
506
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coherences. Partial coherence expresses the linear relationship (the square of the correlation coefficient) that remains between two time series after the influence of a third time series has been removed by a partial regression analysis. In our approach, we conclude that one time series drives or is causal relative to a pair of time series when (i) the partial spectral coherence between the data pair conditioned on this third time series is zero over the frequency range of high spectral density and high spectral coherence and (ii) only one time series of three considered has this property. If the partial coherence is zero, then formally, the third time series explains the linear relationship (coherence) between the other two. If time series are jointly gaussian, zero coherence between two time series implies that the two time series are not linearly related and more strongly, that they are statistically independent. From this perspective, we say that a channel which explains the linear relationship or interdependence between other time series drives or is causal to those time series. Conversely, if the partial coherence of the two time series conditioned on a third time series is substantially different than zero, then the third time series does not explain the linear relationship between the first two series. In the analysis of human seizure data, we ideally hope to locate a single recording site whose electrical activity will explain all of the spectral coherence that is observed between other data channels. For more details on this approach to causality and some applications to electrophysiological data, see Gersch and Goddard (7), Gersch, Midkiff, and Tharp (8) Gersch (16) and Tharp (17). The analysis generates voluminous data. Corresponding to d data channels, there are C(d, 2) spectral coherence versus frequency pairs and C(d, 3) partial spectral coherence triples where C(n, k) is the number of distinct combinations of n things taken k at a time, and C(n, k) = n!/(n - k)! (k!). The data, therefore, consists of(i) the spectral intensity of each of the 10 channels, (ii) C (10, 2) = 45 coherence pairs, and (iii) three graphs for each distinct partial coherence triple, 3 C (10, 3) = 360. The spectral data was computed from the AR model at integer values of frequency from O-50 Hz. A computer program scanned all of the coherence and partial coherence computations and graphs were printed of the spectral coherences and partial coherences in which the criterion that pairwise coherence was greater than or equal to 0.35 (2 0.35) and that partial coherence was equal to or less than 0.05 (SO.05) at the frequencies of high spectral intensities were satisfied. Using this technique, clear patterns in candidate driving channels emerged and appeared to justify the reasonableness of this approach and the arbitrary selection of the minimal coherence and maximal partial coherence values. For almost invariably, candidate driving channels did not appear to be isolated cases. For example, if Channel 3 was causal to Channels 1 and 2 by this criterion in one situation, we also observed that Channel 3 was causal to other channel pairs. Furthermore, the identification of the driving channels did not seem to be at all sensitive to slight variations in the lower limits of acceptable coherence or the upper limit of acceptable partial coherence. In this sense our driving channel criterion appears to be statistically robust.
SPECTRAL
ANALYSIS
OF SEIZURES
507
The application of spectral analyses methods to time series data is based upon the assumption that the time series is a finite duration sample from a stationary random process. An adhoc technique was used to partially substantiate that assumption for the EEG data analyzed. The order of the AR model computed for the spectral analysis is determined by the data and N, the number of data points available for analysis. In addition to computing the AR model determined by the data and N, we computed the model determined by the data and the number 1.15 N and the corresponding spectral analysis. The rationale for this procedure is that if the time series were stationary and the AR model of the data was stable, a nominal increase in N might slightly increase the order of the AR model but would not significantly change the outcome or the interpretation of the spectral analysis results. Conversely, iFthe time series were not stationary or if the observation interval was not sufficiently long, the AR model order and the spectral analysis results would change significantly with increasing N. This approach was justified by empirical experiments on simulated data and on EEG time series. The statistical reliability of the estimate of coherence in the neighborhood of zero coherence is particularly critical for our application. That estimate is in general less reliable than the estimates at greater values of coherence. The topic of the statistical reliability of AR-modeled spectral analysis has a research status. Parzen (13) conjectured that asymptotically the AR-modeled spectral matrix has a complex Wishart distribution with a = N/p degrees of freedom, where p is the order of the AR model fitted to the data. This assumption leads to the result that in the neighborhood of zero coherence the quantity ug2 is approximately chi-squared distributed, where g2 is the estimate of the coherence squared. In a simulation study of the statistical behavior of AR-modeled spectra and coherence estimates (personal observations), Parzen’s estimate of u in the neighborhood of zero coherence estimates has been found to be conservative. Typically, for d = 10 data channels and N = 500, we found 6
