Surface Laplacian

Chapter 22John JB Allen

Surface Laplacian Transform

Is a Spatial FilterIn fact, is the second spatial derivative of the potentials (change in acceleration over space)

Increases topographical specificityFilters out spatially broad features (shared among electrodes)

Thus a high-pass spatial filter (attenuating low spatial-frequency signals)

Caveats:Only for EEG, not MEG dataBest for 64+ electrodes

Surface Laplacian Transform

Spatially-broad features are likely:Volume conducted from distal sourcesDistributed but highly coherent sources

Estimates potentials at the duraEspecially important for connectivity analyses

Surface Laplacian Transform

AKA CSD or SCDCurrent Source DensityCurrent Scalp DensitySurface Current Density

BUT … not brain sourcesSources and sinks of electrical activity at the level of the skullPreferred term: Surface Laplacian

Identifies the mathematical transform usedOther methods available (e.g., Hjorth)

Surface Laplacian TransformAdvantages

Improves Topographical localizationMinimizes volume-conduction effects (important for connectivity analyses)A reference-independent approach!Requires few parameters or assumptions

No head model required (and assumptions about conductivity of layers)No assumptions about source locations

Surface Laplacian TransformCaveats

More sensitive to radial than tangential dipoles.Thus sources in sulci will be minimized

DisadvantageSpatially-broad activities attenuated or eliminated (e.g., P3b)

ImplicationsResults stem from relatively local and superficial sourcesDo not use surface Laplacian if you expect deep sourcesDo not use if you expect widely-distributed coherent sources

Surface Laplacian TransformImplementation

Apply SL to time-domain signalsPerform frequency-domain transformations subsequentlyFor ERPs, applying SL to single trials equivalent to applying it averageMike sayz… Must apply to all conditions, all subjects

Units are now Units influenced by smoothing parametersBut not relevant if using baseline normalization in time-frequency analyses (dB, percent, Z)

Surface Laplacian TransformComputation

Hjorth: subtract from each electrode the average of neighbors’ activity

SimpleComputationally fastBUT…

Not elegantVolume conduction does not affect all neighbors equally

Instead, compute 3D second-spatial derivative

Surface Laplacian Transform3D second-spatial derivativeEasy to visualize in 2D form: Exercise 22.1

Surface Laplacian Transform3D second-spatial derivative (spherical derivative)Several methods:

Deblurring methods with realistic head modelsSpherical Spline interpolations that make no assumptions about conductivity

Spherical spline method of Perrin et al. (1987, 1989) widely used

Surface Laplacian TransformSpherical spline method requires computation of G and H (weighting) matrices

Where:i, j are electrodesm is constant positive integer for smoothness (2-6; higher number

filters our more low spatial frequencies)P is Legendre polynomial for spherical coordinate distancesn is order term for P (Figure 22.2)

42 1

1

42 1

1

More is better?

No, more is sometimes just more..

“With 64 electrodes, order values above 10 mean that the spatial frequency precision of the Laplacian exceeds the spatial resolution of the EEG cap…”

“…as the order becomes large, only very high spatial frequencies can pass through the filter. This may impede cross-subject averaging and comparisons.”

Surface Laplacian Transform

Where:i, j are electrodesm is constant positive integer for smoothness (2-6; higher number

filters our more low spatial frequencies)P is Legendre polynomial for spherical coordinate distancesn is order term for P (Figure 22.2)cosdist is cosine distance among all pairs of electrodes assuming unit

sphere:

42 1

1

42 1

1

1 2

Surface Laplacian Transform

Figure 22.3 (and helpful auxiliary figure)

Surface Laplacian TransformNow, armed with G & H, compute the Laplacian!

Wherelapi is Laplacian for electrode i and one time point, j is each other

electrodeHij is H Matrix corresponding to electrodes i and jC is data!!!!

λ is smoothing parameter added to diagonal elements of G matrix (suggested value of 10-5)

∑

∑

λ

Surface Laplacian TransformCan use functions or toolboxes

laplacian_perrinX.m

Surface Laplacian TransformCan use functions or toolboxes

laplacian_perrinX.mCSD Toolbox Hjorth

Jürgen Kayser

Surface Laplacian TransformSimulated data (Figure 22.4)

Surface Laplacian TransformNunez vs Perrin!

Spatial correlation = .9798

Surface Laplacian TransformConnectivity – volume-conducted activity will increase connectivity across wide distances

Surface Laplacian TransformConnectivity – volume-conducted activity will increase connectivity across wide distances

Surface Laplacian TransformTool for cleaning noise?

Not only a low-pass spatial filter – it is a band-pass spatial filter’

Removes very low and very high frequenciesBut need many electrodes to see impact on high spatial frequencies

Surface Laplacian TransformTool for cleaning noise?

Not only a low-pass spatial filter – it is a band-pass spatial filter’

Removes very low and very high frequenciesBut need many electrodes to see impact on high spatial frequencies

BUT … it is no substitute for good clean data!Besides … who has 256 channels?

Good Practices in ReportingState the purpose of applying the LaplacianTransform

Reference EffectsAR

LM

CSD

Cz

RestingEyes ClosedAlpha Power

Good Practices in ReportingState the purpose of applying the LaplacianTransform

Increase topographical localizationFacilitate electrode-level connectivity analysesAttenuate volume-conducted features that might overshadow local effects of primary interest

If examined raw and Laplacian, state how results changedBe clear about which algorithm was used

And specify any parameters that were changed from default values (and WHY!)