Source Estimation in EEG

1. Source Estimation in EEG The forward and inverse problems Christophe Phillips, Ir, Dr Cyclotron Research Centre, University of Liège, Belgium

2. Agenda • Introduction Why , what for, where, how... • Part I : the forward problem From sources to electrodes • Part II : the inverse problem From electrodes to sources

3. EEG Recordings Electroencephalography (EEG) is ‘simply’ about recording electromagnetic signals produced by neuronalactivity : EEG signal is spread in space and time.

4. Neurone Dendrites Cell body Synaptic terminals Axon EEG signal: origin • Head anatomy: • gray matter, • white matter, • CSF, • bone, air, skin, mucle, etc.

5. Cell body About 106 synapses must be simultaneously and effectively active to produce an evoked response. Dendrites Synaptic terminals Axon About 40 to 200 mm2 of active cortex. EEG signal: origin • From a distance,postsynaptic potential (PSP)looks like acurrent dipoleoriented along the dentrite. • Pyramidal cells have parallel dendrites, oriented perpendicular to the cortical surface. • Typical dipole strength : ~20 fA m(1fA = 10-15A) • Current-dipole moments required to be measured outside the head : ~10 nA m(1nA = 10-9A)

6. PSP AP 100 mV 10 mV 1 ms 10 ms EEG signal: origin • What about action potentials ? • large but brief potential compared to PSP • modelled by a current-quadrupole •  the field decreases with distance as 1/r3, compared to 1/r2 for the PSP dipole. EEG signals are produced in large part by synaptic current flow, which is approximately dipolar.

7. Fitted haemodynamic response function • EEG • fMRI Signal change EEG vs fMRI Recordings Difference between haemodynamic and electromagnetic signals produced by neuronal activity as recorded by :

8. Preconditions for signal detection EEG signal fMRI signal • Activation of a neural population must besynchronous in time. • Active neural population must bespatially organised (parallel fibers). • The sources must be in an ‘open-field’ configuration. • Neural activity needs not be synchronous in time. • Geometrical orientation of the sources is totally irrelevant.

9. Signal Sensitivity EEG signal fMRI signal • Measured signal is sensitive in relative timing and amplitude of activity. • Critical neural activity needs not be extended in time. • Measured signal amplitude influenced by both duration and amplitude of activity. • Signal detected only if net haemodynamic changes

10. Forward Problem Inverse Problem Source Localisation in EEG

11. Head anatomy Neurone Dendrites Cell body Synaptic terminals Axon Head model : conductivity layout Source model : current dipole Solution byMaxwell’s equations Forward Problem

12. v, potential at the electrodes, (Nel x 1) , source(s) location, [rx , ry , rz]’ , source(s) orientation & amplitude [jx , jy , jz]’ f, solution of the forward problem , additive noise, (Nel x 1) where Inverse Problem Solving is an “ill-posed” problem, i.e. the inverse problem does NOT have a unique solution, (von Helmholtz, 1853).

13. Source Estimation in EEG Part I : The forward problem

14. Agenda Part I : Solving the forward problem • Maxwell’s equations • Analytical solution • Numerical solution • Pseudo-sphere approach • Conclusion

15. Maxwell’s equations (1873) Ohm’s law : Continuity equation :

16. No propagation phenomenon, i.e. changes in the field/sources are «instantaneous». Derivative terms can be discarded. Quasistatic approximation • Maxwell’s equations can be simplified because : • EEG frequencies are genrally below 100hz. • Cellular electrical phenomena contain mostly frequencies below 1kHz.

17. With • quasi-static approximation of Maxwell’s equations • and dipolar current sources conductivity V electric potential current source where Simplified Maxwell’s equation i.e. a “simple” mathematical relationship linking current sources and electric potential, depending on the conductivity of the media.

18. From find Solving the forward Problem where f(.) will depend on the conductivity s of the media (and the boundary conditions). The head model is the conductivity layout adopted !

19. From find Solving the forward Problem • f(.) can be estimated • analytically, i.e. an exact solution ‘formula’ exists, • numerically, i.e. numerical methods provide an approximation of the solution.

20. Agenda Part I : Solving the forward problem • Maxwell’s equations • Analytical solution • Numerical solution • Pseudo-sphere approach • Conclusion

21. From find Analytical solution • f(.) can be estimated analytically only for particular cases: • higly symmetrical geometry, e.g. spheres, cubes, concentric spheres, etc. • homogeneous isotropic conductivity These are very restricted head models!

22. Analytical solution, cont’d Human head is not spherical neither is its conductivity homogeneous and isotropic...

23. Agenda Part I : Solving the forward problem • Maxwell’s equations • Analytical solution • Numerical solution • Pseudo-sphere approach • Conclusion

24. From find Numerical solution, 1 • f(.) can be estimated numerically for ANY conductivity layout! • The most general method is the “Finite Element Method”, or FEM: • conductivity can be arbitrary, i.e. anisotropic and inhomogeneous, • potential is estimated throughout the volume.

25. Numerical solution, FEM • Principles of the FEM: • The (head) volume is tesselated into small volume elements on which Maxwell’s equation is solved locally. • The conductivity is defined for every volume elements individually. • Drawbacks: • How to build up the model and define the conductivity at each element ? • Computation time is huge !

