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A reference-independent
measure of potential is helpful for studying the multichannel EEG. The
potentials integrated over the surface of the body is a constant, i.e.
inactive
across time, regardless of the activity and distribution of brain
electric
sources. Therefore, the average reference, the mean of all recording
channels
at each time point, may be used to approximate an inactive reference.
However,
this approximation is valid only with accurate spatial sampling of the
scalp fields.
Accurate sampling requires a sufficient electrode density and full
coverage of
the head's surface. If electrodes are concentrated in one region of the
surface, such as just on the scalp, then the average is biased toward
that
region. Differences from the average will then be smaller in the center
of the
region, e.g. the vertex, than at the periphery. In this paper, we
illustrate
how this polar average reference effect (PARE) may be created by both
the
inadequate density and the uneven distribution of EEG electrodes. The
greater
the coverage of the surface of the volume conductor, the more the
average
reference approaches the ideal inactive reference.

1.
Introduction

The electroencephalogram
(EEG) is measured with a differential amplifier, such that the voltage
time
series reflects the difference in electrical potential between two
electrodes.
It is often assumed that one electrode site is an inactive reference,
such that
the potential at that site is constant across time. Yet this assumption
has
been known to be incorrect for many years (Nunez, 1981; Vaughan,1982;
Lehman
and Skrandies, 1984). There is no site on the surface of the human head
that
can be assumed to remain at a constant potential during the activity of
brain
electrical sources of unknown locations and orientations (Nunez,1981).
Rather,
the `reference' sites such as the earlobes, mastoids, or nose, can be
shown to
vary in their electrical potentials during the event-related potential
(ERP)
epoch. These reference sites are influenced differentially by differing
ERP
components, such that the reference error varies unpredictably with
experimental conditions that manipulate ERP components.

Even for sites
traditionally used as an `inactive' reference, such as the earlobes,
the
variation of the potential over time must not be neglected. Not only
may
sources inthe nearby temporal lobe affect this reference site, but also
distant
generators whose electrical fields propagate to the reference site
through
volume conduction.

The study of scalp topography, therefore, requires a
reference-independent measure of the potential field. Some researchers
have
turned to the current density measure, estimated by the Laplacian
derivation
(the second spatial derivative)of the potential surface, which is
reference independent.
However, adequate spatial sampling is required to compute the spatial
derivative, such that current density measures with 19- or 32-channel
recordings will be unstable and inaccurate (Srinivasan et al., 1996;
Junghoeferet
al., 1997). Furthermore, even for high spatial density EEG, the second
spatial
derivative is much more noise vulnerable than the scalp potential
distribution.
Problems result from the inability to interpolate accurately (and thus
to
create accurate derivatives) with inadequate spatial sampling (below
the
spatial Nyquist frequency) and from the errors in determining the
electrode
positions. Furthermore, the current density measure is insensitive to
deep sources,
whereas the potential measure reflects deep and superficial sources.
Because of
these factors, a method is required to estimate the
reference-independent
potential.

An elegant solution to this problem might rely on the introduction
of constraints based on physical principles. Specifically, the surface
integral
of the potential on a surface that completely encompasses all the
active
sources must be zero (Bertrand et al., 1985). The dipolar fields of
each source
are fully represented by the surface potentials. A positive radial
source near
the surface of the right side of the head produces a tight,
high-amplitude
positive field over that side. The opposing field is diffuse and
low-amplitude over
the left side, but with complete measurement the integrals of the two
sides
cancel exactly. The question for practical EEG recording is whether an
average
reference fulfils the criterion of adequate measurement of the surface
integral.

