Hamiltonian formalism in the problem on the modification of the current density profile during the development of tearing instability in a tokamak

The formation of a magnetic island as a result of tearing instability can be interpreted as the bifurcation of an axisymmetric equilibrium configuration at which nested magnetic surfaces are preserved. The modification of the current density profile due to such bifurcation is studied using the Hamiltonian formalism. In the case of a long narrow island, the gradient profile changes to a profile with an extremum on the axis of the magnetic island.     

Model magnetic field configurations with islands

Helical perturbations of the tokamak magnetic field can give rise to magnetic islands in the vicinity of the rational magnetic surfaces at which the pitch of the magnetic field lines coincides with that of the perturbation. The widely known relationship between the magnetic island width and the perturbation amplitude is valid under the assumptions that the island width is small in comparison to the radius of the rational surface and that the perturbation amplitude is constant in the radial direction. The latter assumption indicates that the island width is small in comparison to the radial size of the region where the perturbation current is localized. The calculations carried out for four model magnetic field configurations show that the geometry of the magnetic islands depends on the extent to which the perturbation current is localize and that the width of the magnetic islands is smaller than that calculated from the familiar relationship. The larger the perturbation amplitude, the greater this difference: it may be as large as 25% for the strong perturbations arising during disruptions. The calculations are based on the solution of the geometric problem of constructing the lines of the magnetic field determined by the given distributions of the initial current and perturbation current; the equilibrium equation is not considered. The question of the direction of the perturbation current within the island relative to the direction of the initial unperturbed current is discussed. The perturbation current flowing in an island is directed opposite to the initial current with a radially decreasing density; for this reason, such an island can naturally be called a “negative” island. Together with the formation of negative islands, the formation of “positive” ones is also considered. The latter are shown to form under the following conditions: the perturbation current density should be higher than the density of the current that produces the unperturbed field and the perturbation current itself should be localized in a sufficiently narrow radial layer. The positive islands are smaller in size than negative ones.     

Effects of fragmentation on vascular plant diversity in a Mediterranean forest archipelago

Abstract

We analysed the effects of patch size and isolation on vascular plants in Quercus cerris forest surrounding Rome (Italy). We randomly sampled 96 plots within 18 forest patches with homogeneous environmental variables; the patches ranged from 1.4 ha to 424.5 ha and were divided into four size classes. We performed the analyses at the patch level using linear regression. At the size class level, the analysis of species richness response to fragmentation (area effect) was performed with ANOVA, while the effect on community composition was analysed by means of PERMANOVA. We also investigated which species could be used as indicator species for each size class. Lastly, to evaluate the advantages of conserving several small patches as opposed to few large ones, we used a cumulative area approach ranking forest fragments. The correlation between species richness and patch area was positive, with a significant difference between the “large” and “small” size classes, while analysis on community composition showed that “large” versus “medium” and “large” versus “small” were significantly different. Nemoral species were recognised as indicators in the “large” class, and shrub and edge species in the “small” class. Our results indicate that 10 ha may be a suitable forest size threshold for planning and conservation.     

Species–area and species–sampling effort relationships: disentangling the effects

Species numbers tend to increase with both the area surveyed (species–area relationship, SAR) and the number of samples taken (species–sampling effort relationship, SSER). These two relationships differ in their nature and underlying mechanisms but are not clearly distinguished in field studies. To discriminate the effects of area (spatial extent) and sampling effort (SE) on species richness, several models explicitly involving both variables were proposed and tested against 13 datasets from marine micro‐, meio‐ and macrobenthos. A combination of power SSER and piecewise power SAR terms was found to have the best fit. The effects of area and SE were both significant, but the former one was noticeably weaker. The SSERs were roughly linear in log‐log space, whereas the SARs demonstrated scale‐dependent behavior with a noticeable threshold (slope breakpoint). Species richness was almost area‐independent below this threshold (the “small area effect”, SAE) but followed typical power‐law SAR beyond the threshold. This effect was similar to the “small island effect” but occurred for arbitrarily delineated areas within continuous habitats. Parameters of the SAR curves depended on organism size. The upper limit of the SAE increased from microorganisms to meiofauna to macrofauna. Also, SAR curves for unicellular groups had significantly lower slopes. SAE is supposed to indicate a spatial range of statistical homogeneity in species composition. Its upper limit corresponds to the characteristic size of a local community (a single habitat occupied by a common species pool). Interpretations of SAR and SSER parameters in terms of α‐ and β‐diversity are proposed. Both SAR and SSER slopes obtained from univariate regressions are overestimated. This upward bias depends on sampling design, decreasing for SAR but increasing for SSER with more unequally spaced samples. Both spatial extent and sampling effort should be taken into account to disentangle properly their effects on diversity.     

