Strong connectivity in real directed networks.

In many real, directed networks, the strongly connected component of nodes which are mutually reachable is very small. This does not fit with current theory, based on random graphs, according to which strong connectivity depends on mean degree and degree-degree correlations. And it has important implications for other properties of real networks and the dynamical behavior of many complex systems. We find that strong connectivity depends crucially on the extent to which the network has an overall direction or hierarchical ordering-a property measured by trophic coherence. Using percolation theory, we find the critical point separating weakly and strongly connected regimes and confirm our results on many real-world networks, including ecological, neural, trade, and social networks. We show that the connectivity structure can be disrupted with minimal effort by a targeted attack on edges which run counter to the overall direction. This means that many dynamical processes on networks can depend significantly on a small fraction of edges.

Machine learning classification of conduct disorder with high versus low levels of callous-unemotional traits based on facial emotion recognition abilities.

Conduct disorder (CD) with high levels of callous-unemotional traits (CD/HCU) has been theoretically linked to specific difficulties with fear and sadness recognition, in contrast to CD with low levels of callous-unemotional traits (CD/LCU). However, experimental evidence for this distinction is mixed, and it is unclear whether these difficulties are a reliable marker of CD/HCU compared to CD/LCU. In a large sample (N = 1263, 9-18 years), we combined univariate analyses and machine learning classifiers to investigate whether CD/HCU is associated with disproportionate difficulties with fear and sadness recognition over other emotions, and whether such difficulties are a reliable individual-level marker of CD/HCU. We observed similar emotion recognition abilities in CD/HCU and CD/LCU. The CD/HCU group underperformed relative to typically developing (TD) youths, but difficulties were not specific to fear or sadness. Classifiers did not distinguish between youths with CD/HCU versus CD/LCU (52% accuracy), although youths with CD/HCU and CD/LCU were reliably distinguished from TD youths (64% and 60%, respectively). In the subset of classifiers that performed well for youths with CD/HCU, fear and sadness were the most relevant emotions for distinguishing them from youths with CD/LCU and TD youths, respectively. We conclude that non-specific emotion recognition difficulties are common in CD/HCU, but are not reliable individual-level markers of CD/HCU versus CD/LCU. These findings highlight that a reduced ability to recognise facial expressions of distress should not be assumed to be a core feature of CD/HCU.

Manifold Alignment Aware Ants: A Markovian Process for Manifold Extraction.

The presence of manifolds is a common assumption in many applications, including astronomy and computer vision. For instance, in astronomy, low-dimensional stellar structures, such as streams, shells, and globular clusters, can be found in the neighborhood of big galaxies such as the Milky Way. Since these structures are often buried in very large data sets, an algorithm, which can not only recover the manifold but also remove the background noise (or outliers), is highly desirable. While other works try to recover manifolds either by pushing all points toward manifolds or by downsampling from dense regions, aiming to solve one of the problems, they generally fail to suppress the noise on manifolds and remove background noise simultaneously. Inspired by the collective behavior of biological ants in food-seeking process, we propose a new algorithm that employs several random walkers equipped with a local alignment measure to detect and denoise manifolds. During the walking process, the agents release pheromone on data points, which reinforces future movements. Over time the pheromone concentrates on the manifolds, while it fades in the background noise due to an evaporation procedure. We use the Markov chain (MC) framework to provide a theoretical analysis of the convergence of the algorithm and its performance. Moreover, an empirical analysis, based on synthetic and real-world data sets, is provided to demonstrate its applicability in different areas, such as improving the performance of t-distributed stochastic neighbor embedding (t-SNE) and spectral clustering using the underlying MC formulas, recovering astronomical low-dimensional structures, and improving the performance of the fast Parzen window density estimator.

Establishing a quantitative fluorescence assay for the rapid detection of kynurenine in urine.

The kynurenine metabolite is associated with many diseases and disorders, ranging from diabetes and sepsis to more recently COVID-19. Here we report a fluorescence-based assay for the detection of kynurenine in urine using a specific chemosensor, 3-formyl-4-(ethylthio)-7-(diethylamino)-coumarin. The assay produces a linear response at clinically relevant ranges (1-20 µM), with a limit of detection of 0.7 µM. The average standard addition recoveries of kynurenine in synthetic urine samples are near to 100%, and the relative standard deviation values are less than 8%. The established fluorescence assay for quantitative analysis of kynurenine in urine is facile, sensitive and accurate and holds great potential for low-cost and high-throughput analysis of kynurenine in clinical laboratory settings.

