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” Effect : Blurs the signal

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 reconstruction

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), they lead to incorrect assumption of the structure- to-function relationship in the literature

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) –WROP (weighted resolution optimization) ; then LAURA (local autoregressive averages) (Grave de Peralta Menendez, 1997 ; 1999

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 mathematical modeling Used for both anatomical and functional purposes

34. The end