Properties of the space horopter.

For general points of fixation the space horopter has been found to be a twisted cubic curve, which, when fixation is in the median plane, degenerates into a conic curve inclined to the plane of fixation. The inclination of the plane of this conic is found to be a function of the angles of torsion of the two eyes. If these angles are zero and fixation is horizontal the space horopter reduces to the Vieth-Müller circle. Previous experimental techniques for plotting the horopter curve do not in fact determine the locus in space of those points that will stimulate corresponding points in the two retinae: they are, locate the position of certain chords of the space horopter. The Hering-Hillebrand deviation coefficient is found theoretically to be a linear function of the fixation distance. This had been shown experimentally by previous workers, but without any satisfactory explanation.     

A note on the horopter

By assuming the fixity (but not the symmetry) of corresponding points on the two retinae, it is possible to derive the equation of any horopter when one is known. In particular when, as experiment shows, one horopter is linear, then all horopters must be conics. These have the form given by Ogle, but whereas Ogle leaves one parameter undetermined at each fixation, on our assumption the only arbitrary parameter is determined by the position of the linear horopter.     

The accuracy of DLT extrapolation in three-dimensional film analysis

An analysis of errors arising from the Direct Linear Transformation (DLT) approach to three-dimensional reconstructions from two-dimensional images has been undertaken, the principal factor studied being the number and distribution of control points used in the calibration procedure. Significantly increased error was found to be associated with extrapolation to unknown points outside the control point distribution space. Differences in accuracy between two camera position set-ups and 11 vs 12 DLT parameter solutions were also examined.     

Z曲线的理论研究

DNA序列与正四面体(RT)中的映象点(D格点)具有对应关系．一系列D格点形成晶格(D晶格)．D晶格证明是面心立方晶格,并揭示了D格点的空间分布规律．在此基础上得到Z曲线的一些特性,如Z曲线束具有S4群的对称性。Z曲线均在最大正四面体的内切球内等等．     

Distribution of mapping points of 20 amino acids in the tetrahedral space

Summary Based on the genetic codes and a simple theorem for the geometrical property of the regular tetrahedron, each amino acid is mapped onto a unique point in a 3-dimensional tetrahedral space. The distribution of the 20 mapping points for 20 amino acids is studied in detail. It is found that the mapping points for the hydrophobic and hydrophilic amino acids are distributed at distinct regions in the 3-dimensional space. A plane separating the two kinds of points satisfactorily based on the Fisher's algorithm has been calculated. It is shown that the codons coding for the hydrophobic amino acids are constituted dominantly by the bases of keto group, i.e., G and T. While the codons coding for the hydrophilic amino acids are constituted dominantly by the bases of amino group, i.e., A and C. The biological implication of the mapping points and the separating plane has been discussed in some details.     

Multiple attractors and boundary crises in a tri-trophic food chain

The asymptotic behaviour of a model of a tri-trophic food chain in the chemostat is analysed in detail. The Monod growth model is used for all trophic levels, yielding a non-linear dynamical system of four ordinary differential equations. Mass conservation makes it possible to reduce the dimension by 1 for the study of the asymptotic dynamic behaviour. The intersections of the orbits with a Poincaré plane, after the transient has died out, yield a two-dimensional Poincaré next-return map. When chaotic behaviour occurs, all image points of this next-return map appear to lie close to a single curve in the intersection plane. This motivated the study of a one-dimensional bi-modal, non-invertible map of which the graph resembles this curve. We will show that the bifurcation structure of the food chain model can be understood in terms of the local and global bifurcations of this one-dimensional map. Homoclinic and heteroclinic connecting orbits and their global bifurcations are discussed also by relating them to their counterparts for a two-dimensional map which is invertible like the next-return map. In the global bifurcations two homoclinic or two heteroclinic orbits collide and disappear. In the food chain model two attractors coexist; a stable limit cycle where the top-predator is absent and an interior attractor. In addition there is a saddle cycle. The stable manifold of this limit cycle forms the basin boundary of the interior attractor. We will show that this boundary has a complicated structure when there are heteroclinic orbits from a saddle equilibrium to this saddle limit cycle. A homoclinic bifurcation to a saddle limit cycle will be associated with a boundary crisis where the chaotic attractor disappears suddenly when a bifurcation parameter is varied. Thus, similar to a tangent local bifurcation for equilibria or limit cycles, this homoclinic global bifurcation marks a region in the parameter space where the top-predator goes extinct. The 'Paradox of Enrichment' says that increasing the concentration of nutrient input can cause destabilization of the otherwise stable interior equilibrium of a bi-trophic food chain. For a tri-trophic food chain enrichment of the environment can even lead to extinction of the highest trophic level.     