Using simulated and real evoked potential data, Desmedtet al. (1990)
and Tomberg et al. (1990) demonstrated that the average reference
shifts across
time, due to the lack of coverage of the head's surface. As pointed out
by
Pascual-Marqui and Lehmann (1993), dipole source modeling is not
affected by
the reference problem, as long as the potential fields are described
properly
(i.e. the location of the reference is included in the model). On the
other
hand, the reference bias, i.e. extent that the surface integral
deviates from zero,
can be an important clue to the extent of undersampling of the head
surface
potential field. Furthermore, dipole modeling `in the dark' can be
dangerous: a
careful study of scalp surface topography may be essential to eliminate
unreasonable dipole solutions. Thus, an active reference may seriously
affect
various stages of the inspection, analyses and interpretation of EEG
and ERP
data (Dien, 1998). The least active reference is the one of choice, and
with
inadequate surface sampling, this is not necessarily the average
reference. The
present paper examines the activity of the average reference in
dependence of spatial
sampling and portion of scalp covered by electrodes, and uncovers the
consequences

As will be
demonstrated, the first requirement to minimize the average reference
problem
is a dense spatial sampling. Current research EEG systems provide
measurement with
64, 128, or 256 electrodes (Gevins et al., 1995; Tucker, 1993). The
second
requirement, for electrodes to cover the whole surface, is more
difficult. In
the present paper, using both simulated source configurations as well
as real
data, we show the bias created when the average reference is computed
from less
than the whole surface of the volume conductor. In a first step,
simulated data
are constructed from variable source configurations. Then the
differences
between the potential distribution (referred to the `true'
average-reference)
and the voltage distribution(referred to the average reference based on
the
detectedpotential) are quantified. The results of these simulations
confirm
what has been observed in empirical studies ofthe average reference in
EEG and
ERP recordings (Curranet al., 1993; Chung et al., 1996): the potential
amplitudes aresmaller for electrodes located in the center of the
electrode array
than for those located on the edge. We refer to this bias as the polar
average
reference effect (PARE). Since the unknown reference bias for a single
time
point changes across samples ± as the volume conducted sources affect
the
reference differently over time ± PARE varies overtime. We then
demonstrate
that, although it is reduced in magnitude, the PARE can be shown to
remain in
128-channelERP data sets. Finally, we suggest two possible ways to
correct the
PARE and discuss the consequences of the PARE for the interpretation of
the
distribution of scalppotentials and statistical analyses.

2.
Methods and results

2.1. Simulations of the
PARE with absolute potential fields

With simulations of a
4-sphere model of head conductivity, we can compute the surface
integral and
then measure the bias of the average reference (PARE)
systematically.This
section describes the calculations of various electrode and source
configurations
with respect to the PARE.

Any arbitrary source configuration can be described
as the sum of a number of stationary current dipoles. Hence, it is
sufficient
to consider the effect of one such single current dipole. The head was
modelled
as an isotropic volume conductor of spherical geometry composed of
4homogeneous
layers, each with a different conductivity (Cuffin and Cohen, 1979).
The layers
model scalp, skull, cerebra-spinal fluid (CSF), and brain with

The radius of the head R
was assumed to be
9.20
cm, the radius of the brain 8.15 cm, the thickness of skull and scalp
were each
set to 4.2 mm, and the thickness of the CSF to 2.1mm. Due to the
spherical
symmetry, it is sufficient to vary the location of the sources in the
two
dimensional r/theta(radius/polar angle) plane
and to
restrict the configuration to electrodes in the same plane. The
following
variables remain: position of the source and the orientation of
itsdipole
moment, proportion of the sphere covered with electrodes and spatial
sampling.
Any possible orientation of the dipole moment can be constructed from
the sum
of its radially and its tangentially polar components (theta
orientation).Fig.
1a illustrates an example of such a source and electrode configuration:
the
polar angle q varies from left mastoid (2908) via C3 (2458), Cz (08), C4 (1458 to the right mastoid (1908). A radial dipole has
an orientation of908, is positive and its depth
(radius) is set
at 16 mm below the cortical surface. In this and
the following examples, the electrode
configuration is symmetrical with respect to the vertex. The spatial
sampling
rate can be calculated using the corresponding positions located on the
opposite side of the sphere. There are 8 electrodes on the circle (3608), resulting in a spatial
sampling rate of 8/2pi.