Data extraction for epidemiological research (DExtER): a novel tool for automated clinical epidemiology studies.

The use of primary care electronic health records for research is abundant. The benefits gained from utilising such records lies in their size, longitudinal data collection and data quality. However, the use of such data to undertake high quality epidemiological studies, can lead to significant challenges particularly in dealing with misclassification, variation in coding and the significant effort required to pre-process the data in a meaningful format for statistical analysis. In this paper, we describe a methodology to aid with the extraction and processing of such databases, delivered by a novel software programme; the "Data extraction for epidemiological research" (DExtER). The basis of DExtER relies on principles of extract, transform and load processes. The tool initially provides the ability for the healthcare dataset to be extracted, then transformed in a format whereby data is normalised, converted and reformatted. DExtER has a user interface designed to obtain data extracts specific to each research question and observational study design. There are facilities to input the requirements for; eligible study period, definition of exposed and unexposed groups, outcome measures and important baseline covariates. To date the tool has been utilised and validated in a multitude of settings. There have been over 35 peer-reviewed publications using the tool, and DExtER has been implemented as a validated public health surveillance tool for obtaining accurate statistics on epidemiology of key morbidities. Future direction of this work will be the application of the framework to linked as well as international datasets and the development of standardised methods for conducting electronic pre-processing and extraction from datasets for research purposes.

Positive and negative parenting in conduct disorder with high versus low levels of callous-unemotional traits.

Less is known about the relationship between conduct disorder (CD), callous-unemotional (CU) traits, and positive and negative parenting in youth compared to early childhood. We combined traditional univariate analyses with a novel machine learning classifier (Angle-based Generalized Matrix Learning Vector Quantization) to classify youth (N = 756; 9-18 years) into typically developing (TD) or CD groups with or without elevated CU traits (CD/HCU, CD/LCU, respectively) using youth- and parent-reports of parenting behavior. At the group level, both CD/HCU and CD/LCU were associated with high negative and low positive parenting relative to TD. However, only positive parenting differed between the CD/HCU and CD/LCU groups. In classification analyses, performance was best when distinguishing CD/HCU from TD groups and poorest when distinguishing CD/HCU from CD/LCU groups. Positive and negative parenting were both relevant when distinguishing CD/HCU from TD, negative parenting was most relevant when distinguishing between CD/LCU and TD, and positive parenting was most relevant when distinguishing CD/HCU from CD/LCU groups. These findings suggest that while positive parenting distinguishes between CD/HCU and CD/LCU, negative parenting is associated with both CD subtypes. These results highlight the importance of considering multiple parenting behaviors in CD with varying levels of CU traits in late childhood/adolescence.

A New Framework for Analysis of Coevolutionary Systems-Directed Graph Representation and Random Walks.

Studying coevolutionary systems in the context of simplified models (i.e., games with pairwise interactions between coevolving solutions modeled as self plays) remains an open challenge since the rich underlying structures associated with pairwise-comparison-based fitness measures are often not taken fully into account. Although cyclic dynamics have been demonstrated in several contexts (such as intransitivity in coevolutionary problems), there is no complete characterization of cycle structures and their effects on coevolutionary search. We develop a new framework to address this issue. At the core of our approach is the directed graph (digraph) representation of coevolutionary problems that fully captures structures in the relations between candidate solutions. Coevolutionary processes are modeled as a specific type of Markov chains-random walks on digraphs. Using this framework, we show that coevolutionary problems admit a qualitative characterization: a coevolutionary problem is either solvable (there is a subset of solutions that dominates the remaining candidate solutions) or not. This has an implication on coevolutionary search. We further develop our framework that provides the means to construct quantitative tools for analysis of coevolutionary processes and demonstrate their applications through case studies. We show that coevolution of solvable problems corresponds to an absorbing Markov chain for which we can compute the expected hitting time of the absorbing class. Otherwise, coevolution will cycle indefinitely and the quantity of interest will be the limiting invariant distribution of the Markov chain. We also provide an index for characterizing complexity in coevolutionary problems and show how they can be generated in a controlled manner.

Learning Predictive Statistics: Strategies and Brain Mechanisms.