26. From find Numerical solution, 2 • f(.) can be more easily estimated numericallywith some assumptions: • volume divided into sub-volumes of homogeneous and isotropic conductivity, • potential is only estimated on the surfaces seperating those sub-volumes. • This is the “Boundary Element Method”, or BEM.

27. Numerical solution, BEM The surfaces are tessellated into flat triangles and the potential is approximated on each triangle as a constant or linear function. Example of BEM head model : Brain surface The sources, current dipoles, are placed in the brain volume. Skull surface Scalp surface

28. BEM, Potential approximation On each triangle of the surfaces, the potential function can be approximated by : a linear function between the potential at the vertices (LPV)

29. Homogeneous volumes definition BEM head model Scalp surface with electrode locations BEM application

30. Error increases with sharpness of distribution. BEM fails when the size of surface elements is ‘large’ compared to the sharpness of potential distribution. BEM limitations Superficial dipoles have sharper potential distribution.

31. Agenda Part I : Solving the forward problem • Maxwell’s equations • Analytical solution • Numerical solution • Pseudo-sphere approach • Conclusion

32. Anatomically constrained spherical head models, or pseudo-spherical model. Solutions: Analytical vs. numerical • The head is NOT spherical: •  cannot use the exact analytical solution because of model/anatomical errors. • Realistic model needs BEM solution: •  surfaces extraction •  computationnaly heavy •  errors for superficial sources • Could we combine the advantages of both solutions ?

33. Pseudo-spherical model Scalp (or brain) surface Best fitting sphere: centre and radii (scalp, skull, brain) Spherical transformation of source locations Leadfield for the spherical model

34. Fitted sphere and scalp surface Applications, scalp surface

35. Applications, cortical surface

36. Agenda Part I : Solving the forward problem • Maxwell’s equations • Analytical solution • Numerical solution • Pseudo-sphere approach • Conclusion

37. Conclusion Solving the « Forward Problem » is not very exciting neither easy but it is crucial for any attempt at source estimation. The key elements are the head model and, from it, the solution used. • Model : • simple  define 3 sphere shell • realistic  extract volume surfaces • most realistic  extract volume conductivity • Solution : • analytic  easy and quick but anatomical errors • numeric  slower, more anatomically accurate but numerical erros

38. So far we still have NOT localised anything…

39. Source Estimation in EEG Part II : The inverse problem

40. Agenda Part II : Solving the inverse problem • Introduction • Equivalent current dipole solution • Distributed linear solution • Other solutions • Conclusion

41. v, potential at the electrodes, (Nel x 1) , source location, [rx , ry , rz]’ , source orientation & amplitude [jx , jy , jz]’ f, solution of the forward problem , additive noise, (Nel x 1) where Inverse Problem Solving • Function f is • linear w.r.t. the source orientation & amplitude • non-linear w.r.t. the source location

42. Parameters When Ns sources are present the problem to solve is • For each source, there are 6 parameters : • 3 for the location, [xyz] coordinates, • 3 for the orientation and amplitude, [jx jy jz] components • or • 3 for the location, [xyz] coordinates • 2 for the orientation, [qj] angles • 1 for the amplitude, j intensity/strength

43. Parameters, cont’d • The inverse problem is « ill posed », i.e. in general there is no unique solution: • Number and location of active sources are unknown! • Measurements from just Ne electrodes. To uniquely solve the inverse problem assumptions/constraints on the solution MUST be adopted. Those constraints define the form of the solution !

44. 2V 2V 12 Ω 6 Ω 12 Ω 6 Ω 12 Ω 6 Ω 4 Ω 3V 6V Inverse Problem, example Example : 4 different networks but with the same measurable output: 2V and 4Ω. S1 S2 S3 • If we constrain the solution to have : • the smallest source  solution S3 • the smallest but deeper source  solution S2 • source along the 12Ω resistor solution S1

45. Equivalent Current Dipole • With the ECD solution : • A priorifixed number of sources considered, usually less than 10 •  over-determined but nonlinearproblem •  iterative fitting of the 6 parameters of each source. Problem : How many ECDs a priori ? The number of sources limited : 6xNs < Ne Advantage : Simple focused solution. But is a single (or 2 or 3 or…) dipole(s) representative of the cortical activity ?

46. With the DL solution : • “All” possible (fixed) source locations (>103) considered simultaneously •  largely under-determined but linear problem : •  external constraints required to calculate a unique solution Distributed Linear solution Problem : What should be the constraints ? Advantage : 3D voxelwise results.

47. Agenda Part II : Solving the inverse problem • Introduction • Equivalent current dipole solution • Distributed linear solution • Other solutions • Conclusion

48. which is an over-determined but non-linear problem. Once the location is fixed  The cost function to be optimised is : To be iteratively minimised only w.r.t the , i.e. 3 parameters per source. ECD solution For Ns sources, the problem can be rewritten as

49. For Ns sources, the cost function can be minimised for the using any nonlinear procedure, e.g. gradient descent, simplex algorithm, etc. ECD solution, cont’d At each iteration the leadfield Lmust be recalculated(many times) as the source locations are modifed Once the location of the sources is determined, their intensity is obtained by

50. The iterative optimisation procedure can only find a local minimum • the starting location(s) used can influence the solution found ! Value of parameter Cost function Local minimum Local minimum Global minimum ECD solution, cont’d • For an ECD solution, initialise the dipoles • at multiple random locations and repeat the fitting procedure  focal cluster of solutions ? • at a «guessed» solution spot. 1D example of optimisation problem: