Surface Laplacian Research Articles

Using Smoothed Boundary Method for Simulating Interfacial Transport in Electrochemical Dynamics

The design of electrodes in current Li-ion batteries has long been based on experimental trial and error on electrode characteristics such as thickness, porosity, and composition [1]. Quantitative design criteria have yet to be developed due to the lack of understanding of coupled electronic-ionic transport in complex electrode structures. Physics-based modeling and simulations at electrode microstructural scale will facilitate the design process. Composite electrodes consist of tortuous channels for transport, particle with non-uniform sizes, and irregular particle-electrolyte interfaces. In addition, interfacial transport of Li and electrons along particle-particle contacts can play an important role in the electrochemical processes. Such geometric complexity makes microstructural-level electrochemical simulation a challenge in the research community. Using a diffuse-interface description, we have previously established a framework for electrochemical simulations based on the smoothed boundary method (SBM) [2], which allows efficient simulations of electrochemical dynamics in complex electrodes [3,4]. In this presentation, we describe an extension to the simulation tool that incorporates interfacial transport of Li and electrons on the boundaries between electrode particles or grains, which can enhance or hinder the net transport through the structure. In the SBM, an individual domain is defined by a domain parameter, which equals one inside the particle/grain and equals zero outside. The particle/grain surface is defined by the transition region where the domain parameter is between zero and one. For an agglomerate containing various particles or a polycrystalline electrode, multiple domain parameters can be employed to define individual domains. The surface Laplacian can be calculated, according to the domain parameters, to obtain the fluxes along the boundaries between the particles or grains. Using this method, bulk transport and interfacial transport can be coupled in evaluating the net transport within the electrode. A preliminary simulation of diffusion along grain boundaries of hexagonal grains is shown in Fig 1. We will use this technique to investigate the individual and combined effects of bulk and interfacial transport. Different transport coefficients can be assigned to distinct particles/grains or particle interfaces/grain boundaries to account for the dependence of the coefficients on crystalline orientations of particles/grains. Simulations using this tool will be utilized to examine the effect of tortuosity of interfaces on Li transport in polycrystalline electrode. In the long term, this method will be incorporated into the electrochemical simulation code to examine the role of interfacial Li and electron transport in determining electrode performance, which will provide insights for design and optimization of electrode architecture. Figure 1: SBM simulated concentration profile for grain boundary diffusion in an ideal hexagonal grain structure. Acknowledgement: This material is based upon work supported as part of the Northeastern Center for Chemical Energy Storage, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0012583. [1] Venkatasailanathan Ramadesigan, et al, J. Electrochem. Soc. 159, R31-R45 (2012). [2] Hui-Chia Yu, Hsun-Yi Chen, and Katsuyo Thornton, Modelling Simul. Mater Sci. Eng. 20, 1-41 (2012). [3] B. Orvananos et al., Electrochimca Acta, 137 (2014) 245 [4] B. Orvananos et al.,J Electrochem Soc, 161 (2014) A535 Figure 1

Surface Laplacians (SL) and phase properties of EEG rhythms: Simulated generators in a volume-conduction model

Surface Laplacian (SL) methods offer advantages in spectral analysis owing to the well-known implications of volume conduction. Although recognition of the superiority of SL over reference-dependent measures is widespread, well-reasoned cautions have precluded their universal adoption. Notably, the expected selectivity of SL for superficial rather than deep generators has relegated SL to the role of an add-on to conventional analyses, rather than as an independent area of inquiry, despite empirical findings supporting the consistency and replicability of physiological effects of interest. It has also been reasoned that the contrast-enhancing effects of SL necessarily make it insensitive to broadly distributed generators, including those suspected for oscillatory rhythms such as EEG alpha. These concerns are further exacerbated for phase-sensitive measures (e.g., phase-locking, coherence), where key features of physiological generators have yet to be evaluated. While the neuronal generators of empirically-derived EEG measures cannot be precisely known due to the inverse problem, simple dipole generator configurations can be simulated using a 4-sphere head model and linearly combined. We simulated subdural and deep generators and distributed dipole layers using sine and cosine waveforms, quantified at 67-scalp sites corresponding to those used in previous research. Reference-dependent (nose, average, mastoids reference) EEG and corresponding SL topographies were used to probe signal fidelity in the topography of the measured amplitude spectra, phase and coherence of sinusoidal stimuli at and between “active” recording sites. SL consistently outperformed the conventional EEG measures, indicating that continued reluctance by the research community is unfounded.

