A Proposed Method for the Inverse Problem in Electrocardiology

The inverse problem in electrocardiography is considered. A method is proposed in which the cardiac electrical generator is represented by a set of dipoles, fixed in location and direction in order to reflect the known features of myocardial excitation, but variable in strength. A crucial innovation is that since the dipole directions have been so chosen, the dipole strengths must be constrained nonnegative. Surface potentials are measured in vivo and the dipole strengths inferred. In this process, torso models with a varying degree of realism are used. An 11-dipole set is used and potentials are measured at 126 surface locations. For a particular normal subject, the effect of various variables, such as the torso modeling assumptions, on the dipole strengths is investigated. Condensed results are given for twelve normal subjects and two patients.     

Effects of cavities on EEG dipole localization and their relations with surface electrode positions

Effects of cavities in the human head on EEG dipole localization have been investigated by computer simulation. The human head is represented by a homogeneous spherical conductor including an eccentric spherical cavity which approximates effects of actual cavities inside the head. The homogeneous sphere model is used for assessing the effects caused by neglecting the cavity in the volume conductor model in the inverse dipole fitting procedure. Four electrode configurations have been examined to investigate their relation to the EEG inverse dipole solution. After examination of 2520 dipoles in the brain, the effects of cavities in the human head are found to be negligible when the dipole is located in the cortex or in the subcortex. When the dipole is located in the brain stem, the EEG inverse dipole solution is strongly affected by the cavity and is sensitive to the electrode configuration on the scalp. The EEG inverse dipole solution in the deep brain is sensitive to inhomogeneity in the lower part of the head when a single positive or negative potential pole is observed by the electrodes on the scalp, and at the same time is sensitive to the extent of the scalp covered by the electrodes. In conclusion, the electrodes should cover as much of the upper scalp as possible for deep source localization.     

基于模拟退火法由脑磁图推测电流偶极子参数

利用模拟退火（Ｓｉｍｕｌａｔｅｄ Ａｎｎｅａｌｉｎｇ） 算法,由脑磁图（ ＭＥＧ） 数据反演脑内作为磁源的单电流偶极子参数,可以得到理想的结果。在上述工作的基础上,对脑内多电流偶极子参数的反演,则呈现如下状况：即以少于实际源数目的偶极子为源假设反演,目标函数得不到极小优化。反之,目标函数可以得到极小优化, 但出现多余的伪偶极子, 且这些伪偶极子在多次不同条件的反演结果中,处于不稳定状态。若将多次反演结果中处于不稳定状态的偶极子作为伪偶极子的判据而将其排除,则可以得到一种判断磁源偶极子数目的方法     

Real-Time Localization of Moving Dipole Sources for Tracking Multiple Free-Swimming Weakly Electric Fish

In order to survive, animals must quickly and accurately locate prey, predators, and conspecifics using the signals they generate. The signal source location can be estimated using multiple detectors and the inverse relationship between the received signal intensity (RSI) and the distance, but difficulty of the source localization increases if there is an additional dependence on the orientation of a signal source. In such cases, the signal source could be approximated as an ideal dipole for simplification. Based on a theoretical model, the RSI can be directly predicted from a known dipole location; but estimating a dipole location from RSIs has no direct analytical solution. Here, we propose an efficient solution to the dipole localization problem by using a lookup table (LUT) to store RSIs predicted by our theoretically derived dipole model at many possible dipole positions and orientations. For a given set of RSIs measured at multiple detectors, our algorithm found a dipole location having the closest matching normalized RSIs from the LUT, and further refined the location at higher resolution. Studying the natural behavior of weakly electric fish (WEF) requires efficiently computing their location and the temporal pattern of their electric signals over extended periods. Our dipole localization method was successfully applied to track single or multiple freely swimming WEF in shallow water in real-time, as each fish could be closely approximated by an ideal current dipole in two dimensions. Our optimized search algorithm found the animal’s positions, orientations, and tail-bending angles quickly and accurately under various conditions, without the need for calibrating individual-specific parameters. Our dipole localization method is directly applicable to studying the role of active sensing during spatial navigation, or social interactions between multiple WEF. Furthermore, our method could be extended to other application areas involving dipole source localization.     