Homoclinic and heteroclinic orbits to a cycle in a tri-trophic food chain

 The asymptotic behavior of a tri-trophic food chain model is studied. The analysis is carried out numerically, by finding both local and global bifurcations of equilibria and limit cycles. The existence of transversal homoclinic orbits to a limit cycle is shown. The appearance of homoclinic orbits, by moving through a homoclinic bifurcation point, is associated with the sudden disappearance of a chaotic attractor. A homoclinic bifurcation curve, which bounds a region of extinction, is continued through a two-dimensional parameter space. Heteroclinic orbits from an equilibrium to a limit cycle are computed. The existence of these heteroclinic orbits has important consequences on the domains of attraction. Continuation of non-transversal heteroclinic orbits through parameter space shows the existence of two codimension-two bifurcations points, where the saddle cycle is non-hyperbolic. The results are summarized by dividing the parameter space in subregions with different asymptotic behavior. Received: 25 February 1998 / Revised version: 19 August 1998     

On the use of comparison regions in visualizing stochastic uncertainty in some two-parameter estimation problems

When considering simultaneous inference for two parameters, it is very common to visualize stochastic uncertainty by plotting two-dimensional confidence regions. This allows us to test post hoc null hypotheses about a single point in a simple manner. However, in some applications the interest is not in rejecting hypotheses on single points, but in demonstrating evidence for the two parameters to be in a convex subset of the parameter space. The specific convex subset to be considered may vary from one post hoc analysis to another. Then it is of interest to have a visualization allowing to perform corresponding analyses. We suggest comparison regions as a simple tool for this task.     

Three-dimensional space: locomotory style explains memory differences in rats and hummingbirds

While most animals live in a three-dimensional world, they move through it to different extents depending on their mode of locomotion: terrestrial animals move vertically less than do swimming and flying animals. As nearly everything we know about how animals learn and remember locations in space comes from two-dimensional experiments in the horizontal plane, here we determined whether the use of three-dimensional space by a terrestrial and a flying animal was correlated with memory for a rewarded location. In the cubic mazes in which we trained and tested rats and hummingbirds, rats moved more vertically than horizontally, whereas hummingbirds moved equally in the three dimensions. Consistent with their movement preferences, rats were more accurate in relocating the horizontal component of a rewarded location than they were in the vertical component. Hummingbirds, however, were more accurate in the vertical dimension than they were in the horizontal, a result that cannot be explained by their use of space. Either as a result of evolution or ontogeny, it appears that birds and rats prioritize horizontal versus vertical components differently when they remember three-dimensional space.     

The symplectic structure of the primary visual cortex

We propose to model the functional architecture of the primary visual cortex V1 as a principal fiber bundle where the two-dimensional retinal plane is the base manifold and the secondary variables of orientation and scale constitute the vertical fibers over each point as a rotation–dilation group. The total space is endowed with a natural symplectic structure neurally implemented by long range horizontal connections. The model shows what could be the deep structure for both boundary and figure completion and for morphological structures, such as the medial axis of a shape.     

The assembly of ionic currents in a thalamic neuron. I. The three-dimensional model

We have previously discussed qualitative models for bursting and thalamic neurons that were obtained by modifying a simple two-dimensional model for repetitive firing. In this paper we report the results of making a similar sequence of modifications to a more elaborate six-dimensional model of repetitive firing which is based on the Hodgkin-Huxley equations. To do this we first reduce the six-dimensional model to a two-dimensional model that resembles our original two-dimensional qualitative model. This is achieved by defining a new variable, which we call q. We then add a subthreshold inward current and a subthreshold outward current having a variable, z, that changes slowly. This gives a three-dimensional (v,q,z) model of the Hodgkin-Huxley type, which we refer to as the z-model. Depending on the choice of parameter values this model resembles our previous models of bursting and thalamic neurons. At each stage in the development of these models we return to the corresponding seven-dimensional model to confirm that we can obtain similar solutions by using the complete system of equations. The analysis of the three-dimensional model involves a state diagram and a stability diagram. The state diagram shows the projection of the phase path from v,q,z space into the v,z plane, together with the projections of the curves z = 0 and v = q = 0. The stability of the points on the curve v = q = 0, which we call the v, q nullcurve, is determined by the stability diagram. Taken together the state and stability diagrams show how to assemble the ionic currents to produce a given firing pattern.     