Spatial and temporal resolutions of EEG: Is it really black and white? A scalp current density view

Among the different brain imaging techniques, electroencephalography (EEG) is classically considered as having an excellent temporal resolution, but a poor spatial one. Here, we argue that the actual temporal resolution of conventional (scalp potentials) EEG is overestimated, and that volume conduction, the main cause of the poor spatial resolution of EEG, also distorts the recovered time course of the underlying sources at scalp level, and hence degrades the actual temporal resolution of EEG. While Current Source Density (CSD) estimates, through the Surface Laplacian (SL) computation, are well known to dramatically reduce volume conduction effects and hence improve EEG spatial resolution, its positive impact on EEG temporal resolution is much less recognized. In two simulation studies, we first show how volume conduction and reference electrodes distort the scalp potential time course, and how SL transform provides a much better spatio-temporal description. We then exemplify similar effects on two empirical datasets. We show how the time courses of the scalp potentials mis-estimate the latencies of the relevant brain events and that CSD provides a much richer, and much more accurate, view of the spatio-temporal dynamics of brain activity.

The surface Laplacian technique in EEG: Theory and methods

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 other 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. 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.

Evolution of interfacial curvatures of a bicontinuous structure generated via nonconserved dynamics

While coarsening of spherical particles has been well documented, our understanding of coarsening of complex microstructures is still limited. The first step in developing a theory of coarsening of microstructures with complex morphologies is to study coarsening of microstructures that evolve self-similarly. In this paper, we examine the morphological evolution of a self-similar two-phase bicontinuous structure generated via nonconserved dynamics (i.e., motion by mean curvature) to elucidate the complex dynamics of coarsening. We find that the evolution proceeds with some interfaces evolving toward topological singularities (pinching) while the majority of interfaces flatten. These two processes were also illustrated through the evolution equation for the mean curvature, which has a term that depends solely on the local curvatures, as well as a term that is proportional to the surface Laplacian of the mean curvature. The first term causes an increase in the magnitude of the mean curvature, while the second term causes smoothing of the mean curvature in a manner similar to diffusion of chemical species on a surface. The second term causes a large dispersion in the values of the time derivative of mean curvature at various locations in the structure, characterized neither by the mean curvature nor the Gaussian curvature.

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

Surface Laplacian methodology has been used to reduce the impact of volume conduction and arbitrary choice of EEG recording reference for the analysis of surface potentials. However, the empirical implications of employing these different transformations to the same EEG data remain obscure. This study directly compared the statistical effects of four commonly-used (nose, linked mastoids, average) or recommended (reference electrode standardization technique [REST]) references and their spherical spline current source density (CSD) transformation for a large data set stemming from a well-understood experimental manipulation. ERPs (72 sites) recorded from 130 individuals during a visual half-field paradigm with highly-controlled emotional stimuli were characterized by mid-parietooccipital N1 (125ms peak latency) and event-related synchronization (ERS) of theta/delta (160ms), which were most robust over the contralateral hemisphere. All five data transformations were rescaled to the same covariance and submitted to a single temporal or time-frequency PCA (Varimax) to yield simplified estimates of N1 or theta/delta ERS. Unbiased nonparametric permutation tests revealed that these hemifield-dependent asymmetries were by far most focal and prominent for CSD data, despite all transformations showing maximum effects at mid-parietooccipital sites. Employing smaller subsamples (signal-to-noise) or window-based ERP/ERS amplitudes did not affect these comparisons. Furthermore, correlations between N1 and theta/delta ERS at these sites were strongest for CSD and weakest for nose-referenced data. Contrary to the common notion that the spatial high pass filter properties of a surface Laplacian reduce important contributions of neuronal generators to the EEG signal, the present findings demonstrate that instead volume conduction inherent in surface potentials weakens the representation of neuronal activation patterns at scalp that directly reflect regional brain activity.

Non-invasive electroencephalography signals classification using rough neural network

This paper proposes a rough neural network (RNN) classification method for mental imagery, multi-class EEG data-set. The raw EEG potentials were first spatially filtered by means of a surface Laplacian. Then, every 62.5 ms (16 times per second), the power spectral density (PSD) in the band 8-30 Hz was estimated over the last second of data with a frequency resolution of 2 Hz for the eight centro-parietal channels C3, Cz, C4, CP1, CP2, P3, Pz and P4. As a result, an EEG sample is a 96-dimensional features vector. The features are normalised, and then the classification is performed using RNN. The experiment is performed using the BCI Competition III, Dataset V. The experimental results of RNN is compared with the back-propagation neural network algorithm. This paper obtains marginal improvement for the proposed RNN classification method in terms of classifying accuracy.

