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

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

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

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

EEG signal: origin Dendrites Cell body Axon Synaptic terminals From a distance, postsynaptic potential (PSP) looks like a current dipole oriented along the dentrite. Pyramidal cells have parallel dendrites, oriented perpendicular to the cortical surface. Typical dipole strength : ~20 fA m (1fA = A) Current-dipole moments required to be measured outside the head : ~10 nA m (1nA = A) About 10 6 synapses must be simultaneously and effectively active to produce an evoked response. About 40 to 200 mm 2 of active cortex.

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/r 3, compared to 1/r 2 for the PSP dipole. PSPAP 10 ms 1 ms 10 mV 100 mV EEG signals are produced in large part by synaptic current flow, which is approximately dipolar.

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

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

Open/closed field configuration

Signal Sensitivity 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 EEG signalfMRI signal

Forward Problem Inverse Problem Source Localisation in EEG

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

Inverse Problem is an “ill-posed” problem, i.e. the inverse problem does NOT have a unique solution, (von Helmholtz, 1853). Solving v, potential at the electrodes, (N el x 1), source(s) location, [r x, r y, r z ]’, source(s) orientation & amplitude [j x, j y, j z ]’ f, solution of the forward problem , additive noise, (Nel x 1) where

Inverse Problem, example 12 Ω4 Ω12 Ω 6 Ω 6V 3V 2V Example : To uniquely solve the inverse problem assumptions and/or constraints on the solution must be adopted. 4 different networks but with the same measurable output: 2V and 4Ω

Overview of fMRI Analysis: Statistical Parametric Mapping realignment & motion correction smoothingnormalisation General Linear Model Ü model fitting Ü statistic image parameter estimates anatomical reference smoothing kernel design matrix image data Statistical Parametric Map

Source Estimation in EEG Part I : The forward problem

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

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

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

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

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

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. Fromfind

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

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! Fromfind

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

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

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. Fromfind

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 !

Numerical solution, 2 f(.) can be more easily estimated numerically with 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. Fromfind

Potential estimated on the N S surfaces S i separating the sub-volumes of homogeneous conductivity. Potential generated by the sources in open space. Influence of the potential on all the surfaces (conductivity discontinuities). V is on both side of the integral equation, V cannot be directly and analytically evaluated  use numerical methods, BEM Numerical solution, BEM

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

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

Where the integral over each triangle is calculated analytically, thanks to the approximation for the surface potential (constant or linear function). Solving the BEM equation With the discretisation of the surfaces, the integral over the surfaces are converted in discrete sums. And the BEM equation becomes a “simple sum of known analytical functions”:

Solving the BEM equation, cont’d turns into a linear equation: The ‘simplified’ BEM equation : v, vector of potential at the nodal points j, vector of sources amplitude and orientation, their location is fixed B, self-influence matrix G, direct potential matrix with simply solved as

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

In a “three sphere shell” model, the approximated numerical solution is compared with the exact analytical solution. Two criteria are used for the comparison:  scalp  skull  brain scalp surface brain surface skull surface where  scalp =  brain =1 and  skull =.01, and r scalp = 1, r brain =.9 and r skull =.8 Validation of BEM

CPV : less efficient method CoG & LPV : small errors, especially with a refined mesh for the brain 2 reasons to prefer LPV : accuracy of the solution better spatial modelling of the surfaces N tr  2N ve Uniform tessellation of the surfaces (2 levels compared) Refined tessellation of the brain surface Results of simulations

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

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

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 ? Anatomically constrained spherical head models, or pseudo-spherical model.

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

Dipole: defined by its polar coordinates (R d,IRM,  d,  d ) Fitted sphere: defined by its centre and radius, (c Sph,R Sph ) Pseudo-spherical model, cont’d Scalp surface Fitted sphere R d,IRM R scalp ( d, d ) Direction ( d, d ) c Sph R Sph

Applications, scalp surface Fitted sphere and scalp surface

Applications, cortical surface

BEM vs. Pseudo-Sphere Comparison of the analytical and BEM solutions in a “three sphere shell” model

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

Model : simple  define 3 sphere shell realistic extract volume surfaces most realistic extract volume conductivity 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. Solution : analytic easy and quick but anatomical errors numeric slower, more anatomically accurate but numerical erros

