BR A I N R ES E A RC H 1 1 4 5 ( 2 00 7 ) 2 3 9 –2 47
a v a i l a b l e a t w w w. s c i e n c e d i r e c t . c o m
w w w. e l s e v i e r. c o m / l o c a t e / b r a i n r e s
Research Report
Computing the center of mass for traveling alpha waves in the
human brain
Elías Manjarrez ⁎, Montserrat Vázquez, Amira Flores
Instituto de Fisiología, Benemérita Universidad Autónoma de Puebla, 14 Sur 6301, Col. San Manuel, Apartado Postal 406, Puebla,
Pue. CP 72570, Mexico
A R T I C LE I N FO
AB S T R A C T
Article history:
The phenomenon of traveling waves of the brain is an intriguing area of research, and its
Accepted 26 January 2007
mechanisms and neurobiological bases have been unknown since the 1950s. The present
Available online 2 February 2007
study offers a new method to compute traveling alpha waves using the center of mass
algorithm. Electroencephalographic alpha waves are oscillations with a characteristic
Keywords:
frequency range and reactivity to closed eyes. Several lines of evidence derived from
Alpha rhythm
qualitative observations suggest that the alpha waves represent a spreading wave process
Traveling brain waves
with specific trajectories in the human brain. We found that during a certain alpha wave
EEG topographic maps
peak recorded with 30 electrodes the trajectory starts and ends in distinct regions of the
Center of mass
brain, mostly frontal–occipital, frontal–frontal, or occipital–frontal, but the position of the
Corpus callosum
trajectory at the time in which the maximal positivity of the alpha wave occurs has a
Brain imaging
definite position near the central regions. Thus we observed that the trajectory always
Scanning hypothesis
crossed around the central zones, traveling from one region to another region of the brain. A
similar trajectory pattern was observed for different alpha wave peaks in one alpha burst,
and in different subjects, with a mean velocity of 2.1 ± 0.29 m/s. We found that all our results
were clear and reproducible in all of the subjects. To our knowledge, the present method
documents the first explicit description of a spreading wave process with a singular pattern
in the human brain in terms of the center of mass algorithm.
© 2007 Elsevier B.V. All rights reserved.
1.
Introduction
The electroencephalogram (EEG) represents the collective
electrical response of neurons along the space-time (e.g., see
Nunez, 1981a, 1989; Lopes da Silva, 1991). Numerous factors
contribute to activating neurons in different regions of the
brain, for example, during the application of periodic
sensory stimuli, or during different stages of sleep and
wakefulness. For example, in relaxed wakefulness, the EEG
alpha waves are most prominent over parietal and occipital
sites. The brain mapping of the EEG has been useful to
detect zones on the scalp in which electrical activity is more
prominent.
Shevelev (1988) and Shevelev et al. (1991, 2000) used brain
mapping to analyze zones in which the alpha waves are more
prominent at certain periods of time. They found a clear
sequence of activation of these zones, consistent with a
traveling wave process, as originally proposed by Pitts and
McCulloch (1947) in the visual cortex. In recent studies,
Massimini et al. (2004) reported that brain waves generated
during stages of sleep also exhibit a related behavior of
propagation. Thus, the regions with more prominent activities
⁎ Corresponding author. Fax: +5222 22 295500x7323.
E-mail address: emanjar@siu.buap.mx (E. Manjarrez).
0006-8993/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.brainres.2007.01.114
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change at certain periods of time, following particular
trajectories very similar in all of the subjects tested. In all of
these studies, and in earlier studies (Goldman et al., 1949;
Petsche and Sterc, 1968; Hughes, 1995; Silberstein et al., 2000),
the methods used to analyze the propagation of the traveling
waves have been substantially qualitative, based only in the
observation of a map changing their form at certain times (e.g.,
see Shevelev et al., 2000), or based on the trajectories of the
delays of the peaks recorded with a set of electrodes on the
scalp (e.g., see Massimini et al., 2004), or in terms of large-scale
Fig. 1 – The method based on the center of mass algorithm to compute an electrical traveling wave. (A) Typical EEG-recording
exhibiting alpha bursts. (B) Zoom of a typical alpha burst delimited by the vertical blue lines A. (C) A zoom of positive-alpha
peaks delimited by the blue vertical lines in B. Every vertical line indicates six consecutive times in which the topographical
maps were computed. (D) Topographical maps at the same times indicated in C. (E) The black vector indicates the coordinates of
the recording electrodes. The ping vector indicates the Cartesian coordinates of the center of mass (x (t ), y (t )). (F) Typical
alpha peak (recorded from the FP1 electrode, i = 30) indicating that the amplitude of the EEG at time t corresponds to the
mass m30(t) detected for the electrode 30 at that time. (G) Formula to calculate the center of mass (x (t ), y (t )) of the EEG activity.
(H) Trajectory calculated by the formula in G, in the same times illustrated in C and D.
