Neurophysiological significance of the Neurophysiological significance of the inverse probleminverse problem

its relation its relation to present “source estimate” to present “source estimate”

methodologies methodologies and to future developmentsand to future developments

Neurophysiological significance of the Neurophysiological significance of the inverse probleminverse problem

its relation its relation to present “source estimate” to present “source estimate”

methodologies methodologies and to future developmentsand to future developments

E. Tognoli

Discussion group about Source Estimation

5 November 2004

I. A “poor spatial resolution”I. A “poor spatial resolution”I. A “poor spatial resolution”I. A “poor spatial resolution”

EEG : a poor spatial resolution 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

Let’s localize the sourcesLet’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

Let’s localize the sourcesLet’s localize the sources

Why?Why?II. What distorts the signal?II. What distorts the signal?

Why?Why?II. What distorts the signal?II. What distorts the signal?

Volume conductionVolume 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

AnisotropyAnisotropy

• 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

More of anisotropyMore of anisotropy

• The sinuses: partially filled with… nothing

• Effect : displaces the signal

Brain foldingBrain 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

Brain foldingBrain folding

Source: Van Essen, 1997

Brain foldingBrain folding

““poor spatial resolution”? 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

III. Some solutions ?III. Some solutions ?III. Some solutions ?III. Some solutions ?

Some solutions?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

Some solutions?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)

Some solutions?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)

Some solutions?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

Some solutions?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)

Some solutions?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)

Some solutions?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

IV. AlgorithmsIV. Algorithmspresent methodspresent methods

IV. AlgorithmsIV. Algorithmspresent methodspresent methods

AlgorithmsAlgorithms

• 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

AlgorithmsAlgorithms

• Inverse problem with smooth solution

– More details in the next sessions

V. Head modelV. Head modelpresent and future methodspresent and future methods

V. Head modelV. Head modelpresent and future methodspresent and future methods

Spline/head model ISpline/head model I

• The spline support of the transform/estimation :

( legacy of Laplacian/CSD)

• Hjorth, 1975• Perrin, 1987• Law, 1995• Babiloni, 1998

Remember the folding Remember the folding problemproblem

Spline/head model ISpline/head model I

• The spline support of the transform/estimation :

( legacy of Laplacian/CSD)

• Hjorth, 1975• Perrin, 1987• Law, 1995• Babiloni, 1998

Spline/head model IISpline/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)

Spline/head model IISpline/head model II

Increasing accuracy of the Increasing accuracy of the shellsshells

• Typically now, 4 compartments: scalp, skull, CSF, grey matter

The accuracy of the The accuracy of the estimated conductivitiesestimated conductivities

• Typically, models use average conductivity values (sampled in the literature)

• Idiosyncrasy of the conductivity• In a given compartment, variability of

the conductivity

The accuracy of the The accuracy of the estimated conductivitiesestimated 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

The endThe end