Bicoherence of intracranial EEG in sleep,
wakefulness and seizures
T.H. Bullock1, J.Z. Achimowicz2, R.B. Duckrow3 and S.S. Spencer4,
1 Department of Neurosciences, University of California, San Diego, La Jolla, CA
2 Department of CNS Diagnostics, Polish Air Force Institute of Aviation Medicine, Warsaw, Poland
3 Department of Neurology, University of Connecticut Health Center, Farmington, CT 06030
4 Department of Neurology, Yale University School of Medicine, New Haven, CT 06510
Running title: Bicoherence
of Intracranial EEG
Address for correspondence:
Dept. Neurosciences 0201
University of California, San Diego
La Jolla, CA 92093-0201
The hypothesis that the intracranial EEG has local structure and
short-term non-stationarity is tested with a little-studied measure of
nonlinear phase coupling, the bicoherence in human subdural and
deep temporal lobe probe data from 11 subjects during sleeping,
waking and seizure states. This measure of cooperativity estimates
the proportion of energy in every possible pair of frequency
components, F1, F2 (from 1-50 Hz in this study), that satisfies the definition of quadratic
phase coupling (phase of component at F3 , which is F1+F2, equals phase of F1 + phase of
F2). Derived from the bispectrum, which segregates the nonGaussian energy, autobicoherence uses the frequency components in one channel; cross-bicoherence uses one
channel for F1 and F2 and another for F3. These higher order spectra are used in physical
systems for detection of episodes of nonlinearity and transients, for pattern recognition and
robust classification, relatively immune to Gaussian components and low signal to noise
Bicoherence is found not to be a fixed character of the EEG but quite local and unstable, in
agreement with the hypothesis. Bicoherence can be quite different in adjacent segments as
brief as 1.6 s as well as adjacent intracranial electrodes as close as 6.5 mm, even when the
EEG looks similar. It can rise or fall steeply within millimeters. It is virtually absent in many
analysis epochs of 17s duration. Other epochs show significant bicoherence with diverse form
and distribution over the bifrequency plane. Isolated peaks, periodic peaks or rounded
mountain ranges are either widely scattered or confined to one or a few parts of the plane.
Bicoherence is generally an invisible feature: one cannot usually recognize the responsible
form of nonlinearity or any obvious correlate in the raw EEG.
During stage II/III sleep overall mean bicoherence is generally higher than in the waking state.
During seizures the diverse EEG patterns average a significant elevation in bicoherence but
have a wide variance. Maximum bispectrum, maximum power spectrum, maximum and mean
bicoherence, skewness and asymmetry all vary independently of each other. Crossbicoherence is often intermediate between the two auto-bicoherence spectra but commonly
resembles one of the two. Of the known factors that contribute to bicoherence, transient as
distinct from ongoing wave forms can be more important in our data sets. This measure of
nonlinear higher moments is very sensitive to weak quadratic phase coupling,; this can come
from several kinds of waveforms. New methods are needed to evaluate their respective
contributions. Utility of this descriptor cannot be claimed before more carefully defined and
repeatable brain states are studied.
Key words: Bispectrum; Nonlinearity; Epilepsy; Cooperativity; Subdural; Hippocampus ;
The hypothesis tested is that human intracranial EEGs are
generally highly differentiated and local in respect to measures of
cooperative interactions, at least down to millimeter dimensions,
and fluctuate in time within seconds. This view is quite in contrast to
a common view, derived from scalp recording, of a more global,
more widely coherent EEG with relatively stationary states often
lasting for many seconds. We undertook to test the hypothesis with a measure of higher
moments and nonlinear cooperativity, the bicoherence, derived from the bispectrum,
introduced into oceanography by Hasselman et al. (1963).
The hypothesis was suggested by evidence from voltage/time series ("raw records") and
linear spectra of EEG such as those revealed by Freeman and his associates (Freeman and
Skarda, 1985; Freeman and Viana di Prisco, 1986; Freeman and Baird, 1987; Freeman and
Davis, 1990; Freeman, 1992), among many other authors, and by evidence of the fine
structure of coherence (Bullock, 1989; Bullock and McClune, 1989; Bullock et al., 1995a, b).
The aim of the present paper is to examine the simplest of the higher moments, the
bispectrum and particularly the bicoherence, which is essentially the normalized bispectrum,
since this sign of interaction varies quite independently of the linear power and coherence
spectra and represents a little studied descriptor. A long-range goal is to find descriptors of
dynamical properties of brain function that might discriminate between parts, states, and
ontogenetic stages of the brain, and especially between evolutionary grades such as those of
insects, cephalopods, fish, reptiles and mammals. The goal of seeking descriptors is modest
but it reflects the conviction that our real understanding of the operations taking place in
organized neural assemblies is still primitive and that natural history or "system identification"
is needed. Lopes da Silva (1987) opining that higher order spectra need to be examined,
states "this is particularly true for the spectrum corresponding to the second order
autocovariance function ..." - the bicorrelation function whose frequency domain equivalent is
The bispectrum and its normalized derivative, bicoherence, look at the components of a time
series that deviate from a Gaussian amplitude distribution. Bispectra have been used to
describe particular aspects of ocean waves and various physical time series (Hasselman et
al., 1963; Elgar and Guza, 1985; Nikias and Raghuveer, 1987; Pezeshki et al., 1990; Mendel,
1991; Chandran and Elgar, 1993). Only a few papers have dealt briefly with the EEG
(Dumermuth et al., 1971, 1983; Dumermuth and Gasser, 1978; Dumermuth and Molinari,
1987; Barnett et al., 1971; Whitton et al., 1985; Lopes da Silva, 1987; Sherman and
Zoltowski, 1989; Sherman, 1993; Samar et al., 1993; Sigl and Chamoun, 1994; Kearse et al.,
1994; Eckhorn, 1994), and they mainly point out the availability and uniqueness of this
measure, using principally a few samples of scalp EEG. Some have reported bispectral
information about relationships between harmonic and subharmonic frequency components of
EEG signals (Barnett et al., 1971; Dumermuth et al., 1971, 1975; Lopes da Silva and Storm
van Leeuwen, 1978). Ning and Bronzino (1989) compared the deviation from Gaussianity and
the strength of phase coupling in the rat cortex and hippocampus during slow wave sleep,
REM sleep and quiet wakefulness. Tang and Norcia (1996) used the bispectrum with the
steady state visual evoked potential and reported much higher signal to noise ratios and more
information about interactions than with classical methods.
With these limited exceptions, little information is available on the incidence, character and
frequency distribution of bicoherence and the nonlinearities that contribute to it, or to indicate
how stable or how local or how sensitive this measure is over time, position or brain states. A
commonly negative reaction to bispectral analysis is based, we believe, on an inadequate
sampling and unrealistic expectations. Possibly also it has not been explored because (i) it is
hard to visualize phase coupling between three nonharmonic frequencies, (ii) the kinds of
time series that produce quadratic phase coupling are diverse and (iii) are not well delimited.
Such coupling, in the presence of uncoupled wideband fluctuations, is a property that can
inhere, without being noticed and with different causes. Hence we considered it timely to
undertake a descriptive study to test the motivating hypothesis with this tool.