BR A I N R ES E A RC H 1 1 4 5 ( 2 00 7 ) 2 3 9 –2 47
phase synchronization (Ito et al., 2005), or phase gradient
(Burkitt et al., 2000), or in theoretical studies of coupled phase
oscillators (Ermentrout and Kleinfeld, 2001). The purpose of
the present study was to introduce a new method, based on
the center of mass algorithm to quantify the two-dimensional
trajectories of the traveling alpha wave-positive peaks in the
scalp xy(t) and their velocity v(x,y,t).
The center-of-mass computation has been used for
pooling the spiking responses of neurons in a population
whose members are dispersed over a parameter space
(Salinas and Abbott, 1994). This method, commonly termed
“vector averaging”, has been used in a variety of studies in
the context of spike activity of neurons (Abbott, 1994; Salinas
and Abbott, 1994) but not in the context of EEG slow field
potentials. For example, Demas et al. (2003) employed the
241
center of mass algorithm for the visualization of spontaneous firing activity across the retina. Demas et al. (2003)
used a multi-electrode array to record the activity of single
neurons in in-vitro retinas. The center of mass of spike
activity for a given time was calculated by vector averaging
the positions of all cells with firing rates that exceeded a
threshold of 2 Hz for that time window. They found the
presence of retinal waves on immature postnatal day P9.
Our study is original because the center of mass of EEG
amplitude for a given time was calculated by vector averaging
the position of all recording electrodes. We used the amplitude
of the human EEG instead of the spike activity of single
neurons considered in previous studies of animals. On the
other hand, although the center of mass equation has been
used in the past in other contexts (Abbott, 1994; Mussa-Ivaldi
Fig. 2 – Trajectories of the center of mass calculated for five consecutive alpha positive-peaks recorded from one subject. (A)
Alpha burst and five consecutive alpha peaks indicated by gray rectangles. (B) Symbols to indicate the beginning of the
trajectory (black triangle), the maximum positivity (blue square) and the ending of the trajectory (white diamond superimposed
on a black triangle). (C) Five trajectories associated to the peaks indicated in A.
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and Giszter, 1992; Foreman and Eaton, 1993; Gwen and
Theunissen, 1996; Snippe, 1996; Siegel, 1998; Churchland and
Lisberger, 2001; Yakovenko et al., 2002), there are no studies in
which this equation has been used to visualize EEG traveling
waves.
Our method offers advantages over graphical topographic
displays because besides the identification of one trajectory it
allows its quantification, its instantaneous velocity, and the
performance of a more formal analysis. In this context our
method improves the analysis, thus allowing a possible future
theoretical interpretation of the results in terms of the center
of mass equation. Our method could be used to analyze EEG
traveling waves in different contexts, during visual illusions
(Shevelev et al., 2000) or during sleep (Massimini et al., 2004),
for example.
We suggest that the present method could be useful in
future studies as a tool to characterize changes in the state of
alpha waves, or other electrical wave processes in different
experimental conditions.
2.
Results
We performed EEG recordings (30 channels) in 27 subjects who
were resting in a chair with their eyes closed. We analyzed the
trajectories of the center of mass of the EEG activity for three
Fig. 3 – (A) alpha burst recorded in another subject. Five groups of different alpha peaks are indicated. (B) Five trajectories
superimposed that were calculated from the five consecutive alpha peaks illustrated in A. This figure is to illustrate how the
trajectories were superimposed. (D) Pooled superimposed trajectories calculated for 27 subjects (gray lines). Some trajectories
are indicated in different colors to highlight the common pattern. Note that every trajectory crosses close to the central regions
where the alpha peaks reach their maximal positivity (blue squares). (C) Diagonal lines indicate the approximated zone in
which the maximal positivities are located (blue squares in D).
BR A I N R ES E A RC H 1 1 4 5 ( 2 00 7 ) 2 3 9 –2 47
alpha bursts per subject. Because one alpha burst is composed
of about 4–10 alpha peaks, we computed the trajectories for
about 270 alpha-peaks in 27 subjects. In Fig. 3D we present the
superimposed trajectories calculated for the 27 subjects, but in
Figs. 1, 2, and 3B we illustrate the procedure used to detect
such trajectories in different individuals. Fig. 1B shows a
typical recording of a burst of alpha waves delimited by the
vertical blue lines illustrated in Fig. 1A. Fig. 1B shows that the
alpha burst exhibits clear sinusoidal-like alpha peaks. Fig. 1C
shows a zoom of typical positive-alpha peaks delimited by the
blue vertical lines in Fig. 1B. Every vertical line in Fig. 1C
indicates six consecutive times in which the topographical
maps illustrated in Fig. 1D were obtained. Note the occipital–
frontal propagation of the red maps (i.e., positive maps) from
t1 to t6, thus suggesting the propagation of an EEG wave
process. However, such propagation is only qualitative. In
counterpart, Fig. 1H shows the quantitative analysis of the
trajectory, based on our center of mass algorithm, for the EEG
in the same times illustrated in Figs. 1C–D. Note the similitude
between both propagations (compare Figs. 1D and H). Figs. 1E
to G illustrate the method to calculate the trajectory of the
center of mass of the positive peaks of the EEG alpha activity
(see Methods).
