Background EEG analytic amplitude 1 Walter J Freeman
Origin, structure, and role of background EEG activity.
Part 1. Analytic amplitude;
Walter J Freeman
Clinical Neurophysiology (2004) 115: 2077-2088.
Department of Molecular & Cell Biology, LSA 142
University of California
Berkeley CA 94720-3200 USA
Tel. 1-510-642-4220 Fax 1-510-643-6791
http://sulcus.berkeley.edu
Running title: Background EEG analytic amplitude
Key words: analytic amplitude, gamma EEG oscillations, Hilbert transform, information, order
parameter, stability, synchrony
Acknowledgments
This study was supported by grant MH 06686 from the National Institute of Mental Health, grant NCC 2-1244 from the National Aeronautics and Space Administration, and grant EIA-0130352 from the National Science Foundation to Robert Kozma. Programming was by Brian C. Burke. Essential contributions to surgical preparation and training of animals, data acquisition, and data analysis by John Barrie, Gyöngyi Gaál, and Linda Rogers are gratefully acknowledged, as well as discussions of theory with Harald Atmanspacher, Giuseppe Vitiello, and Ichiro Tsuda.
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Background EEG analytic amplitude 2 Walter J Freeman
Abstract
Objective: To explain the neural mechanisms of spontaneous EEG by measuring the
spatiotemporal patterns of synchrony among beta-gamma oscillations during perception.
Methods: EEGs were measured from 8x8 (5.6x5.6 mm) arrays fixed on the surfaces of primary sensory areas in rabbits that were trained to discriminate visual, auditory or tactile conditioned stimuli (CSs) eliciting conditioned responses (CRs). EEG preprocessing was by (i) band pass filtering to extract the beta-gamma range (deleting theta-alpha); (ii) low-pass spatial filtering (not high-pass Laplacians used for localization), (iii) spatial averaging (not time averaging used for evoked potentials), and (iv) close spacing of 64 electrodes for simultaneous recording in each area (not sampling single signals from several areas); (v) novel algorithms were devised to measure synchrony and spatial pattern stability by calculating variances among patterns in 64-space derived from the 8x8 arrays (not by fitting equivalent dipoles). These methodological differences are crucial for the proposed new perspective on EEG.
Results: Spatial patterns of beta-gamma EEG emerged following sudden jumps in cortical activity called ―state transitions‖. Each transition began with an abrupt phase re-setting to a new
value on every channel, followed sequentially by re-synchronization, spatial pattern stabilization, and a dramatic increase in pattern amplitude. State transitions recurred at varying intervals in the theta range. A novel parameter was devised to estimate the perceptual information in the beta-gamma EEG, which disclosed 2 to 4 patterns with high information content in the CS-CR interval on each trial; each began with a state transition and lasted ~.1 s.
Conclusions: The function of each primary sensory neocortex was discontinuous; discrete spatial patterns occurred in frames like those in cinema. The frames before and after the CS-CR interval had low content.
Significance: Derivation and interpretation of unit data in studies of perception might benefit from using multichannel EEG recordings to define distinctive epochs that are demarcated by state transitions of neocortical dynamics in the CS-CR intervals, particularly in consideration of the possibility that EEG may reveal recurring episodes of exchange and sharing of perceptual information among multiple sensory cortices. Simultaneously recorded, multichannel beta-gamma EEG might assist in the interpretation of images derived by fMRI, since high beta-gamma EEG amplitudes imply high rates of energy utilization. The spatial pattern intermittency provides a tag to distinguish gamma bursts from contaminating EMG activity in scalp recording in order to establish beta-gamma recording as a standard clinical tool. Finally, EEG cannot fail to have a major impact on brain theory.
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Background EEG analytic amplitude 3 Walter J Freeman
Cover Figure Legend: The EEG shows that neocortex processes information in frames like a cinema. The perceptual content is found in the phase plateaus from rabbit EEG; similar content is predicted to be found in the plateaus of human scalp EEG. The phase jumps show the shutter. The resemblance across a 33-fold difference in width of the zones of coordinated activity reveals the self-similarity of the global dynamics that may form Gestalts (multisensory percepts). Upper frame: coordinated analytic phase differences (CAPD) calculated from human EEG in the beta band (12-30 Hz) with 3 mm spacing of 64 electrodes in a linear 189 mm array digitized at 1 ms intervals.
