Rigid spheres in Riemannian spaces

Choice of an appropriate (3+1)-foliation of spacetime or a (2+1)-foliation of the Cauchy space, leads often to a substantial simplification of various mathematical problems in General Relativity Theory. We propose a new method to construct such foliations. For this purpose we define a special family of topological two-spheres, which we call "rigid spheres". We prove that there is a four-parameter family of rigid spheres in a generic Riemannian three-manifold (in case of the flat Euclidean three-space these four parameters are: 3 coordinates of the center and the radius of the sphere). The rigid spheres can be used as building blocks for various ("spherical", "bispherical" etc.) foliations of the Cauchy space. This way a supertranslation ambiguity may be avoided. Generalization to the full 4D case is discussed. Our results generalize both the Huang foliations (cf. \cite{LHH}) and the foliations used by us (cf. \cite{JKL}) in the analysis of the two-body problem.Comment: 23 page

Common Atlas Format and 3D Brain Atlas Reconstructor: Infrastructure for Constructing 3D Brain Atlases

One of the challenges of modern neuroscience is integrating voluminous data of diferent modalities derived from a variety of specimens. This task requires a common spatial framework that can be provided by brain atlases. The first atlases were limited to two-dimentional presentation of structural data. Recently, attempts at creating 3D atlases have been made to offer navigation within non-standard anatomical planes and improve capability of localization of different types of data within the brain volume. The 3D atlases available so far have been created using frameworks which make it difficult for other researchers to replicate the results. To facilitate reproducible research and data sharing in the field we propose an SVG-based Common Atlas Format (CAF) to store 2D atlas delineations or other compatible data and 3D Brain Atlas Reconstructor (3dBAR), software dedicated to automated reconstruction of three-dimensional brain structures from 2D atlas data. The basic functionality is provided by (1) a set of parsers which translate various atlases from a number of formats into the CAF, and (2) a module generating 3D models from CAF datasets. The whole reconstruction process is reproducible and can easily be configured, tracked and reviewed, which facilitates fixing errors. Manual corrections can be made when automatic reconstruction is not sufficient. The software was designed to simplify interoperability with other neuroinformatics tools by using open file formats. The content can easily be exchanged at any stage of data processing. The framework allows for the addition of new public or proprietary content

Inverse Current Source Density Method in Two Dimensions: Inferring Neural Activation from Multielectrode Recordings

The recent development of large multielectrode recording arrays has made it affordable for an increasing number of laboratories to record from multiple brain regions simultaneously. The development of analytical tools for array data, however, lags behind these technological advances in hardware. In this paper, we present a method based on forward modeling for estimating current source density from electrophysiological signals recorded on a two-dimensional grid using multi-electrode rectangular arrays. This new method, which we call two-dimensional inverse Current Source Density (iCSD 2D), is based upon and extends our previous one- and three-dimensional techniques. We test several variants of our method, both on surrogate data generated from a collection of Gaussian sources, and on model data from a population of layer 5 neocortical pyramidal neurons. We also apply the method to experimental data from the rat subiculum. The main advantages of the proposed method are the explicit specification of its assumptions, the possibility to include system-specific information as it becomes available, the ability to estimate CSD at the grid boundaries, and lower reconstruction errors when compared to the traditional approach. These features make iCSD 2D a substantial improvement over the approaches used so far and a powerful new tool for the analysis of multielectrode array data. We also provide a free GUI-based MATLAB toolbox to analyze and visualize our test data as well as user datasets

Cortical Resonance Frequencies Emerge from Network Size and Connectivity

Neural oscillations occur within a wide frequency range with different brain regions exhibiting resonance-like characteristics at specific points in the spectrum. At the microscopic scale, single neurons possess intrinsic oscillatory properties, such that is not yet known whether cortical resonance is consequential to neural oscillations or an emergent property of the networks that interconnect them. Using a network model of loosely-coupled Wilson-Cowan oscillators to simulate a patch of cortical sheet, we demonstrate that the size of the activated network is inversely related to its resonance frequency. Further analysis of the parameter space indicated that the number of excitatory and inhibitory connections, as well as the average transmission delay between units, determined the resonance frequency. The model predicted that if an activated network within the visual cortex increased in size, the resonance frequency of the network would decrease. We tested this prediction experimentally using the steady-state visual evoked potential where we stimulated the visual cortex with different size stimuli at a range of driving frequencies. We demonstrate that the frequency corresponding to peak steady-state response inversely correlated with the size of the network. We conclude that although individual neurons possess resonance properties, oscillatory activity at the macroscopic level is strongly influenced by network interactions, and that the steady-state response can be used to investigate functional networks

