A joint subspace mapping between structural and functional brain connectomes

Understanding the connection between the brain's structural connectivity and its functional connectivity is of immense interest in computational neuroscience. Although some studies have suggested that whole brain functional connectivity is shaped by the underlying structure, the rule by which anatomy constraints brain dynamics remains an open question. In this work, we introduce a computational framework that identifies a joint subspace of eigenmodes for both functional and structural connectomes. We found that a small number of those eigenmodes are sufficient to reconstruct functional connectivity from the structural connectome, thus serving as low-dimensional basis function set. We then develop an algorithm that can estimate the functional eigen spectrum in this joint space from the structural eigen spectrum. By concurrently estimating the joint eigenmodes and the functional eigen spectrum, we can reconstruct a given subject's functional connectivity from their structural connectome. We perform elaborate experiments and demonstrate that the proposed algorithm for estimating functional connectivity from the structural connectome using joint space eigenmodes gives competitive performance as compared to the existing benchmark methods with better interpretability.

Fourier-synthesis approach for static charge-density reconstruction from theoretical structure factors of CaB6

In a pilot study, electron-density (ED) and ED Laplacian distributions were reconstructed for the challenging case of CaB₆ (Pearson symbol cP7) with conceptually fractional B-B bonds from quantum-chemically calculated structure-factor sets with resolutions 0.5 Å^-1 ≤ [sin(θ)/λ]_max ≤ 5.0 Å^-1 by means of Fourier-synthesis techniques. Convergence of norm deviations of the distributions obtained with respect to the reference ones was obtained in the valence region of the unit cell. The QTAIM (quantum theory of atoms in molecules) atomic charges, and the ED and ED Laplacian values at the characteristic critical points of the Fourier-synthesized distributions have been analysed for each resolution and found to display a convergent behaviour with increasing resolution. The presented method(exponent) (ME) type of Fourier-synthesis approach can qualitatively reconstruct all characteristic chemical bonding features of the ED from valence-electron structure-factor sets with resolutions of about 1.2 Å^-1 and beyond, and from all-electron structure-factor sets with resolutions of about 2.0 Å^-1 and beyond. Application of the ME type of Fourier-synthesis approach for reconstruction of ED and ED Laplacian distributions at experimental resolution is proposed to complement the usual extrapolation to infinite resolution in Hansen-Coppens multipole model derived static ED distributions.

EEG Signal Complexity Measurements to Enhance BCI-Based Stroke Patients' Rehabilitation

The second leading cause of death and one of the most common causes of disability in the world is stroke. Researchers have found that brain-computer interface (BCI) techniques can result in better stroke patient rehabilitation. This study used the proposed motor imagery (MI) framework to analyze the electroencephalogram (EEG) dataset from eight subjects in order to enhance the MI-based BCI systems for stroke patients. The preprocessing portion of the framework comprises the use of conventional filters and the independent component analysis (ICA) denoising approach. Fractal dimension (FD) and Hurst exponent (Hur) were then calculated as complexity features, and Tsallis entropy (TsEn) and dispersion entropy (DispEn) were assessed as irregularity parameters. The MI-based BCI features were then statistically retrieved from each participant using two-way analysis of variance (ANOVA) to demonstrate the individuals' performances from four classes (left hand, right hand, foot, and tongue). The dimensionality reduction algorithm, Laplacian Eigenmap (LE), was used to enhance the MI-based BCI classification performance. Utilizing k-nearest neighbors (KNN), support vector machine (SVM), and random forest (RF) classifiers, the groups of post-stroke patients were ultimately determined. The findings show that LE with RF and KNN obtained 74.48% and 73.20% accuracy, respectively; therefore, the integrated set of the proposed features along with ICA denoising technique can exactly describe the proposed MI framework, which may be used to explore the four classes of MI-based BCI rehabilitation. This study will help clinicians, doctors, and technicians make a good rehabilitation program for people who have had a stroke.

Structure-function models of temporal, spatial, and spectral characteristics of non-invasive whole brain functional imaging

We review recent advances in using mathematical models of the relationship between the brain structure and function that capture features of brain dynamics. We argue the need for models that can jointly capture temporal, spatial, and spectral features of brain functional activity. We present recent work on spectral graph theory based models that can accurately capture spectral as well as spatial patterns across multiple frequencies in MEG reconstructions.

Comprehensive Optimization of the Tripolar Concentric Ring Electrode Based on Its Finite Dimensions Model and Confirmed by Finite Element Method Modeling

The optimization performed in this study is based on the finite dimensions model of the concentric ring electrode as opposed to the negligible dimensions model used in the past. This makes the optimization problem comprehensive, as all of the electrode parameters including, for the first time, the radius of the central disc and individual widths of concentric rings, are optimized simultaneously. The optimization criterion used is maximizing the accuracy of the surface Laplacian estimation, as the ability to estimate the Laplacian at each electrode constitutes primary biomedical significance of concentric ring electrodes. For tripolar concentric ring electrodes, the optimal configuration was compared to previously proposed linearly increasing inter-ring distances and constant inter-ring distances configurations of the same size and based on the same finite dimensions model. The obtained analytic results suggest that previously proposed configurations correspond to almost two-fold and more than three-fold increases in the Laplacian estimation error compared with the optimal configuration proposed in this study, respectively. These analytic results are confirmed using finite element method modeling, which was adapted to the finite dimensions model of the concentric ring electrode for the first time. Moreover, the finite element method modeling results suggest that optimal electrode configuration may also offer improved sensitivity and spatial resolution.

