Details of simulated annealing algorithm to estimate parameters of multiple current dipoles using biomagnetic data

Improved Dipole Source Localization from Simultaneous MEG-EEG Data by Combining a Global Optimization Algorithm with a Local Parameter Search: A Brain Phantom Study

Dipole localization, a fundamental challenge in electromagnetic source imaging, inherently constitutes an optimization problem aimed at solving the inverse problem of electric current source estimation within the human brain. The accuracy of dipole localization algorithms is contingent upon the complexity of the forward model, often referred to as the head model, and the signal-to-noise ratio (SNR) of measurements. In scenarios characterized by low SNR, often corresponding to deep-seated sources, existing optimization techniques struggle to converge to global minima, thereby leading to the localization of dipoles at erroneous positions, far from their true locations. This study presents a novel hybrid algorithm that combines simulated annealing with the traditional quasi-Newton optimization method, tailored to address the inherent limitations of dipole localization under low-SNR conditions. Using a realistic head model for both electroencephalography (EEG) and magnetoencephalography (MEG), it is demonstrated that this novel hybrid algorithm enables significant improvements of up to 45% in dipole localization accuracy compared to the often-used dipole scanning and gradient descent techniques. Localization improvements are not only found for single dipoles but also in two-dipole-source scenarios, where sources are proximal to each other. The novel methodology presented in this work could be useful in various applications of clinical neuroimaging, particularly in cases where recordings are noisy or sources are located deep within the brain.

SANA: cross-species prediction of Gene Ontology GO annotations via topological network alignment

Topological network alignment aims to align two networks node-wise in order to maximize the observed common connection (edge) topology between them. The topological alignment of two protein-protein interaction (PPI) networks should thus expose protein pairs with similar interaction partners allowing, for example, the prediction of common Gene Ontology (GO) terms. Unfortunately, no network alignment algorithm based on topology alone has been able to achieve this aim, though those that include sequence similarity have seen some success. We argue that this failure of topology alone is due to the sparsity and incompleteness of the PPI network data of almost all species, which provides the network topology with a small signal-to-noise ratio that is effectively swamped when sequence information is added to the mix. Here we show that the weak signal can be detected using multiple stochastic samples of "good" topological network alignments, which allows us to observe regions of the two networks that are robustly aligned across multiple samples. The resulting network alignment frequency (NAF) strongly correlates with GO-based Resnik semantic similarity and enables the first successful cross-species predictions of GO terms based on topology-only network alignments. Our best predictions have an AUPR of about 0.4, which is competitive with state-of-the-art algorithms, even when there is no observable sequence similarity and no known homology relationship. While our results provide only a "proof of concept" on existing network data, we hypothesize that predicting GO terms from topology-only network alignments will become increasingly practical as the volume and quality of PPI network data increase.

Design of coaxial coils using hybrid machine learning

A coil system to generate a uniform field is urgently needed in quantum experiments. However, general coil configurations based on the analytical method have not considered practical restrictions, such as the region for coil placement due to holes in the center of the magnetic shield, which could not be directly applied in most of the quantum experiments. In this paper, we develop a coil design method for quantum experiments using hybrid machine learning. The algorithm part consists of a machine learner based on an artificial neural network and a differential evolution (DE) learner. The cooperation of both learners demonstrates its higher efficiency than a single DE learner and robustness in the coil optimization problem compared with analytical proposals. With the help of a DE learner, in numerical simulation, a machine learner can successfully design coaxial coil systems that generate fields whose relative inhomogeneity in a 25 mm-long central region is ∼10^-6 under constraints. In addition, for experiments, a coil system with 0.069% inhomogeneity of the field, designed by a machine learner, is constructed, which is mainly limited by machining the precision of the circuit board. Benefitting from machine learning's high-dimension optimization capabilities, our coil design method is convenient and has potential for various quantum experiments.

Simulated Annealing, Its Variants and Engineering Applications

This paper presents a review of research reported on simulated annealing (SA). Different cooling/annealing schedules are summarized. Variants of SA are delineated. Recent applications of SA in engineering are reviewed.

Multicompare tests of the performance of different metaheuristics in EEG dipole source localization

We study the use of nonparametric multicompare statistical tests on the performance of simulated annealing (SA), genetic algorithm (GA), particle swarm optimization (PSO), and differential evolution (DE), when used for electroencephalographic (EEG) source localization. Such task can be posed as an optimization problem for which the referred metaheuristic methods are well suited. Hence, we evaluate the localization's performance in terms of metaheuristics' operational parameters and for a fixed number of evaluations of the objective function. In this way, we are able to link the efficiency of the metaheuristics with a common measure of computational cost. Our results did not show significant differences in the metaheuristics' performance for the case of single source localization. In case of localizing two correlated sources, we found that PSO (ring and tree topologies) and DE performed the worst, then they should not be considered in large-scale EEG source localization problems. Overall, the multicompare tests allowed to demonstrate the little effect that the selection of a particular metaheuristic and the variations in their operational parameters have in this optimization problem.

