视觉系统初级信息加工的一种数学模型——<Ⅰ>模型的空间性质

本文用广义Gabor(EQ)函数作为视觉系统进行初级信息加工的核函数.所谓广义Gabor函数,就是把原始Gabor函数中的一个独立变量扩展到三个(即空间变量从一维推广到二维,再加时间变量),并相应地引入一些参数.本文中EG取二大类形式:各向同性系统中表达式和各向异性系统表达式.当EG取某一特定形式,而且其中参数取某些特定值时,本模型可分别定性描述电生理实验中发现的几种主要感受野类型(Kuffler,Hubel和Wiesel型等).在空间频率域中引入对数变换假设后,EG经付里叶变换后可很好地合成人眼调制传递函数(MTF)     

A geometrical model for the perceived line of orientation based on subjective evaluations and human VEP data

A geometric model in proposed based on subjective dissimilarity estimates among differently oriented line segments in the frontal plane and on visual evoked potentials (VEP) recorded in respond to abrupt changes in the orientation of such a line segment. The orientations are represented by points constituting a clised planar curve, with interpoint distances corresponding to interstimulus dissimilarities. The angle of the radius-vertor connecting the center of the configuration with its circumference encodes the orientation of the line stimuli in the visual field, and two Cartesian axes drawn through the center are interpreted as two orientation-opponent channels in the neuronal network processing line orientation. The first channel (the ordinate axis) gives maximal positive and maximal negative responses to, respectively, vertical and horizontal orientations, whereas the second channel (the abscissa axis) gives maximal positive and maximal negative responses to, respectively, 45 and 135 degrees orientations. The VEP analysis shows that the activity of the first channel affects the interpeak amplitudes of both the P1-N2 (P130-N180) and N2-P2 (N180-P230) components, whereas the second channel affects only the late component N180-P230. We propose the hypothesis according to which outputs of the two channels are connected linearly (which is reflected in the city-block metric obtained when distances are computed directly from neuronal activity), but the growth rate of the overall output is inhibited nonlinearly when it reaches large absolute values. This inhibition effectively transforms the "true" city-block metric of the line orientation space into the Euclidean metric one obtains when distances are computed from subjective dissimilarity estimates. VEP amplitudes as intermediate characteristics of visual processing between neuronal activity and subjective estimates represent intermediate metric (between city-block and Euclidean).     

The reconstruction of the perceptual spaces of brightness and color on the basis of visual evoked potentials and their comparison with the data from behavioral trials

The visual evoked potentials to a change in color stimuli were studied. The amplitude of the N85 component was correlated with color discrimination. In cases when the brightness of one color was fixed and that of the other one changed, the amplitude dynamics of N85 was V-shaped with the minimum corresponding to the point of equal brightness of both colors. The N85 amplitude in this point serves as a measure of discrimination between stimuli chromaticity. The perceptual color space in rabbit (possessing two cone pigments) was constructed in accordance with the amplitudes of N85 to color change. This space represented a hypersphere in the four-dimensional Euclidean space. The perceptual spaces for brightness and color reconstructed on the basis of conditioning probabilities and N85 amplitudes evoked by replacement of colored and achromatic stimuli were shown to coincide. This suggests the common mechanism of the vector color coding.     

Application of the Gabor Transform for Analysis of Electromyographic Signals of the Intestine in the Low-Frequency Region

The Gabor transform allows quantitative estimation of the non-stationarity of the electromyographic signal in the low-frequency region with the maxim permissible time–frequency resolution. The calculation of the parameters of the Gabor transform was conducted on different time and frequency intervals to estimate the slow-wave activity of the intestine. It was demonstrated that the efficient size of the 32-s time window, which provides the efficient resolution of the frequency spectrum at 0.01 Hz, is suitable for the accurate study of the change in the frequency of slow waves. The ability to construct the dependence of the change in the frequency of slow waves of electromyograms on time with the specified accuracy was demonstrated.     

The Geometry of Path Coefficients and Correlations

In the present paper, a generalized cosine formula is obtained for the difference between two angles in a k-dimensional Euclidean space, which is utilized to derive one of the principles of path analysis, namely, the correlation between two random variables is the sum of all connecting paths between them.     