When immersed in a new environment, we are challenged to decipher initially incomprehensible streams of sensory information. However, quite rapidly, the brain finds structure and meaning in these incoming signals, helping us to predict and prepare ourselves for future actions. This skill relies on extracting the statistics of event streams in the environment that contain regularities of variable complexity from simple repetitive patterns to complex probabilistic combinations. Here, we test the brain mechanisms that mediate our ability to adapt to the environment's statistics and predict upcoming events. By combining behavioral training and multisession fMRI in human participants (male and female), we track the corticostriatal mechanisms that mediate learning of temporal sequences as they change in structure complexity. We show that learning of predictive structures relates to individual decision strategy; that is, selecting the most probable outcome in a given context (maximizing) versus matching the exact sequence statistics. These strategies engage distinct human brain regions: maximizing engages dorsolateral prefrontal, cingulate, sensory-motor regions, and basal ganglia (dorsal caudate, putamen), whereas matching engages occipitotemporal regions (including the hippocampus) and basal ganglia (ventral caudate). Our findings provide evidence for distinct corticostriatal mechanisms that facilitate our ability to extract behaviorally relevant statistics to make predictions.SIGNIFICANCE STATEMENT Making predictions about future events relies on interpreting streams of information that may initially appear incomprehensible. Past work has studied how humans identify repetitive patterns and associative pairings. However, the natural environment contains regularities that vary in complexity from simple repetition to complex probabilistic combinations. Here, we combine behavior and multisession fMRI to track the brain mechanisms that mediate our ability to adapt to changes in the environment's statistics. We provide evidence for an alternate route for learning complex temporal statistics: extracting the most probable outcome in a given context is implemented by interactions between executive and motor corticostriatal mechanisms compared with visual corticostriatal circuits (including hippocampal cortex) that support learning of the exact temporal statistics.

Learning pharmacokinetic models for in vivo glucocorticoid activation.

To understand trends in individual responses to medication, one can take a purely data-driven machine learning approach, or alternatively apply pharmacokinetics combined with mixed-effects statistical modelling. To take advantage of the predictive power of machine learning and the explanatory power of pharmacokinetics, we propose a latent variable mixture model for learning clusters of pharmacokinetic models demonstrated on a clinical data set investigating 11ß-hydroxysteroid dehydrogenase enzymes (11ß-HSD) activity in healthy adults. The proposed strategy automatically constructs different population models that are not based on prior knowledge or experimental design, but result naturally as mixture component models of the global latent variable mixture model. We study the parameter of the underlying multi-compartment ordinary differential equation model via identifiability analysis on the observable measurements, which reveals the model is structurally locally identifiable. Further approximation with a perturbation technique enables efficient training of the proposed probabilistic latent variable mixture clustering technique using Estimation Maximization. The training on the clinical data results in 4 clusters reflecting the prednisone conversion rate over a period of 4 h based on venous blood samples taken at 20-min intervals. The learned clusters differ in prednisone absorption as well as prednisone/prednisolone conversion. In the discussion section we include a detailed investigation of the relationship of the pharmacokinetic parameters of the trained cluster models for possible or plausible physiological explanation and correlations analysis using additional phenotypic participant measurements.

Learning predictive statistics from temporal sequences: Dynamics and strategies.

Human behavior is guided by our expectations about the future. Often, we make predictions by monitoring how event sequences unfold, even though such sequences may appear incomprehensible. Event structures in the natural environment typically vary in complexity, from simple repetition to complex probabilistic combinations. How do we learn these structures? Here we investigate the dynamics of structure learning by tracking human responses to temporal sequences that change in structure unbeknownst to the participants. Participants were asked to predict the upcoming item following a probabilistic sequence of symbols. Using a Markov process, we created a family of sequences, from simple frequency statistics (e.g., some symbols are more probable than others) to context-based statistics (e.g., symbol probability is contingent on preceding symbols). We demonstrate the dynamics with which individuals adapt to changes in the environment's statistics-that is, they extract the behaviorally relevant structures to make predictions about upcoming events. Further, we show that this structure learning relates to individual decision strategy; faster learning of complex structures relates to selection of the most probable outcome in a given context (maximizing) rather than matching of the exact sequence statistics. Our findings provide evidence for alternate routes to learning of behaviorally relevant statistics that facilitate our ability to predict future events in variable environments.

Indefinite Proximity Learning: A Review.