Lower frame: CAPD calculated from rabbit EEG in the gamma band (20-50 Hz) with 8x8 0.79 mm spacing 5.6x5.6 mm array digitized at 2 ms intervals.
The human EEG data from a normal subject awake with eyes closed were provided by Mark D. Holmes in the EEG & Clinical Neurophysiology Laboratory, Harborview Medical Center, University of Washington, Seattle WA and Sampsa Vanhatalo, Department of Clinical Neurophysiology, University of Helsinki, Finland, using 64 of the 256-channel recording System 200 provided by Don Tucker, Electrical Geodesics Incorporated, Riverfront Research Park, Eugene OR.
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Background EEG analytic amplitude 4 Walter J Freeman
1. Introduction
Synchrony of firing of widely distributed neurons in large numbers is necessary for emergence of spatial structure in cortical activity by reorganization of unpatterned background activity. The dendritic currents regulate the firing. The same currents are largely responsible for local field potentials and EEG. The firing is grouped in time by oscillations in dendritic current in the beta (12-30 Hz) and gamma (30-80 Hz) ranges that arise from negative feedback among excitatory and inhibitory neurons (Freeman, 1975, 2004; Freeman and Burke, 2003; Appendix 2.4 in Part 2) with facilitation by the molecular dynamics of synaptic receptors (Traub et al., 1996; Whittington et al., 2000; Kopell et al., 2000). The synchrony between pairs of EEG records can be measured by any of a variety of methods (Lachaux et al., 1999; Le Van Quyen et al., 2001; Quiroga et al., 2002), including the phase difference of oscillations in which they share the same frequency. The synchrony among multiple EEG records can be estimated by measuring the phase of each signal with respect to the phase of the spatial ensemble average at a shared frequency and calculating the standard deviation (SD) of the spatial distribution of the phase (Freeman, Burke X
and Holmes, 2003; Part 2).
While conceptually simple, the approach-using phase has formidable obstacles. EEG signals are highly irregular, in a word, chaotic. Oscillations at fixed frequencies such as 10 Hz and 40 Hz are not the rule. Temporal spectra in log-log coordinates often reveal a linear decrease in log power with log frequency (called a ―1/f‖ power-law scaling), though that is not the rule either, owing to
peaks in temporal power spectral density (PSD). The Fourier transform evaluates phase at each T
frequency but only on average over epochs long enough to measure the frequency. Rapid changes in phase can be seen with the Hilbert transform, but they make sense only with prior band pass filtering. The more narrow the pass band is, the more likely filtering is to cause ringing from spikes that introduce spurious oscillations near the center of the pass band.
Two features of the EEG from high-density 2-D epicortical arrays offered an alternative approach. First, the level of covariance among the EEG signals from arrays up to 1 cm in width was high; the fraction of the variance in the first component of principal components analysis (PCA) usually exceeded 95%. Yet the amplitude of that component, however chaotic the wave form might be, varied with electrode location in the array so as to constitute a spatial pattern of amplitude modulation (AM) of the shared wave form. By this measure, EEG signals from arrays showed a high degree of synchrony. In contrast, when EEG signals were compared over distances >> 1 cm, the first component of PCA averaged ~50% of the total variance (Freeman, Gaál and Jornten, 2003). Second, the spatial patterns of phase modulation (PM) across the array showed epochs of low SD that were bracketed by brief episodes of high variance (Freeman and X
Rogers, 2002; 2003), showing that the high synchrony was episodic. The AM patterns are the focus of Part 1; PM patterns are described in Part 2, and amplitude pattern classification is taken up in Part 3.