A graph convolutional neural network for gene expression data analysis with multiple gene networks

Spectral graph convolutional neural networks (GCN) are proposed to incorporate important information contained in graphs such as gene networks. In a standard spectral GCN, there is only one gene network to describe the relationships among genes. However, for genomic applications, due to condition- or tissue-specific gene function and regulation, multiple gene networks may be available; it is unclear how to apply GCNs to disease classification with multiple networks. Besides, which gene networks may provide more effective prior information for a given learning task is unknown a priori and is not straightforward to discover in many cases. A deep multiple graph convolutional neural network is therefore developed here to meet the challenge. The new approach not only computes a feature of a gene as the weighted average of those of itself and its neighbors through spectral GCNs, but also extracts features from gene-specific expression (or other feature) profiles via a feed-forward neural networks (FNN). We also provide two measures, the importance of a given gene and the relative importance score of each gene network, for the genes' and gene networks' contributions, respectively, to the learning task. To evaluate the new method, we conduct real data analyses using several breast cancer and diffuse large B-cell lymphoma datasets and incorporating multiple gene networks obtained from "GIANT 2.0" Compared with the standard FNN, GCN, and random forest, the new method not only yields high classification accuracy but also prioritizes the most important genes confirmed to be highly associated with cancer, strongly suggesting the usefulness of the new method in incorporating multiple gene networks.

Gradients of connectivity as graph Fourier bases of brain activity

The application of graph theory to model the complex structure and function of the brain has shed new light on its organization, prompting the emergence of network neuroscience. Despite the tremendous progress that has been achieved in this field, still relatively few methods exploit the topology of brain networks to analyze brain activity. Recent attempts in this direction have leveraged on the one hand graph spectral analysis (to decompose brain connectivity into eigenmodes or gradients) and the other graph signal processing (to decompose brain activity "coupled to" an underlying network in graph Fourier modes). These studies have used a variety of imaging techniques (e.g., fMRI, electroencephalography, diffusion-weighted and myelin-sensitive imaging) and connectivity estimators to model brain networks. Results are promising in terms of interpretability and functional relevance, but methodologies and terminology are variable. The goals of this paper are twofold. First, we summarize recent contributions related to connectivity gradients and graph signal processing, and attempt a clarification of the terminology and methods used in the field, while pointing out current methodological limitations. Second, we discuss the perspective that the functional relevance of connectivity gradients could be fruitfully exploited by considering them as graph Fourier bases of brain activity.

Functional network estimation using multigraph learning with application to brain maturation study

Although most dramatic structural changes occur in the perinatal period, a growing body of evidences demonstrates that adolescence and early adulthood are also important for substantial neurodevelopment. We were thus motivated to explore brain development during puberty by evaluating functional connectivity network (FCN) differences between childhood and young adulthood using multi-paradigm task-based functional magnetic resonance imaging (fMRI) measurements. Different from conventional multigraph based FCN construction methods where the graph network was built independently for each modality/paradigm, we proposed a multigraph learning model in this work. It promises a better fitting to FCN construction by jointly estimating brain network from multi-paradigm fMRI time series, which may share common graph structures. To investigate the hub regions of the brain, we further conducted graph Fourier transform (GFT) to divide the fMRI BOLD time series of a node within the brain network into a range of frequencies. Then we identified the hub regions characterizing brain maturity through eigen-analysis of the low frequency components, which were believed to represent the organized structures shared by a large population. The proposed method was evaluated using both synthetic and real data, which demonstrated its effectiveness in extracting informative brain connectivity patterns. We detected 14 hub regions from the child group and 12 hub regions from the young adult group. We show the significance of these findings with a discussion of their functions and activation patterns as a function of age. In summary, our proposed method can extract brain connectivity network more accurately by considering the latent common structures between different fMRI paradigms, which are significant for both understanding brain development and recognizing population groups of different ages.

Excitability and propagation of the electrical impulse in Venus flytrap; a comparative electrophysiological study of unipolar electrograms with myocardial tissue

Mammalian heart cells and cells of leaves of Dionaea muscipula share the ability to generate propagated action potentials, because the excitable cells are electrically coupled. In the heart the propagated action potential causes synchronized contraction of the heart muscle after automatic generation of the impulse in the sinus node. In Dionaea propagation results in closure of the trap after activation of trigger hairs by an insect. The electrical activity can be recorded in the extracellular space as an extracellular electrogram, resulting from transmembrane currents. Although the underlying physiological mechanism that causes the electrogram is similar for heart and Dionaea cells, the contribution of the various ions to the transmembrane current is different. We recorded extracellular electrograms from Dionaea leaves and compared the recorded signals with those known from the heart. The morphology of the electrograms differed considerably. In comparison to activation in mammalian myocardium, electrograms of Dionaea are more temporally and spatially variable. Whereas electrograms in healthy myocardium recorded at some distance from the site of activation reveal a simple biphasic pattern, Dionaea activation showed positive, negative or biphasic deflections. Comparison of patch clamp data from plant cells and cardiomyocytes suggests a role of temperature and ion concentrations in extracellular space for the diversity of morphologies of the Dionaea electrograms.

Sparse linear discriminant analysis for multiview structured data

Classification methods that leverage the strengths of data from multiple sources (multiview data) simultaneously have enormous potential to yield more powerful findings than two-step methods: association followed by classification. We propose two methods, sparse integrative discriminant analysis (SIDA), and SIDA with incorporation of network information (SIDANet), for joint association and classification studies. The methods consider the overall association between multiview data, and the separation within each view in choosing discriminant vectors that are associated and optimally separate subjects into different classes. SIDANet is among the first methods to incorporate prior structural information in joint association and classification studies. It uses the normalized Laplacian of a graph to smooth coefficients of predictor variables, thus encouraging selection of predictors that are connected. We demonstrate the effectiveness of our methods on a set of synthetic datasets and explore their use in identifying potential nontraditional risk factors that discriminate healthy patients at low versus high risk for developing atherosclerosis cardiovascular disease in 10 years. Our findings underscore the benefit of joint association and classification methods if the goal is to correlate multiview data and to perform classification.

A rotation based regularization method for semi-supervised learning

In manifold learning, the intrinsic geometry of the manifold is explored and preserved by identifying the optimal local neighborhood around each observation. It is well known that when a Riemannian manifold is unfolded correctly, the observations lying spatially near to the manifold, should remain near on the lower dimension as well. Due to the nonlinear properties of manifold around each observation, finding such optimal neighborhood on the manifold is a challenge. Thus, a sub-optimal neighborhood may lead to erroneous representation and incorrect inferences. In this paper, we propose a rotation-based affinity metric for accurate graph Laplacian approximation. It exploits the property of aligned tangent spaces of observations in an optimal neighborhood to approximate correct affinity between them. Extensive experiments on both synthetic and real world datasets have been performed. It is observed that proposed method outperforms existing nonlinear dimensionality reduction techniques in low-dimensional representation for synthetic datasets. The results on real world datasets like COVID-19 prove that our approach increases the accuracy of classification by enhancing Laplacian regularization.