So far we still have NOT localised anything…

Source Estimation in EEG Part II : The inverse problem

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

Inverse Problem Function f is linear w.r.t. the source orientation & amplitude non-linear w.r.t. the source location Solving v, potential at the electrodes, (N el x 1), source location, [r x, r y, r z ]’, source orientation & amplitude [j x, j y, j z ]’ f, solution of the forward problem , additive noise, (Nel x 1) where

Parameters When N s sources are present the problem to solve is For each source, there are 6 parameters : 3 for the location, [ x y z ] coordinates, 3 for the orientation and amplitude, [ j x j y j z ] components or 3 for the location, [ x y z ] coordinates 2 for the orientation, [  angles 1 for the amplitude, j intensity/strength

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 N e electrodes. To uniquely solve the inverse problem assumptions/constraints on the solution MUST be adopted. Those constraints define the form of the solution !

Inverse Problem, example 12 Ω 4 Ω 12 Ω 6 Ω 6V 3V 2V Example : 4 different networks but with the same measurable output: 2V and 4Ω. 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 S1S2S3

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

Problem :What should be the constraints ? Advantage : 3D voxelwise results. Distributed Linear solution With the DL solution : “All” possible (fixed) source locations (>10 3 ) considered simultaneously  largely under-determined but linear problem :  external constraints required to calculate a unique solution

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

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

ECD solution, cont’d Global minimum Local minimum Value of parameter Cost function The iterative optimisation procedure can only find a local minimum the starting location(s) used can influence the solution found ! 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:

The location (and orientation) of the sources can be fixed a priori, e.g. using fMRI activation map. the ECD fitting becomes a simple over- determined linear problem  simple fitting of 3 (or 1) parameters for each source : strength (and orientation) This is the “Seeded ECD” (sECD) approach. Seeded ECD solution

ECD applications Epileptic patient: EEG recorded from 21 electrodes FDG-PET with electrode markers EEG data EEG power

ECD applications, cont’d PET scan used to: - generate a pseudo-sphere model - define the electrodes location

ECD applications, cont’d First peak, above F4 Second and third peak

ECD applications, cont’d Second source Third source First source Hypometabolic region

ECD applications, cont’d Original vs. Fitted maps

ECD limitations How many dipoles ? The more sources, the better the fit… in a mathematical sense !!! Is a dipole, i.e. a punctual source, the right model for a patch of activated cortex ? What about the influence of the noise ? Find the confidence interval. (Is the sECD a good approach ? Given that you find what you put in.)

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

With the DL solution : “All” possible (fixed) source locations (>10 3 ) considered simultaneously  largely under-determined but linear problem :  external constraints required to calculate a unique solution Problem : What should be the constraints ? Advantage : three-dimensional voxelwise results. Distributed Linear solution

Orientation ( cortical sheet) Location (in gray matter) Local spatial coherence “Hard” (anatomical) : included in lead field matrix L : v = L j spatially informed basis functions (sIBF) B s : j=B s k s Other source priors weighting matrix H : reduced weight at priors “Soft” (functional or other probabilistic) : 4 Example of “Soft” priors: -location prior as found by other modalities (PET/fMRI), or defined “by hand” -long/short distance spatial coherence between/inside areas - depth constraints Constraints and priors

Spatially Informed Basis Functions (sIBF) : The “hard” anatomical priors (location in gray matter G and spatial coherence D ) enter as constraints of the covariance structure of the source power (over space) C, and are used to motivate the selection of a spatial basis set ( B s ) that maximises the information between the sources and their projection on that set. Mathematically: C is constructed from G and D as : C= G t/2 D t D G 1/2 B s is obtained from the singular value decomposition of C 1/2 : U S W t = svd(D G 1/2 ) Columns of W corresponding to normalised eigenvalues S 2 greater than unity are retained to form the basis set B s. Constraints and priors, cont’d

The linear instantaneous problem: v = L j +  with var (  ) = can be solved using a Weighted Minimum Norm (WMN) approach: which can also be expressed as, where Sources priors Sources likelihood Constrained solution Weighted Minimum Norm solution