Efficient learning of a data analysis task strongly depends on the data representation. Most methods rely on (symmetric) similarity or dissimilarity representations by means of metric inner products or distances, providing easy access to powerful mathematical formalisms like kernel or branch-and-bound approaches. Similarities and dissimilarities are, however, often naturally obtained by nonmetric proximity measures that cannot easily be handled by classical learning algorithms. Major efforts have been undertaken to provide approaches that can either directly be used for such data or to make standard methods available for these types of data. We provide a comprehensive survey for the field of learning with nonmetric proximities. First, we introduce the formalism used in nonmetric spaces and motivate specific treatments for nonmetric proximity data. Second, we provide a systematization of the various approaches. For each category of approaches, we provide a comparative discussion of the individual algorithms and address complexity issues and generalization properties. In a summarizing section, we provide a larger experimental study for the majority of the algorithms on standard data sets. We also address the problem of large-scale proximity learning, which is often overlooked in this context and of major importance to make the method relevant in practice. The algorithms we discuss are in general applicable for proximity-based clustering, one-class classification, classification, regression, and embedding approaches. In the experimental part, we focus on classification tasks.

The benefits of modeling slack variables in SVMs.

In this letter, we explore the idea of modeling slack variables in support vector machine (SVM) approaches. The study is motivated by SVM+, which models the slacks through a smooth correcting function that is determined by additional (privileged) information about the training examples not available in the test phase. We take a closer look at the meaning and consequences of smooth modeling of slacks, as opposed to determining them in an unconstrained manner through the SVM optimization program. To better understand this difference we only allow the determination and modeling of slack values on the same information--that is, using the same training input in the original input space. We also explore whether it is possible to improve classification performance by combining (in a convex combination) the original SVM slacks with the modeled ones. We show experimentally that this approach not only leads to improved generalization performance but also yields more compact, lower-complexity models. Finally, we extend this idea to the context of ordinal regression, where a natural order among the classes exists. The experimental results confirm principal findings from the binary case.

Spatial-temporal modelling of fMRI data through spatially regularized mixture of hidden process models.

Previous work investigated a range of spatio-temporal constraints for fMRI data analysis to provide robust detection of neural activation. We present a mixture-based method for the spatio-temporal modelling of fMRI data. This approach assumes that fMRI time series are generated by a probabilistic superposition of a small set of spatio-temporal prototypes (mixture components). Each prototype comprises a temporal model that explains fMRI signals on a single voxel and the model's "region of influence" through a spatial prior over the voxel space. As the key ingredient of our temporal model, the Hidden Process Model (HPM) framework proposed in Hutchinson et al. (2009) is adopted to infer the overlapping cognitive processes triggered by stimuli. Unlike the original HPM framework, we use a parametric model of Haemodynamic Response Function (HRF) so that biological constraints are naturally incorporated in the HRF estimation. The spatial priors are defined in terms of a parameterised distribution. Thus, the total number of parameters in the model does not depend on the number of voxels. The resulting model provides a conceptually principled and computationally efficient approach to identify spatio-temporal patterns of neural activation from fMRI data, in contrast to most conventional approaches in the literature focusing on the detection of spatial patterns. We first verify the proposed model in a controlled experimental setting using synthetic data. The model is further validated on real fMRI data obtained from a rapid event-related visual recognition experiment (Mayhew et al., 2012). Our model enables us to evaluate in a principled manner the variability of neural activations within individual regions of interest (ROIs). The results strongly suggest that, compared with occipitotemporal regions, the frontal ones are less homogeneous, requiring two HPM prototypes per region. Despite the rapid event-related experimental design, the model is capable of disentangling the perceptual judgement and motor response processes that are both activated in the frontal ROIs. Spatio-temporal heterogeneity in the frontal regions seems to be associated with diverse dynamic localizations of the two hidden processes in different subregions of frontal ROIs.

Emerging opportunities and challenges for the future of reservoir computing.

Reservoir computing originates in the early 2000s, the core idea being to utilize dynamical systems as reservoirs (nonlinear generalizations of standard bases) to adaptively learn spatiotemporal features and hidden patterns in complex time series. Shown to have the potential of achieving higher-precision prediction in chaotic systems, those pioneering works led to a great amount of interest and follow-ups in the community of nonlinear dynamics and complex systems. To unlock the full capabilities of reservoir computing towards a fast, lightweight, and significantly more interpretable learning framework for temporal dynamical systems, substantially more research is needed. This Perspective intends to elucidate the parallel progress of mathematical theory, algorithm design and experimental realizations of reservoir computing, and identify emerging opportunities as well as existing challenges for large-scale industrial adoption of reservoir computing, together with a few ideas and viewpoints on how some of those challenges might be resolved with joint efforts by academic and industrial researchers across multiple disciplines.