Algebraic relationship between the structural network's Laplacian and functional network's adjacency matrix is preserved in temporal lobe epilepsy subjects

The relationship between anatomic and resting state functional connectivity of large-scale brain networks is a major focus of current research. In previous work, we introduced a model based on eigen decomposition of the Laplacian which predicts the functional network from the structural network in healthy brains. In this work, we apply the eigen decomposition model to two types of epilepsy; temporal lobe epilepsy associated with mesial temporal sclerosis, and MRI-normal temporal lobe epilepsy. Our findings show that the eigen relationship between function and structure holds for patients with temporal lobe epilepsy as well as normal individuals. These results suggest that the brain under TLE conditions reconfigures and rewires the fine-scale connectivity (a process which the model parameters are putatively sensitive to), in order to achieve the necessary structure-function relationship.

Examining brain maturation during adolescence using graph Laplacian learning based Fourier transform

BACKGROUND

Longitudinal neuroimaging studies have demonstrated that adolescence is a crucial developmental period of continued brain growth and change. Motivated by both achievements in graph signal processing and recent evidence that some brain areas act as hubs connecting functionally specialized systems, we propose an approach to detect these regions from a spectral analysis perspective. In particular, as the human brain undergoes substantial development throughout adolescence, we evaluate functional network difference among age groups from functional magnetic resonance imaging (fMRI) measurements.

NEW METHODS

We treated these measurements as graph signals defined on the parcellated functional brain regions and proposed a graph Laplacian learning based Fourier transform (GLFT) to transform the original graph signals into the frequency domain. Eigen-analysis was conducted afterwards to study the behaviors of the corresponding brain regions, which enabled the characterization of brain maturation.

RESULT

We first evaluated our method on the synthetic data and then applied it to resting state and task fMRI data from the Philadelphia Neurodevelopmental Cohort (PNC) dataset, comprised of normally developing adolescents from 8 to 22 years of age. The method provided an accuracy of 94.9% in distinguishing different adolescent stages and we detected 13 hubs from resting state fMRI and 16 hubs from task fMRI related to brain maturation.

COMPARISON WITH EXISTING METHODS

The proposed GLFT demonstrated its superiority over conventional graph Fourier transform and alternative graph Fourier transform with high predictive power.

CONCLUSION

The method provides a powerful approach for extracting brain connectivity patterns and identifying hub regions.

Characterizing and Comparing Phylogenetic Trait Data from Their Normalized Laplacian Spectrum

The dissection of the mode and tempo of phenotypic evolution is integral to our understanding of global biodiversity. Our ability to infer patterns of phenotypes across phylogenetic clades is essential to how we infer the macroevolutionary processes governing those patterns. Many methods are already available for fitting models of phenotypic evolution to data. However, there is currently no comprehensive nonparametric framework for characterizing and comparing patterns of phenotypic evolution. Here, we build on a recently introduced approach for using the phylogenetic spectral density profile (SDP) to compare and characterize patterns of phylogenetic diversification, in order to provide a framework for nonparametric analysis of phylogenetic trait data. We show how to construct the SDP of trait data on a phylogenetic tree from the normalized graph Laplacian. We demonstrate on simulated data the utility of the SDP to successfully cluster phylogenetic trait data into meaningful groups and to characterize the phenotypic patterning within those groups. We furthermore demonstrate how the SDP is a powerful tool for visualizing phenotypic space across traits and for assessing whether distinct trait evolution models are distinguishable on a given empirical phylogeny. We illustrate the approach in two empirical data sets: a comprehensive data set of traits involved in song, plumage, and resource-use in tanagers, and a high-dimensional data set of endocranial landmarks in New World monkeys. Considering the proliferation of morphometric and molecular data collected across the tree of life, we expect this approach will benefit big data analyses requiring a comprehensive and intuitive framework.

A non-parametric analytic framework for within-host viral phylogenies and a test for HIV-1 founder multiplicity

Phylogenetics is a powerful tool for understanding the diversification dynamics of viral pathogens. Here we present an extension of the spectral density profile of the modified graph Laplacian, which facilitates the characterization of within-host molecular evolution of viruses and the direct comparison of diversification dynamics between hosts. This approach is non-parametric and therefore fast and model-free. We used simulations of within-host evolutionary scenarios to evaluate the efficiency of our approach and to demonstrate the significance of interpreting a viral phylogeny by its spectral density profile in terms of diversification dynamics. The key features that are captured by the profile are positive selection on the viral gene (or genome), temporal changes in substitution rates, mutational fitness, and time between sampling. Using sequences from individuals infected with HIV-1, we showed the utility of this approach for characterizing within-host diversification dynamics, for comparing dynamics between hosts, and for charting disease progression in infected individuals sampled over multiple years. We furthermore propose a heuristic test for assessing founder heterogeneity, which allows us to classify infections with single and multiple HIV-1 founder viruses. This non-parametric approach can be a valuable complement to existing parametric approaches.

Evaluation of Bipolar, Tripolar, and Quadripolar Laplacian Estimates of Electrocardiogram via Concentric Ring Electrodes

Surface Laplacian estimates via concentric ring electrodes (CREs) have proven to enhance spatial resolution compared to conventional disc electrodes, which is of great importance for P-wave analysis. In this study, Laplacian estimates for traditional bipolar configuration (BC), two tripolar configurations with linearly decreasing and increasing inter-ring distances (TC_LDIRD and TC_LIIRD, respectively), and quadripolar configuration (QC) were obtained from cardiac recordings with pentapolar CREs placed at CMV1 and CMV2 positions. Normalized P-wave amplitude (NAP) was computed to assess the contrast to study atrial activity. Signals were of good quality (20-30 dB). Atrial activity was more emphasized at CMV1 (NAP ≃ 0.19-0.24) compared to CMV2 (NAP ≃ 0.08-0.10). Enhanced spatial resolution of TC_LIIRD and QC resulted in higher NAP values than BC and TC_LDIRD. Comparison with simultaneous standard 12-lead ECG proved that Laplacian estimates at CMV1 outperformed all the limb and chest standard leads in the contrast to study P-waves. Clinical recordings with CRE at this position could allow more detailed observation of atrial activity and facilitate the diagnosis of associated pathologies. Furthermore, such recordings would not require additional electrodes on limbs and could be performed wirelessly, so it should also be suitable for ambulatory monitoring, for example, using cardiac Holter monitors.