This 40-year-old male had a 30-year history of seizures characterized initially as generalized tonic clonic episodes. At age 13 they ceased, and two types of minor seizures appeared which had been relatively refractory to medication to the time of
SPECTRAL
E-5
ANALYSIS
OF SEIZURES
509
E-6
1(A-C) Electroencephalogram obtained during a clinical seizure characterized by tonic extension of upper extremities followed by automatic behavior (the three tracings are continuous). The depth electrodes are located in both anterior hippocampi (Hipp-La, RH), lateral amygdala (Amy-L*, R,,), mesial frontal basal area (Fron. Bas.-L FB,RFB), mesial frontalcortex 15 mm superior to frontal basal electrodes (Fron-Lr, RF), and anterior cingnlum (Cing-Lc,,, Rci.). The autoregressive spectral analysis was performed on the six segments of EEG underlined, E,, E,, . . ., Eb. Calibrations: Channels l-8,500 @; Channels 9,10,200 @; 2 sec. FIG.
this study. The most common minor episode consisted of a feeling of tightness about the left chest followed by abrupt loss of consciousness. The patient slowly slumped to the ground, frequently elevating the flexed left arm and became flaccid. There were no automatisms or lateralizing postictal signs. The other seizure type consisted of a sudden absence and irrelevant verbalization without loss of tone or abnormal motor activity. Neurological examination was essentially normal. Pneumoencephalograms, cerebral arteriography, and cerebrospinal examinations were normal on several occasions. Following this electroencephalographic study, the patient’s seizures were controlled with the addition of carbamazepine to his other anticonvulsants.
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Electroencephalograms. The computer analysis was performed on a seizure recorded with depth electrodes implanted bilaterally into the frontal parasagittal region and temporal lobes (Fig. 1). Multifocal spikes and spike and slow waves were noted prior to the seizure, particularly in the right cingulate and frontal basal regions and the left frontal and cingulate leads. The onset of the seizure was characterized by an abrupt diminution of the amplitude of the EEG in all leads. High amplitude, rhythmical 9 cjsec spiking appeared in the right cingulate gyrus (Rc,,) and right frontal basal (a,,) region and spread rapidly to both amygdala (RA, LA). This activity persisted for 5 set, abruptly disappeared, and 30 c/set spiking appeared in the right frontal basal (RF& left frontal basal (LFB), and right frontal (RF) areas. Approximately 6 set later, low voltage 10-l 2 cjsec spikes again appeared at R cIN, gradually increased in amplitude (maximum 600 /lV), and spread to involve both right frontal leads and L,,, interrupting and then replacing the fast spiking in these areas. The high amplitude activity at Rc,, persisted throughout the remainder of the seizure, though the frequency gradually slowed. Approximately 25 set after the onset of the seizure, high amplitude 2-3 clsec sharp slow waves appeared at R, and R,, and spread rapidly to LA and LH. This slow activity rapidly spread to all the remaining recording sites, admixing and finally replacing the fast spiking at R,, and the rhythmical IO-12 cjsec sharp waves in the frontal areas. There was negligibIe spread to Rcr,. Approximately 40 set later, the seizure waned and high amplitude, multifocal amorphous delta activity appeared in all leads. Visual analysis of the analog data suggested that the seizure had its onset simultaneously at electrodes Rcr, and RFB. Morphologically distinct ictal rhythms (low voltage 30 c/set spiking) appeared several seconds later in both frontal (LF, RF) and frontal basal (L,,, RFB) leads. The seizure rapidly spread to involve both frontal, frontal basal, and cingulate electrodes. Approximately 30 set after its onset, the ictal activity appeared in the temporal leads with a concomitant decrease of activity in the frontal and cingulate cortex. Six epochs, respectively E,, E,, . . ., Es, were analyzed, the first five were 5 set in duration, the sixth 9.6 set (Fig. I). Epoch 1 ends 6 set before the onset of the seizure, Epoch 2 includes the initial 5 set of the seizure, Epochs 3,4, 5, and 6 begin approximately 15 set, 20 set, 30 set, and 35 set, respectively, after the onset of the seizure.