Isomap transform for segmenting human body shapes

Segmentation of the 3D human body is a very challenging problem in applications exploiting volume capture data. Direct clustering in the Euclidean space is usually complex or even unsolvable. This paper presents an original method based on the Isomap (isometric feature mapping) transform of the volume data-set. The 3D articulated posture is mapped by Isomap in the pose of Da Vinci's Vitruvian man. The limbs are unrolled from each other and separated from the trunk and pelvis, and the topology of the human body shape is recovered. In such a configuration, Hoshen-Kopelman clustering applied to concentric spherical shells is used to automatically group points into the labelled principal curves. Shepard interpolation is utilised to back-map points of the principal curves into the original volume space. The experimental results performed on many different postures have proved the validity of the proposed method. Reliability of less than 2?cm and 3° in the location of the joint centres and direction axes of rotations has been obtained, respectively, which qualifies this procedure as a potential tool for markerless motion analysis.     

Isomap transform for segmenting human body shapes

Segmentation of the 3D human body is a very challenging problem in applications exploiting volume capture data. Direct clustering in the Euclidean space is usually complex or even unsolvable. This paper presents an original method based on the Isomap (isometric feature mapping) transform of the volume data-set. The 3D articulated posture is mapped by Isomap in the pose of Da Vinci's Vitruvian man. The limbs are unrolled from each other and separated from the trunk and pelvis, and the topology of the human body shape is recovered. In such a configuration, Hoshen–Kopelman clustering applied to concentric spherical shells is used to automatically group points into the labelled principal curves. Shepard interpolation is utilised to back-map points of the principal curves into the original volume space. The experimental results performed on many different postures have proved the validity of the proposed method. Reliability of less than 2 cm and 3° in the location of the joint centres and direction axes of rotations has been obtained, respectively, which qualifies this procedure as a potential tool for markerless motion analysis.     

Euclidean sections of protein conformation space and their implications in dimensionality reduction

Dimensionality reduction is widely used in searching for the intrinsic reaction coordinates for protein conformational changes. We find the dimensionality?reduction methods using the pairwise root?mean?square deviation (RMSD) as the local distance metric face a challenge. We use Isomap as an example to illustrate the problem. We believe that there is an implied assumption for the dimensionality‐reduction approaches that aim to preserve the geometric relations between the objects: both the original space and the reduced space have the same kind of geometry, such as Euclidean geometry vs. Euclidean geometry or spherical geometry vs. spherical geometry. When the protein free energy landscape is mapped onto a 2D plane or 3D space, the reduced space is Euclidean, thus the original space should also be Euclidean. For a protein with N atoms, its conformation space is a subset of the 3N‐dimensional Euclidean space R^3N. We formally define the protein conformation space as the quotient space of R^3N by the equivalence relation of rigid motions. Whether the quotient space is Euclidean or not depends on how it is parameterized. When the pairwise RMSD is employed as the local distance metric, implicit representations are used for the protein conformation space, leading to no direct correspondence to a Euclidean set. We have demonstrated that an explicit Euclidean‐based representation of protein conformation space and the local distance metric associated to it improve the quality of dimensionality reduction in the tetra‐peptide and β‐hairpin systems. Proteins 2014; 82:2585–2596. © 2014 Wiley Periodicals, Inc.     

A kinetic model of RNA folding

The RNA folding process is represented as a Markov process with states corresponding to RNA secondary structures and transition probabilities corresponding to transformations of a secondary structure caused by formation or disintegration of a helix. Transition probabilities (kinetic constants) are determined. A notion of a group of structures is introduced, and it allows to reduce the state space. Energetic and kinetic parameters of pseudoknots are estimated. Algorithms for computation of a kinetic ensemble for structures and groups of structures are presented, as well as their modifications that take into account pseudoknots. The described algorithms are implemented as a procedure for prediction of RNA secondary structure that is included in the package DNA-SUN.     

Approximate geodesic distances reveal biologically relevant structures in microarray data

MOTIVATION: Genome-wide gene expression measurements, as currently determined by the microarray technology, can be represented mathematically as points in a high-dimensional gene expression space. Genes interact with each other in regulatory networks, restricting the cellular gene expression profiles to a certain manifold, or surface, in gene expression space. To obtain knowledge about this manifold, various dimensionality reduction methods and distance metrics are used. For data points distributed on curved manifolds, a sensible distance measure would be the geodesic distance along the manifold. In this work, we examine whether an approximate geodesic distance measure captures biological similarities better than the traditionally used Euclidean distance. RESULTS: We computed approximate geodesic distances, determined by the Isomap algorithm, for one set of lymphoma and one set of lung cancer microarray samples. Compared with the ordinary Euclidean distance metric, this distance measure produced more instructive, biologically relevant, visualizations when applying multidimensional scaling. This suggests the Isomap algorithm as a promising tool for the interpretation of microarray data. Furthermore, the results demonstrate the benefit and importance of taking nonlinearities in gene expression data into account.     