The
black solid line in Fig.1b refers to the `ideal' potential distribution
(the
`true' average reference) of the dipole as described in Fig. 1a. The
abscissa
refers to the polar angle of the scalp positions. The proportion of
coverage of
the assumed electrode configuration corresponds to a half circle, i.e.
1808. All other lines in Fig. 1b
show the potential distributions subtracted
by an average reference potential which was computed using different
spatial
sampling rates. As is to be expected, the average reference effect
becomes
smaller as the spatial sampling rate increases. But even for a near
continuous
spatial sampling (graysolid line), a distinct effect remains due to the
insufficient
coverage of the surface of the sphere with electrodes only on the
scalp. For a
30-channel recording (gray dashed line), the effect size amounts to
approximately 16% of the maximal amplitude. This ratio drops to 11% for
a
128-channelrecording (black dashed line) and would remain 8% for
continuous
sampling (gray solid line), where the bias is due only to the
insufficient
coverage. As illustrated in Fig.1c, this effect size, thus decreases
logarithmically with the spatial sampling rate. Fig. 1d shows the
change of the
potential that occurs when the average reference is subtracted.

The generalization
of these effects to arbitrary radial source orientations of comparable
depth is
demonstrated in Fig. 2. Here we consider the locations and orientations
that
are exemplified by the course of the potential distribution prior to,
and
after, the transformation, displayed in Fig.2b.As is to be expected,
the effect
of transformation to average reference exhibits a cosinal distribution.
Consequently, the effect is largest for a dipole pointing towards the
vertex.

Fig. 2c corresponds to
Fig. 1d; it describes the generalized effect of change in absolute
value. While
the abscissa(sensor position) is the same as in Fig. 1d, the ordinate
represents
the dipole orientations indicated by Fig. 2a.The last row of the matrix
illustrated in Fig. 2c (last slice of the figure, dipole orientations .2908) is identical to Fig.1d. The
average across all possible source
orientations is displayed in Fig. 2d and shows a u-shaped polar
distribution in
the amount of potential deviation. Potentials from inferiorly located
sensors
are enlarged in amplitude while potentials from superior sensors appear
smaller
than they really are.

Let us consider this PARE more closely. The change to tangentially oriented dipoles (in Fig. 3a) results in a qualitatively similar distribution (Fig. 3b). Inversion of the polarity will, of course, not alter this distribution. The order of the size of the PARE is considerably larger for inferiorly located tangential sources than for the radial ones, since the tangential sources will give rise to greater potentials in those areas of the head surface that are not covered by the sensor array. The polarization of the supratemporal plane resulting from auditory stimulation is an example of a source distribution extending beyond the typical recording array.

When, in the next step, the average reference effect is calculated for different depths of the dipole locations and different polarities (Fig. 3c), the polar bias again appears(Fig. 3d). Hence, considering the effect across all possible source configurations, the size of the potential will be magnified at sensors in inferior regions, but diminished a tthe superior locations. Any data set that is transformed to average reference is contaminated by the PARE. The magnitude of this effect, however, gets smaller with increasing proportion of

area covered by
the sensory array (Fig. 4) and, also, with increasing
spatial sampling(Fig. 1c).

The consequences of
the PARE may be illustrated for
a 3dimensional arrangement of sources and sensors. In this example, we
simulate
spatially correlated noise, such as is created by the background EEG in
an
averaged ERP study. We assume dipoles are located at 42 different
locations,
the amplitude of which are constant across a hundred trials while the
direction
in space of the moment varies randomly.