We characterized the patterns of propagation described by
the trajectories during consecutive alpha positive-peaks of EEG
bursts recorded in different subjects. Fig. 2 shows the symbols
that we used to describe the trajectories of these consecutive
alpha positive-peaks. Fig. 2A shows one alpha burst recorded
in one subject, and five consecutive peaks from which the
trajectories were calculated. Fig. 2B shows the symbols used in
the present paper. The black triangle indicates the beginning of
the peaks and the associated trajectory. The white diamonds
superimposed on a black triangle indicate the end of the peak.
The blue squares illustrated in Fig. 2B indicate the time in
which one of the 30 recording electrodes detected the maximal
positivity. The position of the center of mass at the time of this
maximal positivity is illustrated in the trajectories of Fig. 2C
with a blue square.
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Fig. 2C illustrates the trajectories of the EEG center of mass
for five consecutive positive-peaks in one alpha burst (Fig. 2A).
The black triangle indicates the start position of the trajectory
(Fig. 2B). Note that the end of trajectory 1 (indicated by a white
diamond superimposed on a black triangle) is the beginning of
trajectory 2, and successively the end of trajectory2 is the start
position for trajectory 3.
Figs. 3A and B show one alpha burst for another subject and
their corresponding successive trajectories (traj3 to traj7)
indicated by different colors. The analysis of every positivepeak with the center of mass formula results in a different
trajectory. Fig. 3B shows the superimposed trajectories using
the same symbols defined in Fig. 2. Note that for this subject
the trajectories crossed from one hemisphere to another, and
that their maximal positivity, indicated by squares of colors is
located close to C3 and CP3. We performed a similar analysis
superimposing the trajectories for all the alpha positive-peaks
recorded in all the subjects, but the maximal positivities were
indicated by blue squares. Fig. 3D shows in gray color 270
superimposed trajectories computed for all 27 subjects. Some
trajectories were indicated with other colors to show their
typical pattern. Note that every trajectory crosses from one
region to another region of the brain, but the maximal
positivity is located in the central regions, as is illustrated in
Figs. 3C and D also shows that the regions of less positivity
(white diamonds) are mostly located in frontal, temporal, and
occipital regions of the brain. This pattern is typical of the
alpha peaks recorded in all the 27 subjects. The common
pattern is that the trajectories travel from one place to another
place of the brain (white diamonds) crossing through the
central regions of the brain, in which there are clear maximal
positivities of the positive-peaks of the alpha waves (blue
squares).
We performed a statistical analysis based on the “method
of surrogate data” to distinguish the trajectories associated
with the alpha waves from the trajectories associated with
noise. Fig. 4 shows trajectories obtained after the shuffling
procedure (see methods). Note that shuffling induces the
Fig. 4 – The same as Figs. 3C–D but after shuffling the EEG data. We employed the method of Zero surrogate data implemented
by Theiler et al. (1992). In this method, the surrogate data sets are constructed by a random shuffle of the original data.
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noisy dispersion of maximal peak locations that are not only
in the central zones but are dispersed across all the brain.
This procedure allows the observation of qualitative and
quantitative differences between trajectories associated to
the alpha waves and the trajectories associated with noise.
Qualitatively we can observe that after shuffling, the trajectories do not necessarily cross through the central zones (Fig.
4). Furthermore, quantitatively we can observe a statistically
significant difference (p < 0.01; Student's t-test) between the
mean magnitude of the vectors associated with the maximal
peaks before shuffling (0.5 ± 0.2 m) and after shuffling (0.14 ±
0.07 m). Therefore, the null hypothesis that the original
trajectories are indistinguishable from the trajectories associated with uncorrelated noise can be rejected. This suggests
that trajectories associated with the alpha waves can be
statistically distinguished from trajectories associated with
noise in the signal.
We also analyzed the velocity of propagation of the center
of mass for every trajectory as a function of time. In particular
we calculated the velocity for the alpha positive-peak-wave
propagation. We obtained a grand average of the velocity (2.1 ±
0.29 m/s) for the alpha positive-peak-wave propagation for all
the subjects (n = 27).
3.
Discussion
The study of traveling waves in the human brain is as old as
the implementation of the first EEG recording techniques,
describing a wave process in the brain electrical activity
(Goldman et al., 1949), or the first theoretical studies of a wave
process in the nervous system (Pitts and McCulloch, 1947).