Using robotics to fold proteins and dock ligands

The problems of protein folding and ligand docking have been explored largely using molecular dynamics or Monte Carlo methods. These methods are very compute intensive because they often explore a much wider range of energies, conformations and time than necessary. In addition, Monte Carlo methods often get trapped in local minima. We initially showed that robotic motion planning permitted one to determine the energy of binding and dissociation of ligands from protein binding sites (Singh et al., 1999). The robotic motion planning method maps complicated three-dimensional conformational states into a much simpler, but higher dimensional space in which conformational rearrangements can be represented as linear paths. The dimensionality of the conformation space is of the same order as the number of degrees of conformational freedom in three-dimensional space. We were able to determine the relative energy of association and dissociation of a ligand to a protein by calculating the energetics of interaction for a few thousand conformational states in the vicinity of the protein and choosing the best path from the roadmap. More recently, we have applied roadmap planning to the problem of protein folding (Apaydin et al., 2002a). We represented multiple conformations of a protein as nodes in a compact graph with the edges representing the probability of moving between neighboring states. Instead of using Monte Carlo simulation to simulate thousands of possible paths through various conformational states, we were able to use Markov methods to calculate the steady state occupancy of each conformation, needing to calculate the energy of each conformation only once. We referred to this Markov method of representing multiple conformations and transitions as stochastic roadmap simulation or SRS. We demonstrated that the distribution of conformational states calculated with exhaustive Monte Carlo simulations asymptotically approached the Markov steady state if the same Boltzman energy distribution was used in both methods. SRS permits one to calculate contributions from all possible paths simultaneously with far fewer energy calculations than Monte Carlo or molecular dynamics methods. The SRS method also permits one to represent multiple unfolded starting states and multiple, near-native, folded states and all possible paths between them simultaneously. The SRS method is also independent of the function used to calculate the energy of the various conformational states. In a paper to be presented at this conference (Apaydin et al., 2002b) we have also applied SRS to ligand docking in which we calculate the dynamics of ligand-protein association and dissociation in the region of various binding sites on a number of proteins. SRS permits us to determine the relative times of association to and dissociation from various catalytic and non-catalytic binding sites on protein surfaces. Instead of just following the best path in a roadmap, we can calculate the contribution of all the possible binding or dissociation paths and their relative probabilities and energies simultaneously.     

The remapping of space in motor learning and human–machine interfaces

Studies of motor adaptation to patterns of deterministic forces have revealed the ability of the motor control system to form and use predictive representations of the environment. One of the most fundamental elements of our environment is space itself. This article focuses on the notion of Euclidean space as it applies to common sensory motor experiences. Starting from the assumption that we interact with the world through a system of neural signals, we observe that these signals are not inherently endowed with metric properties of the ordinary Euclidean space. The ability of the nervous system to represent these properties depends on adaptive mechanisms that reconstruct the Euclidean metric from signals that are not Euclidean. Gaining access to these mechanisms will reveal the process by which the nervous system handles novel sophisticated coordinate transformation tasks, thus highlighting possible avenues to create functional human–machine interfaces that can make that task much easier. A set of experiments is presented that demonstrate the ability of the sensory-motor system to reorganize coordination in novel geometrical environments. In these environments multiple degrees of freedom of body motions are used to control the coordinates of a point in a two-dimensional Euclidean space. We discuss how practice leads to the acquisition of the metric properties of the controlled space. Methods of machine learning based on the reduction of reaching errors are tested as a means to facilitate learning by adaptively changing he map from body motions to controlled device. We discuss the relevance of the results to the development of adaptive human–machine interfaces and optimal control.     

The use of canonical analysis for selection within a population of spring barley

A selection method is described involving the application of canonical analysis to data for a number of measured traits in a composite population of spring barley grown in two seasons. Lines were selected which fell within the Euclidean space defined by the range, for a number of standard cultivars, on each significant canonical variate. Alternatively, limits were fixed for an acceptable range for each trait and transformed to canonical variates. Lines were selected which fell nearest to the centroid of this acceptable space. The method might be used as an alternative to the selection index method where difficulty is often encountered in assigning appropriate economic weightings to the traits measured.     