Localization accuracy of single current dipoles from tangential components of auditory evoked fields

We investigated the localization accuracy of single current dipoles from the tangential components of auditory evoked fields. The tangential fields were measured using planar gradiometers arranged in a way so as to detect two orthogonal field components parallel to a flat plane. Field responses to 1 kHz pure tones were recorded and equivalent current dipoles (ECDs) of N1m peak were estimated based on a locally fitted spherical conductor model. As a measure of localization accuracy, the standard deviation of the coordinates of the ECDs of N1m was obtained from repeated measurements for one subject. The estimated ECDs had a standard deviation of 5.5 mm and their mean location was at the supratemporal plane in the sylvian fissure on the MR image of the subject. In order to investigate the contribution of various errors to the localization accuracy, simulations using a sphere model and experiments using a realistically shaped skull phantom were performed. It was found that the background noise, which consisted of instrumental noise and spontaneous brain fields, was the main source of the errors that could explain the observed standard deviation. Further, the amount of systematic error was much less than the standard deviation due to the background noise. These results suggest that the volume currents in a non-spherical conductor shape such as the temporal region do not produce substantial errors in the localization of current dipoles from tangential components of auditory evoked fields.

Use of the ventricular propagated excitation model in the magnetocardiographic inverse problem for reconstruction of electrophysiological properties

A novel magnetocardiographic inverse method for reconstructing the action potential amplitude (APA) and the activation time (AT) on the ventricular myocardium is proposed. This method is based on the propagated excitation model, in which the excitation is propagated through the ventricle with nonuniform height of action potential. Assumption of stepwise waveform on the transmembrane potential was introduced in the model. Spatial gradient of transmembrane potential, which is defined by APA and AT distributed in the ventricular wall, is used for the computation of a current source distribution. Based on this source model, the distributions of APA and AT are inversely reconstructed from the QRS interval of magnetocardiogram (MCG) utilizing a maximum a posteriori approach. The proposed reconstruction method was tested through computer simulations. Stability of the methods with respect to measurement noise was demonstrated. When reference APA was provided as a uniform distribution, root-mean-square errors of estimated APA were below 10 mV for MCG signal-to-noise ratios greater than, or equal to, 20 dB. Low-amplitude regions located at several sites in reference APA distributions were correctly reproduced in reconstructed APA distributions. The goal of our study is to develop a method for detecting myocardial ischemia through the depression of reconstructed APA distributions.

Fast realistic modeling in bioelectromagnetism using lead-field interpolation

The practical use of realistic models in bioelectromagnetism is limited by the time-consuming amount of numerical calculations. We propose a method leading to much higher speed than currently available, and compatible with any kind of numerical methods (boundary elements (BEM), finite elements, finite differences). Illustrated with the BEM for EEG and MEG, it applies to ECG and MCG as well. The principle is two-fold. First, a Lead-Field matrix is calculated (once for all) for a grid of dipoles covering the brain volume. Second, any forward solution is interpolated from the pre-calculated Lead-Fields corresponding to grid dipoles near the source. Extrapolation is used for shallow sources falling outside the grid. Three interpolation techniques were tested: trilinear, second-order Bézier (Bernstein polynomials), and 3D spline. The trilinear interpolation yielded the highest speed gain, with factors better than x10,000 for a 9,000-triangle BEM model. More accurate results could be obtained with the Bézier interpolation (speed gain approximately 1,000), which, combined with a 8-mm step grid, lead to intrinsic localization and orientation errors of only 0.2 mm and 0.2 degrees. Further improvements in MEG could be obtained by interpolating only the contribution of secondary currents. Cropping grids by removing shallow points lead to a much better estimation of the dipole orientation in EEG than when solving the forward problem classically, providing an efficient alternative to locally refined models. This method would show special usefulness when combining realistic models with stochastic inverse procedures (simulated annealing, genetic algorithms) requiring many forward calculations.

Optimal design of gradient coils in MR imaging: optimizing coil performance versus minimizing cost functions

The "forward" method of an optimal gradient coil design provides a coil that has the minimal cost function value. It is shown in this study that the solution obtained by minimizing the cost function is directly dependent on the specified cost function and generally results in a deviation from the most desirable coil design. In this paper, a gradient coil design approach for obtaining the best achievable coil performance for pre-determined imaging applications is presented. Through this approach, all intermediate coil performance values calculated during an optimization process, using a simulated annealing algorithm, are stored and presented in a three-dimensional data set. Using this three-dimensional data set, a coil designer is able to make a balance between different coil performance parameters and to select a coil that is the most desirable for the pre-determined imaging applications.