Biogeography of body size in Pacific island birds

Many insular vertebrates have undergone rapid and dramatic changes in body size compared to their mainland counterparts. Here we explore the relationship between two well known patterns of island body size – the tendency for large‐bodied species to dwarf and small‐bodied species to get larger on islands, known as the “island rule”, and the scaling of maximum and minimum body size of island assemblages with island area. Drawing on both fossil and modern data, we examined the relationship between body size and island area in Pacific island birds, both within clades and at the island assemblage level. We found that the size of the smallest bird on each island decreased with island area while the maximum body size increased with island area. Similarly, within clades the body size of small‐bodied groups decreased and large‐bodied groups increased from small to large islands, consistent with the island rule. However, the magnitude of size change within clades was not sufficient to explain the overall scaling of maximum size with island area. Instead, the pattern was driven primarily by the evolution of very large, flightless birds on large islands. Human‐mediated extinctions on islands over the past few millennia severely impacted large, flightless birds, to the effect that this macroecological pattern has been virtually erased. After controlling for effects of biogeographic region and island area, we found island productivity to be the best predictor of maximum size in flightless birds. This result, and the striking similarities in maximum body size between flightless birds and island mammals, suggests a common energetic mechanism linking body size and landmass area in both the island rule and the scaling of island body size extremes.     

Schizophrenia--a parameters' game?

Schizophrenia is a severe, currently incurable, relatively common mental condition. Its symptoms are complex and widespread. It structurally and functionally affects cortical and subcortical regions involved in cognitive, emotional and motivational aspects of behavior. Its diagnosis is based on statistical behavior rather than on its actual cause and its treatment is elusive.We elaborate a theoretical paradigm that accounts for some of the most important features of this illness. Our nonlinear mathematical model, built upon recent hypotheses of neural vulnerability and limbic dysregulation, addresses the amygdala—hippocampus—prefrontal interactions and their evolution under perturbation. The dependence of the dynamics on the system's parameters offers an analytical context for the “normality/disease” dichotomy. The concept of bifurcation could be the key to understanding the threshold between these two states.The nonlinearity parameter (Lyapunov number) is responsible in our setup for tuning the limbic vulnerability characteristic to schizophrenia. Studying its effect on the dynamics helps us understand how stressful events and medication can switch the system from a regime of safety to one of instability, and conversely. The approach has potential for pre-symptomatic risk assessments and for long-term predictions.     

A bifurcation model for developmental processes

I have constructed, for developmental processes, a qualitative model similar to the compartment hypothesis in Drosophila, and examined its relevance to vertebrate systems. In this model a polarized “cluster” of interacting cells will be the unit for “bifurcation” of the developmental pathway into two alternative states of “locon” which is the genetic unit controlling this process. The minimum size of the cluster critical for bifurcation and the size of the emerging subclusters will be dictated by the cognate locon. This will obviate the need for an extrinsically imposed threshold of some state variable for the boundary of the two subclusters. However, the orientation of bifurcation will be determined by the polarity of the cluster. A physiological factor of competence will impose a temporal constraint to bifurcation.Thus, combinatorial binary codes for a set of locons, like those originally devised by Kauffman (1973), can be assigned to developmental pathways. One of the clusters emerging from a sequence of bifurcations will have the same code as the mother cluster. It will represent the “developmental sink”, and will not recycle through the bifurcation series originating from the initial mother cluster, because of the difference in spatio-temporal factors incorporated in the size and competence of the individual clusters. If bifurcations are prevented, the mother cluster will be forced along the pathway of the developmental sink.I have applied the model to cases in vertebrate development where commitments to developmental pathways for aperiodic or periodic segmentations may follow a linear temporal sequence, producing, in turn, subclusters of uncommitted, or stem, cells towards the more intensely polarized end of the mother cluster. Such cases include limb, somite and tail formation and several stem cell systems with a finite lifespan. I have discussed some possible experimentation which emerges from the model.     