These two features are relevant to a common anatomical property of axons carrying cortical output. Their axons form a divergent-convergent projection in contrast to better-known topographic mapping of input. Each transmitting neuron broadcasts its activity by axonal branching. Each receiving neuron gets terminals from widely distributed transmitting neurons. Such a projection integrates cortical output spatially as well as temporally. This operation
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Background EEG analytic amplitude 5 Walter J Freeman
approximates the summation of multiple EEG signals in a time window corresponding to an epoch of high phase synchrony. In the present study the EEG signals were temporally filtered in a pass band corresponding to the beta or gamma range. The temporal standard deviation (SD) T
across time points was calculated for the average waveform of the filtered EEG in the epoch lasting T digitizing steps. The SD was divided by the average of the standard deviation of all the T
waveforms (SD) in the epoch. If the EEG signals were completely synchronous, the ratio would T
be unity. To the extent that the EEG signals deviated from the average in amplitude, phase or frequency, they tended to cancel, so that the ratio decreased markedly as the signals approached complete independence. This method for estimating synchrony avoided measuring phase explicitly, so it was well suited for multiple, aperiodic, ―chaotic‖ signals. The aims of this study were to compare the results from this ratio with existing measures of synchrony, and to investigate the value of the new index of spatial pattern stability.
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2. Methods
2.1. Experimental animals and EEG recording
The experimental procedures by which the electrode arrays were surgically implanted, the rabbits were trained, and the EEG signals were recorded and stored have been documented elsewhere (Barrie, Freeman and Lenhart, 1996). The 8x8 electrode spacing of 0.79 mm gave a spatial aperture 5.6x5.6 mm. Signals with analog pass band .1-100 Hz were amplified 10K, digitized at 500 Hz with 12-bit ADC, and stored in a 64x3000 matrix for each 6-s trial. Each subject was trained in an aversive classical conditioning paradigm with 20 trials using a reinforced conditioned stimulus (CS+) and 20 trials of an unreinforced conditioned stimulus (CS-) in each session, all with correct conditioned responses. The data set sufficing for the present statistical analysis consisted of 5 trials from each of 6 rabbits, 2 each with an 8x8 array on the visual, auditory or somatic cortices. The analysis was done with MATLAB software, which has excellent graphics capabilities but is slow in computation; hence analysis was restricted to an adequate subset of the available data.
2.2. Derivation of analytic amplitude and phase
The 64 EEG signals in each trial were preprocessed first by de-meaning to remove channel bias (Fig. A1.01). A spatial low pass filter was applied (Appendix 1.2) to remove channel noise (defined in Fig. A1.02, B). A temporal band pass filter was applied (Appendix 1.1) to get the beta-gamma activity (here 20-80 Hz, Fig. A1.02, A). The entire data set for each session and subject was normalized to unit standard deviation (SD). The EEGs, v(t), j = 1,64, served to j
estimate the output of the excitatory neurons in the forward limb of the negative feedback loop. The spatial ensemble average of the signal amplitude. v(t), of the 64 signals is shown by the blue
curve in Fig. A1.03, A (for a listing of symbols, see Table 1.1). In Hilbert terminology this is the ―real part‖ for plotting along the real axis in the ―polar plot‖ in Fig. A1.03, B. The Hilbert
transform (Appendix 1.3) of v(t) gave an estimate of the output, v’(t), of the inhibitory neurons jj
in the feedback limb of the negative feedback loop, which maintained an oscillation at the same frequency but with approximately a quarter cycle lag behind the output of the excitatory neurons (Freeman, 1975, 2000). The spatial ensemble average, v’(t), is shown by the red curve in Fig.
A1.03, A, which also approximated the negative rate of change of v(t) in blue. It is the ―imaginary
part‖ for plotting along the imaginary axis in Fig. A1.03, B. At each digitizing step the real value, v(t) and imaginary value, v’(t) determined a point in the polar plot in Fig. A1.03, B, which was j
the tip of a vector, V(t), extending to that point from the origin of the polar plot where v(t) = 0 jj
and v’(t) = 0. j
V(t) = v(t) + i v’(t). (1) jjj
Successive pairs of values specified the trajectory of the tip of the average vector, V(t), as it
rotated counterclockwise about the origin of the plane with time. The trajectory is shown for the ensemble averages, v(t) and v’(t). The analytic amplitude for each channel, A(t), was the length j
of the vector, which was given by the square root of the sums of squares of the real and the imaginary parts for each channel. The average was denoted A(t) (Table 1.1).