Furthermore, there are many studies about the origin of the
alpha rhythm (Adrian and Yamagiwa, 1935; Andersen et al.,
1967; Lopes da Silva and Storm van Leeuwen, 1977; Inouye et
al., 1995; Hughes and Crunelli, 2005; Feige et al., 2005) and their
wavelike properties (Nunez, 1974, 1981a,b; Nunez et al., 1994),
thus supporting the idea that the alpha waves and their
propagation do not represent an epiphenomenon and could
have a functional role not yet understood.
The purpose of the present study was to introduce a new
method, based on the center of mass algorithm, to quantify
the trajectory patterns of the traveling alpha waves in the
scalp (x(t), y(t)) and their velocity (dx(t)/dt, dy(t)/dt). The
differences between our method and previous approaches
are that the methods used to analyze the propagation of the
traveling waves have been substantially qualitative, based
only on the observation of a map changing their form (e.g., see
Shevelev et al., 2000), or based on the trajectories of the delays
of the peaks recorded with a set of electrodes on the scalp (e.g.,
see Massimini et al., 2004). We can compare the method used
by Shevelev et al. (2000) (Fig. 1D) with our method based on the
center of mass algorithm (Fig. 1H). Note the similitude
between both methods and that our method is more
quantitative. Earlier methods based on topographical displays
only provide a qualitative description of the trajectories of
propagating waves. However, our method describes trajectories and instantaneous velocity with a straightforward
mathematical algorithm. In this context, our method is
valuable and different from earlier graphical methods.
Our method offers advantages over current graphical
methods that describe trajectories and velocities of the
propagating waves. For example, in the study performed by
Massimini et al. (2004) two different montages were required to
calculate trajectories (with 256 electrodes) and velocity (with 20
electrodes). In their study the mean velocity of wave propagation was measured from the data collected by a row of 20
electrodes placed along the antero-posterior axis by calculating the linear correlation between scalp location in millimeters
and the delay at each electrode. However, in our case, we used
only one montage to calculate both the trajectories and the
instantaneous velocity as a function of time. In this context,
our method is simpler than the graphical methods previously
employed. The procedure employed by Massimini et al. (2004)
cannot be used to calculate the instantaneous velocity of the
trajectories, while our method can compute the instantaneous
velocity of the trajectories only by means of the first derivative
of the center of mass function.
Our method complements two-dimensional voltage topographic mapping of brain potentials to describe propagation of
brain waves. We suggest that in future studies our method
could be used as a tool for the analysis of wave propagation in
different conditions of the brain, both in normal and pathological states.
We calculated a velocity of about 2 m/s for the traveling
positive-alpha-peak wave process. This wave speed is within
the range of previous measurements of wave propagation
velocity on the human scalp (Nunez et al., 1994; Hughes, 1995;
Hughes et al., 1992, 1995; Massimini et al., 2004). In in vivo
preparations in animals the wave speed has also been
measured. For example, during a motor task the propagation
velocity of the traveling waves is about 0.2 m/s both in the
monkey motor cortex (Rubino et al., 2006) and in the cat spinal
cord (Manjarrez et al., 2005). A comparable measure (∼ 0.5 m/s)
can be inferred from a study performed in the visual cortex by
Arieli et al. (1995). Arieli et al. (1995) found coherent oscillatory
activity in the visual cortex of anesthetized cats by means of a
diode array covering a 7 × 7 mm region of the 17 and 18 visual
areas. These authors observed that for waves at 10 Hz there
was a large phase difference of ∼ 14 ms between the oscillatory
activities of areas 17 and 18 (i.e., a speed ∼ 0.5 m/s). However, in
cortical slices obtained from animals, the propagation velocity
of the traveling waves was considerably lower, from 0.01 to
0.1 m/s (Bai et al., 2006; Sanchez-Vives and McCormick, 2000).
These velocities are slower than axonal conductance in cortex,
suggesting that traveling waves are mediated by multiple
synapses in neuronal circuits.
We observed that the trajectories followed by the center
of mass crossed through the central zones, traveling from
one region to another region of the brain. Therefore, it is
tempting to suggest that the trajectories reveal possible
cortico-cortical connections through the corpus callosum. In
this context, it would be interesting to compare trajectory
patterns of normal subjects to subjects with a surgical
transection of the corpus callosum. In general, it would be
interesting to explore the influence of cortical incisions on
the trajectories of the traveling waves. Such study could
extend previous observations about the effects of incisions
on the synchronization patterns and traveling waves in the
brain (Petsche and Rappelsberger, 1970).
BR A I N R ES E A RC H 1 1 4 5 ( 2 00 7 ) 2 3 9 –2 47
The graphical topographic methods have been useful to
characterize traveling waves in a qualitative framework. In
this context, we consider that our quantitative method will
also be useful to describe trajectories. We suggest that bulk
conductivity affects the concentrated point obtained from the
center of mass as well as the trajectories visualized by
graphical topographic inspection. For this reason, for both
methods we must exercise caution before making any
conclusion about the location of the generators of the
traveling waves. Therefore, our method cannot be used to
locate generators; instead it can be used only to describe in a
quantitative manner traveling waves that can be observed by
means of gross topographic maps.
