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Cite as: “R.D. Pascual-Marqui: Discrete, 3D distributed, linear imaging methods of electric neuronal activity. Part 1: exact, zero
error localization. arXiv:0710.3341 [math-ph], 2007-October-17, http://arxiv.org/pdf/0710.3341 ”
Page 1 of 16
Discrete, 3D distributed
linear imaging methods of electric
neuronal activity. Part 1: exact, zero error localization
Roberto D. Pascual-Marqui
The KEY Institute for Brain-Mind Research
University Hospital of Psychiatry
Lenggstr. 31, CH-8032 Zurich, Switzerland
pascualm at key.uzh.ch
1.
Abstract
This paper deals with the EEG/MEG neuroimaging problem: given measurements of
scalp electric potential differences (EEG: electroencephalogram) and extracranial magnetic
fields (MEG: magnetoencephalogram), find the 3D distribution of the generating electric
neuronal activity. This problem has no unique solution. Only particular solutions with
“good” localization properties are of interest, since neuroimaging is concerned with the
localization of brain function. In this paper, a general family of linear imaging methods with
exact, zero
error localization to point-test sources is presented. One particular member of
this family is sLORETA (standardized low resolution brain electromagnetic tomography;
Pascual-Marqui,
Methods Find. Exp. Clin. Pharmacol. 2002, 24D:5-12;
http://www.unizh.ch/keyinst/NewLORETA/sLORETA/sLORETA-Math01.pdf). It is shown here that sLORETA has
no
localization bias in the presence of measurement and biological noise. Another member of
this family, denoted as eLORETA (exact low resolution brain electromagnetic tomography;
Pascual-Marqui 2005:
http://www.research-projects.unizh.ch/p6990.htm), is a genuine inverse solution
(not
merely a linear imaging method)
with exact, zero error localization in the presence of
measurement and structured biological noise.
The general family of imaging methods is
further extended
to include data-dependent (adaptive) quasi-linear imaging methods, also
with the exact, zero error localization property.
2.
The forward equation
Details on the electrophysiology and physics of EEG/MEG generation can be found in
Mitzdorf (1985), Llinas (1988), Martin (1991), Hämäläinen et al (1993), Haalman and Vaadia
(1997) , Sukov and Barth (1998), Dale et al (2000), Baillet et al. (2001). The basic underlying
physics can be studied in Sarvas (1987).
Consider the forward EEG equation:
Eq. 1:
where the vector
contains instantaneous scalp electric potential
differences
measured at
electrodes with respect to a single common reference electrode (e.g., the
reference can be linked earlobes, the toe, or one of the electrodes included in
); the matrix
is the lead field matrix corresponding to
voxels;
is the current