1Department of Molecular and Cell Biology and 2Department of Biophysics, University of California at Berkeley, Berkeley, California 94720-3200
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ABSTRACT |
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Freeman, Walter J. and John M. Barrie. Analysis of Spatial Patterns of Phase in Neocortical Gamma EEGs in Rabbit. J. Neurophysiol. 84: 1266-1278, 2000. Arrays of 64 electrodes (8 × 8, 7 × 7 mm) were implanted epidurally on the surface of the visual, auditory or somatosensory cortex of rabbits trained to discriminate conditioned stimuli in the corresponding modality. The 64 electroencephalographic (EEG) traces at all times displayed a high degree of spatial coherence in wave form, averaging >90% of the variance in the largest principal components analysis component. The EEGs were decomposed with the fast Fourier transform (FFT) to give the spatial distributions of amplitude and phase modulation (AM and PM) in segments 128 ms in duration. Spatial (2-dimensional) and temporal (1-dimensional) filters were designed to optimize classification of the spatial AM patterns in the gamma range (20-80 Hz) with respect to discriminative conditioned stimuli. No evidence was found for stimulus-dependent classification of the spatial PM patterns. Instead some spatial PM distributions conformed to the pattern of a cone. The location and sign (maximal lead or lag) of the conic apex varied randomly with each recurrence. The slope of the phase gradient varied in a range corresponding to that of the conduction velocities reported of axons to extend parallel to the cortical surfaces. The durations and times of recurrence of the phase cones corresponded to those of the optimally classified spatial AM patterns. The interpretation is advanced that the phase cones are manifestations of state transitions in the mesoscopic dynamics of sensory cortices by which the intermittent AM patterns are formed. The phase cones show that the gamma EEG spatial coherence is not due to volume conduction from a single deep-lying dipole generator nor to activity at the site of the reference lead on monopolar recording. The random variation of the apical sign shows that gamma AM patterns are self-organized and are not imposed by thalamic pacemakers. The half-power radius of the phase gradient provides a useful measure of the soft boundary condition for the formation and read-out of cooperative cortical domains responsible for binding sensory information into the context of prior experience in the process of perception.
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INTRODUCTION |
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Five premises are adopted here for
analysis of the spatiotemporal integration of neural activity within
and between areas of neocortex. First, neurons communicate with each
other over short distances by ionic diffusion of synaptic
neurotransmitters/modulators and over long distances only by propagated
action potentials that incur delays. This premise disallows appeals to
instantaneous communication by ephaptic linkages through fields of
electrical current (Bullock 1997; Bullock et al.
1995
; Prechtl et al. 2000
; Terzuolo and
Bullock 1956
), electromagnetic resonance (Adey 1969
, 1993
), or quantum coherence (Hameroff 1998
).
Second, communication between parts of neurons is predominantly by
ionic loop current, which is governed by the cable equation and which,
like diffusion, imposes attenuation of amplitude with distance
(Hodgkin and Rushton 1946
). Local sources of energy for
the action potential surmount this attenuation but at the cost of
distance-dependent delay. Therefore neurons that are distributed in a
network or mass and that fire simultaneously can achieve simultaneity
in the arrival times of their action potentials at selected points in
the network only under severe constraints. Third, spatiotemporal
integration of the activity of afferent axons by the cable-like
dendrites of receiving neurons requires some degree of synchrony in the multiple inputs. This means that some component of the activity of each
of the transmitting axon terminals must be at or near zero time lag
with respect to the activity of all. The span for "zero lag" is
defined here as the modal duration of an action potential, ~1 ms, and
"near" as the duration of a passive membrane time constant, about
±5 ms.
Fourth, synchronization is essential for integration of the
activity of "feature detector" neurons in the process of
perception, whereby the action potentials of relay neurons that are
activated by sensory neurons having receptor fields for features of a
stimulus are combined to form a representation of the entire stimulus. How much synchrony is required and how is it achieved in the face of
axonal delays? This constitutes what has been called the "binding problem" (Hardcastle 1994; Milner 1974
;
Schillen and König 1994
; Singer
1993
; von der Malsburg 1983
). When the problem
is formulated in this way, the description of the neural mechanism
requires measurement of that fraction of the variance of the activity
of individual neurons or local ensembles that is covariant over the whole network or mass. The solution also requires an explanation of how
that fraction of the total variance that is covariant over the ensemble
can constitute or carry a neural signal and a description of a neural
mechanism by which that covariant fraction can be extracted by the
target neurons receiving spatial activity distributions that have been
transmitted by axons with obligatory propagation delays.
The fifth premise is that the activity of cortical neurons is
synchronized by their synaptic interaction in closed loops. Anatomical
evidence shows that each neuron forms synapses on thousands of others
within its axonal arbor and receives axon terminals from thousands of
other neurons within its dendritic arbor, but cell counts indicate that
each neuron connects only with on the order of 1% of the neurons that
lie within its reach. Anatomists such as Sholl (1956)
and Braitenberg and Schüz (1991)
have concluded that the likelihood of a feedback connection between any designated pair of neurons may be as low as one in a million. If the connections of most neurons are as sparse as these estimates indicate and if the
impact of single action potentials is as minute with respect to the
thresholds for neural firing of recipient neurons as estimated by
biophysicists (Amit 1989
), one of the feedback loops by
which each neuron can interact with other neurons is provided by its local neighborhood. This premise complements the significant roles of
interactions among delimited subsets of neurons forming discrete networks in information processing (Arieli et al. 1995
;
Jagadeesh et al. 1992
; Nicolelis 1997
;
Nicolelis et al. 1998
; Tsodyks et al.
