Combining sparsity and rotational invariance in EEG/MEG source reconstruction.

Combining sparsity and rotational invariance
in EEG/MEG source reconstruction
Stefan Haufe a,b,∗, Vadim V. Nikulin c,d, Andreas Ziehe a,b,
Klaus-Robert Mu¨ller a,b,d and Guido Nolte b
aMachine Learning Group, Department of Computer Science, TU Berlin,
Franklinstr. 28/29, D-10587 Berlin, Germany
bIntelligent Data Analysis Group, Fraunhofer Institute FIRST, Kekule´str. 7,
D-12489 Berlin, Germany
cNeurophysics Group, Department of Neurology, Campus Benjamin Franklin,
Charite´ University Medicine Berlin, D-12203 Berlin, Germany
dBernstein Center for Computational Neuroscience, Berlin, Germany
Abstract
We introduce Focal Vector Field Reconstruction (FVR), a novel technique for the
inverse imaging of vector fields. The method was designed to simultaneously achieve
two goals: a) invariance with respect to the orientation of the coordinate system, and
b) a preference for sparsity of the solutions and their spatial derivatives. This was
achieved by defining the regulating penalty function, which renders the solutions
unique, as a global ℓ1-norm of local ℓ2-norms. We show that the method can be
successfully used for solving the EEG inverse problem. In the joint localization of
2-3 simulated dipoles, FVR always reliably recovers the true sources. The competing
methods have limitations in distinguishing close sources because their estimates are
either too smooth (LORETA, Minimum ℓ2-norm) or too scattered (Minimum ℓ1-
Preprint submitted to NeuroImage 13 June 2008

norm). In both noiseless and noisy simulations, FVR has the smallest localization
error according to the Earth Mover’s Distance (EMD), which is introduced here
as a meaningful measure to compare arbitrary source distributions. We also apply
the method to the simultaneous localization of left and right somatosensory N20
generators from real EEG recordings. Compared to its peers FVR was the only
method that delivered correct location of the source in the somatosensory area of
each hemisphere in accordance with neurophysiological prior knowledge.
Key words: EEG/MEG, Inverse Problem, Source Localization, Second-Order
Cone Programming, ℓ1-norm Regularization, Sparsity, Vector Fields, Rotational
Invariance
1 Introduction
Precise localization of neuronal activity is an important aspect for a better
understanding of brain functioning. Several functional imaging methods have
been developed for investigating this issue, including Single Photon Emission
Computed Tomography (SPECT), Positron Emission Tomography (PET) and
functional Magnetic Resonance Imaging (fMRI). These techniques provide
high spatial resolution of brain activity using metabolic indicators such as
blood oxygenation level (fMRI) or the concentration of radioactively marked
substances (SPECT/PET) in the tissue. Due to the slow response of the
metabolism, however, these measures cannot be used to assess rapidly varying
neuronal activity in a range of few milliseconds. Apart from measuring direct
neuronal activity, Electroencephalography (EEG) and Magnetoencephalogra-
∗ Corresponding author.
Email address: haufe@cs.tu-berlin.de (Stefan Haufe).
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phy (MEG) allow very accurate reconstruction of the time course of neuronal
signals with a microsecond precision (Nunez and Srinivasan, 2005). Impor-
tantly both techniques are noninvasive, and do not interfere with neuronal
activities. However, the signal arriving at the sensors contains contributions
from all areas of the brain, as well as external noise. The forward mapping
from cerebral sources to sensors is well-defined and can be described mathe-
matically with the help of a suitable model of the head. Inferring the sources
that lead to a certain measurement, on the other hand, is impossible, as in-
finitely many source configurations will fulfill the forward equation. In other
words, the inverse problem is ill-posed.