RESULTS
In view of the large number of graphs (415/epoch for 10 simultaneous EEG channels for each epoch of data analyzed), we present below an abstraction and interpretation of those results that distinguish left hemisphere, right hemisphere, and interhemispheric performance.
511
SPECTRAL ANALYSIS OF SEIZURES
1. Coherences
The coherences occurring in a frequency region of high spectral intensity are listed in Table 1. Although significant coherence was often noted in more than one frequency region in many channel pairings, the highest coherence only is listed in the table. Prior to the onset of the seizure (El), high coherence is present only between the two frontal areas. At the onset of the seizure (E2), high coherence at 7-8 c/set appears between the frontal areas, cingulate regions, and the amygdala. As the seizure continues (E3), the high coherence between the frontal and cingulate areas persists, particularly at 19-21 c/set, whereas there is negligible involvement of the temporal areas with the exception of R,. Approximately 20 set after the onset of the seizure (E4), the high coherences now appear at 11-15 c/set, and the right temporal lobe again becomes involved. Ten set later (E5), the left temporal lobe, particularly L,, becomes coherent with ipsi- and contralateral temporal and frontal areas at 3 c/set. Channel pairings involving RclN are coherent at 10 c/set and 3 c/set. In E6 high coherences occur between the majority of recording sites at 4 c/set as the seizure propagates to the temporal lobes (Fig. 1). 2. Partial Coherenceand Driving
Table 2 lists all the candidate driving channels in Epochs 2-6 that were identified by the coherence criteria discussed above. A. Epoch 1 (preictal). Only one channel tripling revealed significant driving; Lr drove RF-RCIN. B. Epoch 2 (initial 5 set of seizure). Though high coherences developed between many areas, there was relatively little significant driving. The coherence between L,, 0.63 .
I
I
I
1
1
I
I
I
---RH.R 8 0.36 5 & g0.250
F (R GIN’ E3
-
0
1
5
10
15
20 25 FREQUENCY
30 (C/SEC)
36
40
45
50
FIG. 2. Graph of coherence versus frequency of right hippocampus and right mesial frontal area during Epoch 3. Solid line, coherence; broken line, partial coherence (coherence remaining between RH-RF when influence of right cingulum (RCIN) removed by regression analysis). RcrN “drives” RW and RF at 11 c/set only. There is no driving at 21-22 c/set, the second frequency at which RH and Rr are highly coherent.
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and sites in the contralateral frontal and cingulate areas was explained by the influence of LF. The only driving of two ipsilateral hemispheric areas by a contralateral site was that of R,-RcrN by L,,,. C. Epoch 3 (15 set following onset). The spectra revealed dominant peaks at 19-21 c/set in Rn, LA, and both frontal, frontal basal, and left cingulate areas. A secondary peak at IO-12 cjsec was present in R, and R, and was the frequency of highest spectral density of RCIN. The partial coherence data reveals clear driving of both inter- and intrahemispheric channel pairings by L, at 20-21 cjsec. In the only evidence of right hemisphere driving, RCIN drives Ru and R, at 11 cjsec (Fig. 2). Rn and R,,, are coherent at 21 and 11 c/set and are driven by Lr and LFB at 21 cjsec. D. Epoch 4 (20 set foZZowing onset). The spectral curves are characterized by peaks at 12-14 cjsec. A secondary peak at higher frequencies is present at Ln and L, (38.40 c/set), and LFB, RFB, RF, and R,i, (30-33 cjsec). All areas, with the exception of R C,N,have significant activity below 4 c/set. TABLE EPOCH
EPOCH
1
2
I
SPECTRAL EPOCH
ANALYSIS
513
OF SEIZURES
4
LA
.._
.--
.. .