Which Reference Should We Use for EEG and ERP practice?

Which reference is appropriate for the scalp ERP and EEG studies? This unsettled problem still inspires unceasing debate. The ideal reference should be the one with zero or constant potential but unfortunately it is well known that no point on the body fulfills this condition. Consequently, more than ten references are used in the present EEG-ERP studies. This diversity seriously undermines the reproducibility and comparability of results across laboratories. A comprehensive review accompanied by a brief communication with rigorous derivations and notable properties (Hu et al. Brain Topogr, 2019. https://doi.org/10.1007/s10548-019-00706-y ) is thus necessary to provide application-oriented principled recommendations. In this paper current popular references are classified into two categories: (1) unipolar references that construct a neutral reference, including both online unipolar references and offline re-references. Examples of unipolar references are the reference electrode standardization technique (REST), average reference (AR), and linked-mastoids/ears reference (LM); (2) non-unipolar references that include the bipolar reference and the Laplacian reference. We show that each reference is derived with a different assumption and serves different aims. We also note from (Hu et al. 2019) that there is a general form for the reference problem, the 'no memory' property of the unipolar references, and a unified estimator for the potentials at infinity termed as the regularized REST (rREST) which has more advantageous statistical evidence than AR. A thorough discussion of the advantages and limitations of references is provided with recommendations in the hope to clarify the role of each reference in the ERP and EEG practice.

An Accurate Skeleton Extraction Approach From 3D Point Clouds of Maize Plants

Accurate and high-throughput determination of plant morphological traits is essential for phenotyping studies. Nowadays, there are many approaches to acquire high-quality three-dimensional (3D) point clouds of plants. However, it is difficult to estimate phenotyping parameters accurately of the whole growth stages of maize plants using these 3D point clouds. In this paper, an accurate skeleton extraction approach was proposed to bridge the gap between 3D point cloud and phenotyping traits estimation of maize plants. The algorithm first uses point cloud clustering and color difference denoising to reduce the noise of the input point clouds. Next, the Laplacian contraction algorithm is applied to shrink the points. Then the key points representing the skeleton of the plant are selected through adaptive sampling, and neighboring points are connected to form a plant skeleton composed of semantic organs. Finally, deviation skeleton points to the input point cloud are calibrated by building a step forward local coordinate along the tangent direction of the original points. The proposed approach successfully generates accurately extracted skeleton from 3D point cloud and helps to estimate phenotyping parameters with high precision of maize plants. Experimental verification of the skeleton extraction process, tested using three cultivars and different growth stages maize, demonstrates that the extracted matches the input point cloud well. Compared with 3D digitizing data-derived morphological parameters, the NRMSE of leaf length, leaf inclination angle, leaf top length, leaf azimuthal angle, leaf growth height, and plant height, estimated using the extracted plant skeleton, are 5.27, 8.37, 5.12, 4.42, 1.53, and 0.83%, respectively, which could meet the needs of phenotyping analysis. The time required to process a single maize plant is below 100 s. The proposed approach may play an important role in further maize research and applications, such as genotype-to-phenotype study, geometric reconstruction, functional structural maize modeling, and dynamic growth animation.

Multi-Scale Neural Sources of EEG: Genuine, Equivalent, and Representative. A Tutorial Review

A biophysical framework needed to interpret electrophysiological data recorded at multiple spatial scales of brain tissue is developed. Micro current sources at membrane surfaces produce local field potentials, electrocorticography, and electroencephalography (EEG). We categorize multi-scale sources as genuine, equivalent, or representative. Genuine sources occur at the micro scale of cell surfaces. Equivalent sources provide identical experimental outcomes over a range of scales and applications. In contrast, each representative source distribution is just one of many possible source distributions that yield similar experimental outcomes. Macro sources ("dipoles") may be defined at the macrocolumn (mm) scale and depend on several features of the micro sources-magnitudes, micro synchrony within columns, and distribution through the cortical depths. These micro source properties are determined by brain dynamics and the columnar structure of cortical tissue. The number of representative sources underlying EEG data depends on the spatial scale of neural tissue under study. EEG inverse solutions (e.g. dipole localization) and high resolution estimates (e.g. Laplacian, dura imaging) have both strengths and limitations that depend on experimental conditions. The proposed theoretical framework informs studies of EEG source localization, source characterization, and low pass filtering. It also facilitates interpretations of brain dynamics and cognition, including measures of synchrony, functional connections between cortical locations, and other aspects of brain complexity.

Solving the general inter-ring distances optimization problem for concentric ring electrodes to improve Laplacian estimation

Background

Superiority of noninvasive tripolar concentric ring electrodes over conventional disc electrodes in accuracy of surface Laplacian estimation has been demonstrated in a range of electrophysiological measurement applications. Recently, a general approach to Laplacian estimation for an (n + 1)-polar electrode with n rings using the (4n + 1)-point method has been proposed and used to introduce novel multipolar and variable inter-ring distances electrode configurations. While only linearly increasing and linearly decreasing inter-ring distances have been considered previously, this paper defines and solves the general inter-ring distances optimization problem for the (4n + 1)-point method.

Results

General inter-ring distances optimization problem is solved for tripolar (n = 2) and quadripolar (n = 3) concentric ring electrode configurations through minimizing the truncation error of Laplacian estimation. For tripolar configuration with middle ring radius αr and outer ring radius r the optimal range of values for α was determined to be 0 < α ≤ 0.22 while for quadripolar configuration with an additional middle ring with radius βr the optimal range of values for α and β was determined by inequalities 0 < α < β < 1 and αβ ≤ 0.21. Finite element method modeling and full factorial analysis of variance were used to confirm statistical significance of Laplacian estimation accuracy improvement due to optimization of inter-ring distances (p < 0.0001).