1999
; Zhu and Connors 1999
), which can give rise
to "zero lag" correlations between the gamma oscillations of
neurons separated by millimeter distances (Roelfsema et al.
1997
) despite the axonal transmission times that are expected
to cause phase shifts between oscillations. Usher, Schuster and
Niebur (1993)
and Schillen and König
(1994)
have modeled this phenomenon by assuming that the feedback delay within each target matches the transmission delay between targets in an excitatory feedback network. Traub et al. (1996)
have overcome some of the rigidity of those models by
invoking doublet firing of single neurons (Freeman
1975
), which we and they show is enhanced in states of
high-amplitude gamma.
The fifth premise provides an alternative mechanism for spatiotemporal
integration in cortex through a distinction between the
microscopic functions of individual neurons and the
mesoscopic functions of neural masses in local neighborhoods
(Freeman 2000), reserving the term
macroscopic to refer to the much larger ensembles that are
visualized with whole brain imaging techniques such as functional
magnetic resonance imaging, single photon emission computer tomography
(SPECT), and optical recording (Arieli et al.
1995
). The dendritic currents of cortical neurons give rise to
extracellular fields of potential, which, owing to the laminar architecture of cortex, sum in the volume conductor and constitute the
main source of the EEG. These are the same currents that determine the
firing probabilities of the neurons (Freeman
1975
; Mitzdorf 1985
, 1987
; Nunez
1981
). This causal relation is manifested in the statistical
dependence of pulse probability of cortical neurons conditional on the
amplitude of the EEG in the same local neighborhood (Eeckman and
Freeman 1990
; Freeman 1975
). The relation is
weak, because ~10,000 pulses are required from each individual neuron in the local mass to reveal the statistical regularity (Freeman 1992
). Estimates of the fraction of the variance of individual spike trains that is covariant within an ensemble of simultaneously recorded cortical spike trains are ~0.1% (Abeles
1991
; Aertsen et al. 1989
).
The mesoscopic EEG activity that results from distributed
synaptic interactions in each local neighborhood provides a measure of
the local mean field intensity of that neighborhood. What makes the EEG
important is that while it is the local average of electric potentials
established by the vector sum of currents across the cortical
impedance, that sum is the reflection of the mesoscopic state formed by
the cooperative actions that result from synaptic interactions, mainly
by excitatory synapses on excitatory neurons that are broadly but
sparsely distributed. A counterexample is a compound nerve action
potential formed by the sum of action potentials triggered by an
electrical stimulus of a peripheral nerve. It spreads with time and
distance, forming the "A" and "C" peaks, because the axons do
not interact to bind together the action potentials. The currents sum
in the volume conductor, but they do not reveal a local mean field
because the axons do not interact. Moreover in cortex, the EEG currents
are too weak to influence directly the firing rates of the individual
neurons (Freeman 1962, 1975
, 1992
; Freeman and
Baird 1989
). The cooperative interactions that are revealed by
the EEG typically are aperiodic oscillations in the spectral range from
1 to 100 Hz with intermittent peaks in well known ranges labeled theta,
alpha, beta, gamma, etc. The fast Fourier transform (FFT) (Press
et al. 1988
) of relatively long epochs of recording (>1 s)
yields spectra that show a near linear fall in log power with
increasing log frequency at a slope near 2, a
"1/f2" distribution that is
characteristic of Brownian motion and telegraph noise. Short segments
(<0.1 s) typically reveal at least one spectral peak at a frequency
that differs seemingly at random in successive segments. This variation
reflects the FM and aperiodicity that characterize the neocortical EEG.
Simultaneous recordings of EEG from 8 × 8 electrode arrays
(7 × 7 mm) placed subdurally on the visual, auditory, or
somatosensory cortices of trained rabbits have revealed a high degree
of spatial coherence in these spatial windows as previously reported
(Barrie et al. 1996; Freeman and Viana Di Prisco
1986
). After temporal band-pass filtering to extract the
activity in the gamma range (20-80 Hz in the waking rabbit), on
average, >90% of the variance was found to be contained in the first
component of principal components analysis (PCA) as compared with
~50% when EEGs were recorded simultaneously from multiple areas and
filtered in the same way. The shared gamma wave form in trained rabbits
revealed spatial distributions of AM in short time segments. These
segments were classifiable with respect to conditioned stimuli (CS)
that the rabbits had been trained to discriminate (reinforced CS+ from unreinforced CS
) in the sensory modality corresponding to the cortex
under observation (Barrie et al. 1996
). Therefore they were identified as "AM patterns."
The spatial coherence and the relatively narrow spectral peak of
the carrier wave in the gamma range in these same brief AM patterns
have made it possible using the FFT to measure the phase of the EEG at
64 sites in short time segments with respect to the phase of the
spatial ensemble average of the 64 traces at the shared peak frequency.
The square of the cosine of the phase for each trace gives a basis for
estimating its level of shared variance with respect to the whole. The
present report describes the spatial patterns of phase modulation (PM)
that have been found in the EEGs from the primary sensory cortices, an
interpretation of the mechanism by which the PM patterns form and a
suggestion on how the spatial patterns are transformed by output
pathways and received by the targets of transmission. Note should be
taken that every multichannel EEG segment of any length gives an
amplitude distribution and a phase distribution; the term "pattern"
is reserved here for AM patterns that can be classified with respect to
CS and for PM patterns that correspond to the direction and conduction velocities of axons in a specific tract in the brain (Bressler 1987) or that can be fitted with a cone over time durations
corresponding to the time durations of AM patterns as determined by
classification efficacy (Fig. 10 in Barrie et al. 1996
).
