On the existence of limit cycles in motion field

In this paper we show the existence of limit cycles in the vector field generated by the perspective projection on the image plane of the velocity field of a moving surface. The existence of limit cycles is proved with the Poincaré-Bendixon theorem, in the case of a rotating smooth non-planar surface and illustrated with computer graphics. The structural stability of the limit cycles is also discussed.     

Limit cycles in predator-prey models

The general model of interaction between one predator and one prey is studied. A unimodal function of rate of growth of the prey and concave down functional response of the predator is assumed. In this work it is shown that for a given natural number n there exist models possessing at least 2n + 1 limit cycles. It is also proved, applying the Hopf bifurcation theorem, that a model exists with a logistic growth rate of the prey and concave down functional response that has at least two limit cycles.     

Limit cycles in periodically perturbed population systems

A perturbation method is proposed to calculate approximately the limit cycle type nonequilibrium steady-state resulting from periodic perturbation of coefficients of stable population systems; the two species Lotka-Volterra competition system is explicity studied and the results are formulated for general multi-species population systems. Avoidance of competitive or other types of exclusion of species in a periodic environment is indicated.     

Generalized gradient schemes for the measurement of two-dimensional image motion

This paper describes a procedure for recovering the global velocity of an image by incorporating spatial filtering, and, optionally, temporal filtering, into a scheme that employs a generalized version of the gradient algorithm of motion detection. Motion within a patch is analysed by six parallel channels, each incorporating a different spatiotemporal filter. Advantageous features of this scheme are: (a) global velocity is derived directly, without first computing local velocity at a number of image locations; (b) the filters compute first derivatives rather than second derivatives, making the scheme potentially more resistant to noise than certain other schemes; (c) two of the six filters can be chosen almost completely arbitrarily, and can therefore be tailored to maximize signal reliability, and (d) the measurement of velocity can be made as local or as global as desired by altering the size of the patch that is viewed by the filters. An analogous scheme is derived for the measurement of rotation, as well as expansion or contraction of the image.     

Chaotic neural network applied to two-dimensional motion control

Chaotic dynamics generated in a chaotic neural network model are applied to 2-dimensional (2-D) motion control. The change of position of a moving object in each control time step is determined by a motion function which is calculated from the firing activity of the chaotic neural network. Prototype attractors which correspond to simple motions of the object toward four directions in 2-D space are embedded in the neural network model by designing synaptic connection strengths. Chaotic dynamics introduced by changing system parameters sample intermediate points in the high-dimensional state space between the embedded attractors, resulting in motion in various directions. By means of adaptive switching of the system parameters between a chaotic regime and an attractor regime, the object is able to reach a target in a 2-D maze. In computer experiments, the success rate of this method over many trials not only shows better performance than that of stochastic random pattern generators but also shows that chaotic dynamics can be useful for realizing robust, adaptive and complex control function with simple rules.     

Collective motion enhances chemotaxis in a two-dimensional bacterial swarm

In a dilute liquid environment in which cell-cell interaction is negligible, flagellated bacteria, such as Escherichia coli, perform chemotaxis by biased random walks alternating between run-and-tumble. In a two-dimensional crowded environment, such as a bacterial swarm, the typical behavior of run-and-tumble is absent, and this raises the question whether and how bacteria can perform chemotaxis in a swarm. Here, by examining the chemotactic behavior as a function of the cell density, we showed that chemotaxis is surprisingly enhanced because of cell crowding in a bacterial swarm, and this enhancement is correlated with increase in the degree of cell body alignment. Cells tend to form clusters that move collectively in a swarm with increased effective run length, and we showed analytically that this resulted in increased drift velocity toward attractants. We also explained the enhancement by stochastically simulating bacterial chemotaxis in a swarm. We found that cell crowding in a swarm enhances chemotaxis if the cell-cell interactions used in the simulation induce cell-cell alignment, but it impedes chemotaxis if the interactions are collisions that randomize cell moving direction. Therefore, collective motion in a bacterial swarm enhances chemotaxis.     