---
.44
.--
36
EPOCH
%a
-42
.42
42
.60
In E4 there are four right hemisphere candidate driving sites: RA, RCIN, RF, and R,. R, and R,-,* are highly coherent, not influenced by other areas, and also responsible for the majority of the coherences that appear over the right hemisphere. RcrN
514
THARP
AND
GEBSCH
TABLE
Channel
1
Channel
LB LFB
R
2
Channel
II
3
Frequency
Epoch
2
LF
7
RF
LF
7
LFB
&IN
J-F
7
R*
R GIN
LClN
R*
RF
RFB
R”
LFB
LF
RH
R FB
RH
RF
R”
LN
LF I-1 J-F
&I
RC,N
LF
LFB
RFB
L
LFB
LIN
LF
FB
7 7
0.03 0.03 0.00 0.05 0.00
21 21 21 21 21 21 21 20 21 21 21 21 11
0.48 0.42 0.50 0.48 0.38 0.52 0.60 0.52 0.50 0.56 0.38 0.38 0.46
0.00 0.05 0.03 0.03 0.00 0.05 0.05 0.05 0.03 0.05 0.00 0.03 0.00
RF
12 12 12 12 13 11 13 12 13 15 13 13 13
RF
10 12 15 14 13 11 12
0.48 0.48 0.75 0.60 0.35 0.60 0.48 0.48 0.35 0.54 0.46 0.53 0.48 0.38 0.67 0.61 0.36 0.48 0.35 0.67
0.04 0.00 0.03 0.00 0.04 0.03 0.02 0.04 0.04 0.00 0.00 0.03 0.04 0.04 0.00 0.00 0.00 0.00 0.04 0.04
3 3 3
0.65 0.46 0.46
0.03 0.00 0.03
RF&l
RF
LF
RF
LX
LCIN
R C1N R GIN RC,N RF
LF LF L FB L GIN &IN
RH RH
Epoch
RF
R FB R FL3
RH
LF
RFB
RH
RA R GIN &IN R GIN R GIN R CU.4 R GIN RF
RA
R l=B
RF
RCIN
RFB
RF
L
RF
L FB L Lax L
RF
RF RH
RH
L FB L* L
LF
LXN
LF
RC,,
LFB
RH
RF L FB RF
L LF L
RFB RFB RF RA RA
R FB R FB RH RClN
R cm R GIN &IN
RH RH RH RF
LA
3
4
RF
LFB
LB Epoch
LA
Partial Coherence
0.46 0.40 0.60 0.40 0.54
Epoch
RH
Coherence
5
SPECTRAL
TABLE
Channel
LFB L FB L FB RF LF RF
R LIB LA L.4 R.4 RH LB LF RF
ANALYSIS
II-continued
1
Channel
2
Channel
RF
LF
RF
LF
R GIN R GIN
LF
&IN R GIN
LFB
RF
LFB
RF
RH
R GIN R FB R FB R FB
RF &IN
R GIN
RA
L GIN R GIN LA LF LH R GIN R FB L GIN LH
LF
3 10 3 10 10 10 10 3 3 3 3 3 10 10 10
&IN
R UN Epoch
&IN
RFB R FB R* RA RA
RH RH RH RN
LH
RF
&IN
LH
RFB R FB
LH
LF
LH
LClN
L"
LA
LA
LF
L GIN
RH
RF
LF LF
RA
L GIN R FB
RA
LF
LFB
LH
RF
LH
LA LClN R” RH
L FB
R FB R GIN L GIN LF
Frequency
LFE
RF
LFB
3
LF
RH RH hi
RFB RA
515
OF SEIZURES
LFB
L FB LFB LFB
L
FB
LFB LB
L
LN
LA
LF
L FB
L”
L GIN
LFB
Coherence
Partial coherence
0.73 0.75 0.46 0.65 0.77 0.65 0.42 0.46 0.48 0.46 0.38 0.42 0.60 0.46 0.42
0.00 0.00 0.00 0.03 0.00 0.00 0.00 0.03 0.00 0.00 0.03 0.03 0.00 0.00 0.00
0.42 0.70 0.56 0.42 0.57 0.70 0.48 0.80 0.90 0.90 0.75 0.52 0.57 0.86 0.86 0.38 0.52 0.50 0.48 0.84 0.90 0.55 0.73 0.84 0.86 0.88 0.88 0.86
4 4 7 7 4 4 4 4 4 4 4 4 4 4 4 7 8 7 4 4 4 4 4 4 4 4 4 4
6
516
THARP AND GERSCH
TABLE
II-continued
Channel 1
LF L RA RH
R L,’ LFB
Channel 2
Channel 3
Frequency
Coherence
Partial coherence
0.86 0.70 0.48 0.73 0.53 0.55 0.52
drives many areas, particularly the temporal region, whereas RF influences particularly those channel pairings including the frontal areas. R, also drives some of the pairings including RA and RN; however, R cIN is clearly responsible for the high coherence between RF-R, and RF-R,. The high coherence of RCiN with R, and R, is not lowered significantly when partialed on any other channel, including RF. This suggests a relationship of R cIN with the right temporal lobe differing from that of RCrN with the frontal region or contralateral sites. Less ubiquitous driving is manifested by L,, and occurs only with channel pairings including LF. The left temporal lobe was not participating in the driving of the seizure at this time. E. Epoch 5 (30 set following seizure onset). During this epoch the spectral curves contain significant peaks at 3 c/set and 10 clsec. A single peak at 3 cjsec was present in the temporal lobe leads, and at 10 c/set in the cingulate leads. The coherence and partial coherence data was quite complex. A clear pattern of driving could be developed if(i) the results were separated into two groups on the basis of the frequency at which the maximum coherence occurred (3 c/set and 10 cjsec) and (ii) channel pairings with high coherences that are not influenced by conditioning on a third channel are identified. The frontal, frontal basal, and cingulate leads are coherent at 10 cjsec. LF-LFB, R FB are highly coherent and not “driven” by a third channel at LFB-&IN, and &INthis frequency. LFB, LF, and RCIN are the major driving channels at this frequency. R,,, drives several channel pairings but only