These AM and PM patterns provide crucial evidence that the spatially
coherent wave form of the epipial EEG is not due to a deep-lying
generator having a broad point spread function (PSF, a term used in
optics to denote the distribution in a plane of observation of light
from a point source), nor is it due to oscillation in potential at the
reference lead in monopolar recording but instead to an array of
equivalent point dipoles in a plane at some depth below the plane of
recording (Freeman 1975). Direct measurement of the PSF
is an arduous task that has only been done for the olfactory bulb, but
equivalent information can be obtained from the spatial spectrum of the
EEG, and this has been done for the bulb, prepyriform, visual,
auditory, and somatic cortices in rabbits and the superior temporal
gyrus in humans (Freeman et al. 1999
).
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METHODS |
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Chronic implantation of an epidural electrode array
Eighteen female New Zealand White (NZW) rabbits (2.5-4.5
kg, ~2 yr old) were implanted (aseptically, full surgical anesthesia by 4% isoflurane/O2 mixture) with a
prefabricated electrode array onto the left hemisphere. Arrays were
0.25-mm stainless steel wires in an 8 × 8 square matrix (0.79-mm
interelectrode distance to limit spatial aliasing, 7 × 7-mm
window). Reference and ground leads were placed in the skull adjacent
to the site of each placement over the visual, auditory, somatic, or
olfactory cortex. The arrays were placed in accordance with
descriptions of the locations of the primary sensory neocortices in
rabbits by means of mapping with evoked potentials (Galli et al.
1971; Gould 1986
; Hollander and Halbig
1980
; McMullen and Glaser 1982
). Subjects were
maintained 2-4 yr, then killed with 120 mg/kg pentobarbital, perfused
with 10% formalin, and autopsied. All procedures were conducted
according to protocols approved by the University of California at
Berkeley Animal Care and Use Committee with veterinary supervision by
the Office of Laboratory Animal Care.
Experimental paradigm
For recording, each subject was placed into a restraining
carrier in an electrically shielded, sound-resistant dark chamber. The
electrodes were connected to World Precision Instruments ISO 4/8
differential amplifiers. A pneumograph was attached to the chest, skin
clips for the unconditioned stimulus (US) were placed onto the
left cheek, and white noise (72 dB) was introduced into the chamber.
After 3 wk of familiarization, each subject was trained using classical
aversive conditioning to discriminate between two stimuli corresponding
to its implant site: one of two 10-ms full-field flashes varying only
in intensity (3.6 vs. 2.8 ft.-cd); or binaural (stereo earphones at
72-84 dB above the 72 dB white noise) 100-ms sinusoidal tones (500 vs.
5,000 Hz); or a 3-s air puff onto the cheek versus the hindquarters
(Barrie et al. 1996). One of the conditioned stimuli
(designated CS+) was reinforced with a weak electric shock to the cheek
adjusted initially to elicit an orienting response (Freeman and
Viana Di Prisco 1986
), and the other conditioned stimulus
(designated CS
) on randomly interspersed trials was not reinforced.
For each experiment, 40 6-s artifact-free EEG records were recorded,
filtered with single-pole, first-order RC filters (6 dB/octave falloff)
at 100 Hz (3 dB point) and 0.1 Hz, digitized at 500 Hz, phase corrected
for the time lag introduced by multiplexing (10 µs), and digitally
stored to magnetic disk in binary data files (~16 MB/file). Each
record was divided into a 3-s prestimulus period and a 3-s poststimulus period. The sequence of CS+/CS
trials (20 each) was randomized across
the set of 40 trials, and the time interval between each stimulus was
randomized between 30 and 120 s. Three replicates were made for
each experiment (Barrie et al. 1996
).
Data analysis
All data analysis was done off-line on a Macintosh PowerPC using original software specifically developed for this purpose. Every data set was reviewed and edited for bad recording channels and EEG artifacts (bad connections, animal movement, 60 Hz noise).
EEG records were segmented by either of two methods. The first
method involved using a sliding window to parse each record into
m overlapping, 128-ms EEG segments (where w
indexed each segment) separated by 2-ms intervals (Barrie et al.
1996). Each individual 128-ms EEG segment was then converted
into a root-mean-square (RMS) spatial AM distribution, after band-pass
filtering (at the optimal band-pass filter setting), resulting in a
series of 1 × 64 column vectors (
)
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(1) |
The second method was based on locating stable phase cones in the
spatial distributions of phase of neocortical EEGs. Conic phase
patterns had been found in EEGs of the olfactory bulb (Freeman 1990; Freeman and Baird 1987
). Here each
multichannel EEG record was parsed into m overlapping,
128-ms segments separated by 2-ms intervals. Each segment was
decomposed by FFT (Press et al. 1988
), into 50, 64-dimensional spatiotemporal distributions of phase (
) at fixed frequency intervals
(f) ranging from 2 to 100 Hz. Phase
differences in (
) at high spatial frequencies were
inferred to be due to noise (Fig. 6 in Freeman and Viana Di
Prisco 1986
), so they were attenuated with a spatial filter in
each 8 × 8 phase distribution by transforming the frame into the
frequency domain with the two-dimensional (2-D) FFT (embedding in a
matrix of 32 × 32 zeroes and omitting Hamming and Hanning),
passing the real and imaginary components through digital 2-D highcut
and lowcut filters (Barrie et al. 1996
; Freeman
and Baird 1987
) and using the inverse 2-D FFT to recover the
smoothed values. The two optimal cutoff frequencies in cycles/mm were
determined by repetition to give a tuning curve (Fig.