We have chosen to examine a modest number of human subjects with many intracranial
recordings of EEG at fairly closely spaced electrode loci (6.5 - 10 mm), both subdural and
deep in the temporal lobe. The data were recorded in the course of presurgical evaluation of
epileptic patients who are not responsive to medication and become candidates for surgery.
They provide episodes of normal, alert EEG, sleep and seizures. Preliminary notes have been
published and include some bicoherence plots during event related potentials (Achimowicz
and Bullock, 1993a, b; Bullock and Achimowicz, 1994).
The unique features of higher order spectra, whose simplest
representative is the bispectrum, are used in several fields of signal
processing (Nikias and Raghuveer, 1987). Our use corresponds to
their characterization of the properties of nonlinear systems that
generate time series. We have found the bicoherence more
informative than the bispectrum itself and emphasize it in this study,
together with the separate measures of skewness and asymmetry. Bicoherence measures the
proportion of energy in each pair of frequency components, F1 and F2, of the specified
sample of the time series, and their sum, F3, that satisfies the definition of quadratic phase
coupling, i.e. the phase of F3 is the sum of the phases of F1 and F2. Since bicoherence is the
proportion of the coupled to the uncoupled energy, it is normalized (by the product of the F1
power x F2 power x F3 power) and is independent of the total energy. It is reported and
plotted as the bicoherence squared, a value between zero and one for each pair of
frequencies, F1 and F2, over a range, in our case, of 1-50 Hz, at the resolution of the
computation, for each specified epoch, for each channel (electrode). When the component
frequencies, F1 and F2, come from the same time series (channel; electrode), the results are
auto-spectra (auto-bispectrum and auto-bicoherence); when they come from different
channels (brain sites), the results are cross-spectra (F1 and F2 from one channel; F3 from
another). Equations, statistical evaluation and further details are given in references cited
above, especially Elgar and Guza (1985, 1988), Nikias and Raghuveer (1987), Mendel
(1991), and Sigl and Chamoun (1994).
EEGs from 16 to 80 channels were low-passed by analog filters to 100 Hz, digitized at 200 to
256 Hz, analyzed in 256-point epochs (1 to 1.28 s), using a Kaiser-Bessel window,
normalized per Kim and Powers (1979), overlapped 75% by a sliding window that advances
64 points, averaged over data segments of 2-20 s with a frequency resolution of 0.8-1.5 Hz,
sometimes smoothed with reduction of the resolution to 3 Hz, and plotted to a maximum of 50
This is a further analysis of data kindly provided by the neurologists in the epilepsy programs
at UCSD (V.J. I. ) and Yale University hospitals (S.S.S. & R.B.D.). Two data sets are
(i) The Yale data came from depth probes in the temporal lobe in 4 patients. Data from three
of the subjects ("C, H, W") came in 3 files from 3 seizures for each subject, recorded for ca.
30 s preictally and 30-40 s during the start and first part of the seizures. We used 8-12 depth
electrodes and 5-7 subdural electrodes on the same or the opposite side. Sleep, awake and
seizure periods are identified by the neurologist from the EEG. The fourth Subject ("D") is
represented by six files of 3-10 minutes each, including one seizure and three episodes each
of sleep and wakefulness, identified by EEG inspection. This data comes from 80 intracranial
electrodes, 10 depth and 30 subdural electrodes on each side.
(ii) The UCSD data came from subdural electrode strips on the temporal, frontal and parietal
lobes, from seven subjects. Each strip had a row of 8 electrodes and we normally analyzed
16 channels at a time. Most subjects are represented by one seizure and one or two episodes
of sleep and of alertness, in 5-10 files of 2-3 minutes each. The data is taken from 24 hour-aday recordings in the ward, not under standard EEG laboratory conditions. Much of the
interictal sleeping and waking EEG has isolated spikes.
Plots and evaluations are based on 11 subjects and 55 files. (The term file here means a
recording of a continuous 1-10 minute sample in some identified brain state or transition,
employing 16-80 electrodes.) The raw EEG records, as voltage vs time plots, were
characterized separately and marked by the neurologists to identify the brain state and the
start of the electrical seizures. The 3D plots show auto- and cross-bispectra, bicoherence and
amplitude spectra for two channels, usually adjacent loci. For auto-spectra, the bifrequency
plane is redundant and symmetrical around the diagonal of equal frequency except as values
are rounded differently by the x and y display routines. Cross-spectra have been computed
for pairs of adjacent electrodes only. Cross-spectra are not redundant and can be
asymmetrical around the diagonal since F1 and F2 come from one record and F3 from a
different time series.
A bicoherence plot on the bifrequency plane can be evaluated and compared with other
plots in different ways, besides visual inspection. Two of the simplest are the maximum value
of the highest peak and the mean of all values for all intersections (ca. 2500, at 50 x 50 and 1
Hz resolution). The first underrates the common case of widespread, low mountains; the
second underrates the case of one or a few narrow peaks. We have not undertaken to divide
the plane into areas, like compound frequency bands in order to average over the high-high
pairs and the high-low pairs and the theta-alpha and other pairs, or to present slices of the
bifrequency plane along various transects. These would no doubt be insightful. With a full
page of six 3-dimensional plots for each 4 or 9 or 17 s segment, each pair of electrodes, and
each file, for the 11 subjects, there are >>500 pages of this one kind of readout, before
considering the statistical assessment of means and other analyses.
Statistical evaluation begins with estimation of the bicoherence to be expected from a
stochastic data sample of the same length and power spectrum. Following the procedures of
Elgar and Guza (1988) and Elgar and Sebert (1989), a subprogram converts all values not
significantly above the chance level to that value; hence they are plotted as a smooth plateau
representing the 95% significance level. All peaks that rise above it are considered to be
significantly different from the null hypothesis of no bicoherence. The EEG epochs judged to
have no appreciable bicoherence, i.e. only a few low, scattered ant-hills rising as single points
above the plateau level of 95% significance (0.11 in our usual conditions) have a maximum
single value rarely above 0.15 and a mean bicoherence of all intersections of ca. 0.04
(calculated without the conversion to a plateau level); these are the numbers attributable to
chance with the particular departures from Gaussianity in our time series. A mean
bicoherence of 0.045 or more is usually associated with obvious broad mountains or multiple
narrow peaks of highly significant but only moderate bicoherence and is likely to show a
maximum bicoherence of 0.25 or more, depending on the distribution of significant
We also measured some other variables (Table I): the maximum bispectrum peak,
maximum peak and total area in the amplitude spectrum, total skewness (non-equivalence of
EEG waves around the horizontal, time axis, averaged across epochs and electrodes,
calculated as the sum of the real components of the normalized bispectrum) and total
asymmetry (of EEG deflections around the vertical or voltage axis, calculated as the sum of
the complex components of the normalized bispectrum); the last two are taken either as
absolute magnitude, discarding the sign or without discarding the sign.