Conclusions

Obtained results suggest the potential of using optimization of inter-ring distances to improve the accuracy of surface Laplacian estimation via concentric ring electrodes. Identical approach can be applied to solving corresponding inter-ring distances optimization problems for electrode configurations with higher numbers of concentric rings. Solutions of the proposed inter-ring distances optimization problem define the class of the optimized inter-ring distances electrode designs. These designs may result in improved noninvasive sensors for measurement systems that use concentric ring electrodes to acquire electrical signals such as from the brain, intestines, heart or uterus for diagnostic purposes.

Along-tract analysis of the arcuate fasciculus using the Laplacian operator to evaluate different tractography methods

PURPOSE

We propose a new along-tract algorithm to compare different tractography algorithms in tract curvature mapping and along-tract analysis of the arcuate fasciculus (AF). In particular, we quantified along-tract diffusion parameters and AF spatial distribution evaluating hemispheric asymmetries in a group of healthy subjects.

METHODS

The AF was bilaterally reconstructed in a group of 29 healthy subjects using the probabilistic ball-and-sticks model, and both deterministic and probabilistic constrained spherical deconvolution. We chose cortical ROIs as tractography targets and the developed along-tract algorithm used the Laplacian operator to parameterize the volume of the tract, allowing along-tract analysis and tract curvature mapping independent of the tractography algorithm used.

RESULTS

The Laplacian parameterization successfully described the tract geometry underlying hemispheric asymmetries in the AF curvature. Using the probabilistic tractography methods, we found more tracts branching towards cortical terminations in the left hemisphere. This influenced the left AF curvature and its diffusion parameters, which were significantly different with respect to the right. In particular, we detected projections towards the middle temporal and inferior frontal gyri bilaterally, and towards the superior temporal and precentral gyri in the left hemisphere, with a significantly increased volume and connectivity.

CONCLUSIONS

The approach we propose is useful to evaluate brain asymmetries, assessing the volume, the diffusion properties and the quantitative spatial localization of the AF.

A new approach to the intracardiac inverse problem using Laplacian distance kernel

Background

The inverse problem in electrophysiology consists of the accurate estimation of the intracardiac electrical sources from a reduced set of electrodes at short distances and from outside the heart. This estimation can provide an image with relevant knowledge on arrhythmia mechanisms for the clinical practice. Methods based on truncated singular value decomposition (TSVD) and regularized least squares require a matrix inversion, which limits their resolution due to the unavoidable low-pass filter effect of the Tikhonov regularization techniques.

Methods

We propose to use, for the first time, a Mercer’s kernel given by the Laplacian of the distance in the quasielectrostatic field equations, hence providing a Support Vector Regression (SVR) formulation by following the principles of the Dual Signal Model (DSM) principles for creating kernel algorithms.

Results

Simulations in one- and two-dimensional models show the performance of our Laplacian distance kernel technique versus several conventional methods. Firstly, the one-dimensional model is adjusted for yielding recorded electrograms, similar to the ones that are usually observed in electrophysiological studies, and suitable strategy is designed for the free-parameter search. Secondly, simulations both in one- and two-dimensional models show larger noise sensitivity in the estimated transfer matrix than in the observation measurements, and DSM−SVR is shown to be more robust to noisy transfer matrix than TSVD.

Conclusion

These results suggest that our proposed DSM−SVR with Laplacian distance kernel can be an efficient alternative to improve the resolution in current and emerging intracardiac imaging systems.

Functional brain connectivity is predictable from anatomic network's Laplacian eigen-structure

How structural connectivity (SC) gives rise to functional connectivity (FC) is not fully understood. Here we mathematically derive a simple relationship between SC measured from diffusion tensor imaging, and FC from resting state fMRI. We establish that SC and FC are related via (structural) Laplacian spectra, whereby FC and SC share eigenvectors and their eigenvalues are exponentially related. This gives, for the first time, a simple and analytical relationship between the graph spectra of structural and functional networks. Laplacian eigenvectors are shown to be good predictors of functional eigenvectors and networks based on independent component analysis of functional time series. A small number of Laplacian eigenmodes are shown to be sufficient to reconstruct FC matrices, serving as basis functions. This approach is fast, and requires no time-consuming simulations. It was tested on two empirical SC/FC datasets, and was found to significantly outperform generative model simulations of coupled neural masses.

Event-Related Potential Responses to Task Switching Are Sensitive to Choice of Spatial Filter

Event-related potential (ERP) studies using the task-switching paradigm show that multiple ERP components are modulated by activation of proactive control processes involved in preparing to repeat or switch task and reactive control processes involved in implementation of the current or new task. Our understanding of the functional significance of these ERP components has been hampered by variability in their robustness, as well as their temporal and scalp distribution across studies. The aim of this study is to examine the effect of choice of reference electrode or spatial filter on the number, timing and scalp distribution of ERP elicited during task-switching. We compared four configurations, including the two most common (i.e., average mastoid reference and common average reference) and two novel ones that aim to reduce volume conduction (i.e., reference electrode standardization technique (REST) and surface Laplacian) on mixing cost and switch cost effects in cue-locked and target-locked ERP waveforms in 201 healthy participants. All four spatial filters showed the same well-characterized ERP components that are typically seen in task-switching paradigms: the cue-locked switch positivity and target-locked N2/P3 effect. However, both the number of ERP effects associated with mixing and switch cost, and their temporal and spatial resolution were greater with the surface Laplacian transformation which revealed rapid temporal adjustments that were not identifiable with other spatial filters. We conclude that the surface Laplacian transformation may be more suited to characterize EEG signatures of complex spatiotemporal networks involved in cognitive control.

Identifying network structure similarity using spectral graph theory

Most real networks are too large or they are not available for real time analysis. Therefore, in practice, decisions are made based on partial information about the ground truth network. It is of great interest to have metrics to determine if an inferred network (the partial information network) is similar to the ground truth. In this paper we develop a test for similarity between the inferred and the true network. Our research utilizes a network visualization tool, which systematically discovers a network, producing a sequence of snapshots of the network. We introduce and test our metric on the consecutive snapshots of a network, and against the ground truth. To test the scalability of our metric we use a random matrix theory approach while discovering Erdös-Rényi graphs. This scaling analysis allows us to make predictions about the performance of the discovery process.

