Neurophysiological significance of the inverse problemits relation to present “source estimate” methodologies and to future developments EEG : a poor spatial resolution We are repeated that EEG has a “poor spatial resolution”, although good temporal one : thus, assumptions on structure-to-function are loose Principe of EEG recording: volume currents What you want to know: where (projected on scalp) something is happening What you get: a large ususally bipolar propagation of the “source”

1. Neurophysiological significance of the inverse
problem
its relation
to present “source estimate” methodologies
and to future developments
E. Tognoli
Discussion group about Source Estimation
5 November 2004

2. I. A “poor spatial resolution”

3. EEG : a poor spatial resolution
• Priors :
–We are repeated that EEG has a “poor
spatial resolution”, although good
temporal one : thus, assumptions on
structure-to-function are loose

4. Let’s localize the sources
• Each of these 3 topographic maps
comes from a single dipole
activation of the cortex in a dipole
simulator program.
 Estimate the locations of these 3
single sources

5. Let’s localize the sources

6. Why?
II. What distorts the signal?

7. Volume conduction
• Principe of EEG
recording: volume
currents
• What you want to
know: where
(projected on scalp)
something is
happening
• What you get: a large
ususally bipolar
propagation of the
“source”

8. Anisotropy
• Anisotropy :
density/impedance of
tissues
• distort the topography of
the signal recorded from
the scalp, as compared to
the signal recorded from
the cortex
• Act as a spatial low-pass
filter
• Effect : Blurs the signal

9. More of anisotropy
• The sinuses: partially filled with… nothing
• Effect : displaces the signal

10. Brain folding
• Brain folding : the EEG
signal mainly originates
from pyramidal layers III &
V.
• The orientation of the
active patch of neuron is of
prime importance for the
projection of the activity
onto the electrodes
• Effect : displaces the signal

11. Brain folding

12. Source: Van Essen, 1997
Brain folding

13. “poor spatial resolution”?
• Blurring of the source
– Anisotropy
– Volume conduction
• Displacement of the source
– Cavities
– Brain folding
• The signal is corrupted both in its extent and in its
location

14. III. Some solutions ?

15. Some solutions?
If pretending to do some topographical analysis of
the EEG:
• Because of this corrupted correspondence
between the sources of bioelectrical activity and
their scalp topography, we are lead to work, not in
the space of the electrodes (maps/splines of raw
signals), but in the space of the currents
(maps/splines of estimated sources of raw signal)
• We do not record directly the cortex,
but do as if, with a mathematical

16. Some solutions?
• Solutions has been proposed through either :
– Mathematical transform (eg. Laplacian, CSD)
– Estimation/modeling (minimization of Laplacian in 3D
(voxel-based) models : inverse problem)

17. Some solutions?
• Solutions has been proposed through either :
– Mathematical transform (eg. Laplacian, CSD)
– Estimation/modeling (minimization of Laplacian in 3D
(voxel-based) models : inverse problem)

18. Some solutions?
• The Laplacian is
problematic for spatial
analysis of EEG data (eg.
coherence analysis), since it
projects correlated activities
in 2 (presumably silent)
unrelated locations of a
tangential foci : source and
sink
• Although these data can be
correctly interpreted (at
least for a few sources),

19. Some solutions?
• Solutions has been proposed through either :
– Mathematical transform (eg. Laplacian, CSD)
– Estimation/modeling (minimization of Laplacian in 3D
(voxel-based) models : inverse problem)

20. Some solutions?
• Estimation of sources : 2 approaches
– Dipole: one single (or a few) point-like sources, center
of mass of localized activity) : not useful for
spectral/coherence analysis : information is excessively
reduced
– Smooth current estimates (reconstruction of time series
at many (N’>N) spatial locations by estimating solutions
to the inverse problem)

21. Some solutions?
• No unique solution to the inverse problem
– Under-determination (ill-posed problem)
• Two crucial points for the accuracy of the estimation
– Performance/assumptions of the algorithm (given an
undetermined, noisy signal)
– Accuracy of the head model

22. IV. Algorithms
present methods

23. Algorithms
• Inverse problem with smooth solution
– MN (or MNE - minimum norm estimate, or LE, linear
estimation) (Hamalainen & Ilmoniemi, 1984)
– WMN (weight the contribution of sources regarding depth,
to rule out the bias toward high contribution from sources
close to the surface)
– LORETA (low resolution electromagnetic tomography)-
generalized weighted minimum norm/laplacian : extend the
properties of MN by projecting solutions in true 3D (VOXEL
BASED)(Pascual-Marqui)
– VARETA (variable resolution electric-magnetic
tomography) (Valdes-Sosa)

24. Algorithms
• Inverse problem with smooth solution
– More details in the next sessions

25. V. Head model
present and future methods

26. Spline/head model I
• The spline support of
the
transform/estimation :
( legacy of
Laplacian/CSD)
• Hjorth, 1975
• Perrin, 1987
• Law, 1995
• Babiloni, 1998

27. Remember the folding problem

28. Spline/head model I
• The spline support of
the
transform/estimation :
( legacy of
Laplacian/CSD)
• Hjorth, 1975
• Perrin, 1987
• Law, 1995
• Babiloni, 1998

29. Spline/head model II
• Configuration of sulci and gyri (orientation of
active cortical columns)
• Realistic (MRI-based) models
– FEM: extracts volumes of homogenous
conductivity.
– BEM : extracts boundaries between shells
(typically, scalp, skull, CSF, brain)

30. Spline/head model II

31. Increasing accuracy of the shells
• Typically now, 4 compartments: scalp, skull,
CSF, grey matter

32. The accuracy of the estimated
conductivities
• Typically, models use average conductivity
values (sampled in the literature)
• Idiosyncrasy of the conductivity
• In a given compartment, variability of the
conductivity

33. The accuracy of the estimated
conductivities
• Anisotropy is a major contributor :
• We can estimate (voxel-by-voxel) the
conductivity with EIT (Electrical Impedance
Tomography)
– inject small current wave of known properties
– record resulting waves
– since current take the path of least impedance, it’s
possible to compute, from the resulting wave
(shape distortion and temporal variations) the
distribution of conductivities.
• Minimal equipment, then extensive

34. The end