1) to identify the value giving the most
conic-like phase segments. A low cutoff spatial filter was used to
minimize the risk of what appeared to be an artifact from excessive
highcut filtering and to take advantage of the usefulness already
demonstrated of spatial band-pass filtering (Freeman and Baird
1987
). Each spatially filtered phase distribution was then fitted by nonlinear regression with a 2-D conic surface
(
) in planar coordinates
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(2) |
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(3) |
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Conic apices were separated into two categories: apices with <20%
residual (")
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(4) |
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(5) |
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Once all of the individual phase patterns resembling cones were located
and marked (by Eqs. 4 and 5), the
search for phase segments began. This strategy had two parts: the
location of the initial site of nucleation (this was always a cone with
R < 20%) and the location of subsequent temporal
points (always consecutive phase cones with R < 40%)
which were indicative of the temporal duration of a cortical event.
After locating the initial point, called the site of nucleation, two
distances were calculated
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(6) |
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(7) |
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(8) |
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To determine whether the demarcated phase segments recurred in specific
time intervals with respect to the CS, a histogram ()
of the phase segments was composed for the incidence in the 6-s trials
over the 40 trials in each session
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(9) |
Two other important values obtained by this analysis included the EEG
phase velocity (S) in M/s
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(10) |
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(11) |
After an experiment had been segmented (by either fixed-length windows
separated by fixed-length time intervals (Barrie et al.
1996) or by locating segments demarcated by stable phase cones) into a matrix of spatial AM pattern vectors, a method of
cross-classification was employed to determine whether or not the CS
patterns were significantly different from the CS+ patterns. Briefly,
for contiguous temporal epochs, all patterns within that epoch were
grouped as either type A (CS
) or B (CS+). Each type was further
subdivided into equal groups A1,
A2, B1,
B2; the first and second halves of group A/B
became subgroups A1/B1 and A2/B2, respectively. The patterns within the
first two subgroups (A1,
B1) were averaged to form centroids and the
Euclidean distance was measured in 64 space from the two centroids to
each of the patterns remaining in the second two subgroups
(A2, B2). If a patterns was
closer to its centroid than to the opposite centroid (i.e., a pattern from subgroup A2 was closer to centroid
A1 than it was to centroid B1), then it classified correctly. This
classification was repeated by calculating centroids from the second
two subgroups and classifying the patterns from the first subgroups. A
binomial probability pattern was used to determine whether or not a
certain level of classification was significant. This method yielded a
time series of probability values showing the times when the CS
and
CS+ spatial AM patterns could or could not be separated.
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RESULTS |
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Paleocortical versus. neocortical EEGs
Since the goal of this research was to assess a method, originally
used to characterize the EEG from the olfactory system (Freeman
and Baird 1987), for parsing the neocortical EEG, a comparison of the characteristics of the olfactory and neocortical EEG was compiled as a reference. The olfactory EEG was strongly correlated to
the respiratory rhythm of each rabbit (Fig.
2A, top). It was punctuated by a sequence of high-frequency (20-80 Hz), high-amplitude bursts typically referred to as gamma oscillations (Fig. 2B,
left). Such olfactory bursts were time-locked to the
inhalation phase of the respiratory cycle and were separated by
1/f-type (20-100 Hz), broad spectrum interbursts. Previous
research utilized the presence of these gamma bursts as an EEG marker
for stimulus-induced, endogenously generated events (Freeman
1978
, 1991
). A cursory comparison of the olfactory and visual
cortical EEG revealed significant differences. The visual EEG was not
correlated to the respiratory rhythm (Fig. 2A,
bottom), did not have a distribution of gamma oscillations,
and only contained an evoked potential marking the arrival time (3,000 ms) of the afferent volleys of action potentials into the cortex from
the periphery. An examination of the average frequency spectra of
visual cortical neuroactivity further demonstrated that (Fig.
2B, right) log power was uniformly distributed
across the temporal spectrum as a
1/fm function of log frequency.
This relation also held for the spatial spectra (Barrie et al.
1996
).
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Measurement of AM patterns in neocortices
One property that the olfactory and neocortical EEGs shared was the presence of a common carrier wave at any one instant. This meant that the frequency spectrum of the EEG using the FFT was spatially coherent across the cortex (over distances covered by the 7 × 7 mm arrays) at any one time. Coherence was revealed by a complex wave form shared on the multiple EEG traces having a common instantaneous frequency, that changed continually and unpredictably. Short time segments (64-256 ms) had prominent peak frequencies, which changed in successive segments, reflecting a continuous modulation in frequency and/or phase. Power spectral analysis of longer segments usually revealed a broad 1/fm power spectral density, in which log power decreased linearly with increasing log frequency at slope m ~ 2.
Spatial coherence observed in plots of multiple time series (Fig.
3A) was supported by the
incorporation of >90% of the variance in the first component of PCA
(Barrie et al. 1996). The frequency spectra of averages
of EEG segments had a 1/f distribution, but the spectra for
individual EEG segments typically centered around one frequency
relative to the overall spectra (Barrie et al. 1996
). That frequency component revealed spatial AM (AM) over the 64 channels
in a spatial distribution. Owing to the high covariance of the 64 traces the same AM distribution was found in the root mean square
amplitudes. Each AM distribution could be represented by a contour plot
(Fig. 3B) and also by a point in n space, where n was the number of electrodes and EEG traces. Similar AM
distributions formed clusters of points in n space, and
multiple AM distributions gave a set of clusters. Assignment of each
segment to a class was based on the Euclidean distance of its point to
the center of gravity of the nearest cluster.