Data reduction includes the following. (1) Bar graphs condense the many 3D sheets, each
for two loci, into a single 3-D graph, with a time axis, for each of the files (usually 60-120 s of
EEG from 17 seizures, 15 awake episodes, and 23 sleep episodes). Each bar shows the
mean of one of the measured variables for one electrode and 17 s epoch. To avoid
congestion we group 8-12 electrodes and a succession of 6-8 epochs (90-120 s) into one
figure for that file, subject, brain state and part of the brain. A montage of six such variables
on a sheet can represent, for example, one seizure and its preictal epochs, permitting
comparison with other places, times or brain states. Although this has reduced the data to
manageable numbers of plots, it blurs the distribution of the bicoherence in frequency space;
therefore we usually add the following step.
(2) Contour or topographic 2-D plots of the bifrequency plane with pseudocolor or shading
showing the elevations of bicoherence, each represent an electrode locus and a time
segment (ca. 2-20 s). They reveal the distribution of peaks in the bifrequency plane (F1 from
0-50 Hz by F2 from 0-50 Hz, with 1 to 3 Hz resolution). We arrange a number of plots of
several loci (in columns) and several successive segments (in rows) on one sheet as an array
of 2D plots mapping space and time, e.g. for 6 loci 10 mm apart and 5 epochs of 9s each.
(3) Statistics are computed, for each of 8 variables, including 95% confidence intervals for
fluctuations around different mean bicoherence values (Elgar and Sebert, 1989 and S. Elgar,
personal communication), for each of the files (brain states, episodes, subjects). Correlations
between the variables are computed, to see which are closely covarying and which can vary
A number of experiments were conducted with artificial data of known composition to check
our programs, test for leakage of energy into adjacent frequencies and learn the sensitivity of
the displays to relevant parameters and proportions of noise. In the extreme case of just one
intersection having quadratically coupled waves and a pink noise level 5 times larger (signalto-noise ratio = 0.2), the bicoherence at this intersection was 0.15; the 95% significance level
was 0.046 under those conditions. Skewness was 5 times and asymmetry was 2.6 times
larger than the 95% significance level. Thus the method is quite sensitive to phase coupled
signals much smaller than the "noise", i.e. the activity which is not quadratically phase
coupled. This can explain why similar looking EEG records may differ in that one shows
bicoherence and the other lacks any appreciable bicoherence. In another test, one frequency
(e.g. 13 Hz) was amplitude modulated by another (e.g. 3 Hz) with pink noise at signal-to-noise
ratio = 50. No bicoherence results unless some of the modulating frequency is added to the
AM (amplitude modulated) signal; in this condition 1% AM gives a high bicoherence of 0.8 .
Our observations are arranged in the following way. Bicoherence is
the principal focus. Bispectra, which combine into one mixture the
bicoherence, skewness, asymmetry and absolute power, are
relatively neglected in this report. We first describe the autoDiscussion
bicoherence of normal, alert subjects, its temporal stability and
spatial differentiation. Then we report findings on the crossbicoherence spectra. Finally, the three brain states and major brain regions are compared.
Auto-bicoherence spectra of normal, awake EEG
In some _2 min samples of EEG, from many to most of the spectra computed for successive
9 or 17 s segments show virtually no bicoherence. This is important, not only as a finding:
quadratic phase coupling is not typically or generally present. It is also a good control against
prevalent, spurious sources of bicoherence. In every such sample, however, a number of
segments have significant elevations somewhere in the bifrequency plane. For example in
Subject K, file 2 (right temporal cortex; awake), 85 out of 128 spectra from the eight 17 s
segments in 16 electrodes show little to almost no bicoherence, while 43 spectra have
substantial elevations (Figs. 1, 2).
Figure 1. Bispectra and bicoherence spectra of EEG recorded from
subdural electrodes, plotted on the plane of F1 and F2. First and
second columns are auto-spectra of two electrode channels; third
column shows cross-spectra between them. A, Upper row of 3-D
graphs, bispectra and, as profiles on the right edge, ordinary
amplitude spectra. (The units of the bispectrum are arbitrary; the
dimension is volts3/Hz2 each spectrum is plotted to its own
maximum, as also true for the amplitude spectra.) Lower row of
graphs, spectra of bicoherence, squared. The dimension is the
proportion that quadratically phase coupled energy at each
intersection plus the energy at F3 whose phase is the sum of the
phases of F1 and F2 bears to the total energy at these three
frequencies. Values are estimated by computing for each intersection
at the resolution of 1.5 Hz, from 256 points at 200 points per second
(= 1.28 seconds of EEG) and repeating 50 times, advancing 64
points each time (= 75% overlap, before averaging; total 3392 points
= 16.96 s). Bicoherence values of probability p >0.05 are converted
to the 95% significance level to make a plateau. Note that these loci
and this 17 s segment show moderately strong bicoherence in certain
areas of the bifrequency plane and that these two adjacent loci have
distinctly different distributions of auto-bicoherence peaks and
mountain ranges on that plane. Cross-bicoherence is more like
channel 2 than 3. Such plots were made for each pair of adjacent
electrodes (1 & 2, 2 & 3, 3 & 4, ...) and each segment of time for the
EEG samples of 1 to 6 min. Subject K, awake, right temporal cortex,
subdural. Raw EEG at the bottom. B, the same Subject, asleep, right
sub-frontal cortex, bicoherence only, electrodes 14 and 15. Note that
these two electrodes have, respectively, virtually none and quite
strong auto-bicoherence. This difference can be due to less quadratic
phase coupling in locus 14 or to more wideband Gaussian
fluctuation diluting it. Casual inspection cannot decide. Crossbicoherence again resembles the lefthand autospectrum, perhaps
because it contributes both F1 and F2 while locus 15 contributes
only F3. Raw EEG at the bottom.
In subdural recordings auto-bicoherence plots can be virtually flat, meaning no bicoherence,
or have low, scattered ant-hills or one or a few prominent peaks or great mountain ranges.
Elevations can be widespread or confined to the high-high corner (high F1 and high F2) or to
the low-low corner or to the mid-F1 and mid-F2 region or the mid-high or mid-low or any
region. Often there is only one main mountain but there can be two with a deep valley;
sharper peaks can be single or several or many (following paragraphs). Variety of pattern is
the main finding. In one example (Subject K, file 2) only one or two plots have a single peak of
>0.2 height (these are duplicated in our displays by the diagonal symmetry). Overall autobicoherence (total area under the 3D curves) in a number of sites is high (15 out of 72), some
are very high (7/72); 6/72 are low; 1/72 is very low. The rest (43/72) are distributed among the
medium grades, of dubious or lower significance, statistically.
Figure 2. Raw EEG from 10 s of sleep in Subject H, in 13
depth electrodes (EEG) in a row horizontally through the
temporal pole, amygdala and hippocampus, plus the autobispectra (with amplitude spectra on the right edge) and the
auto-bicoherence spectra for electrode channels 1, 2, 5 and
6. Note that there is virtually no bicoherence in three of
these channels, and only a few small peaks in channel 6.
Bispectra are more difficult to read because they include
absolute power and all forms of non-Gaussianity and they
vary by orders of magnitude. Values for 7 of the
measurements recorded are shown in fine print ("Max
PSD" = highest peak of the amplitude spectrum;
"Variance" = total area of the amplitude spectrum; Sum of
bicoherence" is the mean of all bicoherence values for all
intersections before converting to the 95% significance
plateau; "Skewness" and "Asymmetry" are the algebraic
means for all frequencies.)