0.5 22 A(t) = [v (t) + v’(t)] (2) j jj
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Background EEG analytic amplitude 7 Walter J Freeman
The waveform of the average analytic amplitude, A(t), over the 64 channels is shown in Fig.
A1.03, C. The analytic phase, P(t), for the j-th channel was given by the arctangent of the ratio j
of the imaginary part to the real part (Appendix 1.3).
P(t) = atan [v’(t) /v(t)] (3) jj j
An example of the spatial average analytic phase, P(t), is shown by the blue sawtooth curve in
frame D. Whereas the tip of the vector in Fig. A1.03, B followed a trajectory with a continuously varying length, A(t), the analytic phase was discontinuous, because each time v(t) went to zero, the tangent went to infinity, and the analytic phase jumped from +?，！ to –?/2. These jumps are
known as ―branch points‖. They can be removed for each channel by adding ? radians to P(t) or j
P(t) each time the vector rotates across the imaginary axis (frame B) giving a ramp (the red curve). This procedure is called ―unwrapping‖. The unwrapped phase, p(t) on each channel, j, or j
the average p(t) was marked by repeated jumps known as ―phase slip‖ above or below the mean difference. The successive differences, (p(t), of the unwrapped phase show the phase slip in the j
analytic phase differences without the branch points (see also Fig. A1.11). Dividing (p(t) by the j
digitizing step, (t, approximated ?, the rate of change of p(t); dividing ? by 2? gave the jjj
analytic frequency in Hz.
Fig A1.04. The raster plot shows the successive differences of the unwrapped analytic phase, (p(t), j
changing with time (left abscissa) and channel (right abscissa). The 8 columns of 8 rows are aligned
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to show the near-coincidence of the sudden jumps and dips given by fast-forward and backward
rotation of the vector in Fig. 1, B). When the jumps or dips are aligned with the right abscissa, they
occur with nearly zero lags among them. When they form lines that deviate from the direction of the
right abscissa (as most clearly at about -250 ms) there is a phase gradient across the array. These
gradients are detailed in a companion report (Part 2).
Fig. A1.04 (also in the cover illustration, lower frame) shows a raster plot of the 64 curves, (p(t), j
over the same time segment as in Fig. A1.03. The phase slip that was revealed by upward or downward deviations from the mean differences, (p(t), tended to occur synchronously across j
the entire 8x8 array, here plotted in a compressed display of (p(t) in the order of channel j
number. The coordination of the jumps was measured by the spatial standard deviation, SD(t), X
of (p(t) (Fig. 1.01, A, black curve). Superimposing the spatial ensemble average of A(t) (from j
Fig. A1.03, C) showed that maximal SD tended to occur when analytic amplitude fell to low X
values (Fig. 1.01,A, light [blue] curve).