those including RFB, suggesting that interhemispheric linear relationships including R FB are due to their connections to R CIN. L,, drives RF-R,,, and RF-RF,. Coherences between intrahemispheric pairs on the left are not affected by right hemisphere channels. The analysis of the coherence and partial coherence data occurring at 3 cjsec reveals that LF and R, are the major driving channels. The coherence between L, and R, is not altered significantly by conditioning on a third channel, but neither is a driving channel in any channel tripling. RH, however, drives many inter- and intrahemispheric channel pairings that include R, and LA.
SPECTRALANALYSISOFSEIZURES
517
F. Epoch 6 (35 set following seizure onset). The analysis of data occurring for durations of 5 set and 9.6 set after E5 yielded similar results. The data in Table 1 and Table 2, E6, corresponds to the analysis of the 9.6~set epoch. The analysis revealed that high coherence (greater than 0.60) developed between all channel pairings except for L,-RF and pairings including R,, (Table 1). The spectral curves revealed a single peak at 2-4 c/set in all channels, with a lesser peak at 6-8 c/see+ particularly in the cingulate and frontal regions. The temporal leads were all highly coherent with each other and not significantly influenced by conditioning on nontemporal lobe channels. L,, LFB, and L, were responsible for much of the significant driving that occurred at 4 c/set (Table 2). L,, in particular drove several channel pairings that included ipsi- and contralateral frontal and cingulate channels. SUMMARYOFRESULTS
The spectral density curves and coherence data suggest the major areas of brain involvement during the seizure, and the partial coherence analysis with the identification of driving channels provides more specific data to dissect the details of the evolution of the seizure. If high coherence between two areas is considered as participation of those areas in the seizure, the evolution of the ictal activity can be traced from the frontal and cingulate regions to the right temporal lobe and finally to both temporal lobes. Both amygdalae are involved during the initial ictal epoch (E2); their linear relationship to other brain areas decreases during the next 15 set (E3) and increases again, initially involving R, (E4), as the seizure spreads into the temporal lobes. Visual analysis of the analog data is consistent with the computer analysis. The partial coherence analysis of the initial 5 set of the seizure (E2) fails to reveal a single dominant driving site. The driving of R,-R oIN by LCIN and the driving of several interhemispheric pairings by LF were suggestive evidence for lateralization of the seizure pacemaker to the left hemisphere. The high coherences between other brain sites without evidence of driving, however, raises the possibility of multiple, relatively independent, or rapidly changing pacemakers or a more complex, diffuse interrelationship between the frontal and cingulate regions. Rapidly changing pacemakers or multiple independent ones would militate against the emergence of definitive driving channels. Fifteen seconds after the onset of the seizure (E3), two distinct seizure processes can be identified. Faster ictal activity at 21 c/set is present in the frontal and cingulate areas and the right hippocampus, with clear driving by L,. The spectra reveal peaks at IO-12 c/set in R,, R,, and R,,,, and the partial coherence analysis shows driving of Rn-RF at 11 c/set by RCIN. Five seconds later (E4), the faster activity has waned and the ictal discharge at 12-l 5 c/set has now become the dominant frequency. The major driving sites during
518
THARPANDGERSCH