Adaptive Spatial Filter Based on Similarity Indices to Preserve the Neural Information on EEG Signals during On-Line Processing

This work presents a new on-line adaptive filter, which is based on a similarity analysis between standard electrode locations, in order to reduce artifacts and common interferences throughout electroencephalography (EEG) signals, but preserving the useful information. Standard deviation and Concordance Correlation Coefficient (CCC) between target electrodes and its correspondent neighbor electrodes are analyzed on sliding windows to select those neighbors that are highly correlated. Afterwards, a model based on CCC is applied to provide higher values of weight to those correlated electrodes with lower similarity to the target electrode. The approach was applied to brain computer-interfaces (BCIs) based on Canonical Correlation Analysis (CCA) to recognize 40 targets of steady-state visual evoked potential (SSVEP), providing an accuracy (ACC) of 86.44 ± 2.81%. In addition, also using this approach, features of low frequency were selected in the pre-processing stage of another BCI to recognize gait planning. In this case, the recognition was significantly (p<0.01) improved for most of the subjects (ACC≥74.79%), when compared with other BCIs based on Common Spatial Pattern, Filter Bank-Common Spatial Pattern, and Riemannian Geometry.

A Polyakov Formula for Sectors

We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmic singularity at the origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw-Sommerfeld's heat kernel. We also compute the zeta-regularized determinant of rectangular domains of unit area and prove that it is uniquely maximized by the square.

Covariate-assisted spectral clustering

Biological and social systems consist of myriad interacting units. The interactions can be represented in the form of a graph or network. Measurements of these graphs can reveal the underlying structure of these interactions, which provides insight into the systems that generated the graphs. Moreover, in applications such as connectomics, social networks, and genomics, graph data are accompanied by contextualizing measures on each node. We utilize these node covariates to help uncover latent communities in a graph, using a modification of spectral clustering. Statistical guarantees are provided under a joint mixture model that we call the node-contextualized stochastic blockmodel, including a bound on the misclustering rate. The bound is used to derive conditions for achieving perfect clustering. For most simulated cases, covariate-assisted spectral clustering yields results superior both to regularized spectral clustering without node covariates and to an adaptation of canonical correlation analysis. We apply our clustering method to large brain graphs derived from diffusion MRI data, using the node locations or neurological region membership as covariates. In both cases, covariate-assisted spectral clustering yields clusters that are easier to interpret neurologically.

Improving the Accuracy of Laplacian Estimation with Novel Variable Inter-Ring Distances Concentric Ring Electrodes

Noninvasive concentric ring electrodes are a promising alternative to conventional disc electrodes. Currently, the superiority of tripolar concentric ring electrodes over disc electrodes, in particular, in accuracy of Laplacian estimation, has been demonstrated in a range of applications. In our recent work, we have shown that accuracy of Laplacian estimation can be improved with multipolar concentric ring electrodes using a general approach to estimation of the Laplacian for an (n + 1)-polar electrode with n rings using the (4n + 1)-point method for n ≥ 2. This paper takes the next step toward further improving the Laplacian estimate by proposing novel variable inter-ring distances concentric ring electrodes. Derived using a modified (4n + 1)-point method, linearly increasing and decreasing inter-ring distances tripolar (n = 2) and quadripolar (n = 3) electrode configurations are compared to their constant inter-ring distances counterparts. Finite element method modeling and analytic results are consistent and suggest that increasing inter-ring distances electrode configurations may decrease the truncation error resulting in more accurate Laplacian estimates compared to respective constant inter-ring distances configurations. For currently used tripolar electrode configuration, the truncation error may be decreased more than two-fold, while for the quadripolar configuration more than a six-fold decrease is expected.

Improving the accuracy of Laplacian estimation with novel multipolar concentric ring electrodes

Conventional electroencephalography with disc electrodes has major drawbacks including poor spatial resolution, selectivity and low signal-to-noise ratio that are critically limiting its use. Concentric ring electrodes, consisting of several elements including the central disc and a number of concentric rings, are a promising alternative with potential to improve all of the aforementioned aspects significantly. In our previous work, the tripolar concentric ring electrode was successfully used in a wide range of applications demonstrating its superiority to conventional disc electrode, in particular, in accuracy of Laplacian estimation. This paper takes the next step toward further improving the Laplacian estimation with novel multipolar concentric ring electrodes by completing and validating a general approach to estimation of the Laplacian for an (n + 1)-polar electrode with n rings using the (4n + 1)-point method for n ≥ 2 that allows cancellation of all the truncation terms up to the order of 2n. An explicit formula based on inversion of a square Vandermonde matrix is derived to make computation of multipolar Laplacian more efficient. To confirm the analytic result of the accuracy of Laplacian estimate increasing with the increase of n and to assess the significance of this gain in accuracy for practical applications finite element method model analysis has been performed. Multipolar concentric ring electrode configurations with n ranging from 1 ring (bipolar electrode configuration) to 6 rings (septapolar electrode configuration) were directly compared and obtained results suggest the significance of the increase in Laplacian accuracy caused by increase of n.

Characterizing and Comparing Phylogenies from their Laplacian Spectrum

Phylogenetic trees are central to many areas of biology, ranging from population genetics and epidemiology to microbiology, ecology, and macroevolution. The ability to summarize properties of trees, compare different trees, and identify distinct modes of division within trees is essential to all these research areas. But despite wide-ranging applications, there currently exists no common, comprehensive framework for such analyses. Here we present a graph-theoretical approach that provides such a framework. We show how to construct the spectral density profile of a phylogenetic tree from its Laplacian graph. Using ultrametric simulated trees as well as non-ultrametric empirical trees, we demonstrate that the spectral density successfully identifies various properties of the trees and clusters them into meaningful groups. Finally, we illustrate how the eigengap can identify modes of division within a given tree. As phylogenetic data continue to accumulate and to be integrated into various areas of the life sciences, we expect that this spectral graph-theoretical framework to phylogenetics will have powerful and long-lasting applications.

