Claudio G. Carvalhaes

Approximations for the period of the simple pendulum based on the arithmetic-geometric mean

We use the arithmetic-geometric mean to derive approximate solutions for the period of the simple pendulum. The fast convergence of the arithmetic-geometric mean yields accurate solutions. We also discuss the invention of the pendulum clock by Christiaan Huygens in 1656–1657.

The surface Laplacian technique in EEG: Theory and methods

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

Exploring non-signalling polytopes with negative probability

Bipartite and tripartite EPR–Bell type systems are examined via joint quasi-probability distributions where probabilities are permitted to be negative. It is shown that such distributions exist only when the no-signalling condition is satisfied. A characteristic measure, the probability mass, is introduced and, via its minimization, limits the number of quasi-distributions describing a given marginal probability distribution. The minimized probability mass is shown to be an alternative way to characterize non-local systems. Non-signalling polytopes for two to eight settings in the bipartite scenario are examined and compared to prior work. Examining perfect cloning of non-local systems within the tripartite scenario suggests defining two categories of signalling. It is seen that many properties of non-local systems can be efficiently described by quasi-probability theory.

Using phase to recognize English phonemes and their distinctive features in the brain

The neural mechanisms used by the human brain to identify phonemes remain unclear. We recorded the EEG signals evoked by repeated presentation of 12 American English phonemes. A support vector machine model correctly recognized a high percentage of the EEG brain wave recordings represented by their phases, which were expressed in discrete Fourier transform coefficients. We show that phases of the oscillations restricted to the frequency range of 2–9 Hz can be used to successfully recognize brain processing of these phonemes. The recognition rates can be further improved using the scalp tangential electric field and the surface Laplacian around the auditory cortical area, which were derived from the original potential signal. The best rate for the eight initial consonants was 66.7%. Moreover, we found a distinctive phase pattern in the brain for each of these consonants. We then used these phase patterns to recognize the consonants, with a correct rate of 48.7%. In addition, in the analysis of the confusion matrices, we found significant similarity–differences were invariant between brain and perceptual representations of phonemes. These latter results supported the importance of phonological distinctive features in the neural representation of phonemes.

A spline framework for estimating the eeg surface laplacian using the euclidean metric

This letter develops a framework for EEG analysis and similar applications based on polyharmonic splines. This development overcomes a basic problem with the method of splines in the Euclidean setting: that it does not work on low-degree algebraic surfaces such as spherical and ellipsoidal scalp models. The method’s capability is illustrated through simulations on the three-sphere model and using empirical data.

EEG classification by ICA source selection of Laplacian-filtered data

We studied the performance of a double-spatial filtering method for classification of single-trial electroencephalography (EEG) data that couples the spherical surface Laplacian (SL) and independent component analysis (ICA). This method was evaluated in the context of a binary classification experiment with brain states driven by mental imagery of auditory and visual stimuli. A statistically significant improvement was achieved with respect to the rates provided by raw data and by data filtered by either SL or ICA.

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

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