Enhancing the performance of motor imagery EEG classification using phase features.

An electroencephalogram recognition system considering phase features is proposed to enhance the performance of motor imagery classification in this study. It mainly consists of feature extraction, feature selection and classification. Surface Laplacian filter is used for background removal. Several potential features, including phase features, are then extracted to enhance the classification accuracy. Next, genetic algorithm is used to select sub-features from feature combination. Finally, selected features are classified by extreme learning machine. Compared with "without phase features" and linear discriminant analysis on motor imagery data from 2 data sets, the results denote that the proposed system achieves enhanced performance, which is suitable for the brain-computer interface applications.

Comparison of different spatial transformations applied to EEG data: A case study of error processing

The purpose of this paper is to compare the effects of different spatial transformations applied to the same scalp-recorded EEG data. The spatial transformations applied are two referencing schemes (average and linked earlobes), the surface Laplacian, and beamforming (a distributed source localization procedure). EEG data were collected during a speeded reaction time task that provided a comparison of activity between error vs. correct responses. Analyses focused on time–frequency power, frequency band-specific inter-electrode connectivity, and within-subject cross-trial correlations between EEG activity and reaction time. Time–frequency power analyses showed similar patterns of midfrontal delta–theta power for errors compared to correct responses across all spatial transformations. Beamforming additionally revealed error-related anterior and lateral prefrontal beta-band activity. Within-subject brain–behavior correlations showed similar patterns of results across the spatial transformations, with the correlations being the weakest after beamforming. The most striking difference among the spatial transformations was seen in connectivity analyses: linked earlobe reference produced weak inter-site connectivity that was attributable to volume conduction (zero phase lag), while the average reference and Laplacian produced more interpretable connectivity results. Beamforming did not reveal any significant condition modulations of connectivity. Overall, these analyses show that some findings are robust to spatial transformations, while other findings, particularly those involving cross-trial analyses or connectivity, are more sensitive and may depend on the use of appropriate spatial transformations.

Surface Laplacian of scalp electrical signals and independent component analysis resolve EMG contamination of electroencephalogram

The serious impact of electromyogram (EMG) contamination of electroencephalogram (EEG) is well recognised. The objective of this research is to demonstrate that combining independent component analysis with the surface Laplacian can eliminate EMG contamination of the EEG, and to validate that this processing does not degrade expected neurogenic signals. The method involves sequential application of ICA, using a manual procedure to identify and discard EMG components, followed by the surface Laplacian. The extent of decontamination is quantified by comparing processed EEG with EMG-free data that was recorded during pharmacologically induced neuromuscular paralysis. The combination of the ICA procedure and the surface Laplacian, with a flexible spherical spline, results in a strong suppression of EMG contamination at all scalp sites and frequencies. Furthermore, the ICA and surface Laplacian procedure does not impair the detection of well-known, cerebral responses; alpha activity with eyes-closed; ERP components (N1, P2) in response to an auditory oddball task; and steady state responses to photic and auditory stimulation. Finally, more flexible spherical splines increase the suppression of EMG by the surface Laplacian. We postulate this is due to ICA enabling the removal of local muscle sources of EMG contamination and the Laplacian transform being insensitive to distant (postural) muscle EMG contamination.

Dipolar estimates of the cortical map.

Various methods based on anatomical or mathematical models have been developed to estimate cortical potentials. Among them, the most popular are the surface Laplacians (SL) and the Electrical Source Imaging (ESI) approaches. In this paper, we develop an informed method named dipolar cortical mapping (DCM), aiming to find a balance between ESI methods based on anatomical models and methods without strong anatomical priors, such as surface Laplacians. Our method only uses easily available information on the electrode position and is based on a physiologically parametrized family of interpolating functions. Simulation results show that DCM competes with previously proposed surface Laplacians and with the model based Minimum Norm Estimates (MNE) computed with a Boundary Element Model (BEM).