Fig 1.01. The spatial standard deviation of phase differences across the array, SD(t), (black curve) was X
adopted as a standard for comparing other methods of indexing the degree of synchrony. A. SD(t) X
and analytic amplitude, A(t). B. SD(t) and q(t). X
Table 1. Symbol List for Part 1 and Part 2
General
N Number of electrodes, channels and signals in high-density EEG recording; when
subscripted, N, number of qualifying phase cones
(t digitizing interval in ms
w window length in T ms, also expressed as window order T in number of bins at (t
t time of midpoint of moving window in ms
T elapsed time within the moving window, specified as window order in number of bins AM amplitude modulation in space of a beta or gamma carrier wave
PM phase modulation in space of a beta or gamma carrier wave
SD(t) standard deviation of a variable in j-th channel over a time window, w, centered at t j,T
SD(t) average of SD(t) over 64 channels over a time window, w, centered at t Tj,T
SD(t) standard deviation of a variable over an array of channels at one point in time, t X
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Background EEG analytic amplitude 9 Walter J Freeman
Fourier method
V(t) amplitude of the Fourier components within a moving window i
f frequency in Hz of the EEG from the Fourier components by nonlinear regression of i
the i-th component on the j-th channel in a window ~(t) phase in radians of the EEG from the Fourier components by nonlinear regression of i,j
the i-th component on the j-th channel in a window ~(t) shuffled phase in radians in a control by randomization k
PSD temporal power spectral density T
PSDspatial power spectral density X
f cut-off frequency in Hz of a temporal filter o
f cut-off frequency in c/mm of a spatial filter, usually Gaussian x
Hilbert method
v(t) EEG from the j-th channel after spatial and temporal band pass filtering, also the real j
part of the Hilbert transform
v(t) spatial ensemble average of v(t) over N channels j
v’(t) the imaginary part of the EEG from the j-th channel after spatial and temporal band j
pass filtering and the Hilbert transform v’(t) spatial ensemble average of v’(t) over N channels j
V(t) vector given by the real and imaginary parts j
A(t) analytic amplitude from the Hilbert transform at the j-th channel j
A(t) mean analytic amplitude over N channels P(t) analytic phase in radians for the j-th channel from the Hilbert transform by the j
MATLAB atan2 function without unwrapping P(t) mean analytic phase in radians over P(t) from N channels j
p(t) analytic phase in radians from the Hilbert transform by the MATLAB atan function i
after unwrapping
p(t) mean unwrapped phase over p(t) from N channels i
(p(t) successive analytic phase differences were calculated from p(t) by the atan function jj
after unwrapping
?(t) time-varying instantaneous frequency in rad/s from (p(t) divided by (t j
CAPD coordinated analytic phase differences (Freeman, Burke and Holmes, 2003)
Measures derived from analytic amplitude in Part 1. 2A(t) analytic amplitude squared waveform on the j-th channel at intervals of (t j22A(t) spatial ensemble average of A(t) over the 64 channels in the window, w. of length T j2SD(t) standard deviation of the j-th signal A(t) in the window, w j,Tj,T
SD mean of the N values of SD(t) in window, w Tj,T2SD standard deviation of the mean wave form A(t) in window, w T
R(t) ratio of the SD of the mean signal to the mean SD of the N signals, a measure of eTT
synchrony among a collection of aperiodic ―chaotic‖ wave forms, giving an indirect
estimate of the order parameter, k 2A(t) a normalized spatial pattern of amplitude that is formed by N channels of EEG
designating a point in N-space that is evaluated by an Nx1 vector
D(t) change in normalized spatial pattern given by Euclidean distance between successive e22points separated by (t, given by the vector length between A (t) and A (t-1)
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Background EEG analytic amplitude 10 Walter J Freeman
k feedback gain coefficient between the i-th and j-th populations; an estimator of the i,j
order parameter that is the intensity of synaptic interaction in populations of cortical
neurons and that is symbolized in models by the nonlinear gain
(k an estimator of a change in order parameter and k with (t that is approximated by D i,je2E(t) an estimator of free energy that is approximated by A(t) from the square of the EEG
current, i(t), estimated from the potential difference, v(t), established by its passage
across fixed tissue specific resistance, r: i = v / r 2(E change in free energy in (t at t, approximated by A(t) 2 2H pragmatic information provided by a pattern, A, whereH = - (E / (k ~ A/ D eee
Parameters of Cones fitted to phase by Fourier [~(t)] or Hilbert [P(t)] method in Part 2 i,jj
x, y the coordinates of the phase cone from the center of the array at x, y in mm oo
；(t) height of the fitted cone above the plane of fit (the pial surface of the cortex) in o
radians
；(t) value of phase at the j-th electrode from the fitted cone in radians j
?(t) gradient of phase cone in radians/mm at time, t in ms ? average phase gradient over the duration of a stable phase cone in radians/mm k
W spatial wavelength in mm/radian x
faverage frequency over the N-th qualifying phase cone in Hz from Fourier method N
? average analytic frequency over N-th qualifying phase cone in rad/s from Hilbert N
method
W temporal wavelength in ms/radian t
； phase velocity in m/s
D half-power diameter of phase cone in mm x
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