Spatio-temporal EEG source localization using simulated annealing

The estimation of multiple dipole parameters in spatio-temporal source modeling (STSM) of electroencephalographic (EEG) data is a difficult nonlinear optimization problem due to multiple local minima in the cost function. A straightforward iterative optimization approach to such a problem is very susceptible to being trapped in a local minimum, thereby resulting in incorrect estimates of the dipole parameters. In this paper, we present and evaluate a more robust optimization approach based on the simulated annealing algorithm. The complexity of this approach for the STSM problem was reduced by separating the dipole parameters into linear (moment) and nonlinear (location) components. The effectiveness of the proposed method and its superiority over the traditional nonlinear simplex technique in escaping local minima were tested and demonstrated through computer simulations. The annealing algorithm and its implementation for multidipole estimation are also discussed. We found the simulated annealing approach to be 7-31% more effective than the simplex method at converging to the true global minimum for a number of different kinds of three-dipole problems simulated in this work. In addition, the computational cost of the proposed approach was only marginally higher than its simplex counterpart. The annealing method also yielded similar solutions irrespective of the initial guesses used. The proposed simulated annealing method is an attractive alternative to the simplex method that is currently more common in dipole estimation applications.

A robust reconstruction of sparse biomagnetic sources

Inequalty constraints are introduced to a normalized minimum-L1-norm estimator, which gives a sparse solution of the biomagnetic inverse problem. The constraints have a numeric tolerance to take into account the measurement ambiguity caused by noise. Computer simulation and phantom-data analysis show how the solution is improved by the constraints with a moderate tolerance; the improvement is examined in noisy conditions such that signal-to-noise ratios (SNR's) are lower than 10 dB.

Studies on the performance of circular and elliptical Z-gradient coils using a simulated annealing algorithm

Elliptical z-gradient coils with different ellipticities are optimized using a simulated annealing algorithm. This study demonstrates that when the coil ellipticity was changed from 1.0 to 1.67, the gradient strength was increased by 21% and the coil inductance was reduced by about 34%. At the same time, the gradient inhomogeneity was increased by a factor between 15 to 47%. In these examples we also observed that when the coil ellipticity was increased from 1.0 to 1.11, the gradient inhomogeneity was reduced by a factor between 16 to 19%. This study provides a quantitative assessment of the advantages and disadvantages in coil performance obtained by using elliptical z-gradient coils.

Selective minimum-norm solution of the biomagnetic inverse problem

A new multidipole estimation method which gives a sparse solution of the biomagnetic inverse problem is proposed. This solution is extracted from the basic feasible solutions of linearly independent data equations. These feasible solutions are obtained by selecting exactly as many dipole-moments as the number of magnetic sensors. By changing the selection, we search for the minimum-norm vector of selected moments. As a result, a practically sparse solution is obtained; computer-simulated solutions for Lp-norm (p = 2, 1, 0.5, 0.2) have a small number of significant moments around the real source-dipoles. In particular, the solution for L1-norm is equivalent to the minimum-L1-norm solution of the original inverse problem. This solution can be uniquely computed by using Linear Programming.

Multiple current dipole estimation using simulated annealing

A method for estimating electrical current distribution in the human brain using a multiple current dipole model is presented. A cost function for estimating multiple dipoles is proposed and a simulated annealing algorithm is used to obtain an acceptable solution. Computer simulation is used to evaluate the effectiveness of this method.

Relationship between dipole parameter estimation errors and measurement conditions in magnetoencephalography

The relationship between dipole parameter estimation errors and measurement conditions in magnetoencephalography is determined by computer simulation. The model uses a single current dipole in a spherical homogeneous medium. Dipole parameters are estimated using a moving dipole procedure. Signal-to-noise ratio (SNR) is defined as the square-root of the ratio of the average signal power to the average noise power over all measurement points. At SNR > 20, accurate estimation can be carried out independently of dipole depth and coil size. At SNR < 20, dipole depth influences estimation error. When the dipole is located near the center of the sphere, the measurement region should include both extrema of the magnetic field to minimize estimation error. However, when the dipole is not so deep, the position of the measurement region does not influence estimation error. When SNR < 4, estimation error increases as coil size increases. Coil size minimizing estimation error is determined by the ratio of environmental magnetic field noise to electrical noise. For a constant size of measurement region, increasing the number of measurement points decreases estimation error to a certain level. This error level depends on SNR. The number of measurement points required to minimize estimation error also depends on SNR.