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The AM patterns were modified by classically conditioning the animals
to respond to stimuli in the cortex of the corresponding modality; for
example, animals with implants over the primary visual cortex (V1)
(Hollander and Halbig 1980; Hughes and Vaney 1982
; Thompson et al. 1950
) received visual
stimuli. As the animals were trained to discriminate between one
reinforced conditioned stimulus (CS+) and another unreinforced (CS
),
new AM patterns appeared and prior patterns were subtly altered. After
completion of training, the AM patterns reappeared on delivery of the
corresponding stimuli to which the animals had learned to respond. The
locations and durations in time of the AM patterns in neocortical EEGs
were not visible as an AM over time. The most effective method for detecting them was to pass a moving window, 64-256 ms in duration, in
2-ms steps along the set of 20 CS+ trials and 20 CS
trials, and to
determine when the AM patterns differed significantly between the CS
and the CS+ trials. The 64- to 256-ms duration sliding window was only
used for data exploration. The 128-ms window length was most effective
and was used throughout this manuscript. The shorter window lengths did
not have enough spectral resolution for successful phase segment
tracking. The longer window lengths yielded unsatisfactory results,
possibly because the longer windows included EEGs that were unrelated
to the activity being tracked (i.e., noise was included in the FFT
decomposition of the EEG). The binomial probability of a difference by
chance offered a convenient measure for when the two sets of
distributions differed. As shown in Fig.
4, the AM patterns distinguishing CS+ and
CS
trials occurred shortly after the arrival of the CS and at
irregular intervals a few hundred milliseconds thereafter.
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This statistical assay, which served effectively to classify AM patterns from visual, auditory, somatic, or prepyriform cortical EEGs, did not serve to locate the AM patterns in time on single trials. The decline in significance of AM pattern separation with elapsed time from the stimulus presentation indicated that the locations in time of successive events varied across trials. The cortical distribution of phase modulation (PM) at the same frequency was given by the FFT. The PM distributions were measured in the same stepped windows to determine whether they could be localized on single trials.
Measurement of PM patterns in neocortices
The discovery of PM patterns was based on the finding of spatially
coherent oscillation in multichannel EEGs with spectra in
simultaneously recorded short segments having local maxima at the same
frequency somewhere in the gamma range. We restricted the selection of
the local spectral peak to the gamma range of 20-80 Hz because this
was the spectral range in which the AM patterns could be classified
with respect to CS (Fig. 9 in Barrie et al. 1996). The
spatial coherence was demonstrated by use of PCA to extract the
dominant component for pattern classification on every data set (Fig. 8 in Barrie et al. 1996
) to show that the results of
classification were the same using RMS, FFT, or PCA amplitudes. All of
the cortical areas showed that the dominant component captured between
90 and 98% of the total variance despite the lack of zero lag
synchrony (Fig. 3A), with no significant differences between areas. The 64 spectra of the EEG traces correspondingly had peaks at
the same frequency even as that frequency shifted unpredictably from
one segment to the next. Therefore when that frequency was identified
or when any other frequency was selected, the 64 phase values defined
at that frequency provided a spatial phase distribution constituting PM
about the phase of the spatial ensemble average. Comparisons could then
be made between PM distributions from differing cortices in search of
varieties of PM patterns. Stable patterns of phase had already been
identified in the prepyriform cortical EEGs, which were clearly
determined by the velocities and directions of propagated action
potentials in the lateral olfactory tract (Boudreau and Freeman
1963
; Bressler 1987
, 1995
). Comparable phase patterns were sought in EEG segments from the primary sensory neocortices, first by visual inspection of contour plots of phase in
search of regularities and next by attempts to find distinctive locations of peaks of phase lead on CS+ versus CS
trials on the assumption of topographic organization of activity driven by thalamic pacemakers. Peaks were occasionally found, but the locations varied unpredictably and without relation to CS, and when they were present in
the array, the sign varied seemingly at random.
Then a plane was fitted to the 8 × 8 phase values in search of
broad phase gradients such as those found in array recordings of scalp
EEGs (Walter 1953) and magnetoencephalograms (MEGs)
(Llinás and Ribary 1993
), on the premise that
large wave fronts might be approximated by a plane with no curvature
over the local 7 × 7 mm covered by the epidural arrays. The phase
gradients in some segments did appear to conform to a plane, but the
orientation and steepness varied randomly. The patterns were eventually
resolved by fitting a cone to the 64 values of phase, giving
consistently lower residuals than a plane fitted to the same phase data
and allowing a close fit to the phase extrema that happened to be located within the arrays. The FFT of the 64 traces (128-ms windows stepped at 2-ms intervals) yielded the spatial distributions of amplitude (Fig. 3B) and phase (Fig. 3C) at
frequencies in the gamma range. Conic functions were fit (in planar
coordinates) to each phase distribution (Fig. 3D) after
band-pass filtering (0.2-0.5 c/mm) of the phase values, by which the
standard error of measurement of phase was minimized (Fig. 6 in
Freeman and Viana Di Prisco 1986
). A phase cone apex was
flagged (Fig. 5A) when the
variance incorporated by the fitted conic surface exceeded 80% of the
total variance. That conic pattern was tracked in successive window
steps (defined by Eqs. 4 and 5) as illustrated in
Fig. 5B. PM patterns lasting 25 steps (50 ms) were labeled
as stable phase cones. The distributions that could be
fitted with a plane could be explained as having conic apices located
well outside the arrays. No significant differences were seen in PM
patterns from the three neocortical sensory areas. The relative
constancy of the sign of the phase extremum, and of its location, and
of the frequency at which the phase was defined, together constituted evidence for stationarity of the dynamic states over the time durations
shown in Fig. 7A. Stationarity is defined here to mean cortical dynamics holding in an attractor basin between two successive state transitions, irrespective of the oscillations within that basin,
including the trajectories of entry and exit.