In deep temporal lobe recordings, including the hippocampus, bicoherence values can
exceed 0.8 of the maximum possible in the highest peaks; mean values over all the
bifrequency intersections are usually <0.1 but can be highly significant even when <0.06, as
explained above. Illustrating a different method of counting than the previous paragraph, in an
example from Subject C (preictal 30 s just before seizure 2), of the 12 electrodes, numbers 3,
8, 11 and 12 had virtually no auto-bicoherence (means = 0.044 to 0.052); no.9 was very small
(0.062); nos.1, 2 and 4 were considerable (0.082, 0.071, 0.085, resp.); nos.5, 6 and 7 were
relatively high (0.13, 0.094, 0.068 with a maximum peak of 0.59); and no. 10 had the highest
mean (0.13). This subject shows power spectral peaks at 7, 14, 21, 28 and 35 Hz and
bicoherence peaks in some channels but not in others.
Among the variety of patterns, 3 or 4 of 72 auto-bicoherence plots in this file of this subject
show the "bed of nails" or hairbrush pattern (Figs. 3, 4). Narrow peaks have a constant
spacing of ca. 6-10 Hz, filling most of the bifrequency plane but sparing the high-high corner 1 high peaks on each side of the diagonal axis of symmetry. They may or may not correspond
to the 3-7Hz spikes in the raw EEG. More striking instances of this periodic pattern were
found in some other subject's files, e.g. "H," correlating with regular seizure spikes at the
same frequency in the raw data. Another occasional pattern is a ridge of elevated
bicoherence parallel to the F1 axis, meaning quadratic phase coupling between a certain low
frequency and all other frequencies at least to 50 Hz.
Figure 3. Raw data from an early part of seizure 1 in
Subject H (ca. 0.5 min after Fig. 2), plus the autobicoherence spectra for the same segment for 8 of the
channels. Channels 9-12 have virtually no bicoherence
although channel 10 might be expected to. The highest
spectra have a "bed of nails" pattern with a 6 Hz modulus,
covering a wide area of the bifrequency plane,.but sparing
the high-high corner This is not the same as the
conspicuous 4 Hz waves in the EEG. This periodic pattern
of bicoherence peaks has been seen only in a few seizures.
Figure 4. Topographic plots - like the 3-D plots of
previous figures, viewed from above - of six adjacent
electrode loci and five successive epochs of 9 s each, to
show the distribution of auto-bicoherence in the
bifrequency plane and along the length of the depth probe
at 6 mm intervals and in time. Subject H, seizure 1, same
data as Fig. 2; note the symmetry around the diagonal from
upper left (low-low frequency) to lower right (high-high
frequency). White = values below the 95% significance
level; darkest gray = 0.6. Note only occasional small hills
in the low-low corner in a few loci before the seizure starts
in the fourth row and brings extensive bicoherence with the
"bed of nails" pattern.
Table I shows the means and 95% confidence intervals for a variety of measures in the
waking state, pooling a number of electrodes and time segments. Visual inspection suggests
apparent correlates of strong bicoherence in the wave forms of the raw EEG (voltage vs time
= V/t plots) in some plots but other segments show that these apparent correlates are not
real. A crude way of testing this is to compare two adjacent channels (electrode loci), with
respect to the overall mean of bicoherence, e.g. approximately equal or greater in one than in
the other, and then independently estimating by visual inspection of the activity in the raw
EEG V/t plot, some candidate or suspected feature. A representative example is K02 (awake,
with interictal spikes in all channels): the first segment of 17 s, has 5 strikingly opposite trends
of the two estimates out of 14 pairs ("5/14"), i.e. one increases and the other decreases;
segment 2 (seconds 17-34) has four; segment 3 has 5; segment 4 has four - in sum, 18 out of
56 change in unexpected directions. The majority that change in the same direction may
mean there is a weak tendency for positive correlations of raw EEG features and
bicoherence, but more importantly, the strong minority indicate that visible features are not
reliable indicators of bicoherence. Instead of comparing simultaneous channels, we also
looked at changes in the same electrode in separate time segments. In 15 out of 45 the pairs
are conspicuously divergent in direction of change. It appears to be impossible to find a
consistent relation between time domain EEG pattern, by inspection, and bicoherence
Table I. Sample of auto- and cross-spectra for selected measures in Subject "D" in two
waking episodes of 3 min. each. Each value is the mean (large numbers) or 95%
confidence interval (small numbers) for 20 epochs of 9 s each, pooled from 10 electrodes
6.5 to 10 mm apart in one of the 8 brain regions: rpt, right posterior temporal depth probe,
traversing amygdala and hippocampus; lpt, same on left; rfp, right frontopolar subdural
strip; lfp, same on left; rat, right anterior temporal subdural strip; lat, same on left; rpo,
right parieto-occipital subdural strip; lpo, same on left. mxp, maximum of normal
amplitude spectrum, arbitrary units; mxbisp, maximum of bispectrum (1/Hz2); mxbico,
maximum bicoherence, squared; mnbico, mean bicoherence, squared (value for nosignificant bicoherence is ca.0.03); skew, skewness, incl. signs; asym, asymmetry, incl.
signs. Note: the two depth probes and eight subdural strips have significantly different
mean bicoherence and some other measures, with some significant differences between
episodes "Awake 1" and "2". Cross-spectra show almost as much bicoherence as autospectra but much more skewness and asymmetry. N = 207 for each value.
Neighboring channels that show large differences in bicoherence without any discernible
basis in the EEG are illustrated in Figure 5. This is especially true when the bicoherence
mountain is in the high-high corner (>30 Hz), because we cannot see this component in the
EEG, since it has such low power.
Figure 5. Raw data and topographic plots for a 13 s
segment for the first 5 electrode loci in the right deep
temporal lobe probe in Subject D (10 mm apart, #1 in the
tip of the temporal pole in fusiform grey matter,#2 in
fusiform white matter, #3 in amygdala, #4 and #5 on the
ventricular surface of hippocampus). Awake, session 1,
with a few interictal spikes and large delta waves in the
temporal pole. See Fig. 4 for explanation of plots; they are
rotated here so that the low-low frequency corner is lower,
left. Note that adjacent electrodes can have almost no autobicoherence in one and extensive mountain ranges of autobicoherence in the other (loci 3 and 4) - with no obvious
difference in the V/t record. Similarly for successive
epochs, the next 13 s epoch in channel 4 (not shown)
appears spikier and perhaps more asymmetrical but has
only the tiniest hill of bicoherence in the low-low corner.
Once more: such differences can mean changes in the
absolute amount of quadratic phase coupling or in the
uncoupled fraction; in the latter case this would have to
involve all frequencies, which is unlikely in Fig.5.