this period are the right mesial frontal area and, as in E3, the right cingulate area. The apparent linearity between sites in the frontal lobes can be explained by their connection to the right frontal lobe and coherence between the right temporal lobe and frontal areas by their connection, particularly, to the right cingulate. Two distinct ictal processes are again apparent 10 set later (E5). Ten cjsec activity is present in the frontal and cingulate areas but the major driving site has shifted to the left frontal and frontal basal regions. The slower activity (3 cjsec) is driven by R, and LF. This is the first indication of driving by a temporal lobe channel. Analysis of the final epoch (E6) reveals that the seizure has spread into the temporal lobes with a concomitant change in the morphology and frequency of the analog activity. During this epoch it is not possible to define a single source or to lateralize the seizure to a hemisphere by visual analysis of the EEG. The computer analysis reveals that the ictal activity at 4 c/set, driven by L, and R, in the previous epoch, is now driven by L,, L,, and L,,. The analysis reveals three distinct seizure patterns: (i) a 21 cjsec rhythm that appears transiently in the early phases of the seizure (E3), with driving by Lr, (ii) an intermediate frequency rhythm (10-l 5 c/set) that is noted initially in E3 with driving by RCIN, becomes the dominant activity in E4, and is again driven predominantly by RF and &IN and wanes in E5, with the pacemaker now located in L,, LFB, and RCIN, and (iii) a slow ictal rhythm at 3-4 cjsec that first appears in E5, with the pacemakers located predominantly at R,, and L,, and becomes the dominant activity in E6 with the pacemaker shifting to the left hemisphere, i.e., L,. LH, and LFB. DISCUSSION
The present study describes the application of a spectral coherence analysis method for the identification of causality in human ictal EEG data. Several earlier attempts to apply spectral coherence measurements to cerebral electrical activity have been reported. Brazier (2,3) described changes in the percentage coherence between areas in the temporal lobes and between activity from these depth sites and the scalp during changing levels of consciousness, seizures, and following intravenous barbiturates. In a later study, coherences and phase information were employed to trace the propagation of seizures through the brain of humans (4). Larsen (29) demonstrated differences in the coherence between the occipital regions prior to photic stimulation, which could be correlated with the type of photic response obtained. Walter et al. (20) employed graphic displays to illustrate the evolution of the spectral intensity and coherence of the EEG recorded from the occipital areas in normal humans, but did not attempt to explain the coherence that developed between two areas of the brain, i.e., were the two sites directly related or was there a “shared-trigger or pacemaker?’ (2). Walter and Adey (21) used multivariate spectral analysis and a principal component analysis to investigate the concept of biophysical generators, but their approach does not appear to have been applied to practical EEG analysis or to
SPECTRAL
ANALYSIS
OF SEIZURES
519
have been further pursued. A technique for identifying causality, purportedly useful in the analysis of economic time series, was developed by Granger (22). Granger’s definition of causality is based entirely on the predictability of time series and not on phase lead or lag. A critical comparison of our technique with Granger’s appears in Gersch (16). Applications of our technique are discussed in Gersch and Goddard (7) and Gersch, Midkiff, and Tharp (8). Our approach to the concept of causality in time series is developed from considerations of extensions of concepts of regression and partial regression in the multivariate analysis of random variables. A single time series is identified to be causal relative to other time series if it uniquely explains the pair-wise linear relationships between other time series over a relevant interval in the frequency domain. That is, if the influence of a driving or causal time series is removed from the other time series by a regression analysis, the residual or partialed time series are pairwise uncorrelated, and hence, incoherent over that frequency interval. This definition lends itself to explicit statistical tests, to multivariate extensions of the definitions of causality, to the case in which a linear combination of time series may be causal relative to other time series, and to the physical plausability arguments that