1. Depth and Surface EEG: Generation and Propagation
ME (Signal Processing), IISc: Neural Signal Processing, Spring 2014
Depth and Surface EEG: Generation and Propagation
Kaushik Majumdar
Indian Statistical Institute Bangalore Center
ME (Signal Processing), IISc: Neural Signal Processing

2. ME (Signal Processing), IISc: Neural Signal Processing
Baillet et al., IEEE Sig. Proc. Mag., Nov 2001, p. 16
Head, Brain and Signal
ME (Signal Processing), IISc: Neural Signal Processing
ME (Signal Processing), IISc: Neural Signal Processing

3. ME (Signal Processing), IISc: Neural Signal Processing
Mountcastle, Brain, 120: , 1997
Six Layer Brain
Excitatory post synaptic potential (EPSP)
ME (Signal Processing), IISc: Neural Signal Processing

4. Chemical-Electrical-Sequence
ME (Signal Processing), IISc: Neural Signal Processing
ME (Signal Processing), IISc: Neural Signal Processing

5. Electroencephalogram (EEG) and Electrocorticogram (ECoG)
LFP
Surface electrode
Depth electrode
ME (Signal Processing), IISc: Neural Signal Processing

6. ME (Signal Processing), IISc: Neural Signal Processing
LFP, ECoG and EEG
Comparison of SNR in EEG and ECoG for four patients P1 – P4. The ratio of SNR varies from 21 to 173. Ball et al., NeuroImage, 46: 708 – 716, 2009.
Buzsaki et al., Nat. Rev. Neurosci., 13: 407 – 420, 2012
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7. ME (Signal Processing), IISc: Neural Signal Processing
Signal Acquisition
Depth EEG
Scalp EEG
Curtsey: Freiburg Seizure Prediction Project, Germany
International system.
ME (Signal Processing), IISc: Neural Signal Processing

8. Information Propagation from Cortex to Scalp
Local field potential (LFP) is the most information-rich among all the functional brain signals.
LFP is also strongly correlated with blood oxygen level dependent (BOLD) functional magnetic resonance imaging (fMRI) signal.
ME (Signal Processing), IISc: Neural Signal Processing

9. ME (Signal Processing), IISc: Neural Signal Processing
EEG Forward Problem
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10. 3-Layer Realistic Head Model
Hallez et al., 2007
3-Layer Realistic Head Model
Brain
Skull
Head
ME (Signal Processing), IISc: Neural Signal Processing

11. Different Model Paradigms
Source Models
Head Models
Dipole Source Model
Distributed Source Model
Boundary Elements Method (previous slide)
Finite Elements Method
Finite Difference Method
Hallez et al., 2007
Majumdar, IEEE TBME, 56(4): 1228 – 1235, 2009
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12. ME (Signal Processing), IISc: Neural Signal Processing
Dipole Orientations
In a dipole source model number of dipoles must have to be fixed beforehand.
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13. Mathematical Formulation
Hallez et al., 2007
Mathematical Formulation
EEG
Lead filed matrix or gain matrix
Dipoles
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14. ME (Signal Processing), IISc: Neural Signal Processing
Hallez et al., 2007
Tissue Impedance
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15. Poisson’s Equation for the Head
Kybic et al., Phys. Med. Biol., 51: 1333 – 1346, 2006
Poisson’s Equation for the Head
ME (Signal Processing), IISc: Neural Signal Processing

16. Potential at any Single Scalp Electrode Due to All Dipoles
r is the position vector of the scalp electrode
rdip - i is the position vector of the ith dipole
di is the dipole moment of the ith dipole
ME (Signal Processing), IISc: Neural Signal Processing

17. EEG Gain Matrix Calculation
For detail of potential calculations see Geselowitz, Biophysical J., 7, 1967, 1-11.
ME (Signal Processing), IISc: Neural Signal Processing

18. Gain Matrix : Elaboration
The whole purpose of the forward problem or head modeling is to determine the gain matrix, which will have to be inverted in some sense during solving the inverse problem.
ME (Signal Processing), IISc: Neural Signal Processing