Limit cycles in a chemostat model for a single species with age structure

We model an age-structured population feeding on an abiotic resource by combining the Gurtin-MacCamy [Math. Biosci. 43 (1979) 199] approach with a standard chemostat model. Limit cycles arise by Hopf bifurcations at low values of the chemostat dilution rate, even for simple maternity functions for which the original Gurtin-MacCamy model has no oscillatory solutions. We find the exact location in parameter space of the Hopf bifurcations for special cases of our model. The onset of cycling is largely independent of both the form of the resource uptake function and the shape of the maternity function.     

The geometry and motion of reaction-diffusion waves on closed two-dimensional manifolds

Chemical or biological systems modelled by reaction diffusion (R.D.) equations which support simple one-dimensional travelling waves (oscillatory or otherwise) may be expected to produce intricate two or three-dimensional spatial patterns, either stationary or subject to certain motion. Such structures have been observed experimentally. Asymptotic considerations applied to a general class of such systems lead to fundamental restrictions on the existence and geometrical form of possible structures. As a consequence of the geometrical setting, it is a straightforward matter to consider the propagation of waves on closed two-dimensional manifolds. We derive a fundamental equation for R.D. wave propagation on surfaces and discuss its significance. We consider the existence and propagation of rotationally symmetric and double spiral waves on the sphere and on the torus. On leave of absence from: Department of Mathematics, Glasgow College of Technology, Cowcaddens Road, Glasgow G4 0BA, Scotland, UK     

Glacial cycles drive rapid divergence of cryptic field vole species

Understanding the factors that contribute to the generation of reproductively isolated forms is a fundamental goal of evolutionary biology. Cryptic species are an especially interesting challenge to study in this context since they lack obvious morphological differentiation that provides clues to adaptive divergence that may drive reproductive isolation. Geographical isolation in refugial areas during glacial cycling is known to be important for generating genetically divergent populations, but its role in the origination of new species is still not fully understood and likely to be situation dependent. We combine analysis of 35,434 single‐nucleotide polymorphisms (SNPs) with environmental niche modeling (ENM) to investigate genomic and ecological divergence in three cryptic species formerly classified as the field vole (Microtus agrestis). The SNPs demonstrate high genomic divergence (pairwise F_ST values of 0.45–0.72) and little evidence of gene flow among the three field vole cryptic species, and we argue that genetic drift may have been a particularly important mechanism for divergence in the group. The ENM reveals three areas as potential glacial refugia for the cryptic species and differing climatic niches, although with spatial overlap between species pairs. This evidence underscores the role that glacial cycling has in promoting genetic differentiation and reproductive isolation by subdivision into disjunct distributions at glacial maxima in areas relatively close to ice sheets. Future investigation of the intrinsic barriers to gene flow between the field vole cryptic species is required to fully assess the mechanisms that contribute to reproductive isolation. In addition, the Portuguese field vole (M. rozianus) shows a high inbreeding coefficient and a restricted climatic niche, and warrants investigation into its conservation status.     

Effect of multiple freeze-thaw cycles on intervertebral dynamic motion characteristics in the porcine lumbar spine

Fresh frozen spine specimens are commonly used in biomechanical investigations of the spine. Since many study designs require staged preparation and testing, the effect of multiple freeze-thaw cycles on motion behavior should be understood. The objective of this study was to investigate the effect of multiple freeze-thaw cycles on the biomechanical parameters measured during dynamic pure moment loading. Ten porcine lumbar motion segments were harvested immediately after death and potted in acrylic fixtures. Specimens were tested in continuous pure moment flexion-extension, lateral flexion, and rotation cycles up to a limit of +/-5Nm. Moment-angular displacement data were analyzed and parameters quantified including range of motion, elastic zone, transitional zone (neutral region) size and slope, and width of the hysteresis loop. All specimens were tested at baseline and after each of three subsequent cycles of freezing and thawing. The transitional zone size decreased and the transitional zone slope increased during flexion-extension and lateral bending after the initial freeze-thaw cycle. These parameters were not altered after subsequent cycles. No significant change was observed in the elastic zone or width of hysteresis loop. Although freezing porcine spine specimens increased the stiffness in the neutral region of motion, up to three subsequent cycles of freezing and thawing did not further affect these motion characteristics. This suggests that data obtained from porcine spines which have been frozen and thawed multiple times are stable after initial freezing.     