we think are essential in order to conclude that causality is present in time series. The results of the spectral analysis of a sequence of epochs during the propagation of a seizure are reported in this paper and reveal the migration of the driving sites, the appearance of new pacemakers concomitant with the cessation of driving in others, and the separate pacemakers for concomitant ictal activity at different frequencies. Though a seizure may be initiated in a relatively small neuronal aggregate, it will drive other areas to which it is anatomically related, either through long association pathways or short intracortical connections. Such pacemaker activity has been demonstrated by Petsche et al. (23) and Gogolak et al. (24) in the analysis of septal theta activity, and the concept was applied by Brazier (4) to the analysis of seizures. These secondary sites may then develop their own inherent rhythmicity and, in turn, will drive other populations of neurons within their sphere of influence even when the site of the seizure initiation has ceased discharging. This pattern of selfpropagation has been called the “will-o’-the-wisp” response (25). The almost simultaneous recruitment of many widespread areas, however, often leads to difficulties in localizing these pacemakers in analog data even with electrodes implanted near the participating neurons. The computer analysis was unable to locate a single site of seizure initiation in this patient, but it did delineate more clearly the propagation of the seizure and also revealed complexities not apparent in the analog data, such as the separate pathways of spread of ictal activity of different frequencies. Similarly, this technique appears to offer potential for inferring mechanisms for the initiation and evolution of the less understood generalized or “cortico-reticular” epilepsies (1). Current research techniques for locating the site of seizure generation in humans include the indirect method of intracarotid drug infusion (14) or depth electrode implantation with subsequent visual analysis. The limitations of these
520
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AND
GERSCH
latter experimental approaches and the inadequacies of a simple morphological analysis have been reviewed by Ajmone Marsan (26). Conjectures about the participation of thalamic nuclei in generalized seizure activity may be tested using the spectral analysis methods of identifying causality in time series. Our current approach has been to emphasize the computer analysis of epileptic data that could be understood by more conventional EEG techniques. The objective is to gain confidence and experience in the spectral analysis method and to clarify the new information obtained by this approach, which was not available in the visual examination of the EEG. The patient analyzed in this study was treated medically; therefore, confirmation of the site of seizure initiation by surgical removal of the involved tissue could not be obtained. We do not feel that this technique has been perfected to a degree such that the site of surgical intervention can be determined by the computer analysis. Considerably more experience is required before the technique can reliably supplement existing analysis techniques used for the identification of seizure pacemakers or foci in patients being evaluated for seizure surgery. REFERENCES 1. GLOOR, P. Generalized cortico-reticularepilepsies. Someconsiderations on the pathophysiologq of generalized bilaterally synchronous spike and wave discharge. Epilepsia 9, 249-263 (1968). 2. BRAZIER, M. Studies of the EEG activity of limbic structures in men. Electroencephalo~r. C/in. Neurophysiol. 25309-318 (1968). 3. BRAZIER, M. Electrical activity recorded simultaneously from the scalp and deep structures in the human brain. J. Nerv. Merit. Dis. 147, 31-39 (1968). 4. BRAZIER, M. Spread of seizure discharges in epilepsy: Anatomical and electrophysiological considerations. Exp. Neuvol. 36,263-212 (1972). 5. WALTER, D. AND BRAZIER, M. Advances in EEG analysis. Eiectroencephalogr. C/in. Ncurophysiul.
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27 (1968).