Inflection, canards and excitability threshold in neuronal models

A technique is presented, based on the differential geometry of planar curves, to evaluate the excitability threshold of neuronal models. The aim is to determine regions of the phase plane where solutions to the model equations have zero local curvature, thereby defining a zero-curvature (inflection) set that discerns between sub-threshold and spiking electrical activity. This transition can arise through a Hopf bifurcation, via the so-called canard explosion that happens in an exponentially small parameter variation, and this is typical for a large class of planar neuronal models (FitzHugh–Nagumo, reduced Hodgkin–Huxley), namely, type II neurons (resonators). This transition can also correspond to the crossing of the stable manifold of a saddle equilibrium, in the case of type I neurons (integrators). We compute inflection sets and study how well they approximate the excitability threshold of these neuron models, that is, both in the canard and in the non-canard regime, using tools from invariant manifold theory and singularity theory. With the latter, we investigate the topological changes that inflection sets undergo upon parameter variation. Finally, we show that the concept of inflection set gives a good approximation of the threshold in both the so-called resonator and integrator neuronal cases.     

An analytic approach to developing transport threshold models of neoclassical tearing modes in tokamaks

Transport threshold models of neoclassical tearing modes in tokamaks are investigated analytically. An analysis is made of the competition between strong transverse heat transport, on the one hand, and longitudinal heat transport, longitudinal heat convection, longitudinal inertial transport, and rotational transport, on the other hand, which leads to the establishment of the perturbed temperature profile in magnetic islands. It is shown that, in all these cases, the temperature profile can be found analytically by using rigorous solutions to the heat conduction equation in the near and far regions of a chain of magnetic islands and then by matching these solutions. Analytic expressions for the temperature profile are used to calculate the contribution of the bootstrap current to the generalized Rutherford equation for the island width evolution with the aim of constructing particular transport threshold models of neoclassical tearing modes. Four transport threshold models, differing in the underlying competing mechanisms, are analyzed: collisional, convective, inertial, and rotational models. The collisional model constructed analytically is shown to coincide exactly with that calculated numerically; the reason is that the analytical temperature profile turns out to be the same as the numerical profile. The results obtained can be useful in developing the next generation of general threshold models. The first steps toward such models have already been made.     

Structure of Lennard–Jones fluid near large spherical particles: further test of the “universality” of adjustable parameter in perturbation density functional theory

Grand canonical Monte Carlo simulation is used to study the density profiles of Lennard–Jones (LJ) fluid next to a large hard sphere (mimicking a colloidal particle) of various sizes. The LJ fluid in the inhomogeneous system thus maintains equilibrium with the bulk LJ fluid. The chosen density and potential parameters for the bulk fluid correspond to the conditions situated at “dangerous” regions of the phase diagram, i.e. near the critical temperature or close to the gas–liquid coexistence curve. The aim of present extensive simulations is to provide exact data for the broad range of the bulk parameters against which the “universality” of adjustable parameter associated with a perturbation density functional approximation (DFA) can be tested. Here the term “universality” means independence of this parameter on the particular external field responsible for the generation of a non-uniform density profile of the fluid. It is shown that the “universality” of this parameter associated with a third order+second order perturbation DFA holds also in the present case of a large spherical particle as a source of external potential, similarly as established in previous studies dealing with other interaction potential and other external fields [J. Chem. Phys., 122, 064503 (2005); J. Chem. Phys., 123 124708 (2005)]. This DFA can be used as input into a recently proposed framework for the calculation of interparticle potential of mean force (PMF).     