We have looked at the measures of asymmetry and skewness in the raw EEG, as well as
spikes and other abrupt transients. Sometimes, as in the intense electrical seizure of files
Csz2 and Csz3, a lot of conspicuously asymmetric and skewed waves do not give more than
moderate peaks of bicoherence. This of course, can mean that an even greater increase took
place in the uncoupled components, providing this involved the same frequencies, i.e. was
quite wideband. In view of the signal to noise ratios tested with simulated data, and the
statements of Elgar and Sebert (1989) on the relative immunity of bicoherence to background
noise, this explanation is unlikely. In some channels it appears as though one or two large
spikes, skewed (around the voltage axis) can make a large bicoherence over many
intersections of the bifrequency plane, but we have not tested this hypothesis, for example by
surgical excision of these spikes from the time series.
The table of correlations (Table II) compares each variable with others in nine data sets.
Note that four classes of pairwise correlation are clear. (i) Some show a consistently high
correlation, e.g. the height of the maximum amplitude peak and the total amplitude variance in
all frequencies. (ii) Some show a medium to high correlation, varying by several tenths in
different data sets, e.g. mean bicoherence and maximum bicoherence, mean bicoherence
and asymmetry, mean bicoherence and skewness. (iii) Some show only a low to medium
correlation, e.g. maximum bicoherence and maximum bispectrum, maximum amplitude and
maximum bispectrum; and (iv) some are extremely variable, fluctuating from very high to very
low, e.g. mean bicoherence and maximum bispectrum, mean bicoherence and total amplitude
variance, skewness and asymmetry. The principal finding is that, although correlated on the
average, variables such as bispectrum, bicoherence, skewness, asymmetry and power are
essentially independent and do not predict each other.
Table II. Correlations of some pairs of variables in a sampling of data sets. A
selection of 25 of the 182 possible pairs of measured variables, arranged in the
order of their mean correlations. Data sets from four subjects: H, W and C with
deep temporal probes, include 30 s of preictal and 60 s of ictal recording for three
seizures in each. Some are single files (file names with numbers), representing
single seizures, to illustrate consistent correlations of some pairs of variables and
inconsistent correlations for others. Columns without numbered names are pooled
from three episodes in each subject. Subject K is an example of the patients with
subdural electrodes during a waking period (K02) and a seizure (K04). Most sets
include 9-15 unselected electrodes; the two columns with "sel" in the file name
include only those electrodes that show seizure activity; this does not notably
affect the correlations. Skewness and asymmetry can be either positive or
negative; the sign was retained in computing means for Subject K but discarded
for H, W and C. Since most values are positive, this only raises the mean
fractionally and better represents the departure from Gaussianity although it
destroys some of the variability, assures the correlations are positive and
seemingly more significant.
Time course of bicoherence
The part of the motivating hypothesis concerning fine temporal structure was tested by
computing the bispectrum and bicoherence for epochs of various lengths from 1.6 to 17
seconds, then plotting the successive values vs time. Short segments raise the threshold of
significance or "bias" plateau proportionally to the segment length (under our usual conditions
0.11 with 10 s, 0.22 with 5s, 0.34 with 3.8 s). However, the maxima of peaks also rise almost
as much. Time segments of 1.6 s are of limited use for evaluating changes with time, given
our usual computing epoch of 1.28 s, since in addition to the rise in the significance plateau,
the rise in span of the 95% confidence interval for fluctuations around the presumed true
bicoherence, or its estimate, the mean value, makes no value significantly different from the
mean or from any other.
If bicoherence actually changes more slowly than the sample duration (segment length), each
maximum and minimum would be defined by more than one point. In fact we find even at the
shortest useable sampling time (3.8 s) that peaks and valleys generally consist of a single
point and the highest peaks and deepest valleys are significant. This means bicoherence can
fluctuate widely within a few seconds. EEG segments that look stationary to the eye in the V/t
record can be highly differentiated in respect to quadratic phase coupling. Peaks are higher
and more numerous and the average over many points is higher for the 3.8 than for the 7.6
and 15.2 s segments .
Thus it is common to see low and high bicoherence in adjacent time segments of the same
channel, of whatever length, even the shortest. Apparent rhythmicity at ca. 5-10 s periods is
common when the segment length is long (9-17 s), but usually disappears with shorter
Separate samples of the same state (tens of minutes or hours apart) can be similar in the
range of bicoherence statistics and patterns or they can be distinctly different. This is treated
further, below, under Brain states.
Figure 6. Bicoherence fluctuation with time. Maximum
bicoherence anywhere in the bifrequency plane for a
certain electrode analyzed in time segments of three
lengths, from a two minute record during a seizure; Subject
D. Ordinates are not to the same scale. Note the two 95%
statistical levels: the 95% significance level gives a bias
level that would contain 95% of the chance bicoherence
values in stochastic time series. It tests the null hypothesis
that there is no real bicoherence. For the three curves here,
these levels are, from top to bottom: 0.33, 0.14, 0.06. The
other, shown as "error" bars, gives a 95% confidence
interval within which fluctuations around the true value, or
the mean as an approximation, are expected by chance. It
tests the hypothesis that there are no real departures from
the mean. The "error" bars are for the one mean value and
for 18, 42 and 100 degrees of freedom, from top to bottom,
interpolated from values given in Elgar and Sebert (1989).
Most fluctuations are insignificant but some suggest that
beyond the chance fluctuations, bicoherence can change
abruptly - within a few seconds. This is not expected if the
main cause of quadratic phase coupling is an ongoing or
slowly changing wave form, such as a skewed or
asymmetrical wave. A wide variety of curves is found,
some with apparent oscillations, some with a single cycle
of a slow wave with a period >1 min.
Bicoherence can be very local. Although the majority of neighboring electrode sites are quite
similar at a given moment, low and high bicoherence plots are often side by side, in adjacent
channels (Fig. 5). For example, in "K" 7 out of 32 examined in this way are different in mean
bicoherence by 20% or more. Interstrip pairs are no more likely to show such abrupt
differences (8/32) than pairs within the same strip.
Influence of distance
To examine the relation between bicoherence at a given locus and that at other loci closer
and farther apart, we chose to compute the correlations between each pair of loci in rows of
ten electrodes in each subdural strip or depth probe. The value chosen for each locus was the
mean bicoherence value over the whole bifrequency plane for a number of successive data
segments in a record with 4.9 min of EEG. The data segments could be 15.36 s long and
hence 19 numbers in sequence, computed for correlation as they fluctuated up and down. Or
they could be 3.84 s long and hence 76 numbers in sequence; or they could be intermediate
in length. This method overlooks differences in detailed pattern of bicoherences in successive
time segments. Another method would have been to compare the means of the crossbicoherence spectra for pairs of different separation. This could prove to give quite different
results from the correlation method. Fig. 7 shows correlation plots of several deep temporal
and subdural strips during sleep.
Figure 7. Correlation between loci vs interelectrode
distance. Columns each represent a temporal lobe depth
probe or a subdural strip of 10 electrodes, 10 mm apart, in
right and left, approximately symmetrical sets. The
abscissa is electrode number = position in the row. Each
curve plots the correlations between one of the electrodes
and each of the others, from closest to farthest. The
common electrode is indicated by an arrow (numbered in
column one) and, being self-correlated, is always 1.0. The
ordinates are not to the same scale. The value for each
electrode locus was the correlation coefficient of 19
successive means of the bicoherences for all intersections
over the bifrequency plane for each 12.8 s data segment in
a record with 4.9 min of sleeping EEG; Subject D. Note
that although the average behavior is a certain slope falling
with distance, a large variability is manifest between the
individual pairs and often the highest correlation, apart
from the self-correlated point, is not the closest pair. These
curves, too, fluctuate with time.