6. WALTER, D. Spectral analysis for electroencephalograms: Mathematical determination of neurophysiological relationships from records of limited duration. Exp. Neural. 8, 155-181 (1963). 7. GERSCH,W. AND GODDARD, G. Locating the site of epileptic focus by spectral analysis methods. Science
169,701-702(1970).
8. GERXH, W., MIDKIFF, A., AND THARP, B. Alpha rhythm generators. In “Computers in Biomedicine,” pp. 26-29. Suppl. to the Proceedings of the Fifth Hawaii International Conference on System Sciences, Honolulu, Hawaii, 1972. 9. GERSCH, W. Spectral analysis of EEGs by autoregressive decomposition of time series. Math. Biosci. 7,205-222 (1970). IO. AKAIKE, H. Statistical predictor identification. Ann. Inst. Statist. Math. 21,243-247 (1969). II. AKAIICE, H. On a semi-automatic power spectrum estimation procedure, In “Proceedings ofthe Third Hawaii International Conference on System Sciences,” pp. 974-977. Honolulu, Hawaii, 1970.
AKAIKE, H. “Information Theory and an Extension of the Maximum Likelihood Principle.” Research Memo 46, Inst. of Stat. Math., Tokyo (1971). To appear in Probl. ControEZnf: Theory. Akademiai Kiado, Budapest. 13. PARZEN, E. Multiple time series modeling. In “Multivariate Analysis” (P. Krishnaish, Ed.) Vol. 2, pp. 389-406. Academic Press, New York, 1969. 12.
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14. KROMER, R. “Asymptotic Properties of the Autoregressive Spectral Estimator.” Tech. Rept. No. 13, Dept. of Statistics, Ph.D. dissertation, Stanford University, 1969. 15. GERSCH, W. “Statistical Performance of the Autoregressive Model Method for Multidimensional Spectral Anaiysis.” Paper presented at NSF Regional Conference on Time Series Analysis, Chapel Hill, North Carolina, 1973. 16. GERSCH,W. Causality or driving in electrophysiological signal analysis. Math. Biosc. 14, 177196 (1972). 17. THARP, B. Autoregressive spectral analysis-a unique technique for the study of human seizure activity. In “Computers in Biomedicine,” pp. 2629. Suppl. to the Proceedings of the Fifth Hawaii International Conference on System Sciences, Honolulu, Hawaii, 1972. 18. GERSCH, W. A note on trapezoidal digital filter design. Comput. Biomed. Res. 6, 281-285 (1973)
19. LARSEN, L. Mechanisms ofepiiepticsusceptibility to photic activation. Case study of the photoconvulsive response. Epilepsia 10,473-480 (1969). 20. WALTER, D., RHODES,J., BROWN, D., AND ADEY, W. Comprehensive spectral analysis of human EEG generators in posterior cerebral regions. Electroencephalogr. Clin. Neurophysiol. 20,224237 (1966).
21. WALTER, D. AND ADEY, W. Analysis of brain-wave generators as multiple statistical time series. IEEE Trans. Biomed. Eng. 12,8-13 (1965). 22. GRANGER, C. Investigating causal relations by economic models and cross-spectral methods. Econometrics 3,424-438 (1969). 23. PETSCHE, H., GOGOLAK, G., AND VAN ZW~ETEN, P. Rhythmicity of septal cell discharges at various levels of reticular excitation. Electroencephalogr. Clin. Neurophysiol. 19,25-33 (1965). 24. GOGOLAK, G., STUMPF, C., PETSCHE,H., AND STERC,J. The firing pattern of septal neurons and the form of the hippocampal theta wave. Bruin Res. 7,201-207 (1968). 25. UDVARHELYI, G. AND WALKER, A. Dissemination of acute focal seizures to the monkey. I. From cortical foci. Arch. Neural. Chicago 12,333-356 (1965). 26. AJMONE MARSAN, C. Pathophysiology of the EEG pattern characteristic of petit ma1 epilepsy, a critical review of some of the experimental data. In “The Physiopathogenesis of the Epilepsies” (H. Gastaut, H. Jasper, J. Bancaud, and A. Waltregny, Eds.), pp. 237-248. Charles C Thomas, Springfield, Ill., 1969.