Weakly ionized plasma in a supersonic plasma flow

The structure of the ion acoustic precursor of a shock wave in a weakly ionized collision-dominated plasma is studied numerically. It is shown that the simultaneous action of the nonlinearity, dispersion, and dissipation leads to the formation of an oscillating profile of the ion density in the precursor. There exist regimes in which the charged-particle density decreases abruptly and simultaneously the number of maxima in its profile within the precursor becomes smaller as the shock wave velocity increases in a jumplike manner. This effect is analogous to the corresponding hydrodynamic effect in narrow shallow channels (the “Houston's horse” effect). In the stage preceding this jumplike process, local regions may appear in which the degree of plasma ionization is elevated. Such plasma “bunches” give rise to the strong reverse action of the charged particles on the neutral component, resulting in the “stretching” of the precursor. This phenomenon is resonant in character and occurs in a narrow range of shock wave velocities.     

Periodic oscillations and backward bifurcation in a model for the dynamics of malaria transmission

A deterministic ordinary differential equation model for the dynamics of malaria transmission that explicitly integrates the demography and life style of the malaria vector and its interaction with the human population is developed and analyzed. The model is different from standard malaria transmission models in that the vectors involved in disease transmission are those that are questing for human blood. Model results indicate the existence of nontrivial disease free and endemic steady states, which can be driven to instability via a Hopf bifurcation as a parameter is varied in parameter space. Our model therefore captures oscillations that are known to exist in the dynamics of malaria transmission without recourse to external seasonal forcing. Additionally, our model exhibits the phenomenon of backward bifurcation. Two threshold parameters that can be used for purposes of control are identified and studied, and possible reasons why it has been difficult to eradicate malaria are advanced.     

Inference on Collapsibility in Generalized Linear Models

GREENLAND and MICKEY (1988) derived a closed-form collapsibility test and confidence interval for IxJxK contingency tables with qualitative factors, and presented a small simulation study of its performance. We show how their method can be extended to regression models linear in the natural parameter of a one-parameter exponential family, in which the parameter of interest is the difference of “crude” and “adjusted” regression coefficients. A simplification of the method yields a generalization of the test for omitted covariates given by HAUSMAN (1978) for ordinary linear regression. We present an application to a study of coffee use and myocardial infarction, and a simulation study which indicates that the simplified test performs adequately in typical epidemiologic settings.     

Slow Passage Through a Hopf Bifurcation in Excitable Nerve Cables: Spatial Delays and Spatial Memory Effects

It is well established that in problems featuring slow passage through a Hopf bifurcation (dynamic Hopf bifurcation) the transition to large-amplitude oscillations may not occur until the slowly changing parameter considerably exceeds the value predicted from the static Hopf bifurcation analysis (temporal delay effect), with the length of the delay depending upon the initial value of the slowly changing parameter (temporal memory effect). In this paper we introduce new delay and memory effect phenomena using both analytic (WKB method) and numerical methods. We present a reaction–diffusion system for which slowly ramping a stimulus parameter (injected current) through a Hopf bifurcation elicits large-amplitude oscillations confined to a location a significant distance from the injection site (spatial delay effect). Furthermore, if the initial current value changes, this location may change (spatial memory effect). Our reaction–diffusion system is Baer and Rinzel’s continuum model of a spiny dendritic cable; this system consists of a passive dendritic cable weakly coupled to excitable dendritic spines. We compare results for this system with those for nerve cable models in which there is stronger coupling between the reactive and diffusive portions of the system. Finally, we show mathematically that Hodgkin and Huxley were correct in their assertion that for a sufficiently slow current ramp and a sufficiently large cable length, no value of injected current would cause their model of an excitable cable to fire; we call this phenomenon “complete accommodation.”     