Using Repeated Measurements to Predict Cardiovascular Risk in Patients With Type 2 Diabetes Mellitus.

The QRISK cardiovascular disease (CVD) risk assessment model is not currently optimized for patients with type 2 diabetes mellitus (T2DM). We aim to identify if the abundantly available repeatedly measured data for patients with T2D improves the predictive capability of QRISK to support the decision-making process regarding CVD prevention in patients with T2DM. We identified patients with T2DM aged 25 to 85, not on statin treatment and without pre-existing CVD from the IQVIA Medical Research Data United Kingdom primary care database and then followed them up until the first diagnosis of CVD, ischemic heart disease, or stroke/transient ischemic attack. We included traditional, nontraditional risk factors and relevant treatments for our analysis. We then undertook a Cox's hazards model accounting for time-dependent covariates to estimate the hazard rates for each risk factor and calculated a 10-year risk score. Models were developed for males and females separately. We tested the performance of our models using validation data and calculated discrimination and calibration statistics. The study included 198,835 (180,143 male with 11,976 outcomes and 90,466 female with 8,258 outcomes) patients. The 10-year predicted survival probabilities for females was 0.87 (0.87 to 0.87), whereas the observed survival estimates from the Kaplan-Meier curve for all female models was 0.87 (0.86 to 0.87). The predicted and observed survival estimates for males were 0.84 (0.84 to 0.84) and 0.84 (0.83 to 0.84) respectively. The Harrell's C-index of all female models and all male models were 0.71 and 0.69 respectively. We found that including time-varying repeated measures, only mildly improved CVD risk prediction for T2DM patients in comparison to the current practice standard. We advocate for further research using time-varying data to identify if the involvement of further covariates may improve the accuracy of currently accepted prediction models.

Unsupervised multimodal modeling of cognitive and brain health trajectories for early dementia prediction.

Predicting the course of neurodegenerative disorders early has potential to greatly improve clinical management and patient outcomes. A key challenge for early prediction in real-world clinical settings is the lack of labeled data (i.e., clinical diagnosis). In contrast to supervised classification approaches that require labeled data, we propose an unsupervised multimodal trajectory modeling (MTM) approach based on a mixture of state space models that captures changes in longitudinal data (i.e., trajectories) and stratifies individuals without using clinical diagnosis for model training. MTM learns the relationship between states comprising expensive, invasive biomarkers (ß-amyloid, grey matter density) and readily obtainable cognitive observations. MTM training on trajectories stratifies individuals into clinically meaningful clusters more reliably than MTM training on baseline data alone and is robust to missing data (i.e., cognitive data alone or single assessments). Extracting an individualized cognitive health index (i.e., MTM-derived cluster membership index) allows us to predict progression to AD more precisely than standard clinical assessments (i.e., cognitive tests or MRI scans alone). Importantly, MTM generalizes successfully from research cohort to real-world clinical data from memory clinic patients with missing data, enhancing the clinical utility of our approach. Thus, our multimodal trajectory modeling approach provides a cost-effective and non-invasive tool for early dementia prediction without labeled data (i.e., clinical diagnosis) with strong potential for translation to clinical practice.

Exploitation of pairwise class distances for ordinal classification.

Ordinal classification refers to classification problems in which the classes have a natural order imposed on them because of the nature of the concept studied. Some ordinal classification approaches perform a projection from the input space to one-dimensional (latent) space that is partitioned into a sequence of intervals (one for each class). Class identity of a novel input pattern is then decided based on the interval its projection falls into. This projection is trained only indirectly as part of the overall model fitting. As with any other latent model fitting, direct construction hints one may have about the desired form of the latent model can prove very useful for obtaining high-quality models. The key idea of this letter is to construct such a projection model directly, using insights about the class distribution obtained from pairwise distance calculations. The proposed approach is extensively evaluated with 8 nominal and ordinal classifiers methods, 10 real-world ordinal classification data sets, and 4 different performance measures. The new methodology obtained the best results in average ranking when considering three of the performance metrics, although significant differences are found for only some of the methods. Also, after observing other methods of internal behavior in the latent space, we conclude that the internal projections do not fully reflect the intraclass behavior of the patterns. Our method is intrinsically simple, intuitive, and easily understandable, yet highly competitive with state-of-the-art approaches to ordinal classification.