End-stopping and the aperture problem: two-dimensional motion signals in macaque V1

Our perception of fine visual detail relies on small receptive fields at early stages of visual processing. However, small receptive fields tend to confound the orientation and velocity of moving edges, leading to ambiguous or inaccurate motion measurements (the aperture problem). Thus, it is often assumed that neurons in primary visual cortex (V1) carry only ambiguous motion information. Here we show that a subpopulation of V1 neurons is capable of signaling motion direction in a manner that is independent of contour orientation. Specifically, end-stopped V1 neurons obtain accurate motion measurements by responding only to the endpoints of long contours, a strategy which renders them largely immune to the aperture problem. Furthermore, the time course of end-stopping is similar to the time course of motion integration by MT neurons. These results suggest that cortical neurons might represent object motion by responding selectively to two-dimensional discontinuities in the visual scene.     

Bacterial chemotaxis: a field in motion

Many different types of studies are being combined to provide an increasingly detailed picture of the bacterial chemotaxis system. The structures of periplasmic receptors and a cytoplasmic response regulator, along with structures of domains of a membrane receptor, a receptor-modifying enzyme and a cytoplasmic histidine kinase, have been determined. These structures provide a basis for other work which is likely to open up new structural avenues.     

An optimal control strategy for two-dimensional motion camouflage with non-holonimic constraints

Motion camouflage is a stealth behaviour observed both in hover-flies and in dragonflies. Existing controllers for mimicking motion camouflage generate this behaviour on an empirical basis or without considering the kinematic motion restrictions present in animal trajectories. This study summarises our formal contributions to solve the generation of motion camouflage as a non-linear optimal control problem. The dynamics of the system capture the kinematic restrictions to motion of the agents, while the performance index ensures camouflage trajectories. An extensive set of simulations support the technique, and a novel analysis of the obtained trajectories contributes to our understanding of possible mechanisms to obtain sensor based motion camouflage, for instance, in mobile robots.     

Estimation of food consumption from field observations of fish feeding cycles

The available methods of estimating food consumption by fish require that experiments be performed on confined animals and that experimental results may be validly applied to free fish. A method is described by which food consumption in periodically feeding fish may be estimated without performing laboratory experiments. A relatively simple input–output model of stomach contents is fitted to the observed time trajectory of stomach fullness, and food consumption is calculated from the estimated model parameters. Feeding is considered to be restricted to a distinct feeding period, and the rate of feeding during that period can be either constant or linearly decreased with the quantity of food already present in the stomach.
The method is applied to three examples. The model appears robust, and generally provides very similar food consumption estimates to those obtained from methods requiring an independent estimate of gastric evacuation rate. However, the described method is sensitive to violation of the assumption that feeding occurs only during a discrete feeding period.     

