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Cortical Patch Basis Model for Spatially Extended Neural Activity
1740
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 53, NO. 9, SEPTEMBER 2006
Cortical Patch Basis Model for Spatially Extended
Neural Activity
Tulaya Limpiti*, Student Member, IEEE, Barry D. Van Veen, Fellow, IEEE, and Ronald T. Wakai
AbstractA new source model for representing spatially dis-
tributed neural activity is presented. The signal of interest is
modeled as originating from a patch of cortex and is represented
using a set of basis functions. Each cortical patch has its own set
of bases, which allows representation of arbitrary source activity
within the patch. This is in contrast to previously proposed cor-
tical patch models which assume a specic distribution of activity
within the patch. We present a procedure for designing bases
that minimize the normalized mean squared representation error,
averaged over different activity distributions within the patch.
Extension of existing algorithms to the basis function framework is
straightforward and is illustrated using linearly constrained min-
imum variance (LCMV) spatial ltering and maximum-likelihood
signal estimation/generalized likelihood ratio test (ML/GLRT).
The number of bases chosen for each patch determines a tradeoff
between representation accuracy and the ability to differentiate
between distinct patches. We propose choosing the minimum
number of bases that satisfy a constraint on the normalized mean
squared representation accuracy. A mismatch analysis for LCMV
and ML/GLRT is presented to show that this is an appropriate
strategy for choosing the number of bases. The effectiveness of
the patch basis model is demonstrated using real and simulated
evoked response data. We show that signicant changes in per-
formance occur as the number of basis functions varies, and that
very good results are obtained by allowing modest representation
error.
Index TermsCortical bases, cortical patches, extended sources,
maximum-likelihood, MEG inverse problem, minimum variance
beamformer.
I. I
NTRODUCTION
E
LECTROENCEPHALOGRAPHIC
and
magnetoen-
cephalographic (EEG/MEG) techniques provide temporal
resolution on the order of milliseconds, the same time scale as
the underlying neural activity. This is a signicant advantage
relative to other functional neuroimaging modalities which
are limited by the time scales of metabolic or hemodynamic
processes. However, obtaining high spatial resolution is dif-
cult due to the inherent ill-posed nature of the EEG/MEG
inverse problem and relatively low signal-to-noise ratio (SNR)
Manuscript received August 13, 2005; revised January 29, 2006. This work
was supported in part by the National Institutes of Health (NIH) under Grant
R01HL0631742 and Grant R01NS037740. Asterisk indicates corresponding
author.
*T. Limpiti is with the Department of Electrical and Computer Engineering,
University of Wisconsin-Madison, 1415 Engineering Drive, Madison, WI
53706 USA (e-mail: tlimpiti@cae.wisc.edu).
B. D. Van Veen is with the Department of Electrical and Computer Engi-
neering, University of Wisconsin-Madison, Madison, WI 53706 USA (e-mail:
vanveen@engr.wisc.edu).
R. T. Wakai is with the Department of Medical Physics, University of Wis-
consin-Madison, WI 53706 USA (e-mail: rtwakai@facstaff.wisc.edu).
Digital Object Identier 10.1109/TBME.2006.873743
typical of EEG/MEG data. A wide variety of signal processing
algorithms have been proposed to address various aspects of
the inverse problem [1]. These algorithms are usually based
on a mathematical source model that describes the measured
activity. For example, parametric methods typically assume
that the measured activity is well described by a small number
of equivalent current dipoles of unknown strength and location,
while nonparametric or imaging methods tessellate the cortical
surface with a very large number of dipoles, typically 10 000 or
more, at known locations. Imaging methods employ the most
general source model, but this exibility leads to an extremely
ill-posed estimation problem that requires prior information or
strong regularization [1].
The equivalent current dipole is a reasonable model for focal
activity [2][4], but is inadequate for spatially distributed ac-
tivity since it cannot represent the spatial extent of the activity.
Use of the dipole model can also lead to signicant bias in
the localization of spatially distributed source activity. Recogni-
tion of the limitations of the dipole model have led researchers
to propose alternative source models for spatially extended ac-
tivity, including multipole expansions [5][8] and cortical patch
models [9][14]. Multipole expansions generally require high
SNR and are sensitive to errors in the forward model [6]. Cor-
tical patch models normally assume that the distribution of spa-
tial activity within the patch is known in advance [10][13]. This
can lead to signicant representation errors, so [10] presents a
procedure for determining the patch size that best ts the data,
assuming uniform activity within the patch. In this case, the
source model is a nonlinear function of patch size, so identi-
fying the best patch size requires an iterative search.
In this paper, we extend the cortical patch concept to develop
a source model that represents arbitrary spatially distributed ac-
tivity within the patch as a linear function of a small number of
unknown parameters. This model describes the distribution of
activity within each patch using a set of local basis functions.
The coefcients of the basis functions describe the activity dis-
tribution and are estimated from the data. Thus, in contrast to
previously proposed patch models, our patch basis model does
not assume the distribution of activity within the patch is known
a priori; both focal and spatially distributed activity are repre-
sented by the basis functions. The design of the patches and cor-
responding basis functions depends only on the forward model
and anatomy of the brain. Basis design is independent of the
measured data.
Furthermore, it is straightforward to extend most dipole based
source localization and/or estimation algorithms to the patch
basis model because the unknown activity distribution enters
into the source model in a linear fashion. We illustrate this exten-
sion using two common algorithms: linearly constrained min-
imum variance (LCMV) spatial ltering (see, e.g., [15] and [16])
and maximum-likelihood (ML) detection and estimation (see,
e.g., [17] and [18]). These algorithms are complementary in that
0018-9294/$20.00 © 2006 IEEE LIMPITI et al.: CORTICAL PATCH BASIS MODEL FOR SPATIALLY EXTENDED NEURAL ACTIVITY
1741
the ML approach assumes the signal of interest is in the mean
of the data, while LCMV exploits the spatial correlation matrix.
Both approaches can be used to either detect which patches con-
tain signicant activity or estimate the coefcients of the bases
for the patch [19], [20].
The number of basis functions used to describe each patch
determines the complexity of the source model and the degree
to which arbitrary activity within the patch can be accurately
represented. As the number of bases per patch increases, the
representation accuracy increases, but there is also increased
ability to represent noise and source activity originating exte-
rior to the patch of interest. Consequently, choosing the number
of bases involves a tradeoff between the ability to represent sig-
nals from within the patch and the ability to differentiate activity
from other regions. We propose using the minimum number of
bases that satisfy a constraint on the mean squared representa-
tion accuracy. Different algorithms have different tradeoffs be-
tween modeling error or mismatch and differentiation or reso-
lution, and often these tradeoffs are dependent on the data itself
through SNR. For example, the effect of mismatch is generally
more signicant at high SNR. We illustrate this tradeoff by an-
alyzing the effect of mismatch for the LCMV and ML methods.
One of the conclusions of this analysis is that the mean square
representation accuracy is an appropriate metric for choosing
the number of bases. The mismatch analysis also applies to con-
ventional dipole and xed patch models.
The remainder of this paper is organized as follows. Section II
develops our patch basis source model and proposes a metric
for choosing the number of basis functions. Section III extends
the LCMV and GLRT/ML approaches to the patch basis source
model and analyzes the effect of differing numbers of basis
fun