Generalized Learning Vector Quantization With Log-Euclidean Metric Learning on Symmetric Positive-Definite Manifold.

In many classification scenarios, the data to be analyzed can be naturally represented as points living on the curved Riemannian manifold of symmetric positive-definite (SPD) matrices. Due to its non-Euclidean geometry, usual Euclidean learning algorithms may deliver poor performance on such data. We propose a principled reformulation of the successful Euclidean generalized learning vector quantization (GLVQ) methodology to deal with such data, accounting for the nonlinear Riemannian geometry of the manifold through log-Euclidean metric (LEM). We first generalize GLVQ to the manifold of SPD matrices by exploiting the LEM-induced geodesic distance (GLVQ-LEM). We then extend GLVQ-LEM with metric learning. In particular, we study both 1) a more straightforward implementation of the metric learning idea by adapting metric in the space of vectorized log-transformed SPD matrices and 2) the full formulation of metric learning without matrix vectorization, thus preserving the second-order tensor structure. To obtain the distance metric in the full LEM learning (LEML) approaches, two algorithms are proposed. One method is to restrict the distance metric to be full rank, treating the distance metric tensor as an SPD matrix, and readily use the LEM framework (GLVQ-LEML-LEM). The other method is to cast no such restriction, treating the distance metric tensor as a fixed rank positive semidefinite matrix living on a quotient manifold with total space equipped with flat geometry (GLVQ-LEML-FM). Experiments on multiple datasets of different natures demonstrate the good performance of the proposed methods.

Influence and influenceability: global directionality in directed complex networks.

Knowing which nodes are influential in a complex network and whether the network can be influenced by a small subset of nodes is a key part of network analysis. However, many traditional measures of importance focus on node level information without considering the global network architecture. We use the method of trophic analysis to study directed networks and show that both 'influence' and 'influenceability' in directed networks depend on the hierarchical structure and the global directionality, as measured by the trophic levels and trophic coherence, respectively. We show that in directed networks trophic hierarchy can explain: the nodes that can reach the most others; where the eigenvector centrality localizes; which nodes shape the behaviour in opinion or oscillator dynamics; and which strategies will be successful in generalized rock-paper-scissors games. We show, moreover, that these phenomena are mediated by the global directionality. We also highlight other structural properties of real networks related to influenceability, such as the pseudospectra, which depend on trophic coherence. These results apply to any directed network and the principles highlighted-that node hierarchy is essential for understanding network influence, mediated by global directionality-are applicable to many real-world dynamics.

A two-step inverse solution for a single dipole cardiac source.

Introduction: The inverse problem of electrocardiography noninvasively localizes the origin of undesired cardiac activity, such as a premature ventricular contraction (PVC), from potential recordings from multiple torso electrodes. However, the optimal number and placement of electrodes for an accurate solution of the inverse problem remain undetermined. This study presents a two-step inverse solution for a single dipole cardiac source, which investigates the significance of the torso electrodes on a patient-specific level. Furthermore, the impact of the significant electrodes on the accuracy of the inverse solution is studied. Methods: Body surface potential recordings from 128 electrodes of 13 patients with PVCs and their corresponding homogeneous and inhomogeneous torso models were used. The inverse problem using a single dipole was solved in two steps: First, using information from all electrodes, and second, using a subset of electrodes sorted in descending order according to their significance estimated by a greedy algorithm. The significance of electrodes was computed for three criteria derived from the singular values of the transfer matrix that correspond to the inversely estimated origin of the PVC computed in the first step. The localization error (LE) was computed as the Euclidean distance between the ground truth and the inversely estimated origin of the PVC. The LE obtained using the 32 and 64 most significant electrodes was compared to the LE obtained when all 128 electrodes were used for the inverse solution. Results: The average LE calculated for both torso models and using all 128 electrodes was 28.8 ± 11.9 mm. For the three tested criteria, the average LEs were 32.6 ± 19.9 mm, 29.6 ± 14.7 mm, and 28.8 ± 14.5 mm when 32 electrodes were used. When 64 electrodes were used, the average LEs were 30.1 ± 16.8 mm, 29.4 ± 12.0 mm, and 29.5 ± 12.6 mm. Conclusion: The study found inter-patient variability in the significance of torso electrodes and demonstrated that an accurate localization by the inverse solution with a single dipole could be achieved using a carefully selected reduced number of electrodes.