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The locations and signs of the apices of successive stable phase cones varied unpredictably without relation to the CS. The duration of identified stable phase cones averaged 80 ms with a long tailed distribution. Neocortical phase velocities in M/s (Eq. 10) had skewed distributions from 0.5-4.0 M/s (Fig. 5C). The modal value of 0.6-2.1 M/s was consistent with measurements reported on the conduction velocity of large groups of cortical axons parallel to the pia. No attempt succeeded to define edges of the coherent domains, so an appropriate criterion of size was adopted: the half power diameter across the apex (± cos 45°), using Eq. 11. Diameter values were skewed from 5 to 20 mm or more with a mode near 0.5 cm (Fig. 5D).
The dependence of phase lag in radians on the distance between
recording electrodes in millimeters was first found without spatial
filtering by calculating the absolute phase differences between
channels at all frequencies in many windows and many trials. In prior
measurements of PM patterns in the olfactory bulb, the conic basis
function led to an invariant measure, which was the steepness of the
phase gradient expressed in M/s (Freeman and Baird
1987). The gradients were measured in radians/mm, and they varied with frequency, but when they were converted to M/s by use of
the frequency of the oscillation (Eq. 10), they converged to the
conduction velocity estimated for the axons of the bulbar axon
collaterals running parallel to the surface of the bulb. A similarly
invariant relationship was found in EEGs from the prepyriform cortex
(Boudreau and Freeman 1963
), so it was sought in the EEG
data from the neocortices. Unlike the invariant relation for the
olfactory bulb, the phase velocity of the neocortical EEG increased on
average with increasing frequency. The breakdown by frequency of the
distributions of phase velocities on conversion to M/s using Eq. 10 is shown in Fig. 6
(top) and Table 1 for three frequency bands. The modal phase velocities were consistent with the
conduction velocities reported for short intracortical axons over
distances
3 mm, as distinct from corticocortical, callosal, and
efferent axons that are much faster (Langdon and Sur
1990
; Lohmann and Roerig 1994
; Swadlow
1994
; Swadlow and Weyand 1981
). In the
neocortical EEGs, a new invariant was found in the modal half-power
diameter (Eq. 11) of the identifiable stable phase cones across neocortical locations and frequency bands (Table 1; Fig. 6,
bottom).
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Stable phase cones identified by these criteria were found at irregular
intervals throughout each trial (Fig.
7A). Pre- and poststimulus
time histograms of the stable phase cones showed higher than average
probabilities of cones after the time (3,000 ms) of stimulus arrival
(Fig. 7B). The FFT of histograms calculated from all
neocortices yielded peaks in the 4- to 80-Hz range (Fig. 7,
C-E), indicating a degree of rhythmicity in the theta range for the arrival and duration times of stable phase cones. The AM
patterns were derived from segments demarcated by stable phase cones in
visual, auditory and somatic EEGs as in Fig. 3B. These AM
patterns also revealed a significant level of correct classification of
CS+ and CS trials, and in comparison to the fixed steps, the AM
patterns accompanying stable phase cones gave higher levels of
significance later in trials (Fig. 8,
A-C), indicating that detection of phase cones improved the
identification of the locations of behaviorally significant EEG events
in the CS-CR interval.
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DISCUSSION |
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EEGs were measured by decomposing them with appropriate basis
functions, which in the present case were cosines giving the phase and
amplitude of frequencies in the gamma range. Whereas accurate
measurement of amplitude was relatively easy, measurement of the phase
was subject to large errors of measurement, owing to the brief duration
of segments, the small number of cycles in each segment, the strong
tendency to FM and AM about center frequencies in both time and space,
and the mix of multiple frequencies in spectra of most segments,
especially from neocortices with their tendency to
"1/f2" spectra. The standard
error of measurement was estimated by measuring cosines at unit
amplitude and zero phase embedded in varying levels of random numbers
to simulate white noise (Freeman and Viana Di Prisco
1986). The results gave standard deviations averaging ±60°
for raw EEGs, which were reduced to ±6° after application of
appropriately designed spatial and temporal filters.
The requirement for validating the measurements of spatiotemporal AM
patterns was met by relating them to behavior. This criterion led step
by step (Barrie et al. 1996; Freeman 1975
,
1992
; Freeman and Schneider 1982
) to
decomposition of EEG time series with cosine basis functions,
measurement of AM and PM patterns having the same instantaneous
frequency in the arrays, and classification of the AM distributions
with respect to sets of CS+ and CS
trials after discriminative
conditioning. The PM patterns consistently failed to classify with
respect to locations and signs of apices, but the AM patterns extracted
from the segments demarcated as stable phase cones did classify well
above chance levels at intermittent times between the CS and CR onsets.
The main problem encountered in correlation of neocortical EEGs with
behavior (as compared with olfactory EEGs) was temporal segmentation.
The identification of spatial AM patterns in olfactory EEGs was
facilitated by the prominent respiratory wave in the theta range and
the associated temporal AM of the gamma activity giving "bursts" of
oscillation. In contrast, visual inspection of the neocortical EEGs
gave no indication of where the stationary segments might start or end,
either by any near-periodic wave in the theta and alpha ranges or by
any temporal AM in the gamma range. Therefore a fixed-duration segment
was stepped along the multiple EEG traces on every trial, and
classifications of AM or PM patterns from CS+ and CS trials were made
between segments on different trials all from the same time step with
respect to stimulus onset. The results showed that the selection of AM
patterns that was guided by determination of the temporal locations of the stable phase cones improved the reliability of segment
classification in the CS-CR intervals. The consistency and reliability
of the results from olfactory, visual, auditory, and somatosensory EEGs led to the conclusion that the radial phase gradients are not an
artifact of the measurement algorithms and that they manifest an
important biophysical property of the neural masses that generate the
gamma oscillations of the EEG.