The plasticity of correspondence: After-effects,illusions and horopter shifts in depth perception

Retinal disparity is the cue for stereoscopic depth perception. Disparity detection begins with cortical single units driven binocularly from the two eyes. A previous paper (Nelson, 1975) has shown that inhibitory and facilitatory interactions are essential to insure successful disparity detection, particularly with repeating stimulus patterns, and that such a system will display all the appropriate properties of sensory fusion. This paper shows that most depth illusions occur as by-products of the same domain interactions. Such illusion effects fall into two classes: those caused by shifts in the distribution of activity along the appropriate sensory domain (here, the disparity domain) and those caused by changes in the average activity level within the domain. Profile shifts cause depth contrast illusions. The most important profile level change is an activity lowering due to disparity domain inhibition. This adversely affects fusional range (Panum's area). It is postulated that all domain interactions persist following cessation of stimulation. Persistent profile shifts cause depth after-effect illusions; persistent profile lowering is responsible for threshold elevation after-effects.Sensory fusion, the coding errors seen in illusions, the induced effect, and widespread failure to perceive depth from disparity input show that retinal correspondence is not stable in the normal individual. Yet horopter research has attempted to specify one set of retinal points as corresponding. Not surprisingly, horopter research shows systematic shifts in retinal correspondence linked to eye position. Small, simple, tonic modulations of the domain interactions responsible for so many other stereopsis system properties provide a satisfactory cortical mechanism for horopter changes.     

A test of improved force field parameters for urea: molecular-dynamics simulations of urea crystals

Molecular-dynamics (MD) simulations of urea crystals of different shapes (cubic, rectangular prismatic, and sheet) have been performed using our previously published force field for urea. This force field has been validated by calculating values for the cohesive energy, sublimation temperature, and melting point from the MD data. The cohesive energies computed from simulations of cubic and rectangular prismatic urea crystals in vacuo at 300 K agreed very well with the experimental sublimation enthalpies reported at 298 K. We also found very good agreement between the melting points as observed experimentally and from simulations. Annealing the crystals just below the melting point leads to reconstruction to form crystal faces that are consistent with experimental observations. The simulations reveal a melting mechanism that involves surface (corner/edge) melting well below the melting point, and rotational disordering of the urea molecules in the corner/edge regions of the crystal, which then facilitates the translational motion of these molecules.     

Normalization in the fitting of data by iterative methods. Application to tracer kinetics and enzyme kinetics

1. The normalization of biochemical data to weight them appropriately for parameter estimation is considered, with reference particularly to data from tracer kinetics and enzyme kinetics. If the data are in replicate, it is recommended that the sum of squared deviations for each experimental variable at each time or concentration point is divided by the local variance at that point. 2. If there is only one observation for each variable at each sampling point, normalization may still be required if the observations cover more than one order of magnitude, but there is no absolute criterion for judging the effect of the weighting that is produced. The goodness of fit that is produced by minimizing the weighted sum of squares of deviations must be judged subjectively. It is suggested that the goodness of fit may be regarded as satisfactory if the data points are distributed uniformly on either side of the fitted curve. A chi-square test may be used to decide whether the distribution is abnormal. The proportion of the residual variance associated with points on one or other side of the fitted curve may also be taken into account, because this gives an indication of the sensitivity of the residual variance to movement of the curve away from particular data points. These criteria for judging the effect of weighting are only valid if the model equation may reasonably be expected to apply to all the data points. 3. On this basis, normalizing by dividing the deviation for each data point by the experimental observation or by the equivalent value calculated by the model equation may both be shown to produce a consistent bias for numerically small observations, the former biasing the curve towards the smallest observations, the latter tending to produce a curve that is above the numerically smaller data points. It was found that dividing each deviation by the mean of observed and calculated variable appropriate to it produces a weighting that is fairly free from bias as judged by the criteria mentioned above. This normalization factor was tested on published data from both tracer kinetics and enzyme kinetics.     