The advantages of the surface Laplacian in brain–computer interface research

Brain–computer interface (BCI) systems frequently use signal processing methods, such as spatial filtering, to enhance performance. The surface Laplacian can reduce spatial noise and aid in identification of sources. In BCI research, these two functions of the surface Laplacian correspond to prediction accuracy and signal orthogonality. In the present study, an off-line analysis of data from a sensorimotor rhythm-based BCI task dissociated these functions of the surface Laplacian by comparing nearest-neighbor and next-nearest neighbor Laplacian algorithms. The nearest-neighbor Laplacian produced signals that were more orthogonal while the next-nearest Laplacian produced signals that resulted in better accuracy. Both prediction and signal identification are important for BCI research. Better prediction of user's intent produces increased speed and accuracy of communication and control. Signal identification is important for ruling out the possibility of control by artifacts. Identifying the nature of the control signal is relevant both to understanding exactly what is being studied and in terms of usability for individuals with limited motor control.

Alpha power gates relevant information during working memory updating.

Human working memory (WM) is inherently limited, so we must filter out irrelevant information in our environment or our mind while retaining limited important relevant contents. Previous work suggests that neural oscillations in the alpha band (8-14 Hz) play an important role in inhibiting incoming distracting information during attention and selective encoding tasks. However, whether alpha power is involved in inhibiting no-longer-relevant content or in representing relevant WM content is still debated. To clarify this issue, we manipulated the amount of relevant/irrelevant information using a task requiring spatial WM updating while measuring neural oscillatory activity via EEG and localized current sources across the scalp using a surface Laplacian transform. An initial memory set of two, four, or six spatial locations was to be memorized over a delay until an updating cue was presented indicating that only one or three locations remained relevant for a subsequent recognition test. Alpha amplitude varied with memory maintenance and updating demands among a cluster of left frontocentral electrodes. Greater postcue alpha power was associated with the high relevant load conditions (six and four dots cued to reduce to three relevant) relative to the lower load conditions (four and two dots reduced to one). Across subjects, this difference in alpha power was correlated with condition differences in performance accuracy. In contrast, no significant effects of irrelevant load were observed. These findings demonstrate that, during WM updating, alpha power reflects maintenance of relevant memory contents rather than suppression of no-longer-relevant memory traces.

Sintering at Particle Scale: An Eulerian Computing Framework to Deal with Strong Topological and Material Discontinuities

This work presents a numerical modelling approach of particle packing consolidation, at the particle scale, based on specific numerical methods implemented in a high-performance computing framework. Typically, the sintering process triggers several mass transport paths, thermally activated, that are driven by geometrical as well as physical gradients and laplacians. Computing precisely such major characteristics is of paramount importance but represents a real scientific challenge, which have not been fully solved yet but which must however be tackled to gain precious insights into sintering mechanisms which are seldom accessible at this scale. An Eulerian-based formulation is then proposed here to deal with the strong topological changes related to particle sintering. Also, a specific attention is paid to the precise and robust computation of high-order derivatives which are known to control the physics of surface solid diffusion, namely the surface laplacian of the curvature. Besides, the hydrostatic pressure gradient is known to control the volume diffusion path, it results from the coupled fluid-solid mechanical equilibrium, including surface tension, which must be solved precisely. Furthermore, a mesh adaptation technique allows the particles surface description to be improved, while the number of mesh elements is kept reasonable. Once verified on test-cases, this numerical approach is applied to several 3D granular packings undergoing micro-structural changes under combined surface and volume diffusion.

The Influence of the Exterior Surface on Grain Boundary Mobility Measurements

Polycrystalline materials are materials made up of a typically large number of crystals (``grains''), whose boundaries (``grain boundaries'') define an interior network which effectively determines the microstructure of the specimen. As the network is not at equilibrium, it evolves in time to reduce the overall energy of the specimen. Since the grain boundaries couple to the exterior surface of the specimen, the evolution of the microstructure is influenced by the motion of the exterior surface. In thin specimens, this effect can be appreciable. Within the framework of the classical Mullins' model, grain boundaries move by mean curvature motion, $V_n=A\kappa$, and the exterior surface evolves by surface diffusion, $V_n=-B\Delta_s\kappa$, where $V_n$ and $\kappa$ denote the normal velocity and the mean curvature of the respective evolving surfaces, and $\Delta_s$ is the surface Laplacian. Accurately estimating the kinetic parameters, $A$ and $B$, constitutes an important characterization of material properties. A classical way to determine the “reduced mobility,” $A$, is to make measurements based on a specially constructed “half-loop bicrystalline geometry.” In this geometry there are two grains; one grain is embedded within the other and recedes at a roughly constant rate. This rate may be used to estimate $A$ if the effects of the exterior surface may be neglected. Here we focus on the effects of the exterior surface on grain boundary mobility measurements in the context of the half-loop bicrystalline geometry. Asymptotics, based on a small grain boundary surface energy to exterior surface energy ratio and a thin specimen aspect ratio, are used to predict the coupled grain boundary and exterior surface motion, as well as a leading order correction to mobility measurements.