Compositional dissimilarity as a robust measure of ecological distance

The robustness of quantitative measures of compositional dissimilarity between sites was evaluated using extensive computer simulations of species' abundance patterns over one and two dimensional configurations of sample sites in ecological space. Robustness was equated with the strength over a range of models, of the linear and monotonic (rank-order) relationship between the compositional dissimilarities and the corresponding Euclidean distances between sites measured in the ecological space. The range of models reflected different assumptions about species' response curve shape, sampling pattern of sites, noise level of the data, species' interactions, trends in total site abundance, and beta diversity of gradients.The Kulczynski, Bray-Curtis and Relativized Manhattan measures were found to have not only a robust monotonic relationship with ecological distance, but also a robust linear (proportional) relationship until ecological distances became large. Less robust measures included Chord distance, Kendall's coefficient, Chisquared distance, Manhattan distance, and Euclidean distance.A new ordination method, hybrid multidimensional scaling (HMDS), is introduced that combines metric and nonmetric criteria, and so takes advantage of the particular properties of robust dissimilarity measures such as the Kulczynski measure.We thank M. P. Austin for encouraging this study, and I. C. Prentice, E. Van der Maarel, and an anonymous reviewer for helpful comments. E. M. Adomeit provided technical assistance.     

基于独立元分析和联合小波熵检测多导联ECG信号的QRS

QRS波群的准确定位是ECG信号自动分析的基础。为提高QRS检测率,提出一种基于独立元分析（ICA）和联合小波熵（CWS）检测多导联ECG信号QRS的算法。ICA算法从滤波后的多导联ECG信号中分离出对应心室活动的独立元;然后对各独立元进行连续小波变换（CWT）,重构小波系数的相空间,结合相空间中的QRS信息对独立元排序;最后检测排序后独立元的CWS得到QRS信息。实验对St.Petersburg12导联心率失常数据库及64导联犬心外膜数据库测试,比较本文算法与单导联QRS检测算法和双导联QRS检测算法的性能。结果表明,该文算法的性能最好,检测准确率分别为99.98%和100%。     

Thinking Outside the Euclidean Box: Riemannian Geometry and Inter-Temporal Decision-Making

Inter-temporal decisions involves assigning values to various payoffs occurring at different temporal distances. Past research has used different approaches to study these decisions made by humans and animals. For instance, considering that people discount future payoffs at a constant rate (e.g., exponential discounting) or at variable rate (e.g., hyperbolic discounting). In this research, we question the widely assumed, but seldom questioned, notion across many of the existing approaches that the decision space, where the decision-maker perceives time and monetary payoffs, is a Euclidean space. By relaxing the rigid assumption of Euclidean space, we propose that the decision space is a more flexible Riemannian space of Constant Negative Curvature. We test our proposal by deriving a discount function, which uses the distance in the Negative Curvature space instead of Euclidean temporal distance. The distance function includes both perceived values of time as well as money, unlike past work which has considered just time. By doing so we are able to explain many of the empirical findings in inter-temporal decision-making literature. We provide converging evidence for our proposal by estimating the curvature of the decision space utilizing manifold learning algorithm and showing that the characteristics (i.e., metric properties) of the decision space resembles those of the Negative Curvature space rather than the Euclidean space. We conclude by presenting new theoretical predictions derived from our proposal and implications for how non-normative behavior is defined.     

Perception of timbral analogies.

Recent studies have investigated the structure of perceptual relations among musical instrument timbres by multidimensional scaling (MDS) techniques. These studies have employed both acoustically produced tones and digitally synthesized imitations and hybrids of acoustic instrument tones. The analyses of dissimilarity ratings for all pairs of a set of tones are usually represented as geometrical structures in a two- or three-dimensional Euclidean space in which the shared 'perceptual' axes are shown to have a qualitative correspondence to acoustic properties such as spectral energy distribution, onset characteristics and degree of change in spectral distribution over the duration of the tone. The present study took as a point of departure a MDS analysis for complex, synthetic tones with the aim of testing whether musician and non-musician listeners used the relations defined by the perceptual space to perform an analogies task of the sort: timbre A is to timbre B as timbre C is to which of two possible timbres, D or D'? A parallelogram model was used to select the D timbres: if the relation between A and B is represented as a vector with both magnitude and direction components, then the appropriate D should form a vector with C having similar magnitude and direction in the timbre space. Aside from conceptual difficulties with the task for both non-musicians and composers, choices for both groups provide support for the parallelogram model indicating a capacity in listeners to perceive abstract relations among the timbres of complex sounds without specific training in such a task.     