Several inferences follow from this conclusion. First, the common wave
form on the 64 electrodes cannot be ascribed to activity at the site of
the reference electrode on monopolar recording or to volume conduction
from a deep dipole generator far under the array, not only because of
the AM (which might conceivably be explained by local variations on
cortical specific resistance) but also because of the phase modulation
(the reactive component of the cortical impedance vector is much too
low to give the observed range of differences) (Freeman
1975). Nor is it due to decorrelation with distance over a
distribution of noise generators smeared by volume conduction
(Elul 1972
) because that would not give radial phase
gradients. Nor does it manifest entrainment of coupled oscillators because the phase gradients persist through the segments, probably owing to the sparseness (Braitenberg and Schüz
1991
) of the local connection densities, which fails to support
convergence of activity into synchrony.
Second, the findings can explain the zero time lag correlation reported
between the pulse trains of neurons separated by distances less than
the array size (Brosch et al. 1995; Engel et al.
1992
; Gray 1994
; König and Schillen
1991
; Roelfsema et al. 1997
; Singer 1993
), when that relation has been derived between pulse trains of pairs of neurons by time ensemble averaging over multiple trials. When the technique is used because there are too few pulses on any one
trial, the unpredictable variation in location and sign of the conic
apices randomizes the phase relations on repeated single trials, and
the resulting average can only approach zero.
Third, the random variation of the sign of the apices cannot be explained by an intracortical or thalamic pacemaker because those drivers could only give apices with phase lead whether acting by excitation or inhibition. It is compatible with a symmetry breaking state transition such as a saddle node bifurcation. The property of mesoscopic states that makes them interesting is the capacity they give to an ensemble of neurons for rapid changes in the global spatiotemporal distributions of organization and function of ensembles. Some well-known examples are the transitions between waking and sleeping states, between vocalizing and swallowing, and between walking and running by which the neurons distributed in the brain and spinal cord shift their firing from one coordinated pattern to the next globally in a few milliseconds.
We postulate that in primary sensory cortices a state transition is
enabled by a volley of action potentials on afferent axons, which is
initiated by a sensory event and gated by thalamic relay nuclei. Gated
input transiently increases the level of activity in the cortex. Owing
to the asymmetry of the sigmoid curve governing the output of the
cortical neurons receiving the volley (Freeman 1992),
the forward gain (sensitivity of the trigger zones of the excited
neurons) is increased. Owing to the predominance of intracortical excitatory synapses to other excitatory neurons, the triggered output
goes mainly to excite those other neurons and sensitize them as well.
In the ensemble, some of the activity returns to the initially excited
neurons, still further increasing their activity and sensitivity. This
constitutes regenerative feedback at the mesoscopic level that is
equivalent to the nonlinear local response of axons approaching their
thresholds (Hodgkin and Rushton 1964
). Just as
microscopic axons are bistable (either above or below threshold)
(Izus et al. 1998
), we infer that mesoscopic ensembles
of cortical neurons are bistable or polystable. Above some threshold an
ensemble can become unstable and jump into a high-intensity oscillatory
state. Owing to the amplitude-dependent gain of the population
(Freeman 1992
), the pattern generated in the new stable
state that has been triggered by the input is dominated by the
intracortical synaptic connections, which are subject to modification
by learning during previous experience (Emery and Freeman
1969
), particularly at the excitatory synapses on the dendritic
spines of excitatory neurons as described by Rall (1995)
and others. Hence the AM patterns could reflect the past and present contexts imposed by behavioral constraints, not the specific forms imposed by the sensory input volley. This could explain the fact that
the AM patterns are not invariant with respect to the stimuli but are
modified by new learning, such as that which occurs with changes in
reinforcement contingencies (Freeman 1991
;
Freeman and Grajski 1987
).
Fourth, evidence from physical distributed systems shows that state
transitions do not start simultaneously throughout the systems but
begin at a site of nucleation and spread radially, as in the
formation of a snowflake around a dust particle. The velocity of spread
indicated by the radial phase gradient is compatible with the
conduction velocities reported for axons running parallel to the pia
(Langdon and Sur 1990; Lohmann and Roerig
1994
; Swadlow 1994
; Swadlow and Weyand
1981
). Those axons extending more than a millimeter
(Hellwig 1993
; Read et al. 1997
) could
play an important role, because there is insufficient time to allow for
serial multisynaptic transmission over the predominantly short axons
(Sholl 1956
) to the distances covered by observed phase
cones. We infer that the phase cones manifest a group property, the
velocity of a state transition, not a transfer of information
(Freeman 1990
). Physicists call the spread of change at
different velocities in media anomalous dispersion. An
example is hitting a metal rod on one end with a hammer; the sound wave
gets to the other end of the rod before the impulse does. Anomalous
dispersion in the cerebral cortex may account for the rapidity with
which state transitions can spread over large distances faster than can
be achieved by serial synaptic transmission, relying instead on small
percentages of long axons in corticocortical projections and on the
extreme sensitivity of the cortical ensemble as it approaches the
border of its existing basin of attraction, analogous to the "local
response" that sensitizes axons brought near threshold
(Hodgkin and Rushton 1946
). We conclude that the radial
phase patterns in the EEG provide strong evidence that AM patterns form
by self-organizing cortical state transitions that direct the cortices
through a landscape of attractors, each yielding a reproducible AM
pattern (Freeman 1992
; Freeman et al. 1995
).