A two dimensional field theory for motion computation

The local extraction of motion information from brightness patterns by individual movement detectors of the correlation-type is considered in the first part of the paper. A two-dimensional field theory of movement detection is developed by treating the distance between two adjacent photoreceptors as a differential. In the first approximation of the theory we only consider linear terms of the time interval between the reception of a contrast element and its delayed representation by the detector and linear terms of the spatial distances between adjacent photoreceptors. As a result we may neglect terms of higher order than quadratic in a Taylor series development of the brightness pattern. The responses of pairs of individual movement detectors are combined to a local response vector. In the first approximation of the detector field theory the response vector is proportional to the instantaneous pattern velocity vector and linearly dependent on local properties of the moving pattern. The linear dependence on pattern properties is represented by a two by two tensor consisting of elements which are nonlinear, local functional of the moving pattern. Some of the properties of the tensor elements are treated in detail. So, for instance, it is shown that the off-diagonal elements of the tensor disappear when the moving pattern consists of x- and y-dependent separable components. In the second part of the paper the tensor relation leading to the output of a movement detector pair is spatially integrated. The result of the integration is an approximation to a summation of the outputs of an array of detector pairs. The spatially integrated detector tensor relates the translatory motion vector to the resultant output vector. It is shown that the angle between the motion vector and the resultant output vector is always smaller than ±90° whereas the angle between the motion vector and local response vectors, elicited by detector pairs, may cover the entire angular range. In the discussion of the paper the limits of the field theory for motion computation as well as its higher approximations are pointed out in some detail. In a special chapter the dependence of the detector response on the pattern properties is treated and in another chapter questions connected with the so called aperture problem are discussed. Furthermore, properties for compensation of the pattern dependent deviation angle by spatial physiological integration are mentioned in the discussion.     

Assessment of segmental wall motion abnormalities using contrast two-dimensional echocardiography in awake mice

Murine models of cardiac disease are becoming an important tool for studying pathophysiological processes. Development of methods to accurately assess ventricular function are therefore important. The purpose of this study was to evaluate the feasibility of echocardiographic assessment of segmental wall motion abnormalities in a murine model of myocardial infarction. Two-dimensional contrast (C+) and noncontrast (C-) echocardiography were performed in 76 awake mice 2 days before and 2 days after left coronary ligation. The short-axis images obtained with two-dimensional echocardiography and corresponding postmortem cross-sectional histological samples stained with Evans blue dye were each divided into 16 segments, and all matched segments were examined for correlation between wall motion abnormalities and myocardial hypoperfusion. With the use of contrast enhancement, the number of visualized segments was significantly increased (base: C- 86%, C+ 98%; midpapillary: C- 57%, C+ 89%; apex: C- 30%, C+ 74%). Agreement between echocardiographically assessed regional wall motion abnormalities and pathologically determined hypoperfusion in basal, midpapillary, and apical levels were 90%, 93%, and 93%, respectively. Agreement between echocardiographically normal wall motion and pathologically normal findings in basal, midpapillary, and apical levels were 99%, 88%, and 71%, respectively. Thus echocardiographic assessment of segmental wall motion in awake mice was feasible and the accuracy was improved with the use of a contrast agent.     

Obstacle bypass in protein motion along DNA by two-dimensional rather than one-dimensional sliding

Site-specific DNA-binding proteins locate their target sites by facilitated diffusion. Several proteins have been shown to slide along DNA in vitro. However, whereas sliding is often envisaged as one-dimensional tracking of the DNA major groove, such a mechanism would not allow linear diffusion over long distances in vivo, where short stretches of free DNA are delimited by bound proteins. I propose a two-dimensional sliding mechanism, in which the protein diffuses freely on the cylindrical DNA surface, and I present experiments that can distinguish between one- and higher-dimensional diffusion along the DNA contour length. At 100 mm NaCl, translocation of EcoRI restriction endonuclease between sites on two DNA helices connected by a Holliday junction is as efficient as between sites on the same helix, indicating a three-dimensional mechanism. At 25 mm NaCl, translocation between sites on the same DNA helix is more efficient, indicating a role for sliding at low ionic strength. Obstacles attached to the major groove of one face of the DNA helix did not interfere with sliding, regardless of their orientation relative to the cleavage sites. This result is compatible with two-dimensional but not one-dimensional sliding. As illustrated by Monte-Carlo simulation, two-dimensional sliding may not only allow proteins to move around nucleosomes in vivo but also reduce the redundancy of their search for the target site.