Analysis of kinematic invariances of multijoint reaching movement

 There is a no unique relationship between the trajectory of the hand, represented in cartesian or extrinsic space, and its trajectory in joint angle or intrinsic space in the general condition of joint redundancy. The goal of this work is to analyze the relation between planning the trajectory of a multijoint movement in these two coordinate systems. We show that the cartesian trajectory can be planned based on the task parameters (target coordinates, etc.) prior to and independently of angular trajectories. Angular time profiles are calculated from the cartesian trajectory to serve as a basis for muscle control commands. A unified differential equation that allows planning trajectories in cartesian and angular spaces simultaneously is proposed. Due to joint redundancy, each cartesian trajectory corresponds to a family of angular trajectories which can account for the substantial variability of the latter. A set of strategies for multijoint motor control following from this model is considered; one of them coincides with the frog wiping reflex model and resolves the kinematic inverse problem without inversion. The model trajectories exhibit certain properties observed in human multijoint reaching movements such as movement equifinality, straight end-point paths, bell-shaped tangential velocity profiles, speed-sensitive and speed-insensitive movement strategies, peculiarities of the response to double-step targets, and variations of angular trajectory without variations of the limb end-point trajectory in cartesian space. In humans, those properties are almost independent of limb configuration, target location, movement duration, and load. In the model, these properties are invariant to an affine transform of cartesian space. This implies that these properties are not a special goal of the motor control system but emerge from movement kinematics that reflect limb geometry, dynamics, and elementary principles of motor control used in planning. All the results are given analytically and, in order to compare the model with experimental results, by computer simulations. Received: 6 April 1994/Accepted in revised form: 25 April 1995     

Quasi-harmonic method for studying very low frequency modes in proteins

A quasi-harmonic approximation is described for studying very low frequency vibrations and flexible paths in proteins. The force constants of the empirical potential function are quadratic approximations to the potentials of mean force; they are evaluated from a molecular dynamics simulation of a protein based on a detailed anharmonic potential. The method is used to identify very low frequency (～1 cm^?1) normal modes for the protein pancreatic trypsin inhibitor. A simplified model for the protein is used, for which each residue is represented by a single interaction center. The quasi-harmonic force constants of the virtual internal coordinates are evaluated and the normal-mode frequencies and eigenvectors are obtained. Conformations corresponding to distortions along selected low-frequency modes are analyzed.     

From spiking neuron models to linear-nonlinear models

Neurons transform time-varying inputs into action potentials emitted stochastically at a time dependent rate. The mapping from current input to output firing rate is often represented with the help of phenomenological models such as the linear-nonlinear (LN) cascade, in which the output firing rate is estimated by applying to the input successively a linear temporal filter and a static non-linear transformation. These simplified models leave out the biophysical details of action potential generation. It is not a priori clear to which extent the input-output mapping of biophysically more realistic, spiking neuron models can be reduced to a simple linear-nonlinear cascade. Here we investigate this question for the leaky integrate-and-fire (LIF), exponential integrate-and-fire (EIF) and conductance-based Wang-Buzsáki models in presence of background synaptic activity. We exploit available analytic results for these models to determine the corresponding linear filter and static non-linearity in a parameter-free form. We show that the obtained functions are identical to the linear filter and static non-linearity determined using standard reverse correlation analysis. We then quantitatively compare the output of the corresponding linear-nonlinear cascade with numerical simulations of spiking neurons, systematically varying the parameters of input signal and background noise. We find that the LN cascade provides accurate estimates of the firing rates of spiking neurons in most of parameter space. For the EIF and Wang-Buzsáki models, we show that the LN cascade can be reduced to a firing rate model, the timescale of which we determine analytically. Finally we introduce an adaptive timescale rate model in which the timescale of the linear filter depends on the instantaneous firing rate. This model leads to highly accurate estimates of instantaneous firing rates.