Investigating large-scale brain dynamics using field potential recordings: Analysis and interpretation

New technologies to record electrical activity from the brain on a massive scale offer tremendous opportunities for discovery. Electrical measurements of large-scale brain dynamics, termed field potentials, are especially important to understanding and treating the human brain. Here, our goal is to provide best practices on how field potential recordings (EEG, MEG, ECoG and LFP) can be analyzed to identify large-scale brain dynamics, and to highlight critical issues and limitations of interpretation in current work. We focus our discussion of analyses around the broad themes of activation, correlation, communication and coding. We provide best-practice recommendations for the analyses and interpretations using a forward model and an inverse model. The forward model describes how field potentials are generated by the activity of populations of neurons. The inverse model describes how to infer the activity of populations of neurons from field potential recordings. A recurring theme is the challenge of understanding how field potentials reflect neuronal population activity given the complexity of the underlying brain systems

Improving the Generalization Ability of Neuro-Fuzzy Systems by e-Insensitive Learning

A new learning method tolerant of imprecision is introduced and used in neuro-fuzzy modelling. The proposed method makes it possible to dispose of an intrinsic inconsistency of neuro-fuzzy modelling, where zero-tolerance learning is used to obtain a fuzzy model tolerant of imprecision. This new method can be called e-insensitive learning, where, in order to fit the fuzzy model to real data, the e-insensitive loss function is used. e-insensitive learning leads to a model with minimal Vapnik-Chervonenkis dimension, which results in an improved generalization ability of this system. Another advantage of the proposed method is its robustness against outliers. This paper introduces two approaches to solving e-insensitive learning problem. The first approach leads to a quadratic programming problem with bound constraints and one linear equality constraint. The second approach leads to a problem of solving a system of linear inequalities. Two computationally efficient numerical methods for e-insensitive learning are proposed. Finally, examples are given to demonstrate the validity of the introduced methods

An ε-Insensitive Approach to Fuzzy Clustering

Fuzzy clustering can be helpful in finding natural vague boundaries in data. The fuzzy c-means method is one of the most popular clustering methods based on minimization of a criterion function. However, one of the greatest disadvantages of this method is its sensitivity to the presence of noise and outliers in the data. The present paper introduces a new varepsilon-insensitive Fuzzy C-Means (varepsilonFCM) clustering algorithm. As a special case, this algorithm includes the well-known Fuzzy C-Medians method (FCMED). The performance of the new clustering algorithm is experimentally compared with the Fuzzy C-Means (FCM) method using synthetic data with outliers and heavy-tailed, overlapped groups of the data

A Fuzzy If-Then Rule-Based Nonlinear Classifier

This paper introduces a new classifier design method that is based on a modification of the classical Ho-Kashyap procedure. The proposed method uses the absolute error, rather than the squared error, to design a linear classifier. Additionally, easy control of the generalization ability and robustness to outliers are obtained. Next, an extension to a nonlinear classifier by the mixture-of-experts technique is presented. Each expert is represented by a fuzzy if-then rule in the Takagi-Sugeno-Kang form. Finally, examples are given to demonstrate the validity of the introduced method

Iteratively reweighted least squares classifier and its l2- and l1-regularized Kernel versions

This paper introduces a new classifier design method based on regularized iteratively reweighted least squares criterion function. The proposed method uses various approximations of misclassification error, including: linear, sigmoidal, Huber and logarithmic. Using the represented theorem a kernel version of classifier design method is introduced. The conjugate gradient algorithm is used to minimize the proposed criterion function. Furthermore, .1-regularized kernel version of the classifier is introduced. In this case, the gradient projection is used to optimize the criterion function. Finally, an extensive experimental analysis on 14 benchmark datasets is given to demonstrate the validity of the introduced methods

Kernel Ho-Kashyap classifier with generalization control

This paper introduces a new classifier design method based on a kernel extension of the classical Ho-Kashyap procedure. The proposed method uses an approximation of the absolute error rather than the squared error to design a classifier, which leads to robustness against outliers and a better approximation of the misclassification error. Additionally, easy control of the generalization ability is obtained using the structural risk minimization induction principle from statistical learning theory. Finally, examples are given to demonstrate the validity of the introduced method