Typically there is a decline with distance, falling to half in about 10-30 mm. This estimate is
made ignoring the 1.0 peak of self-correlation. Some curves are even steeper and some
cannot be reliably distinguished from a plateau. Deep valleys of low correlation can be 20-30
mm wide. Peaks of similar width can occur at any distance within our range of 90 mm.
Exceptionally we see a secondary maximum at long distances, after a steep valley. When we
calculated for more numerous short time segments (3.84 s), the curves were not more jagged
but quite similar to the 15 s curves.
Cross-bicoherence has only been examined between adjacent electrodes and only for the
difference interactions, where F1 and F2 come from one channel and F3 from the other. Such
spectra are found in each of three categories. (i) They can look roughly intermediate between
the two auto-bicoherence plots, in the distribution of peaks over the bifrequency plane. (ii)
They can be lower in overall bicoherence than either of the auto-bicoherence plots. (iii) Quite
commonly the cross-bicoherence closely resembles one or the other auto-bicoherence
channel (Fig. 1), usually the one providing F1 and F2 and the specific pattern of the plot
differs, depending on which file of the pair provides F1 and F2 and which provides F3. Crossbicoherence is never higher than either of the auto-spectra.
Frequency-specific examination of cross-bicoherence shows great variety among spectra,
with respect to given peaks, i.e. some peaks are like those in one or the other auto-spectrum,
some are intermediate, some smaller than in either auto-spectra. In a sample of 64 crossspectra from pairs of 72 auto-spectra, 14 cross-spectra have clear or strong peaks in the
high-high frequency corner, compared to the auto-spectra where such peaks are much more
common (34/72). Cross-spectra show low-low frequency peaks in 23/64, not much less
commonly than in auto-spectra (29/72).
Cross-bicoherence can be asymmetrical around the equal frequency diagonal. The sample of
64 cross- spectra have 12 with obviously asymmetrical peaks and many more with less
conspicuous ones. Most peaks are roughly symmetrical.
An exclusion "rule" is more commonly true than not, namely if there is a strong peak in the
low-low frequency area, then there is no large peak in the high-high frequency area, and vice
versa. This "rule" has 5 exceptions out of 64 in the foregoing sample. Auto-spectra show more
common exceptions (14 / 72).
Sleep & awake states. The statistics on all five subjects for whom suitable records and
statistics are available (D, K, Ma, Me, R; >140 electrode loci) show a significant and
consistent difference between waking and sleeping (stage II/III) states in the EEG samples of
several minutes each - in all, 20 separate sleep episodes and nearly as many awake. In the
sleep state, both in subdural and hippocampal loci, mean auto-bicoherence of all frequency
pairs, epochs and electrodes ranges between 0.05 and 0.14, whereas in the waking state it is
0.03-0.10 and in each subject nearly always higher in sleep, on average by ca. 0.04. This is
generally 2 or 3 times the 95% confidence intervals (Table III). It should be recalled that the
mean value when there is virtually no significant bicoherence is ca. 0.04. Thus the greater
value in sleep is relatively large. Evidence is suggestive that the fraction of nonlinearly
coupled energy is highest in the EEG range <4 Hz in sleep whereas in the awake state the
contribution from gamma band oscillations is more significant seizure state. The seizure state
usually causes an increase in auto-bicoherence (Table IV) but there are many epochs after
the "start of seizure" with no visible change in bicoherence, compared to the preceding, preictal EEG. The distribution of high bicoherence is quite variable between seizures and
electrodes, e.g. sometimes mainly in high frequencies, sometimes mainly in low.
Table III. Comparison of the awake, asleep and seizure states. Subject K; two episodes
of each state; 16 subdural electrodes on frontal and parietal and anterior temporal lobes.
Means (boldface) and 95% confidence intervals (small italics) for measures of the
maximum power (maxp) in arbitrary units, maximum bispectrum (maxbis), maximum
bicoh (maxbic) between zero and one, mean bicoherence (mnbic), skewness, (skew),
asymmetry (asym) in the auto- and cross-spectra. From 6 files of ca. 2 min each,
computed for 10 s epochs, pooling the values calculated for all epochs and for each of
the 16 electrodes, except for the two rows of seizure measures from selected electrodes
that show some stages of seizure activity during some of the epochs. Note that almost
the only difference between the two episodes of the waking state and two of sleep is that
mean bicoherence is higher in sleep. The ictal state in this sample shows distinctive
characters only if the affected electrodes, judged from the raw record, are selected; they
show principally an increase in bicoherence and asymmetry, besides the obvious
increase in power and bispectrum.
Table IV. Comparison of mean bicoherence in three brain states in several subjects.
Means and 95% confidence intervals from many electrodes and 17 s epochs [except D =
9 s]. Sample files from each subject. All electrodes are pooled except in seizure files
marked with * - in which electrodes are selected that show seizure activity has affected
No one pattern of distribution of bicoherence over the bifrequency plane is consistent among
the 20 seizure episodes in 11 subjects analyzed (Figs. 3, 4). Often there is a wide scatter of
small peaks; sometimes there is a narrow mountain range parallel to one of the axes (like the
ridge mentioned above). Some seizure files show a protracted period without spread of the
electrical seizure beyond a few, or even one channel and the other loci may be subnormal in
EEG power, with little or no bicoherence. Some stages of seizure activity in some seizures
show little bicoherence. Even during conspicuous epochs of 4 Hz spike and dome waves, we
observe in some files that mean bicoherence can be very low. Some seizures, which happen
to be preceded by unusually high preictal bicoherence, show an actual overall reduction of
bicoherence during the seizure. If we look more closely at the distribution of peaks over the
bifrequency plane, there is no typical signature of seizures or of classical stages such as the
spike and dome pattern. Of the several measures provided by the bispectral analysis, mean
bicoherence is not always the one most clearly showing the change from the preictal to the
ictal EEG. Sometimes maximum bicoherence, wherever it occurs in the bifrequency plane, or
maximum bispectrum or skewness or asymmetry or total power shows the change more
clearly (Fig. 8).
Figure 8. Comparison of several measures in bar graphs.
Means from 8 electrodes for successive 9 s epochs of the
ca. 60 s sample of EEG; first two columns of bars are preictal; seizure starts with third column. Subject C, seizure 3.
Skewness is averaged absolute magnitude, discarding sign.
Each measure has disadvantages; e.g. maximum autobicoherence undervalues wide, low hills; mean autobicoherence undervalues narrow peaks. Note all these
measures can jump abruptly from one electrode to the next
and from one epoch to the next.
Among three of the deep temporal lobe subjects from whom we have three seizures each, the
seizures are quite diverse between the subjects and somewhat less so between the episodes
in one subject. They progress through stages of distinct character. Some stages may never
involve some electrode loci. Subject C is a case where the seizure is accompanied by
substantial increases in maximum and mean auto- bicoherence, maximum and mean crossbicoherence, skewness and asymmetry besides large increases in maximum and total power.