Nevertheless, although the volume conductor is involved in
the graphical topographic displays of the traveling waves, some
interesting studies have been published in the past (Goldman
et al., 1949; Petsche and Sterc, 1968; Hughes, 1995; Shevelev et
al., 2000; Burkitt et al., 2000) and recently (Massimini et al.,
2004). These studies have suggested that traveling waves have
a functional role in brain function (see also Rubino et al., 2006).
In this context, our method also could be useful to obtain a
more formal analysis of the traveling waves. For example, the
studies performed by Massimini et al. (2004) or Shevelev et al.
(2000) or Rubino et al. (2006) could be analyzed in a more formal
context by means of our method, thus allowing a possible
future theoretical interpretation (and/or simulation) of the
results in terms of the center of mass equation.
We consider that our mathematical method is important
because the center of mass of a distribution of masses does
not always coincide with its intuitive graphical geometric
center, and one can exploit this freedom. Furthermore, earlier
methods based on graphical topographic displays only provide
a qualitative description of the trajectories by means of the
subjective observation of the researcher. However, our study
provides a quantitative method to describe trajectories of the
propagating waves, and their instantaneous velocity.
3.1.
Functional implications
Since Pitts and McCulloch in 1947, and Goldman in 1949, the
physiological basis of the traveling alpha waves is yet unclear.
In 1947, Pitts and McCulloch speculated that the alpha rhythm
represents a spreading wave process which reads information
from the visual cortex. This conjecture is now known as the
“scanning hypothesis” and has been a matter of discussion of
many psychophysical and EEG studies with the idea that the
scanning hypothesis links EEG alpha activity with rhythmically spreading waves in the visual cortex (see references in
Shevelev et al., 2000). For example, to give support to this
hypothesis Shevelev et al. (2000) performed psychophysical
and EEG experiments in humans. These authors found that
under flicker stimulation through the closed lids at the
frequency of the alpha rhythm, all the subjects perceived
illusory visual objects (a ring, a spiral, a spiral spring, or a grid).
With this stimulation-protocol they found that the illusory
visual perception of the ring and spiral objects was significantly associated with the occipital–frontal trajectories, and
the grid illusion with the left occipital to right frontal
trajectories. In this context, Shevelev et al. (2000) suggested
that the illusions could be produced by the impact between
245
the traveling alpha activity of cortical neurons and the EEG
responses evoked by the flicker stimulation through the visual
pathway (as in a Lissajous figure produced in an oscilloscope
by the combination of two sinusoids). Interestingly, a similar
phenomenon of spiral dynamics (induced by noise) has been
observed in a FitzHugh–Nagumo model of a square twodimensional lattice of NxN coupled cells (Garcia-Ojalvo and
Schimansky-Geier, 1999). Therefore, in the context of the
experimental evidence, it is tempting to speculate that the
traveling waves follow pathways related to sensory processes
through a mechanism based on the scanning hypothesis.
The idea that traveling waves could be related with
physiological processes is not exclusive of the alpha waves
(∼ 10 Hz), it has also been discussed in the context of slow
traveling waves (∼0.8 Hz) during sleep in humans (Massimini
et al., 2004), or in the context of high frequency oscillations in
the beta range (10–45 Hz) during movement preparation and
motor execution in monkeys (Rubino et al., 2006). Propagating
waves have also been observed in different animal preparations during visual (Arieli et al., 1995; Prechtl et al., 2000),
olfactory (Friedrich and Korsching, 1997; Freeman and Barrie,
2000), somatosensory (Nicolelis et al., 1995; Petersen et al.,
2003; Ferezou et al., 2006), and spinal motor (Manjarrez et al.,
2005) processing. Some theoretical and experimental studies
(see references in Bai et al., 2006) suggest that the propagating
waves are generated by a simple mechanism based on the
spatial distribution of phase gradient of coupled local oscillators (where each local oscillator is defined as a group of tightly
coupled oscillating neurons). Bai et al. (2006) performed
interesting experiments in rat neocortical slices. They found
that cortical network oscillations of about 25 Hz are organized
at two levels: locally, oscillating neurons are tightly coupled to
form local oscillators, and globally the coupling between local
oscillators is weak, allowing abrupt spatial phase lags and
propagating waves with multiple initiation sites. Therefore Bai
et al. (2006) characterized an oscillation where spatial coupling
is weak enough to identify local oscillators but strong enough
to form propagating waves. In this context we suggest that the
propagating alpha traveling waves may also have a functional
role that remains to be revealed.
4.
Experimental procedure
4.1.
Participants
Recordings were performed in 27 subjects (healthy righthanded volunteers of both sexes, age 20–24 years) with a
Synamps EEG amplifier of 32 channels (NeuroScan, Inc.