Simplified form of equation: only one source constraints H is employed, and the corresponding hyperparameter is simply used to take into account the noise contained in the data With known and C , and spatial basis function B s such that j=B s k s, the solution of the WMN problem is Noise regularisation noise variance- covariance matrix balancing between model and constraint fit WMN solution, cont’d

Under Gaussian assumptions for the distributions of j and , the WMN solution is connected to the Bayesian estimate. The conditional expectation (or posterior mean) of the sources is j given by Thenwhere where is the prior covariance of the sources. WMN & Bayesian estimate

By defining and using an EM algorithm, j,  i and µ i can be jointly estimated. Hierarchical “parametric empirical Bayes” approach In the context of a 2-level hierarchical parametric empirical Bayes (PEB) approach, the source localisation problem can be expressed as:

As C j (and C  ) is defined as a linear combination of users defined covariance basis functions, the true source (and noise) covariance matrix can be more precisely approximated. Interpretation: With the 2-level approach, the unknown parameters j are assumed to have zero mean (due to the shrinkage priors at the 2 nd level) and some variance C j. To render some location more likely to be active, the local variance can be increased: the activity at some location is less constrained (larger variance), thus more likely to be different from zero. Hierachical PEB, cont’d Hierarchical PEB, cont’d

The 2-level hierachical model can be collapsed into: Because of the 2nd level shrinkage prior, there are no “fixed effects” and only “random effects”. The covariance partitioning implied is The only unkowns are the hyperparameters  i and  i, which can be estimated through a ReML procedure. Restricted Maximum Likelihood solution

Simple Model: oriented dipoles on an horizontal plane - within an 3-shell spherical model - leadfield L calculated for a set of 27 electrodes spread uniformly on the upper hemisphere. sIBF: constrained by - assumed grey matter density - spatial coherence (Gaussian function with  = 2*«grid size»). Simulation: model

Data sets: - distributed source sets (3 to 5 adjacent dipoles) - at 200 random locations, - extended over 64 time bins - source amplitude follows a bell shape - added white noise to achieve 3 different SNRs. SNR=Inf SNR=4 SNR=12 SNR=100 Simulation: Data

Soft priors: -depth weighting and spatial smoothness (the singular values obtained during the extraction of the sIBF) -accurate location priors (centered at the source location) -inaccurate location priors, «close» or «distant» from the source location Source set Accurate location prior Inaccurate location priors Simulation: Priors

i)without location priors ii)with accurate location priors iii)with inacurate location priors, close or distant iv)with both accurate and inaccurate (close or distant) location priors iv) i) ii) iii) EM-IBF solution: Simulation: tests

i)without location priors ii)with accurate location priors iii)with inacurate location priors, close or distant iv)with both accurate and inaccurate (close or distant) location priors EM-WMN solution: i)ii) iii) iv) Simulation: tests, cont’d

Maximum Smoothness (MS) solution: i)without noise regularisation, i.e. as if noise-free data ii)with EM noise estimation i) ii) Simulation: tests, cont’d

LE: maximum LE required to recover at least 80% of the sources within this bound Simulation: Localisation Error

RMSE: mean of the Frobenius norm of the (scaled) difference between the original and reconstructed source Simulation: Root Mean Square Error

Mean value and standard deviation of the hyper- parameters relative to the noise component. Actual variance of the added noise component : low SNR 2.3 medium SNR.26 high SNR.0037 Simulation: Hyper-parameter estimate

Mean value and standard deviation of the other hyper-parameters.

Simple right wrist electrical stimulation 59 electrodes, ERP data ERP at electrode CP5 Stim P45 N20 Somatosensory ERP

Somatosensory ERP, cont ’d nose left ear right ear Scalp mapping of the average N20 peak 0 1 Somatosensory ERP, cont’d

Reconstruction of N20 without (left) and with (below) location prior. Location prior Somatosensory ERP, cont’d

Hierachical PEB and ReML (EM) seem a good approach to EEG source localisation as: LE and RMSE greatly reduced by the introduction of accurate location priors accurate and inaccurate location priors used simultaneously  no effect of inaccurate location priors on the source reconstruction accurate noise variance estimate, on average Interestingly, the seeded ECD approach is simply an over constrained DL solution. Advantages of DL solutions

The solution calculated totally relies on the constraints employed, therefore if only inaccurate priors are used  wrong/biased solution in general, without location priors  distributed/blurred solution  spurious sources, depending on the threshold used Using a single constraint is not advisable... Limitations of DL solutions

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

Other solutions ECD and DL solutions are not the only approaches to source localisation in EEG. Other possible approaches: MUSIC (MUltiple Signal Classification), Beamforming/SAM (Synthetic Aperture Magnetometry), L1-norm,...