Subject D, however, shows no average difference from the 6 min of preictal bicoherence to
the 2'20" of seizure state and the 4'30" immediately after the seizure. Overall, the mean
bicoherence of all electrodes and epochs during seizures ranges among subjects between
0.07 and 0.14, usually no higher than in slow wave sleep in the same subjects. If only those
electrodes are chosen that show clear seizure activity, the mean bicoherence is hardly any or
no greater than the mean of all electrode loci.
Most of the nine seizure starts in Subjects C, H and W, as determined by inspection of the
raw EEG (V/t), are presaged by small but distinct increases in bicoherence, usually scattered
or in the low-low corner of the bifrequency plane. This grows in successive 9s samples to
large mountain ranges of bicoherence over many or all intersections (pairs of frequencies),
except, usually, the high-high corner. That corner is occupied in the last epochs of the second
seizure in Subject C (Csz2, 25+ seconds after seizure start), and in earlier epochs in the third
seizure (Csz3 ca. 10 s after start.)
In some seizures (3 out of the nine in C, H and W) a conspicuous "bed of nails" (Figs. 3, 4) or
hair- brush pattern of narrow bicoherence peaks in the low-low, mid-low and high-low regions
coincides with quite regular and asymmetrical spikes at 3-5 Hz in the EEG and sharp teeth in
the power spectra and a hair-brush in the 3D bispectrum. This means there are deep valleys
of low bispectrum and bicoherence between the peaks at intersections of harmonics of 3-5 Hz
up to at least the 6th or 7th harmonic. The impression is that when the pattern is a rounded
mountain range or with fewer, irregular peaks and shallow valleys, i.e. a distribution of
bicoherence over all frequency pairs, including nonharmonic ones, it is simply due to less
regular intervals between asymmetrical EEG waves.
In seizures as in interictal states, neighboring electrodes can have extremely different
bicoherence (6- 10 mm apart, center-to-center; each contact is quite large, being 1 mm wide
bands around a 1 mm shaft), e.g. one electrode may show a high mountain or tall hair-brush
and an adjacent one no bicoherence. This difference can sometimes correspond to a
difference in the EEG between high amplitude paroxysmal spikes in one channel and normal
background activity in the next. But quite often we can not see anything that would explain the
observed difference in bicoherence, between different channels or adjacent epochs.
In order to look for possible differences between loci or larger brain regions, such as the
lobes, which were sampled by eight or ten electrodes each, cluster analyses were used
combining three measures: mean bicoherence, asymmetry and skewness. As indicated
above, these can vary independently although there is some average correlation. Fig. 9 is an
example, comparing two regions of the same lobe, each represented by 8 electrodes and 4
epochs of 17 s each, i.e. 32 values. Such cluster plots show that there can be much overlap
but also substantial differences between regions and lobes, given enough loci and time
samples. We cannot, however, make any general statement about the consistency of such
differences with our limited data set. The method remains an available, untried means to
distinguish right and left, and cortical lobes. In the single subject with both subdural and deep
temporal electrodes, the averages of many deep temporal loci are consistently different from
those of many subdural cortical loci, in all three states: in the hippocampal loci mean and
maximum bicoherence are always higher, on the average, but none of the other measures is
Figure 9. Comparison of two
brain regions in the same
subject at the same time, in an
awake, interictal state, by
cluster analysis; . Left, eight
right subfrontal electrodes;
right, eight right lateral frontal
electrodes; four 17 s epochs.
Each point represents the mean
bicoherence over all
intersections for one electrode
for one epoch. Above, sample
EEG from one of the loci in
each cluster. Such cluster
differences can be found
between brain regions but are
not yet known to be
The most important conclusion from these findings is that a great
deal more spatiotemporal information is carried in the intracranial
EEG than the eye or familiar analyses can see. In many subdural
and depth samples of 1.6 to 17 seconds we find episodically a form
of cooperativity measured by quadratic phase coupling between
pairs of frequency components, in the same or in different
channels. Though commonly absent or nearly so, it is now and then prominent. Abrupt
changes occur in the space of a few millimeters and a time span of one or two seconds.
These numbers are based on our data set, which employed gross electrodes. It seems likely
that the real dimensions, if we used semimicroelectrodes within the tissue, would be found to
lie within tens of micrometers and milliseconds. Since different pairs of frequency components
can be different in bicoherence and each can fluctuate rapidly, the picture resembles the
bouncing light bars of an audio-amplifier frequency equalizer plus spatial differentiation in
Some EEG features like asymmetry and skewness clearly contribute to bicoherence, but most
the features responsible for local or episodic bicoherence are usually not recognizable by
visual inspection of the EEG, even when bicoherence is prominent. Highly similar looking
segments of EEG a few seconds or a few millimeters apart may have either no appreciable
auto-bicoherence or a substantial proportion of it. When present, it can be distributed in the
bifrequency plane in any of a wide variety of patterns: periodic or aperiodic, solitary peaks like
obelisks or broad mountain ranges, located in one or another or several parts of the plane,
e.g. the high-high frequency region or high-low or low and intermediate or any other region. It
is important to say that narrow, sharp peaks are not usual; relatively wide hills or mountains,
including many contiguous frequency pairs are more common. Likewise, peaks at harmonic
and subharmonic intersections are neither the rule nor particularly common.
Once accustomed to the idea that another and unfamiliar variable belongs within the scope of
EEG dynamics, the important finding becomes its local, patchy distribution, fluctuating
temporal status and complex 3-dimensional pattern on the bifrequency plane at each place
and time. The patterns of these variables may turn out to characterize brain structures,
ontogenetic stages, physiological or cognitive states or phylogenetic levels of complexity.
Such characterization will not come easily, both because of the wide variety of wave forms
that cause quadratic phase coupling and the high variance in time and space. The limited
number of samples examined so far mainly emphasize great variability within a given
structure and state. The evidence, in short, is in agreement with the original hypothesis that
human cortical EEG is generally highly differentiated in respect to nonlinear cooperative
interactions. This emphasis is the same as that reached in a separate study of the distribution
of linear coherence between loci as well as successive time segments (Bullock and McClune,
1989; Bullock et al., 1995a, b), a measure quite independent of the bicoherence between
frequencies. This emphasis on the local differentiation of what has been called the microEEG
is not in conflict at all with the findings from scalp recording where a macrostructure exists
that we could not predict from the sampling of microstructure.
