The surface Laplacian technique in EEG: Theory and methods

Abstract

This paper reviews the method of surface Laplacian differentiation to study EEG. We focus on topics that are helpful for a clear understanding of the underlying concepts and its efficient implementation, which is especially important for EEG researchers unfamiliar with the technique. The popular methods of finite difference and splines are reviewed in detail. The former has the advantage of simplicity and low computational cost, but its estimates are prone to a variety of errors due to discretization. The latter eliminates all issues related to discretization and incorporates a regularization mechanism to reduce spatial noise, but at the cost of increasing mathematical and computational complexity. These and several others issues deserving further development are highlighted, some of which we address to the extent possible. Here we develop a set of discrete approximations for Laplacian estimates at peripheral electrodes and a possible solution to the problem of multiple-frame regularization. We also provide the mathematical details of finite difference approximations that are missing in the literature, and discuss the problem of computational performance, which is particularly important in the context of EEG splines where data sets can be very large. Along this line, the matrix representation of the surface Laplacian operator is carefully discussed and some figures are given illustrating the advantages of this approach. In the final remarks, we briefly sketch a possible way to incorporate finite-size electrodes into Laplacian estimates that could guide further developments.

Issues and considerations for using the scalp surface Laplacian in EEG/ERP research: A tutorial review

“…While the loss of information at border locations may be a sacrifice for finite difference methods (e.g., Hjorth, 1975), surface Laplacian estimates are readily available for these and at any other location by the use of spherical splines and other continuous interpolations (e.g., Nunez and Srinivasan, 2006; Pascual-Marqui et al, 1988; Perrin et al, 1989; reviewed by Carvalhaes and de Barros, 2015). …”

“…The biophysical principle of volume conduction relates current sources generated within the brain to the macroscopic potentials observable at scalp according to Poisson’s equation (e.g., Carvalhaes and de Barros, 2015; Nunez and Srinivasan, 2006; Tenke and Kayser, 2012). A surface Laplacian (often also termed Laplacian, scalp current density [SCD], current source density [CSD]), is a mathematical simplification of this equation as a vector form of Ohm’s law, relating current generators within an (isotropic) electrical conductor to the (negative) second spatial derivative of the field potential at each electrode.…”

“… 3 It is important to distinguish between an analytic SL, which is a theoretical estimate usually obtained with forward simulations (e.g., Babiloni et al, 1996; Nunez, 1981; Nunez and Srinivasan, 2006), and various empirical estimates of this variable, which include spherical spline interpolations (e.g., Carvalhaes and de Barros, 2015). Likewise, the term CSD has also been used in other contexts, including laminar intracranial recordings from cortical depths arrays or inverse solutions (e.g., LORETA), which has resulted in considerable confusion (a historical context of the EEG surface Laplacian by Paul L. Nunez is available at http://ssltool.sourceforge.net/history.html).…”

Neuronal generator patterns at scalp elicited by lateralized aversive pictures reveal consecutive stages of motivated attention

“…Notwithstanding the recognized merits of multivariate data decomposition approaches for affective ERP research (e.g., Delplanque et al, 2006; Olofsson et al, 2008; Pourtois et al, 2008), these techniques do not resolve the interpretational ambiguity of ERP signals caused by the EEG reference (e.g., Junghöfer et al, 2006a; Kayser and Tenke, 2010) or their spatial smearing due to volume conduction (e.g., Tenke and Kayser, 2012). However, these limitations can be conveniently overcome by incorporating a surface Laplacian, or current source density (CSD; e.g., Perrin et al, 1989), transformation of surface potentials in the data processing pipeline, which renders a unique, reference-free representation of radial current flow (sinks and sources) underlying the scalp-recorded EEG (e.g., Carvalhaes and de Barros, 2015; Nunez and Srinivasan, 2006; Tenke and Kayser, 2012). Compared to ERPs, CSDs provide higher spatial and temporal resolution (i.e., a more distinct time course; Burle et al, 2015).…”

Hemifield-dependent N1 and event-related theta/delta oscillations: An unbiased comparison of surface Laplacian and common EEG reference choices

“…Acting as spatial high pass filter (second spatial derivative), it yields sharper topographies compared to those of scalp potentials (e.g., Nunez and Srinivasan, 2006). At the same time, the value of the surface Laplacian is not restricted to its mathematical filter properties but is related to physical principles rendering a unique, reference-free representation of current generators underlying an EEG topography that has – unlike surface potentials – physical meaning (Carvalhaes and de Barros, 2015; Tenke and Kayser, 2012). Despite the recognized theoretical advantages of the surface Laplacian, EEG researchers at large have been surprisingly reluctant to embrace these methods in their research, partly because direct comparisons and simulation studies are scarce (e.g., Kayser and Tenke, 2006a; Nunez et al, 1997, 1999).…”