Sterling, VA) during eyes-closed wakeful rest. All participants
gave written informed consent. Topographic maps in the time
domain were created using Scan 4.2 Software from NeuroScan.
We recorded data with a sampling rate of 500 Hz.
4.2.
Methods
Electroencephalographic alpha waves were characterized by
their characteristic frequency range (around 10 Hz) and
reactivity to closed eyes. We selected alpha bursts and
computed their power spectra. We applied the center of mass
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algorithm to these bursts of activity in which the power spectra
of EEG recordings exhibited a clear dominant power peak in the
alpha band. We analyzed the trajectories of the center of mass
of the EEG activity for three alpha bursts per subject (one alpha
burst is composed of about 4–10 alpha peaks). We computed
the trajectories for about 270 alpha-peaks in 27 subjects.
4.2.1.
The center of mass
Our technique is straightforward and can be explained in
detail by means of the center of mass equation. In physics
the center of mass for a system of particles is a specific point
at which, for many purposes, the system's mass behaves as if
it were concentrated. The center of mass is a function only of
the positions and masses of the particles that comprise the
system. In an analogy, in our case, for each instant of time
we consider the EEG voltage amplitudes as the masses mi(t)
(Fig. 1F) and the positions (ai, bi) of all the 30 electrodes (Fig.
1E) are considered as the positions of the particles. We
defined the “positive mass values” at time t, by means of the
functions: mi(t) > 0, from m1(t) to m30(t), which were obtained
from the EEG activity recorded with the electrodes located in
positions: (a1, b1) to (a30, b30). For example, Fig. 1F shows a
peak of EEG activity recorded with the electrode located in
position FP1 (i = 30). Note that the mass values mi(t) can be
computed for different times and for different electrodes.
Therefore, the mass mi(t) > 0 values represent the amplitudes
of the positive EEG potential at time t, and their units are in
volts.
We calculated the coordinates of the center of mass of the
electrical activity recorded on the scalp with the following
equations (see also Fig. 1G):
xðtÞ ¼ ðm1 ðtÞa1 þ m2 ðtÞa2 þ N þ m30 ðtÞa30 Þ=ðm1 ðtÞ þ m2 ðtÞ þ N
þ m30 ðtÞÞ
yðtÞ ¼ ðm1 ðtÞb1 þ m2 ðtÞb2 þ N þ m30 ðtÞb30 Þ=ðm1 ðtÞ þ m2 ðtÞ þ N
þ m30 ðtÞÞ
where x(t) and y(t) are functions representing coordinates of
the center of mass (ping vector (x(t), y(t)) in Fig. 1E), and a1 to
a30, and b1 to b30 are fixed coordinates of the 30 recording
electrodes (blue circles in Fig. 1E).
The trajectory of the center of mass was determined by
coordinates (x(t), y(t)) at time t. Fig. 1H illustrates a typical
trajectory of the center of mass for one group of alpha-positive
peaks delimited by the blue lines in Fig. 1C.
We defined the velocity (v(t)) of propagation of the center of
mass of EEG activity by the first derivative of the center of
mass function:
vðtÞ ¼ ðdðxðtÞÞ=d t; dðyðtÞÞ=d tÞ
The trajectories of the center of mass for the EEG activity
and velocity were computed using custom software written in
MATLAB (The MathWorks, Inc., Natick, MA).
4.3.
Statistical analysis
We employed the method of Zero surrogate data implemented
by Theiler et al. in 1992. In this method, the surrogate data sets
are constructed by a random shuffle of the original data.
Algorithm Zero surrogate address the following null hypothesis: the original time series is indistinguishable from
uncorrelated noise (i.e., in our case, the original trajectories
associated with the alpha waves are indistinguishable from
the trajectories associated with the uncorrelated noise). In this
method the same measure (i.e., the center of mass) is applied
to the original data and to the surrogates. If the results are
different, the null hypothesis can be rejected. We calculated
the mean magnitude of the center of mass vector associated
with the maximal peaks before and after shuffling. We used a
Student's t-test (p < 0.01) to reject the null hypothesis that the
mean magnitude of this center of mass vector before shuffling
is indistinguishable from the mean magnitude of this measure
after shuffling (i.e., surrogated uncorrelated noise).
Acknowledgments
We thank Robert Simpson for proof reading the English
manuscript and Alma López for technical assistance. This
work was partly supported by the following grants: CONACyT
J36062-N (E.M), Fondo-Ricardo J. Zevada (A.F) and PIFI-FOMESPROMEP-BUAP and VIEP-BUAP (E.M), México.
REFERENCES
Abbott, L.F., 1994. Decoding neuronal firing and modelling neural
networks. Q. Rev. Biophys. 27, 291–331.
Adrian, E.D., Yamagiwa, K., 1935. The origin of the Berger rhythm.
