Beta Null Spikes, Phase Cones, and Vortices
Walter J Freeman
http://sulcus.berkeley.edu/Null_Spikes_Apices/
ElectroCorticoGraphic fields of potential at the surface of the visual cortex, recorded
with a 6x6 mm electrode array with spacing of 0.79 mm from a rabbit trained to
discriminate visual conditioned stimuli
00. Summary description of DVD
cinematographic representations of ECoG
(00SummaryDescription,DVD,ECoG,BetaPhaseTransition.doc)
There are three pairs of DVDs, each giving
a perspective view and a contour plot of the three main state variables: the
amplitude of the filtered ECoG in that pass band, the analytic amplitude, and
the analytic phase.
01. Summary of Time Functions: to be Viewed Concomitantly with
Spatial Displays
(01F152X12test20.602to21.8sec20to25HzPhaseTransition.jpg)
A. PSD by the multitaper method showing peaks in theta but not beta
or gamma ranges.
Long time segments (³6 sec) give PSD with power-law distributions, which reflects the scale-free dynamics of
cerebral cortex (Freeman and Breakspear, 2007). The optimal pass band is 5 Hz,
as shown by the symbol Ô – Ô at the frequency band selected for analysis
in the mid-beta range.
B. The superimposed 62 ECoG filtered signals show maximal spatial
coherence in two high-amplitude bursts of oscillation separated by minima
referred to as beats in Rayleigh
noise. The null spike at the minimum in the envelope of the oscillations 21.098
sec is an example of a phase transition. Not all Rayleigh beats manifest phase
transitions. The line segment shows the time range of the movies.
C. Log10 analytic
power shows a succession of down spikes, the deepest of which are referred to
as null spikes (Freeman, 2009). The
latencies from the arbitrary start of the display are in sec
D. The analytic frequency in rad/sec is approximated by the ratio of
the change in phase, DF in
radians, in each time step to the duration of the digitizing step, Dt, in sec. The spikes reflect
indeterminacy when the analytic amplitude is low; the flat segments between
spikes show times of minimal temporal and spatial variance of the analytic
phase. The length of the flat segment determines the duration of a frame.
A0. Perspective, Electric Field of a Burst Demarcated by Null
Spikes
(A0F152X12test21.0to21.6sec20to25HzFilteredECoGFlapping.mov)
The
movie shows a perspective image of the scalar electric potential field in the
mid-beta 20-25 Hz range from 21 sec to 21.6 sec (0.6 sec), with the amplitude
color-coded. The task for analysis is to identify the stable components of
phase and analytic power (analytic amplitude squared) of multiple bursts of
oscillations at differing center frequencies and mean phase. The tendency for
rotation prevents use of PCA and ICA, which assume time-invariance of the
signals over space, i.e., spatial stationarity.
A1. Electric Potential Field: Complex Rotating Amplitude Patterns
(Vortices)
(A1F152X12test21.0to21.sec20to25HzRotationECoG,CCW,Flat.mov)
The movie
shows the view from above with amplitude minus the spatial average, color-coded
to show the rotation of the field in complex patterns. The widespread
coordination is strong evidence that the neural field underlying the scalar
fields of potential is a spatiotemporal continuous vector field that influences
and includes every neuron in the region of phase-lagged synchrony in the
cortex. Every point has both amplitude and vectorial directions of temporal and
spatial change.
B0. Examples are Shown of Null Spikes, often in Clusters
(B0F152X12test21.0to21.6sec20to25HzLog10AnalyticPowerPersp.mov)
This
movie shows a perspective representation of the log10 analytic power from 21
sec to 21.6 sec (0.6 sec) across the three clusters of down spikes. The color-coding shows the fluctuating
amplitude and phase of the filtered signal (not the analytic amplitude or
power). The temporal and spatial localization of the down spikes is
apparent.
B1. Contour Plots Show the Locations of Null Spikes (Blue)
(B1F152X12test21to21.6sec20to25HzPhaseExplosionCCW.avi)
The same data as in B0F are shown as seen from vertically over the
surface and color-coded to represent power: blue is minimum. The spatial
localization is limited by the interelectrode distances averaging 0.79 mm in an
approximately 6x6 mm array placed on the surface of the rabbit visual cortex.
C0. A Phase Cone Appears Shortly after a Null Spike in the
Analytic Phase
(C0F152X12test21.0to21.8sec20to25HzPhasePersp.mov)
The phase
gradient is radially symmetric about a center point that closely conforms to
the location of the preceding null spike. The maximum lead of the analytic
phase designates the apex of a cone with downward slope, indicating a phase lag
to the periphery called an explosion.
The approximate co-location of the null spike and the apex of the following
phase cone (either maximal lead = explosion, or maximal lag = implosion) is
seen only in frames in which a single burst has one dominant frequency.
C2. The Conic Apex is near
the Location of the Null Spike and a Phase Discontinuity
(C1F152X12test21to21.6sec20to25HzPhaseExplosionCW.avi)
The
same data as in C0F are presented as seen from vertically above the cortical
surface, color-coded with maximum +¹ and minimum –¹ radians. The features
of note are the evidence for phase discontinuity in the temporal vicinity of
the null spike near 21.1 sec and the array numbers ordinate 6 and abscissa 5.
References
Freeman W.J. [2004a] Origin, structure,
and role of background EEG activity. Part 1. Analytic
amplitude. Clin. Neurophysiol. 115:
2077-2088.
http://repositories.cdlib.org/postprints/1006
Freeman W.J. [2004b] Origin, structure,
and role of background EEG activity. Part 2. Analytic
phase. Clin. Neurophysiol. 115:
2089-2107.
http://repositories.cdlib.org/postprints/1486.
Freeman WJ, Breakspear M [2007] Scale-free
neocortical dynamics. Encyclopedia for Computational
Neuroscience, Izhikevich E (ed.).
http://www.scholarpedia.org/article/Scale-free_neocortical_dynamics
Freeman WJ (in press) Evidence for a spatiotemporal
singularity in percept formation by cerebral cortex.
Proc ICCN2009, 15-19 Nov, Hangzhou, China. ICCN2009,WJFreeman
Singularity