Application of boundary element method for more accurate localization of EEG dipole sources

The proposed modification of boundary element method (BEM) makes it possible to solve direct EEG problem taking into account individual head geometry. This modification decreases computational cost thus enabling practical usage of BEM for localization of EEG dipole sources. The method developed was applied for error estimation of dipole localization using spherical approximation with the aid of BESA software. It was shown that the localization error of BESA for real head geometry could reach 2 cm.     

Determination of Interspin Distance Distributions by cw-ESR Is a Single Linear Inverse Problem

Cw-ESR distance measurement method is extremely valuable for studying the dynamics-function relationship of biomolecules. However, extracting distance distributions from experiments has been a highly technique-demanding procedure. It has never been conclusively identified, to our knowledge, that the problems involved in the analysis are ill posed and are best solved using Tikhonov regularization. We treat the problems from a novel point of view. First of all, we identify the equations involved and uncover that they are actually two linear first-kind Fredholm integral equations. They can be combined into one single linear inverse problem and solved in a Tikhonov regularization procedure. The improvement with our new treatment is significant. Our approach is a direct and reliable mathematical method capable of providing an unambiguous solution to the ill-posed problem. It need not perform nonlinear least-squares fitting to infer a solution from noise-contaminated data and, accordingly, substantially reduces the computation time and the difficulty of analysis. Numerical tests and experimental data of polyproline II peptides with variant spin-labeled sites are provided to demonstrate our approach. The high resolution of the distance distributions obtainable with our new approach enables a detailed insight into the flexibility of dynamic structure and the identification of conformational species in solution state.     

Dipole characterization of single neurons from their extracellular action potentials

The spatial variation of the extracellular action potentials (EAP) of a single neuron contains information about the size and location of the dominant current source of its action potential generator, which is typically in the vicinity of the soma. Using this dependence in reverse in a three-component realistic probe + brain + source model, we solved the inverse problem of characterizing the equivalent current source of an isolated neuron from the EAP data sampled by an extracellular probe at multiple independent recording locations. We used a dipole for the model source because there is extensive evidence it accurately captures the spatial roll-off of the EAP amplitude, and because, as we show, dipole localization, beyond a minimum cell-probe distance, is a more accurate alternative to approaches based on monopole source models. Dipole characterization is separable into a linear dipole moment optimization where the dipole location is fixed, and a second, nonlinear, global optimization of the source location. We solved the linear optimization on a discrete grid via the lead fields of the probe, which can be calculated for any realistic probe + brain model by the finite element method. The global source location was optimized by means of Tikhonov regularization that jointly minimizes model error and dipole size. The particular strategy chosen reflects the fact that the dipole model is used in the near field, in contrast to the typical prior applications of dipole models to EKG and EEG source analysis. We applied dipole localization to data collected with stepped tetrodes whose detailed geometry was measured via scanning electron microscopy. The optimal dipole could account for 96% of the power in the spatial variation of the EAP amplitude. Among various model error contributions to the residual, we address especially the error in probe geometry, and the extent to which it biases estimates of dipole parameters. This dipole characterization method can be applied to any recording technique that has the capabilities of taking multiple independent measurements of the same single units.     

Why forces between proteins follow different Hofmeister series for pH above and below pI

The relative effectiveness of different anions in crystallizing proteins follows a reversed Hofmeister sequence for pHpI. The phenomenon has been known almost since Hofmeister's original work but it has not been understood. It is here given a theoretical explanation. Classical electrolyte and double layer theory deals only with electrostatic forces acting between ions and proteins. Hydration and hydration interactions are dealt with usually only in terms of assumed hard core models. But there are, at and above biological salt concentrations, other non-electrostatic (NES) ion-specific forces acting that are ignored in such modeling. Such electrodynamic fluctuation forces are also responsible for ion-specific hydration. These missing forces are variously comprehended under familiar but generally unquantified terms, typically, hydration, hydrogen bonding, pi-electron-cation interactions, dipole-dipole, dipole-induced dipole and induced dipole-induced dipole forces and so on. The many important body electrodynamic fluctuation force contributions are accessible from extensions of Lifshitz theory from which, with relevant dielectric susceptibility data on solutions as a function of frequency, the forces can be extracted quantitatively, at least in principle. The classical theories of colloid science that miss such contributions do not account for a whole variety of ion-specific phenomena. Numerical results that include these non-electrostatic forces are given here for model calculations of the force between two model charge-regulated hen-egg-white protein surfaces. The surfaces are chosen to carry the same charge groups and charge density as the protein. What emerges is that for pHpI (where anions are co-ions) the forces increase in the order NaCl相似文献   