Brain 58, 251–323.
Andersen, P., Andersson, S.A., Junge, K., Lomo, T., Sveen, O.H.,
1967. Physiological mechanism of the slow 10c/sec cortical
rhythmic activity. Electroencephalogr. Clin. Neurophysiol. 23,
394–395.
Arieli, A., Shoham, D., Hildesheim, R., Grinvald, A., 1995.
Coherent spatiotemporal patterns of ongoing activity
revealed by real-time optical imaging coupled with
single-unit recording in the cat visual cortex. J. Neurophysiol.
73, 2072–2093.
Bai, L., Huang, X., Yang, Q., Wu, J.Y., 2006. Spatiotemporal
patterns of an evoked network oscillation in neocortical
slices: coupled local oscillators. J. Neurophysiol. 96,
2528–2538.
Burkitt, G.R., Silberstein, R.B., Cadusch, P.J., Wood, A.W., 2000.
Steady-state visual evoked potentials and traveling waves.
Clin. Neurophysiol. 111, 246–258.
Churchland, M.M., Lisberger, S.G., 2001. Shifts in the population
response in the middle temporal visual area parallel
perceptual and motor illusions produced by apparent motion.
J. Neurosci. 21, 9387–9402.
Demas, J., Eglen, S.J., Wong, R.O.L., 2003. Developmental loss
of synchronous spontaneous activity in the mouse retina
is dependent of visual experience. J. Neurosci. 23,
2851–2860.
Ermentrout, G.B., Kleinfeld, D., 2001. Traveling electrical waves in
cortex: insight from phase dynamics and speculation on a
computational role. Neuron 29, 33–44.
Feige, B., Scheffler, K., Esposito, F., Di Salle, F., Hennig, J., Seifritz, E.,
2005. Cortical and subcortical correlates of
electroencephalographic alpha rhythm modulation.
J. Neurophysiol. 93, 2864–2872.
Ferezou, I., Bolea, S., Petersen, C.C., 2006. Visualizing the cortical
BR A I N R ES E A RC H 1 1 4 5 ( 2 00 7 ) 2 3 9 –2 47
representation of whisker touch: voltage-sensitive dye imaging
in freely moving mice. Neuron 50, 617–629.
Foreman, M., Eaton, R.C., 1993. The direction change concept for
reticulospinal control of goldfish escape. J. Neurosci. 13,
4101–4113.
Freeman, W.J., Barrie, J.M., 2000. Analysis of spatial patterns of
phase in neocortical gamma EEGs in rabbit. J. Neurophysiol. 84,
1266–1278.
Friedrich, R.W., Korsching, S.I., 1997. Combinatorial and
chemotopic odorant coding in the zebrafish olfactory bulb
visualized by optical imaging. Neuron 18, 737–752.
Garcia-Ojalvo, J., Schimansky-Geier, L., 1999. Noise-induced
spiral dynamics in excitable media. Europhys. Lett. 47,
298–303.
Goldman, S., Santelmann, W.F., Vivian, W.E., Goldman, D., 1949.
Traveling waves in the brain. Science 109, 524.
Gwen, A., Theunissen, F.E., 1996. Functional organization of a
neural map in the cricket cercal sensory system. J. Neurosci. 16,
769.784.
Hughes, J.R., 1995. The phenomenon of traveling waves: a review.
Clin. Electroencephalogr. 26, 1–6.
Hughes, S.W., Crunelli, V., 2005. Thalamic mechanisms of EEG
alpha rhythms and their pathological implications.
Neuroscientist 11, 357–372.
Hughes, J.R., Kuruvilla, A., Kino, J.J., 1992. Topographic analysis of
visual evoked potentials from flash and pattern reversal
stimuli: evidence for ‘Traveling Waves’. Brain Topogr. 4,
215–228.
Hughes, J.R., Ikram, A., Fino, J.J., 1995. Characteristics of traveling
waves under various conditions. Clin. Electroencephalogr. 26,
7–22.
Inouye, T., Skinosaki, K., Toi, S., Matsumoto, Y., Hosaka, N., 1995.
Potential flow of alpha activity in the human
electroencephalogram. Neurosci. Lett. 187, 29–32.
Ito, J., Nikolaev, A.R., van Leeuwen, C., 2005. Spatial and temporal
structure of phase synchronization of spontaneous alpha EEG
activity. Biol. Cybern. 92, 54–60.
Lopes da Silva, F.H., 1991. Neural mechanisms underlying brain
waves: from neural membranes to networks.
Electroencephalogr. Clin. Neurophysiol. 79, 81–93.
Lopes da Silva, F.H., Storm van Leeuwen, L.W., 1977. The cortical
source of alpha rhythm. Neurosci. Lett. 6, 237–241.
Manjarrez, E., Quevedo, J., Juarez-Cirne, V., 2005. Propagation of
spinal waves during fictive scratching in the cat. Abstr.-Soc.