The active dipoles in
Fig. 5a were restricted to those 26spatial locations that have positive
z
values. Hence, the area covered by the modelled 128-channel recording
was
considerable. Fig. 5d presents the interpolated distribution of
standard deviations
across all trials, after transformation of the simulated potentials to
average
reference. Despite the considerable coverage by the sensors, a strong
PAR
effect remains; the standard deviation of the most superiorly located
electrode, Cz, amounts to only 80% of the most inferior ones. If all 42
dipoles
are activated, the result is qualitatively the same, but the difference
in
standard deviation increases by another 5%. The reason for this lies in
a secondary
AR effect; active sources that are located below the area covered by
the
sensors affect the inferiorly located sensors to a greater extent than
the ones
located at superior positions. In the present example, the secondary
effect is relatively
small because the electrode configuration, as displayed in Fig. 5b, has
considerable coverage ± even forthe globally distributed source
configuration.
Even then, there are only a few model dipoles that are located beyond
the
covered space.

This effect, however,
will gain increasing weight if coverage is insufficient. In Fig. 5c
only the 82
electrode positions with a positive z value were used in the
calculation of the average reference; these positions were located
above the
upper dashed line of Fig. 5d. The polar distribution is qualitatively
similar,
but larger; the standard deviation at the vertex electrode drops to
only 50% of
that for the most inferiorly located sensors. This area of coverage is
by no
means unrealistic; it reflects that of the international10±20 system.
To make
this point even more explicit, the effect can be illustrated for a
simple focal
active source that varies as a function of time. Assuming the
82-sensorconfiguration
and 42 source locations, as in Fig. 5c, in Fig. 5e, an additional
radial
source, located below the vertex, was consistently activated in all the
trials
with asinusoidal time course and an amplitude that amounted to25% of
the total
amplitude averaged across all 42 noise sources. The signal-to-noise
ratio was
1:4 in one trial and, hence, 2.5 after averaging across the 100 trials.
The
potential distribution shows a pronounced PAR effect (Fig. 5e)with a
signal-to-noise ratio of only 1.25 at the inferior locations as opposed
to 2.5
at the vertex electrode. This dependence of the signal-to-noise ratio
on the
sensor location must not be neglected.

2.2. Examples of the
PARE in dense array ERP recordings

Of course, because the
differences in potential can be reattributed within a common reference
recording array through subtraction, the PARE is independent of the
choice of
reference electrode for recording purposes. Although the PARE decreases
with
the improved spatial sampling of a dense array EEG recording, the
following
example suggests that it can still be observed. The upper column of
Fig.6 (a,b)
illustrates the distribution of a Bereitschaftspotential recorded in a
standard
design: the subject was asked to voluntarily press a button with the
left (left
column in Fig. 6), or right (right column),index finger about every 3
s. Data
was low-pass filtered with40 Hz and backward averaged, using the EMG
onset as a
trigger. Then the mean was calculated for each sensor across an epoch
beginning
500 ms prior to the motor response. The Bereitschaftspotential shows
the
typical frontocentral distribution with contralateral dominance.

The lower columns of
Fig. 6c±f illustrate the scalp distribution of the standard deviations,
averaged over the time interval mentioned above. These standard
deviations of
the averaged ERP primarily reflect the background (response
asynchronous) EEG
of the recording. The standard deviation distributions for left and
right movement
look surprisingly similar, although the lateralization of the potential
distributions differ distinctly. In both cases, the standard deviation
is
smallest in left parietal regions. Higher standard deviations are
detectable in
frontal and right occipital regions .It is tempting to conclude that
the
generators in the sensory motor cortex produce a rather constant
activation across
trials while other areas vary greatly from trial to trial in their
degree of
activation. Unfortunately, these very interesting distributions are
contaminated by the PAR effect. As outlined above, we can not reliably
estimate
the magnitude of the PARE unless the generator structure is known.
Therefore, any
interpretation of the distribution of standard deviation calls for
considerable
caution.