脑源定位技术的精度评估及其在实际中的应用

脑源定位技术旨在通过头皮表面的脑电、脑磁信号来识别大脑内的神经活动源，是研究大脑皮层神经活动、认知过程和病理功能的基础。其毫秒级的时间分辨率可以有效弥补功能核磁共振在低时间分辨率方面的不足。然而，理论分析层面中逆问题的不适定性，以及实践操作层面上不同的记录方式、电极数量和头模型构建等过程带来的误差，给脑源定位的准确性带来极大挑战，也在一定程度上限制了脑源定位方法在神经科学和心理学研究以及临床诊断治疗中的实际应用。因此，理论分析和实践操作层面中的精度评估在脑源定位方法的实际使用中至关重要。针对以上问题，本文在对现有脑源定位方法介绍的基础上，着重分析了脑源定位技术的精度评估方法以及其在基础研究和临床诊断治疗中的实际应用。具体地，本文在理论分析中总结了基于空间分辨率、基于点扩散以及串扰函数的评估方法对于不同脑源定位方法中源的重叠程度和其他源对目标源的影响；在实践操作中介绍了记录方式、电极数量和密度、头部容积传导模型等因素对源定位精度的影响；进一步介绍了脑源定位技术在时频分析、连通性分析中的应用，以及其在临床中的应用，包括癫痫、注意缺陷与多动障碍等脑部疾病。     

A New Approach for Inversion of Large Random Matrices in Massive MIMO Systems

We report a novel approach for inversion of large random matrices in massive Multiple-Input Multiple Output (MIMO) systems. It is based on the concept of inverse vectors in which an inverse vector is defined for each column of the principal matrix. Such an inverse vector has to satisfy two constraints. Firstly, it has to be in the null-space of all the remaining columns. We call it the null-space problem. Secondly, it has to form a projection of value equal to one in the direction of selected column. We term it as the normalization problem. The process essentially decomposes the inversion problem and distributes it over columns. Each column can be thought of as a node in the network or a particle in a swarm seeking its own solution, the inverse vector, which lightens the computational load on it. Another benefit of this approach is its applicability to all three cases pertaining to a linear system: the fully-determined, the over-determined, and the under-determined case. It eliminates the need of forming the generalized inverse for the last two cases by providing a new way to solve the least squares problem and the Moore and Penrose''s pseudoinverse problem. The approach makes no assumption regarding the size, structure or sparsity of the matrix. This makes it fully applicable to much in vogue large random matrices arising in massive MIMO systems. Also, the null-space problem opens the door for a plethora of methods available in literature for null-space computation to enter the realm of matrix inversion. There is even a flexibility of finding an exact or approximate inverse depending on the null-space method employed. We employ the Householder''s null-space method for exact solution and present a complete exposition of the new approach. A detailed comparison with well-established matrix inversion methods in literature is also given.     

Dynamics of granzyme B-induced apoptosis: mathematical modeling

Apoptosis is mediated by an intracellular biochemical system that mainly includes proteins (procaspases, caspases, inhibitors, Bcl-2 protein family as well as substances released from mitochondrial intermembrane space). The dynamics of caspase activation and target cleavage in apoptosis induced by granzyme B in a single K562 cell was studied using a mathematical model of the dynamics of granzyme B-induced apoptosis developed in this work. Also the first application of optimization approach to determination of unknown kinetic constants of biochemical apoptotic reactions was presented. The optimization approach involves solving of two problems: direct and inverse. Solving the direct optimization problem, we obtain the initial (baseline) concentrations of procaspases for known kinetic constants through conditional minimization of a cost function based on the principle of minimum protein consumption by the apoptosis system. The inverse optimization problem is aimed at determination of unknown kinetic constants of apoptotic biochemical reactions proceeding from the condition that the optimal concentrations of procaspases resulting from the solution of the direct optimization problem coincide with the observed ones, that is, those determined by biochemical methods. The Multidimensional Index Method was used to perform numerical solution of the inverse optimization problem.     