19. Boundary Elements Method for Distributed Source ModelIf
on the complement of a smooth surface then
can be completely determined by its values and the values of its derivatives on that surface.
This is utilized to solve the forward problem, with distributed source, by BEM. For detail see Kybic, et al., IEEE Trans. Med. Imag., 24(1): 12 – 28, 2005.
ME (Signal Processing), IISc: Neural Signal Processing

20. Inverse Problem : Peculiarities
Inverse problem is ill-posed, because the number of sensors is less than the number of possible sources.
Solution is unstable, i.e., susceptible to small changes in the input values. Scalp EEG is full of artifacts and noise, so identified sources are likely to be spurious.
ME (Signal Processing), IISc: Neural Signal Processing

21. Geometric Interpretation
||V-BJ||2Wn
||J||2Wp
Convex combination
of the two terms with
λ very small.
minU(J)
ME (Signal Processing), IISc: Neural Signal Processing

22. ME (Signal Processing), IISc: Neural Signal Processing
Derivation
U(J) = ||V – BJ||2Wn + λ||J||2Wp
= <Wn(V – BJ), Wn(V – BJ)> + λ<WpJ, WpJ>
ΔJU(J) = 0 implies (using <AB,C> = <B,ATC>)
J = CpBT[BCpBT + Cn]-1V
where Cp = (WTpWp)-1 and Cn = λ(WTnWn)-1.
ME (Signal Processing), IISc: Neural Signal Processing

23. ME (Signal Processing), IISc: Neural Signal Processing
Derivation
U(J) = ||V – BJ||2Wn + λ||J||2Wp
= <Wn(V – BJ), Wn(V – BJ)> + λ<WpJ, WpJ>
ΔJU(J) = 0 implies (using <AB,C> = <B,ATC>)
J = CpBT[BCpBT + Cn]-1V
where Cp = (WTpWp)-1 and Cn = λ(WTnWn)-1.
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24. ME (Signal Processing), IISc: Neural Signal Processing
Different Types
When Cp = Ip (the p x p identity matrix) it reduces to classical minimum norm inverse solution.
If we derive the current density estimate by the minimum norm inverse and then standardize it using its variance, which is hypothesized to be due to the source variance, then that is called sLORETA.
ME (Signal Processing), IISc: Neural Signal Processing

25. Low Resolution Brain Electromagnetic Tomography (LORETA)
ME (Signal Processing), IISc: Neural Signal Processing
Low Resolution Brain Electromagnetic Tomography (LORETA)
Pascual-Marqui et al., Int. J. Psychophysiol., vol. 18, p. 49 – 65, 1994.

26. Standardized Low Resolution Brain Electromagnetic Tomography (sLORETA)
U(J) = ||V – BJ||2 + λ||J||2
Ĵ = TV, where
T = BT[BBT + λH]+, where
H = I – LLT/LTL is the centering matrix.
Pascual-Marqui at
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27. ME (Signal Processing), IISc: Neural Signal Processing
sLORETA (cont)
Ĵ is estimate of J, A+ denotes the Moore-Penrose inverse of the matrix A, I is n x n identity matrix where n is the number of scalp electrodes, L is a n dimensional vector of 1’s.
Hypothesis : Variance in Ĵ is due to the variance of the actual source vector J.
Ĵ = BT[BBT + λH]+BJ.
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28. ME (Signal Processing), IISc: Neural Signal Processing
Form of H
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29. If # Source = # Electrodes = n
B and BT both will be n x n identity matrix.
with λ = 0
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30. ME (Signal Processing), IISc: Neural Signal Processing
sLORETA Result
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31. EEG Source Modeling at DLPFC
EEG Source Modeling at DLPFC
During behavioral inhibition the right DLPFC is more active than the left.
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32. ME (Signal Processing), IISc: Neural Signal Processing
References
S. Baillet, J. C. Mosher and R. M. Leahy, Electromagnetic brain mapping, IEEE Sig. Proc. Mag., 18(6): 14 – 30, 2001.
H. Hallez et al., Review on solving the forward problem in EEG source analysis, J. Neuroeng. Rehab., 4: 46, Available online at
ME (Signal Processing), IISc: Neural Signal Processing