2.3. PARE correction

As obvious from the
theoretical considerations, the PARE effect is a consequence of a
failure to
fully sample the surface potential on the head. More precisely, it is
the
average potential across the area not covered by electrodes, which is
not
measured. If it could be estimated, the PARE effect might be
compensated for.
The following section examines this possibility. For this purpose, we
again
assume± as in Fig. 5 ± 42 source locations. The source dipoles vary
randomly in
direction and amplitude of their moments across 200 trials. Let us
begin
assuming one active sourc per trial.If there is no measurement error,
the
surface integral, i.e. the average potential across the head's surface,
must be
constant across time. Consequently, the average reference that results
from
averaging a potential across all electrodes using any reference
electrode must
also be constant, provided the sensors cover the head surface
completely. The
average potential for a 129-electrode recording which does not
completely cover
the body surface varies with brain activity. Across the 200-source
configurations,
its standard deviation amounts to about 20% of the standard deviation
of the
potential at a single electrode. (Note that for this simulation study,
the
standard deviations at the individual lelectrode sites are almost
identical).

The standard deviation
of the average divided by that o fone of the individual electrodes can
be
described as the `average reference coefficient'. The examples of the
potential
distribution presented in Fig. 7 depict the amount of activity missed
by the
129 sensor net for a single dipole(Fig. 7a) and a multiple random
source configuration
(Fig.7b). In both cases, this activity is different from zero, causing
the PARE.
The information from the 129 electrodes maybe used to estimate the
potential on
the surface not covered by electrodes using a spherical spline
interpolation.
In a second step, the average reference can be computed for the entire
surface
using the results from the interpolation. This procedure reduces the
standard
deviation of the average reference to only 10% of the standard
deviation across
source configurations at any single electrode, i.e. it halves the error
in
estimating the average reference and, thus, also reduces the PARE.

The possibility of
estimating the true average reference is facilitated if the sources are
deep
and the spatial sampling is high. Fig. 8 illustrates the dependency on
both,
the number of channels and the depths of source generators, to properly
estimate
the average reference. Again, a measure m for
the usefulness of the correction can be based on the average reference
coefficient
m.

Generally, the
average reference coefficient
varies between 16 and 20% (gray dashed lines). Since the proportion of
the area
covered by the sensor array increases with spatial sampling, the
uncorrected
average reference coefficient is a few percent greater for the
129-channel
recording than for the 33 channel net. As the probability of detecting
peaks in
the potential distribution increases with the densit yof the array, the
likelihood that the average reference is different from zero will also
increase. Consequently, the PARE is likely to be greater for higher
numbers of
sensors given the same proportion of area covered. For the uncorrected
average
reference coefficient, the depth of the sources has little influence.
The
measure m for the usefulness of the
correction increases
nearly linearly with the number of sensors. With an average over single
and
multiple random sources of different depths we obtain m. 0% for the
33-channelconfiguration, meaning that PARE correction is not possible.
Using 65
channels the averaged m
became 15% and in
the
case of a 129-channel system the value doubles to 30%.The implication
is that
the PARE is more accurately characterized when the full field
topography is
more accurately estimated with the 128-channel sampling. Instead of
using the
spherical spline interpolation, an alternative approach can be based on
the
estimation of the source configuration and, subsequently, the
calculation of the
average potential from the forward solution of the estimated sources.
Additional gain from this approach is limited.

3.
Discussion

The inaccuracy of the
average reference, described by the polar average reference effect, is
substantial when the number of electrodes is small and when a small
proportion of
the surface of the head is covered. With increasing coverage and a high
electrode density, the PARE is reduced. However, even if the surface of
the
head is covered by an electrode net extending across 2708, this effect is not negligible
.It affects further steps of EEG
analysis. We emphasize again, that the problems arising from the PARE
cannot be
resolved by simply choosing a different type of reference. The only
solution to
minimizing the effect is to use a high density electrode array that
covers as
large a portion of the head surface as possible. Since current source
density
calculations are unstable and inaccurate when based on insufficient
spatial
sampling recordings (e.g. with 19- or32-channel), these transformations
for
solving the reference dependencies should not be used with a small
number of electrodes.
We are currently evaluating to what extent an electrode that is
inserted into
the oral cavity might supplement a 128-channel geodesic sensor net to
improve
the characterization of the electrical fields at the inferior surface
of the
brain-case.

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