We take the existence of bicoherence (defined by the fraction of non-Gaussian energy having
a particular phase relationship among three frequencies: any two, F1, F2, and their sum, F3,
such that the phase of F3 = the phases of F1 + F2) to be a sign of nonlinear cooperativity
among frequency components of the compound field potential that arises from many
generators. Whether the information carried in the cooperativity is itself a code (a signal and
therefore influential in brain functioning) or is only a sign of the interactions among cellular
and subcellular generators and their dynamic fluctuations (Bullock, 1997), in either case it
becomes another descriptor potentially able to distinguish regions, stages, states and levels
of brain evolution. As a quantitative and sensitive descriptor it might contribute to the
understanding of what goes on in the brain, the more so as methods develop to evaluate
different causes of bicoherence, due to transients or to ongoing relations such as special
forms of amplitude or frequency modulation. Research on methods of estimating the relative
contributions to the quadratic phase coupling of different kinds of ongoing wave forms (e.g.
skewed or frequency modulated) and transient events (as in spikes or steps), is under way
but still on the cutting edge (S. Elgar, pers. comm.). The abrupt changes in bicoherence here
reported and the significant narrow peaks and valleys in plots of bicoherence vs time suggest
that a common cause of quadratic phase coupling is episodic transients such as sharp
corners and that this can be more important in some cases than steady state nonlinearity
such as AM or ongoing asymmetry. Simulations have shown us that even a 10 ms pulse
irregularly appearing about once per second causes widespread peaks detectable when the
rms noise is as much as 0.2 the peak of the pulse. Transients can be expected to contribute
all the more to bicoherence if events of the same shape are repeated, even if irregularly, as
opposed to the same number of events of diverse shape.
We are not surprised to find bicoherence in the EEG, knowing that many departures from
Gaussianity occur in the EEG. Several authors have already noted one or more peaks of
bicoherence in small samples of EEG (cited in the Introduction). We were not prepared,
however, for the extensive periods and places where no frequency pairs show any significant
bicoherence or for the extreme diversity of the distribution of elevations and patterns of them
all over the bifrequency plane even in loci a few millimeters apart, or for the rapid changes
with time. We are surprised by the broad mountains as opposed to narrow peaks. We could
not anticipate the curious forms of cross-bicoherence. We expected the finding of generally
higher bicoherence during seizures but did not expect to find more bicoherence in sleep than
in wakefulness. The state labelled wakefulness is probably heterogeneous. Our subjects were
recorded on the ward under somewhat more natural living conditions than the standard EEG
recording conditions. They almost never show periods of obvious alpha or other rhythms. We
had no control of the open or closed state of the eyes or of attention, conversation or
cognitive state and hence should not be surprised by the abrupt changes over time and the
Variability of bicoherence can be due to change in the absolute amount of phase coupling and
also to change in the relative amount of non-quadratically coupled power at the corresponding
frequencies. Thus some EEG segments with obvious skewness or numerous spikes may
show low bicoherence from dilution with uncoupled activity or even phase-coupled activity of
higher moments. Power line (60 Hz) interference should not affect our bicoherence values
because we limit our view to the band below 50 Hz but muscle or other forms of extraneous
spikes surviving the 100 Hz lowpass filter could either add spurious bicoherence or dilute the
fraction of neural bicoherence, depending on their wave forms. If non- neural sources are
adding spurious bicoherence in our data, they must be doing so in improbably short episodes.
We have visually inspected the raw recordings and deleted the occasional bad channel; it is
unlikely that we have missed sustained periods of artifacts. We do believe that some of the
periods or channels of surprisingly low or insignificant bicoherence throughout the bifrequency
plane, compared to the strong bicoherence in neighboring channels with similar and obviously
skewed, asymmetric or spiky wave forms, may be due to dilution by a visible, sustained, low
amplitude "fuzz" in the low bicoherence channels that, if it is responsible, must be wideband
as well as symmetrical and unskewed.
An important finding is the independence of the variables: maximum peak power, total power
(these are highly but not invariably or necessarily correlated in our samples), maximum
bispectrum, maximum and mean bicoherence (usually rather highly correlated in our samples
but of course either could be high when the other is low), skewness and asymmetry - each in
the auto- and in the cross-spectra. Particularly interesting and to some, surprising, is the
independence of bicoherence and both skewness and asymmetry (asymmetry of wave form
around the time axis and the voltage axis, respectively), although the correlations are
moderately high. Skewness and asymmetry are remarkably independent and poorly
correlated with each other in our material.
In spite of the considerable number of electrode sites, epochs, and subjects, our material is
insufficient to claim consistent differences between lobes or regions of the brain even within
this set of individuals. The method may prove useful in some subjects to establish differences
between homologous sites on the two sides or anomalies in particular lobes. As a diagnostic
sign of seizure activity, bicoherence appears to be not as good as inspection of the raw EEG
by a trained eye. As a candidate for automatic detection of seizures, it is doubtfully any better
than simply measuring the total power in the power spectrum. It may, however, be a helpful
complement. The typical differences reported between sleeping, waking and seizure states
are at the moment less diagnostic than naked eye or linear spectral analyses, mainly because
of the episodic nature of bicoherence amid long stretches of none.
As yet unexamined aspects might turn out to be the most significant features of bicoherence
and especially of the bispectrum which we have not yet attempted to characterize
systematically. These unexamined aspects include statistical comparison of the autobicoherence in different brain regions and more controlled brain states in a larger number of
subjects, and of the several other measures of cross- bicoherence (e.g. subtractive, additive,
F1 and F2 from different channels) for the same regions and states. Also in this list are the
possibility of mapping loci with more cross-bicoherence than the general average, any rules
about the function of distance between loci, the frequency-specific distribution of quadratic
phase coupling and of skewness and asymmetry with brain state or region, and the degree of
independence of local coherence and bicoherence. Our brain states are not well controlled
and leave open the more subtle cognitive states. We have not compared the scalp EEG with
the intracranial; suitable data is rare for technical reasons.
This close look at EEG bicoherence, besides the surprises mentioned above, has compelled
us to press the mathematical experts for help in dissecting the relative roles of the known
sources of quadratic phase coupling. These can apparently be subsumed under two rubrics,
with subdivisions. (i) Ongoing rhythms that have modulations of amplitude, frequency or
phase by other rhythms can cause bicoherence, but only if some of the modulating signal is
present in addition to the modulated mixture (note the experiment described under Methods,
with artificial data). Modulations also include those distortions of sine waves that are either
sawtooth (or otherwise asymmetrical around the voltage axis) or scalloped (rounded peaks
and sharp valleys: skewed around the time axis). (ii) Transient events with more or less sharp
corners at their onset, offset or change in slope introduce a wide band of time-locked, phase-
coupled frequencies. Although we have numerical estimates of the skewness and asymmetry,
we do not know their relative contributions to the bicoherence compared to other causes. It
will be valuable to learn how to evaluate the classes of contributions.
In the meantime we can speculate that transient events may play a larger role in the
dynamics and cooperative properties of EEG than has been recognized heretofore. These
considerations also serve to emphasize that bicoherence is not a measure of a simple or a
single operation or form of interaction but a mathematical relation among the artificially
segregated frequency components of one or two time series that can come from a variety of
wave forms, continuous or transient. Interpretation of what it means for brain operations when
bicoherence suddenly appears, in one or another pattern on the bifrequency plane, is
therefore not to be expected, pending new tools. What it means for brain understanding is
two-fold. (i) It can direct attention to certain episodes and loci, certain wave forms and
frequency bands that might yield to closer inspection. (ii) It provides a powerful descriptor, not
predictable from familiar analyses, that might be found to distinguish certain states, stages,
regions or taxa.
Acknowledgement This work was aided by grants from the
National Institute of Neurological Diseases and Stroke. Robert
Guza and Stephen Elgar have provided crucial expert advise and a
number of friends have helped with the manuscript.
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