One strategy to still obtain a unique solution to the inverse problem is to
regularize, i.e. to restrict the search space to a sufficiently simple class of
sources. A common approach is to assume that the measured scalp pattern has
been generated by dipolar (point-like) sources (Scherg and von Cramon, 1986;
Mauguie`re et al., 1997; Komssi et al., 2004; Huttunen et al., 2006). Respective
approaches model a small number of dipoles, where the optimal number has to
be known in advance. The inversion is carried out by solving an overdetermined
nonlinear system in a least-squares sense. Unfortunately, the cost function of
dipole fits is highly nonconvex and the obtained solution depends heavily on
the initialization. Additionally, dipolar sources can be a poor approximation
if, e.g., the true sources are spatially extended and oriented normal to a folded
cortical surface.
An approach related to dipole fitting is dipole imaging. Imaging methods
model a large but fixed number of dipoles. These are arranged in a regular
grid covering the whole brain (or optionally just the cortical areas). Inferring
the dipole current vectors requires solving a heavily underdetermined system,
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in which the fulfillment of the forward equation constitutes only a constraint.
Several methods tackle the underdetermined nature of the problem by incor-
porating additional information. Very often temporal structure in the signal
is used, as for example in beamforming (Veen and Buckley, 1988), subspace
methods like MUSIC and (RAP)MUSIC (Schmidt, 1986; Mosher and Leahy,
1999) and the methods proposed in (Baillet and Garnero, 1997; Huang et al.,
2006; Malioutov et al., 2005; Cotter et al., 2005; Polonsky and Zibulevsky,
2004). The approach of Dale and Sereno (1993) imposes anatomical constraints
obtained from MRI on the sources. A general overview on inverse methods for
EEG and MEG is given by Baillet et al. (2001).
In this paper we focus on the situation in which only the scalp pattern at
one time point is available. In this case, imaging methods have to define an
additional quality criterion in order to obtain a unique solution. Ideally, this
regularizing criterion should encode prior knowledge on how a “good” solution
looks like. We here assume that a) brain sources are focal and we request
b) invariance with respect to rotations of the coordinate system. Standard
Minimum ℓp-norm solutions, weighted or not, are either rotationally invariant
but highly non-focal (p=2) or focal but violating rotational invariance (p=1).
We will propose an alternative consisting of a global ℓ1-norm of local ℓ2-norms
which fulfills both goals simultaneously. Local ℓ2-norms can be calculated both
of the sources (as in “standard” Minimum ℓ2-norm solutions and of their
second order spatial derivatives (as in LORETA). We here suggest to use a
specific combination of the two, relaxing the strict focality requirement in
favor of a more robust “simplicity” requirement.
This paper is organized as follows. In section 2 we will first give an overview
of existing methods and then we will present the mathematical details of our
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method. In section 3 we show illustrative examples in a simple constructed one-
dimensional scenario, followed by detailed simulated examples of EEG inverse
calculations and a case study using real EEG data from electric stimulation of
left and right median nerve. We finally discuss the results and give a conclusion
in sections 4 and 5, respectively.
2 Materials and methods
2.1 Inverse imaging
Let x ∈ RM denote a scalp pattern measured at M EEG or MEG sen-
sors. The current density in the brain is modeled by N dipolar sources di =
(rTi , sTi )T , i ∈ {1, . . . , N}. The locations ri ∈ R3 are kept fixed, so that the
quantities to be inferred are the dipole moment vectors si = (si,x, si,y, si,z)T , i ∈
{1, . . . , N}. Let s ∈ R3N = (sT1 , . . . , sTN)T be the vector containing the stacked
moments. As the relationships between source currents and EEG/MEG mea-
surements are linear, the forward equation just reads x = Ls in both cases.
The matrix L is called lead field matrix. It comprises information about ge-
ometric and conductive properties of the tissue. We will assume L to have
maximal rank, that is for EEG the reference electrode is not included in x
and L. If we require that the estimated solution explains the data exactly, the
inverse solution for an imaging methods can be cast as
sˆ = argmin
s
f(s) s.t. x = Ls, (1)
where f defines the imaging method. The choice of f choice crucially affects the
shape of the estimated source distribution, as there are much less constraints
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