Fifth, the delays in axonal propagation manifested in the phase cones
may serve to delimit the spatial boundaries of neocortical AM patterns
during both construction and read-out. The neocortical neuropil forms a
continuous sheet, as shown by the phenomenon of spreading depression of
Leaõ (Bures et al. 1974), which stops only at the
borders of the neocortex with the archicortex and the callosum. The
submillimeter microscopic architecture of cortex is spatially
coarse-grained by its input projections into cortical columns and
barrels, which are smaller by an order of magnitude than the mesoscopic
AM patterns. The EEGs of cortical areas that are separated by
macroscopic distances lack high spatial coherence, giving evidence that
the mesoscopic AM patterns must have soft boundaries. The obligatory
axonal propagation delays may provide the neocortical dynamic boundary
conditions, which are required to give the different areas a degree of
autonomy, while not freezing them into anatomically fixed arrangements.
Within each area the spatially coherent EEGs manifest cooperative
interactions among millions of neurons but cooperativity must weaken
with phase dispersion. The radial phase gradients can serve to
attenuate mesoscopic synaptic interactions with distance. With no hard
edges, the half-power radius (± cos 45°) can serve to define the
functional boundaries and sizes for local cooperative domains, at the
interface between microscopic neural activity and cortical mesoscopic
states. The cosine values then provide the desired measure of the
degree of relatedness of the gamma activity throughout a temporal
segment of the EEG in a mesoscopic cortical area.
Sixth, the classification of AM patterns with respect to CS reveals
that the classificatory information is homogeneously distributed in
space; no recording channel in an array is any more or less important
than any other (Barrie et al. 1996). This property shows that the information relating to the topographic mapping of the sensory
input has been spatially disseminated by the dynamic operation of
constructing AM patterns. The independence of the mesoscopic pattern
from the details of the input-dependent cortical architecture may be
critical for the integration of multisensory percepts in which the
local space-time gradients peculiar to the retinal, cochlear, and
cutaneous mappings are no longer relevant. This operation also pertains
to the read-out of AM patterns. The olfactory bulb transmits through
the divergent-convergent lateral olfactory tract that imposes a spatial
integral transformation. Each bulbar neuron disseminates its pulses
broadly, and each target neuron integrates over input from a wide area
of the bulb. This operation sums only the transmitted activity having
minimal frequency and phase dispersion, and it attenuates the activity
that is not spatially coherent. In effect, it defines the
self-organized bulbar mesoscopic activity as signal, and the
sense-dependent microscopic activity as noise. Whether the outputs of
neocortical primary sensory cortices likewise undergo spatial integral
transformation is unknown, but the fact that the neurons in the
secondary sensory areas have very large receptor fields gives evidence
that a comparable mesoscopic operation may be performed on the outputs
of neocortices. In effect, every local population sends the same signal
to all targets of a transmitting cortex, in the manner that each
fragment of a broken hologram has the entire picture, and each
receiving target selects what is appropriate to it. An alternative
hypothesis, holding that discrete microscopic networks of selected
neurons embedded in the neuropil have been synthesized by large local
synaptic modifications during learning, suffers from combinatorial
explosions in matching input and output topographic maps, and it does
not readily offer a neural code that is common to all perceptual ports beyond the early stages of sensory information processing but prior to
multisensory percept formation.
The prepyriform cortex provides an important exception because its
phase gradients are not conic but conform instead to the propagation
delays imposed by the directions and velocities of the axonal branches
in the lateral olfactory tract or occasionally to axonal projections in
the opposite direction (Freeman 1973, 1999
). This
finding indicates that this cortex lacks the autonomous dynamics
required for the self-organization of AM patterns and is directly
driven by the bulb (Boudreau and Freeman 1963
;
Bressler 1987
). This difference offers a distinction
between two types of cortex, one that constructs AM patterns, the other
that integrates them selectively. These two types may work pairwise in
reciprocity. They might be sought among neocortical areas by
measurement of their PM patterns, so that their respective roles might
be better understood.
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APPENDIX |
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Variable list
)
1 × 64 RMS amplitude column vector (
)
R)
RMS amplitude of the EEG for each channel
e)
EEG recording channel index
)
spatial phase distribution
w)
window index
m)
number of overlapping, 128 ms EEG windows separated by 2 ms intervals
f)
frequency index
)
1 × 64 cone vector representing the conic pattern regressed onto the
phase pattern
)
conic function regressed onto the phase distribution
")
apices of conic regressions with <20% residual
')
apices of conic regressions with <40% residual
n")
numbers of apices within each segment from conic regressions with
R < 20%
n')
numbers of apices within each segment from conic regressions with
R < 40%
i)
index
j)
index
A1)
distance indicating how far (in mm) an apex (located 2q ms
in time from the initial phase cone) was from the initial site
)
distance indicating how far any two consecutive phase cones were from
each other
q)
number of consecutive time points in a stable phase segment
)
1 × q vector demarcating each stable phase segment
)
time in milliseconds beginning at window w and extending to
window w + q
s)
phase segment index
)
record index
)
number of segments per record (r)
)
histogram of phase segments
S)
EEG phase velocity in M sec1
b)
slope of the conic surface regressed onto the phase distribution
D)
modal diameter in cmdefined as twice the distance from the apex to
the half-power point (the square of cosine 45° = 0.5) of the EEG
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ACKNOWLEDGMENTS |
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The authors thank M. D. Lenhart for invaluable assistance with the surgical procedures and with the experimental paradigm.
This research was supported by National Institute of Mental Health Grant MH-06686 and Office of Naval Research Grant N00014-90-J-4054.
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FOOTNOTES |
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Address for reprint requests: W. J. Freeman, 141 Life Sciences Addition, University of California at Berkeley, Berkeley, CA 94720-3200 (E-mail: wfreeman{at}socrates.berkeley.edu).
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 4 October 1999; accepted in final form 23 May 2000.
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REFERENCES |
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