Solution of the direct and inverse kinematic problems by a common algorithm based on the mean of multiple computations

For the control of the movement of a multijoint manipulator a mental model which represents the geometrical properties of the arm may prove helpful. Using this model the direct and the inverse kinematic problem could be solved. Here we propose such a model which is based on a recurrent network. It is realized for the example of a three-joint manipulator working in a two-dimensional plane, i.e., for a manipulator with one extra degree of freedom. The system computes the complete set of variables, in our example the three joint angles and the two work-space coordinates of the endpoint of the manipulator. The system finds a stable state and a geometrically correct solution even if only a part of these state variables is given. Thus, the direct and the inverse kinematic problem as well as any mixed problem, including the underconstrained case, can be solved by the network.     

FT3D: three-dimensional Fourier analysis on small Unix workstations for electron microscopy and tomo graphic studies

FT3D is a self-contained package of tools for three-dimensionalFourier analysis, written in the C language for Unix workstations.It can evaluate direct transforms of three-dimensional realfunctions, inverse transforms, auto-and cross-correlations andspectra. The library has been developed to support three-dimensionalreconstructions of biological structures from projections obtainedin the electron microscope. This paper discusses some fraturesof the library, which has been implemented in such a way asto profit from the resources of modern workstations. A tableof elapsed times for jobs of different dimensions with differentRAM buffers is reported for the particular hardware used inthe authors' laboratory.     

Transient polarization currents in the squid giant axon.

It has been repeatedly noted that the change of conformation of the molecules that serve as the ion-selective channels for sodium and potassium conductance in the nerve membrane will be accompanied by a change in the dipole moment of the molecule. This time-dependent change of dipole moment will produce transient currents in the membrane. The canonical form for these currents is determined with conventional statistical mechanics formalism. It is pointed out that the voltage dependence of the conductance channel conductance determines the free energy of the system to within a factor that is an unknown function of the voltage. Since the dipole currents do not depend on this unknown function, they are completely determined 0y the observed properties of the conductance system. The predicted properties of these dipole currents, their time constants and strengths, are calculated. By using the observed properties of gating currents, the density of the sodium channels is computed. The predicted properties of the dipole currents are found to compare satisfactorily with the observed properties of gating currents.     

EEG source localization: comparative study of classical and neural network methods.

We address the problem of estimating biopotential sources within the brain, based on EEG signals observed on the scalp. This problem, known as the inverse problem of electrophysiology, has no closed-form solution, and requires iterative techniques such as the Levenberg-Marquardt (LM) algorithm. Considering the nonlinear nature of the inverse problem, and the low signal to noise ratio inherent in EEG signals, a backpropagation neural network (BPN) has been recently proposed as a solution. The technique has not been properly compared with classical techniques such as the LM method, or with more recent neural network techniques such as the Radial Basis Function (RBF) network. In this paper, we provide improved strategies based on BPN and consider RBF networks in solving the inverse problem. We compare the performances of BPN, RBF and a hybrid technique with that of the classical LM method.     

Extremal problems for topological indices in combinatorial chemistry.

Topological indices of molecular graphs are related to several physicochemical characteristics; recently, the inverse problem for some of these indices has been studied, and it has some applications in the design of combinatorial libraries for drug discovery. It is thus very natural to study also extremal problems for these indices, i.e., finding graphs having minimal or maximal index. In this paper, these questions will be discussed for three different indices, namely the sigma-index, the c-index and the Z-index, with emphasis on the sigma-index.     

Stable generation of simple forms

It has been suggested by Thom that the mathematical theory of structural stability gives insight into the stable reproduction of form in biological systems. In this paper a process for the generation of some simple forms is described and it is shown that the forms produced are insensitive to perturbations in the production process. The reason for this stability is found in an analysis of the process in terms of the theory of structural stability and in particular in terms of “catastrophes”. Based on the example of our form generating process, we give a general definition of form and define the pattern recognition problem (which is inverse to the generation problem) in the context of that definition.