Neurosci. 753.14.
Massimini, M., Huber, R., Ferrarelli, F., Hill, S., Tononi, G., 2004. The
sleep slow oscillation as a traveling wave. J. Neurosci. 24,
6862–6870.
Mussa-Ivaldi, F.A., Giszter, S.F., 1992. Vector field approximation: a
computational paradigm for motor control and learning. Biol.
Cybern. 67, 491–500.
Nicolelis, M.A., Baccala, L.A., Lin, R.C., Chapin, J.K., 1995.
Sensorimotor encoding by synchronous neural ensemble
activity at multiple levels of the somatosensory system.
Science 268, 1353–1358.
Nunez, P.L., 1974. Wavelike properties of the alpha rhythm. IEEE
Trans. Biomed. Eng. 21, 473–482.
Nunez, P.L., 1981a. Electric Fields of the Brain: The Neurophysics of
EEG. Oxford Univ. Press, New York.
Nunez, P.L., 1981b. A study of origins of the time dependencies of
247
scalp EEG: II. Experimental support of theory. IEEE Trans.
Biomed. Eng. 28, 281–288.
Nunez, P.L., 1989. Generation of human EEG by a combination of
long and short range neocortical interactions. Brain Topogr. 1,
199–215.
Nunez, P.L., Silberstein, R.B., Cadusch, P.J., Wijesinghe, R.S.,
Westdorp, A.F., Srinivasan, R., 1994. A theoretical and
experimental study of high resolution EEG based on surface
Laplacians and cortical imaging. Electroencephalogr. Clin.
Neurophysiol. 90, 40–57.
Petersen, C.C., Hahn, T.T., Mehta, M., Grinvald, A., Sakmann, B.,
2003. Interaction of sensory responses with spontaneous
depolarization in layer 2/3 barrel cortex. Proc. Natl. Acad. Sci.
U. S. A. 100, 13638–13643.
Petsche, H., Rappelsberger, P., 1970. Influence of cortical incisions
on synchronization pattern and traveling waves.
Electroencephalogr. Clin. Neurophysiol. 28, 592–600.
Petsche, H., Sterc, J., 1968. The significance of the cortex for the
traveling phenomena of brain waves. Electroencephalogr. Clin.
Neurophysiol. 25, 11–22.
Pitts, W., McCulloch, W.S., 1947. How we know universals. The
perception of auditory and visual forms. Bull. Math. Biophys. 9,
127–147.
Prechtl, J.C., Bullock, T.H., Keinfeld, D., 2000. Direct evidence for
local oscillatory current sources and intracortical phase
gradients in turtle visual cortex. Proc. Natl. Acad. Sci. U. S. A. 97,
877–882.
Rubino, D., Robbins, K., Hatsopoulos, N.G., 2006. Propagating
waves mediate information transfer in the motor cortex. Nat.
Neurosci. 9, 1549–1557.
Sanchez-Vives, M.V., McCormick, D.A., 2000. Cellular and network
mechanisms of rhythmic recurrent activity in neocortex. Nat.
Neurosci. 3, 1027–1034.
Salinas, E., Abbott, L.F., 1994. Vector reconstruction from firing
rates. J. Comp. Neurosci. 1, 89–107.
Shevelev, I.A., 1988. Visual recognition and scanning process
based on the EEG alpha wave. Perception 17S, 413.
Shevelev, I.A., Kostelianetz, N.B., Kamenkovich, V.M., Sharaev,
G.A., 1991. EEG alpha wave in the visual cortex: check of the
hypothesis of the scanning process. Int. J. Psychophysiol. 11,
195–201.
Shevelev, I.A., Kamenkovich, V.M., Bark, E.D., Verkhlutov, V.M.,
Sharaev, G.A., Mikhailova, E.S., 2000. Visual illusions and
traveling alpha waves produced by flicker at alpha frequency.
Int. J. Psychophysiol. 39, 9–20.
Siegel, R.M., 1998. Representation of visual space in area 7a
neurons using the center of mass equation. J. Comput.
Neurosci. 5, 365–381.
Silberstein, R.B., Burkitt, G.R., Cadusch, P.J., Wood, A.W., 2000.
Steady-state visual evoked potentials and traveling waves.
Clin. Neurophysiol. 111, 246–258.
Snippe, H.P., 1996. Parameter extraction from population codes: a
critical assessment. Neural Comput. 8, 511–529.
Theiler, J., Eubank, S., Longtin, A., Gldrikian, B., Farmer, J.D., 1992.
Testing for nonlinearity in time series: the method of surrogate
data. Physica 58D, 77–94.
Yakovenko, S., Mushahwar, V., VanderHorst, V., Holstege, G.,
Prochazka, A., 2002. Spatiotemporal activation of lumbosacral
motoneurons in the locomotor step cycle. J. Neurophysiol. 87,
1542–1553.