Functional organization of the human 4D Nucleome

The 4D organization of the interphase nucleus, or the 4D Nucleome (4DN), reflects a dynamical interaction between 3D genome structure and function and its relationship to phenotype. We present initial analyses of the human 4DN, capturing genome-wide structure using chromosome conformation capture and 3D imaging, and function using RNA-sequencing. We introduce a quantitative index that measures underlying topological stability of a genomic region. Our results show that structural features of genomic regions correlate with function with surprising persistence over time. Furthermore, constructing genome-wide gene-level contact maps aided in identifying gene pairs with high potential for coregulation and colocalization in a manner consistent with expression via transcription factories. We additionally use 2D phase planes to visualize patterns in 4DN data. Finally, we evaluated gene pairs within a circadian gene module using 3D imaging, and found periodicity in the movement of clock circadian regulator and period circadian clock 2 relative to each other that followed a circadian rhythm and entrained with their expression.

A Laplacian based image filtering using switching noise detector

This paper presents a Laplacian-based image filtering method. Using a local noise estimator function in an energy functional minimizing scheme we show that Laplacian that has been known as an edge detection function can be used for noise removal applications. The algorithm can be implemented on a 3x3 window and easily tuned by number of iterations. Image denoising is simplified to the reduction of the pixels value with their related Laplacian value weighted by local noise estimator. The only parameter which controls smoothness is the number of iterations. Noise reduction quality of the introduced method is evaluated and compared with some classic algorithms like Wiener and Total Variation based filters for Gaussian noise. And also the method compared with the state-of-the-art method BM3D for some images. The algorithm appears to be easy, fast and comparable with many classic denoising algorithms for Gaussian noise.

How to stop or change a motor response: Laplacian and independent component analysis approach

Response inhibition is an essential control function necessary to adapt one's behavior. This key cognitive capacity is assumed to be dependent on the prefrontal cortex and basal ganglia. It is unresolved whether varying inhibitory demands engage different control mechanisms or whether a single motor inhibitory mechanism is involved in any situation. We addressed this question by comparing electrophysiological activity in conditions that require stopping a response to conditions that require switching to an alternate response. Analyses of electrophysiological data obtained from stop-signal tasks are complicated by overlapping stimulus-related activity that is distributed over frontal and parietal cortical recording sites. Here, we applied Laplacian transformation and independent component analysis (ICA) to overcome these difficulties. Participants were faster in switching compared to stopping a response, but we did not observe differences in neural activity between these conditions. Both stop- and change-trials Laplacian transformed ERPs revealed a comparable bilateral parieto-occipital negativity around 180 ms and a frontocentral negativity around 220 ms. ICA results suggested an inhibition-related frontocentral component which was characterized by a negativity around 200 ms with a likely source in anterior cingulate cortex. The data provide support for the importance of posterior mediofrontal areas in inhibitory response control and are consistent with a common neural pathway underlying stopping and changing of a motor response. The methodological approach proved useful to distinguish frontal and parietal sources despite similar timing and the ICA approach allowed assessment of single-trial data with respect to behavioral data.

Elliptic differential operators on Lipschitz domains and abstract boundary value problems

This paper consists of two parts. In the first part, which is of more abstract nature, the notion of quasi-boundary triples and associated Weyl functions is developed further in such a way that it can be applied to elliptic boundary value problems on non-smooth domains. A key feature is the extension of the boundary maps by continuity to the duals of certain range spaces, which directly leads to a description of all self-adjoint extensions of the underlying symmetric operator with the help of abstract boundary values. In the second part of the paper a complete description is obtained of all self-adjoint realizations of the Laplacian on bounded Lipschitz domains, as well as Kreĭn type resolvent formulas and a spectral characterization in terms of energy dependent Dirichlet-to-Neumann maps. These results can be viewed as the natural generalization of recent results by Gesztesy and Mitrea for quasi-convex domains. In this connection we also characterize the maximal range spaces of the Dirichlet and Neumann trace operators on a bounded Lipschitz domain in terms of the Dirichlet-to-Neumann map. The general results from the first part of the paper are also applied to higher order elliptic operators on smooth domains, and particular attention is paid to the second order case which is illustrated with various examples.

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.

Effects of time lag and frequency matching on phase-based connectivity

The time- and frequency-varying dynamics of how brain regions interact is one of the fundamental mysteries of neuroscience. In electrophysiological data, functional connectivity is often measured through the consistency of oscillatory phase angles between two electrodes placed in or over different brain regions. However, due to volume conduction, the results of such analyses can be difficult to interpret, because mathematical estimates of connectivity can be driven both by true inter-regional connectivity, and by volume conduction from the same neural source. Generally, there are two approaches to attenuate artifacts due to volume conduction: spatial filtering in combination with standard connectivity methods, or connectivity methods such as the weighted phase lag index that are blind to instantaneous connectivity that may reflect volume conduction artifacts. The purpose of this paper is to compare these two approaches directly in the presence of different connectivity time lags (5 or 25 ms) and physiologically realistic frequency non-stationarities. The results show that standard connectivity methods in combination with Laplacian spatial filtering correctly identified simulated connectivity regardless of time lag or changes in frequency, although residual volume conduction artifacts were seen in the vicinity of the "seed" electrode. Weighted phase lag index under-estimated connectivity strength at small time lags and failed to identify connectivity in the presence of frequency mismatches or non-stationarities, but did not misidentify volume conduction as "connectivity." Both approaches have strengths and limitations, and this paper concludes with practical advice for when to use which approach in context of hypothesis testing and exploratory data analyses.

Speed pressure in conflict situations impedes inhibitory action control in Parkinson's disease

The current study investigated the effects of Parkinson's disease (PD) on the ability to resolve conflicts when performance emphasized speed vs. response accuracy. PD patients and healthy controls (HC) completed a Simon task, and a subset of participants provided movement-related potential (MRP) data to investigate motor cortex activation and inhibition associated with conflict resolution. Both groups adjusted performance strategically with speed or accuracy instructions. The groups experienced similar susceptibility to making fast errors in conflict trials, but PD patients were less proficient compared to HC at suppressing incorrect responses, especially under speed pressure. Analysis of MRPs showed attenuated inhibition of the motor cortex controlling the conflicting response in PD patients compared to HC. These results confirm the detrimental effects of PD on inhibitory control mechanisms with speed pressure and also suggest that a downstream effect of inhibitory dysfunction in PD might be due to diminished inhibition of the motor cortex.