A versatile interface model for thermal conduction phenomena and its numerical implementation by XFEM

A general interface model is presented for thermal conduction and characterized by two jump relations. The first one expresses that the temperature jump across an interface is proportional to the interfacial average of the normal heat flux while the second one states that the normal heat flux jump is proportional to the surface Laplacian of the interfacial average of the temperature. By varying the two scalar proportionality parameters, not only the Kapitza resistance and highly conducting interface models can be retrieved but also all the intermediate cases can be covered. The general interface model is numerically implemented by constructing its weak form and by using the level-set method and XFEM. The resulting numerical procedure, whose accuracy and robustness are thoroughly tested and discussed with the help of a benchmark problem, is shown to be efficient for solving the problem of thermal conduction in particulate composites with various imperfect interfaces.

The Joint Use of the Tangential Electric Field and Surface Laplacian in EEG Classification

We investigate the joint use of the tangential electric field (EF) and the surface Laplacian (SL) derivation as a method to improve the classification of EEG signals. We considered five classification tasks to test the validity of such approach. In all five tasks, the joint use of the components of the EF and the SL outperformed the scalar potential. The smallest effect occurred in the classification of a mental task, wherein the average classification rate was improved by 0.5 standard deviations. The largest effect was obtained in the classification of visual stimuli and corresponded to an improvement of 2.1 standard deviations.

On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions

In this paper we consider a geometric motion associated with the minimization of a functional which is the sum of a kinetic part of $p$-Laplacian type, a double well potential $\psi$ and a curvature term. In the case $p=2$, such a functional arises in connection with the image segmentation problem in computer vision theory. By means of matched asymptotic expansions, we show that the geometric motion can be approximated by the evolution of the zero level set of the solution of a nonlinear $p$-order equation. The singular limit depends on a complex way on the mean and Gaussian curvatures and the surface Laplacian of the mean curvature of the evolving front.

Statistical evaluation of coherence estimated from optimally beamformed signals

In this paper we investigate a situation where we want to perform a coherence analysis of two signal sources, one of which is measured directly, and the other is measured through a sensor array affected by noise. To extract the latter signal, we suggest the use of the optimal beamforming with reference. We note, however, that this approach results in a coherence estimate that is noticeably biased, and cannot be evaluated by the known statistical tests. We therefore derive a new statistical test, that allows the evaluation of the biased coherence estimate. We illustrate the applicability of our methodology on the coherence analysis of EEG and EMG signals. We note that the suggested approach has several advantages over the surface Laplacian, which is a spatial filter commonly used in the EEG–EMG coherence analysis.

Detection of correlated sources in EEG using combination of beamforming and surface Laplacian methods

Beamforming offers a way to estimate the solution to the inverse problem in EEG and MEG but is also known to perform poorly in the presence of highly correlated sources, e.g. during binaural auditory stimulation, when both left and right primary auditory cortices are activated simultaneously. Surface Laplacian, or the second spatial derivative calculated from the electric potential, allows for deblurring of EEG potential recordings reducing the effects of low skull conductivity and is independent of the reference electrode location. We show that anatomically constrained beamforming in conjunction with the surface Laplacian allows for detection of both locations and dynamics of temporally correlated sources in EEG. Whole-head 122 channel binaural stimulus EEG data were simulated using a boundary element method (BEM) and realistic geometry forward model. We demonstrate that in contrast to conventional potential-based EEG beamforming, Laplacian beamforming allows to determine locations of correlated source dipoles without any a priori assumption about the number of sources. We also show (by providing simulations of auditory evoked potentials) that the dynamics at the detected source locations can be derived from subsets of electrodes. Deblurring auditory evoked potential maps subdivides EEG signals from each hemisphere and allows for the beamformer to be applied separately for left and right hemispheres.