Tracing personalized health curves during infections

It is difficult to describe host-microbe interactions in a manner that deals well with both pathogens and mutualists. Perhaps a way can be found using an ecological definition of tolerance, where tolerance is defined as the dose response curve of health versus parasite load. To plot tolerance, individual infections are summarized by reporting the maximum parasite load and the minimum health for a population of infected individuals and the slope of the resulting curve defines the tolerance of the population. We can borrow this method of plotting health versus microbe load in a population and make it apply to individuals; instead of plotting just one point that summarizes an infection in an individual, we can plot the values at many time points over the course of an infection for one individual. This produces curves that trace the course of an infection through phase space rather than over a more typical timeline. These curves highlight relationships like recovery and point out bifurcations that are difficult to visualize with standard plotting techniques. Only nine archetypical curves are needed to describe most pathogenic and mutualistic host-microbe interactions. The technique holds promise as both a qualitative and quantitative approach to dissect host-microbe interactions of all kinds.     

Analysis of nucleotide distribution in the genome of Streptomyces coelicolor A3(2) using the Z curve method

The nucleotide distribution of all 33 527 open reading frames (ORFs) (≥300 bp) in the genome of Streptomyces coelicolor A3(2) has been analyzed using the Z curve method. Each ORF is mapped onto a point in a 9-dimensional space. To visualize the distribution of mapping points, the points are projected onto the principal plane based on principal component analysis. Consequently, the distribution pattern of the 33 527 points in the principal plane shows a flower-like shape, in which there are seven distinct regions. In addition to the central region, there are six petal-like regions around the center, one of which corresponds to 7172 coding sequences. The central region and the remaining five petal-like regions correspond to the intergenic sequences and out-of-frame non-coding ORFs, respectively. It is shown that selective pressure produces a remarkable bias of the G+C content among three codon positions, resulting in the interesting phenomenon observed. A similar phenomenon is also observed for other bacterial genomes with high genomic G+C content, such as Pseudomonas aeruginosa PA01 (G+C=66.6%). However, for the genomes of Bacillus subtilis (G+C=43.5%) and Clostridium perfringens (G+C=28.6%), no similar phenomenon was observed. The finding presented here may be useful to improve the gene-finding algorithms for genomes with high G+C content. A set of supplementary materials including the plots displaying the base distribution patterns of ORFs in 12 prokaryotes is provided on the website http://tubic.tju.edu.cn/highGC/.     

Quasi-fixed Points w.r.t. the Height Growth Curve

When height growth curves are linearized, these lines are nearly concurrent after the growth spurt (Masuyama, 1979, 1982b, 1983), i.e. there exists a nearly fixed point, which seems to correspond to the upper limit of the growth of the mean person. On other hand, before the spurt, there seems to exist a nearly fixed point on the mean distance-velocity curve between 4 and 6 years of age. To grasp easily the existence of nearly fixed point, instead of many straight lines surrounding this point, we represent them dually by many points surrounding a straight line.     

Stability switches and double Hopf bifurcation in a two-neural network system with multiple delays

Time delay is an inevitable factor in neural networks due to the finite propagation velocity and switching speed. Neural system may lose its stability even for very small delay. In this paper, a two-neural network system with the different types of delays involved in self- and neighbor- connection has been investigated. The local asymptotic stability of the equilibrium point is studied by analyzing the corresponding characteristic equation. It is found that the multiple delays can lead the system dynamic behavior to exhibit stability switches. The delay-dependent stability regions are illustrated in the delay-parameter plane, followed which the double Hopf bifurcation points can be obtained from the intersection points of the first and second Hopf bifurcation, i.e., the corresponding characteristic equation has two pairs of imaginary eigenvalues. Taking the delays as the bifurcation parameters, the classification and bifurcation sets are obtained in terms of the central manifold reduction and normal form method. The dynamical behavior of system may exhibit the quasi-periodic solutions due to the Neimark- Sacker bifurcation. Finally, numerical simulations are made to verify the theoretical results.     

Straightening Caenorhabditis elegans images

MOTIVATION: Caenorhabditis elegans, a roundworm found in soil, is a widely studied model organism with about 1000 cells in the adult. Producing high-resolution fluorescence images of C.elegans to reveal biological insights is becoming routine, motivating the development of advanced computational tools for analyzing the resulting image stacks. For example, worm bodies usually curve significantly in images. Thus one must 'straighten' the worms if they are to be compared under a canonical coordinate system. RESULTS: We develop a worm straightening algorithm (WSA) that restacks cutting planes orthogonal to a 'backbone' that models the anterior-posterior axis of the worm. We formulate the backbone as a parametric cubic spline defined by a series of control points. We develop two methods for automatically determining the locations of the control points. Our experimental methods show that our approaches effectively straighten both 2D and 3D worm images.