High-Frequency Oscillations Recorded on the Scalp of Patients With Epilepsy Using Tripolar Concentric Ring Electrodes

Epilepsy is the second most prevalent neurological disorder ([Formula: see text]% prevalence) affecting [Formula: see text] million people worldwide with up to 75% from developing countries. The conventional electroencephalogram is plagued with artifacts from movements, muscles, and other sources. Tripolar concentric ring electrodes automatically attenuate muscle artifacts and provide improved signal quality. We performed basic experiments in healthy humans to show that tripolar concentric ring electrodes can indeed record the physiological alpha waves while eyes are closed. We then conducted concurrent recordings with conventional disc electrodes and tripolar concentric ring electrodes from patients with epilepsy. We found that we could detect high frequency oscillations, a marker for early seizure development and epileptogenic zone, on the scalp surface that appeared to become more narrow-band just prior to seizures. High frequency oscillations preceding seizures were present in an average of 35.5% of tripolar concentric ring electrode data channels for all the patients with epilepsy whose seizures were recorded and absent in the corresponding conventional disc electrode data. An average of 78.2% of channels that contained high frequency oscillations were within the seizure onset or irritative zones determined independently by three epileptologists based on conventional disc electrode data and videos.

Integrated Laplacian-based phase unwrapping and background phase removal for quantitative susceptibility mapping

Quantitative susceptibility mapping (QSM) is a recently developed MRI technique that provides a quantitative measure of tissue magnetic susceptibility. To compute tissue magnetic susceptibilities based on gradient echoes, QSM requires reliable unwrapping of the measured phase images and removal of contributions caused by background susceptibilities. Typically, the two steps are performed separately. Here, we present a method that simultaneously performs phase unwrapping and HARmonic (background) PhasE REmovaL using the LAplacian operator (HARPERELLA). Both numerical simulations and in vivo human brain images show that HARPERELLA effectively removes both phase wraps and background phase, whilst preserving all low spatial frequency components originating from brain tissues. When compared with other QSM phase preprocessing techniques, such as path-based phase unwrapping followed by background phase removal, HARPERELLA preserves the tissue phase signal in gray matter, white matter and cerebrospinal fluid with excellent robustness, providing a convenient and accurate solution for QSM. The proposed algorithm is provided, together with QSM and susceptibility tensor imaging (STI) tools, in a shared software package named 'STI Suite'.

Integrated Laplacian-based phase unwrapping and background phase removal for quantitative susceptibility mapping

Quantitative susceptibility mapping (QSM) is a recently developed MRI technique that provides a quantitative measure of tissue magnetic susceptibility. To compute tissue magnetic susceptibilities based on gradient echoes, QSM requires reliable unwrapping of the measured phase images and removal of contributions caused by background susceptibilities. Typically, the two steps are performed separately. Here, we present a method that simultaneously performs phase unwrapping and HARmonic (background) PhasE REmovaL using the LAplacian operator (HARPERELLA). Both numerical simulations and in vivo human brain images show that HARPERELLA effectively removes both phase wraps and background phase, whilst preserving all low spatial frequency components originating from brain tissues. When compared with other QSM phase preprocessing techniques, such as path-based phase unwrapping followed by background phase removal, HARPERELLA preserves the tissue phase signal in gray matter, white matter and cerebrospinal fluid with excellent robustness, providing a convenient and accurate solution for QSM. The proposed algorithm is provided, together with QSM and susceptibility tensor imaging (STI) tools, in a shared software package named 'STI Suite'.

Remarks on the boundary set of spectral equipartitions

Given a bounded open set in R2 (or in a Riemannian manifold), and a partition of Ω by k open sets ωj, we consider the quantity maxj λ(ωj), where λ(ωj) is the ground state energy of the Dirichlet realization of the Laplacian in ωj. We denote by Lk(Ω) the infimum of maxj λ(ω) over all k-partitions. A minimal k-partition is a partition that realizes the infimum. Although the analysis of minimal k-partitions is rather standard when k=2 (we find the nodal domains of a second eigenfunction), the analysis for higher values of k becomes non-trivial and quite interesting. Minimal partitions are in particular spectral equipartitions, i.e. the ground state energies λ(ωj) are all equal. The purpose of this paper is to revisit various properties of nodal sets, and to explore if they are also true for minimal partitions, or more generally for spectral equipartitions. We prove a lower bound for the length of the boundary set of a partition in the two-dimensional situation. We consider estimates involving the cardinality of the partition.

NeuroMap: A Spline-Based Interactive Open-Source Software for Spatiotemporal Mapping of 2D and 3D MEA Data

A major characteristic of neural networks is the complexity of their organization at various spatial scales, from microscopic local circuits to macroscopic brain-scale areas. Understanding how neural information is processed thus entails the ability to study them at multiple scales simultaneously. This is made possible using microelectrodes array (MEA) technology. Indeed, high-density MEAs provide large-scale coverage (several square millimeters) of whole neural structures combined with microscopic resolution (about 50 μm) of unit activity. Yet, current options for spatiotemporal representation of MEA-collected data remain limited. Here we present NeuroMap, a new interactive Matlab-based software for spatiotemporal mapping of MEA data. NeuroMap uses thin plate spline interpolation, which provides several assets with respect to conventional mapping methods used currently. First, any MEA design can be considered, including 2D or 3D, regular or irregular, arrangements of electrodes. Second, spline interpolation allows the estimation of activity across the tissue with local extrema not necessarily at recording sites. Finally, this interpolation approach provides a straightforward analytical estimation of the spatial Laplacian for better current sources localization. In this software, coregistration of 2D MEA data on the anatomy of the neural tissue is made possible by fine matching of anatomical data with electrode positions using rigid-deformation-based correction of anatomical pictures. Overall, NeuroMap provides substantial material for detailed spatiotemporal analysis of MEA data. The package is distributed under GNU General Public License and available at http://sites.google.com/site/neuromapsoftware.