Music It is a «scanning» method: locations are tested one by one, usually according to a fixed grid, and one coefficient is estimated at each location considered. The key ideas are split (generally using SVD) the measured data into 2 components (signal and noise) defining 2 orthogonal sub-spaces in the «data space». if a source is truly located at location x q, then its leadfield L(x q ) is orthogonal to the noise sub-space.

Beamforming/SAM It is also a «scanning» method but the main idea is to build a spatial filter. Given a source j(x o ), the measured signal is An ideal filter T(x o ) would be such that which is impossible to achieve for all the locations.

Beamforming/SAM, cont’d But a «linearly constrained minimum variance» spatial filter is feasible: minimise the variance at the filter output, and satisfy the passband constraint. It is possible to take into account the noise contained in the data. BUT, if the sources are correlated, the minimisation constraint could lead to erroneous results. and the solution is

L1-norm The DL solution presented uses an L2-norm which usually leads to largely spread activation. An L1-norm solution should provide more focal activation maps but then the reconstruction problem is non-linear, and it requires an iterative procedure L1-n : 4 L2-n : 4 L1-n : 7 L2-n : 3.6 Example: 2 sets of supposingly equally fitting solutions and their L1- and L2-norm

Agenda Part II : Solving the inverse problem Introduction Equivalent current dipole solution Distributed linear solution Other solutions Conclusions

Depending on the data available, your knowledge about the activation and the assumptions you can/want to make you should decide wich approach or solution to use. There does NOT exist any ideal and general solution so far…

Special thanks to: Prof. Karl J. Firston (a) Prof. Mick D. Rugg (b) Dr. Pierre Maquet (c) Dr. Jeremie Mattout (a) Dr. Olivier David (a) (a) Wellcome Department of Imaging Neuroscience, University College London, UK. (b) University of California Irvine, USA. (c) Cyclotron Research Centre, University of Liège, Belgium.

Bayes Rule

Bayes rule states: p(q|e)  p(e|q) p(q) –p(q|e) is the a posteriori probability of parameters q given errors e. –p(e|q) is the likelihood of observing errors e given parameters q. –p(q) is the a priori probability of parameters q. Maximum a posteriori (MAP) estimate maximises p(q|e). Maximising p(q|e) is equivalent to minimising the Gibbs potential of the posterior distribution ( H(q|e), where H(q|e)  -log p(q|e) ). The posterior potential is the sum of the likelihood and prior potentials: H(q|e) = H(e|q) + H(q) + c –The likelihood potential ( H(e|q)  -log p(e|q) ) is based upon the sum of squared difference between the images. –The prior potential ( H(q)  -log p(q) ) penalises unlikely deformations. Bayesian Formulation

Expectation-Maximisation algorithm Objective: maximise the likelihood p(y|  ) of the observed data y, conditional on some hyperparameters , in the presence of unobserved variables x : p(y |  ) = x p(x,y |  ) dx log p(y |  ) ≥ F(Q,  )= log p(y |  ) - x Q(x) log(Q(x)/p(x |y,  ) dx = x Q(x) log p(x,y |  ) dx - x Q(x) log Q(x) dx Objective function to maximise (the free energy) : Approximation: introduce a distribution Q(x) to approximate p(x |y,  ). EM then maximises a lower bound on p(y |  ). Expectation Maximisation algorithms maximise F(Q,  ):  E-step: maximise F(Q,  ) w.r.t. Q, keeping  constant.  M-step: maximise F(Q,  ) w.r.t.  keeping Q constant.

EM algorithm, cont’d Interpretation: The maximum in the E-step is obtained when actually Q(x) = p(x |y,  ). The M-step finds the ML estimate of the hyperparameters, i.e. the values of  that maximise p(y |  ) integrating p(x,y |  ) over the parameters using the current estimate of their conditional distribution. Under present assumtions ReML is formely identical to EM, where the E-step would be embedded into the M-step.