CELL SIZE DIFFERENTIATION IN THE BLOOM-FORMING DINOFLAGELLATE GYMNODINIUM CF. NAGASAKIENSE1

Two subpopulations differing essentially by their mean cell size were observed regularly in cultures and natural samples of the naked dinoflagellate Gymnodinium cf. nagasakiense Takayama et Adachi (currently known as Gyrodinium aureolum Hulburt), a species which frequently forms red tides in North European seas. “Large” cells represented the typical forms; they were morphologically similar to cells of the closely related Japanese species G. nagasakiense, which did not form any subpopulation of reduced size. “Small” and “large” cells of G. cf. nagasakiense had the same DNA content, but the nucleus of the former appeared to be much more condensed during interphase. Each cell type was able to divide and had its own growth dynamics; therefore, any intermediary between pure populations of “small” and of “large” cells were observed in culture. The “large” form generated a “small” cell by an atypical budding-like division, whereas the “small” form gave back a “large” form, once it ceased to divide, by simple enlargement of its cell body. Factory inducing cell size differentiation are yet unclear. Neither nitrogen nor phosphorus starvation induced a significant increase in the relative proportion of “small” and budding cells. Although cell size differentiation is associated with the formation of gametes in a variety of dinoflagellates, we demonstrated that “small” cells of G. cf. nagasakiense are able to divide asexually, in contrast to gametes of most other species. The high proliferative power of “small” cells as compared with normal cells suggests that they could play a significant role during red tides of G. cf. nagasakiense; in contrast, cells of the Japanese species G. Nagasakiense could sustain high growth rates with larger cell size because this species generally blooms in waters much warmer than those found in northern Europe.     

Aspects of the Phænogamic Vegetation of Rodriguez,with Descriptions of new Plants from the Island.

As regards the history of its flora, the island of Rodriguez bears a striking resemblance to St. Helena. We read of the latter island that fire, goats, and finally introduced foreign plants well nigh exterminated the indigenous flora; and the same causes have operated and, I regret to say, are still operating in Rodriguez. The result is that the primitive vegetation has been in great part destroyed, leaving the island a field for the rank and rapid growth of common tropical weeds. The old and luxuriant vegetation of two hundred years ago—thus quaintly described by Leguat*, “We could hardly take our eyes off from the little mountains of which the island entirely consists; they are so richly spread with great and tall trees,” and, again, “‘Tis as I have hinted, composed of lovely hills covered all with fine trees whose perpetual verdure is entirely charming”—has now, to a large extent, disappeared, and is represented by only a few species, many of which are confined to the more unfrequented and less accessible places. It is indeed difficult to recognize in the barren and arid Rodriguez of the present day the “little Eden,”“lovely isle,”“earthly paradise” of Leguat. The flora, as it now exists, is an exceedingly fragmentary one; it is therefore a matter of some difficulty to determine its exact limits and to draw conclusions as to its affinities with the floras of other oceanic islands and of adjacent continents. This is the more to be regretted, as, from the geographical position of the island and the physical condition of climate to which it is subject, its flora might be expected, whether taken singly or as part of that of the Mascarene group, to contribute very important data towards the solution of the problem of the distribution of plant‐life in that region. The following general statements may, however, be made:—     

A new image processing technique for determination of cell-generated deformations on substrata

This paper introduces a new image processing technique that determines the displacement field of a given substrate from “null-force” and “force-loaded” images. In this method, fluorescent elements used to track motion, which will be referred to as beads, can be seen in these images by locating the gray value that is normally distributed around their central point. Next comes a two-step process of matching the beads with displacements. The first step matches the beads with a small displacement using the correlation function of the characteristic pixels. Based on results from this initial step, another correlation function determines a pair of beads with a relatively large displacement. The entire matching process is done in this way, gradually working from the small displacement to the large one. Finally, using the cubic spline weight function, the whole displacement field is interpolated and filtered out of those displacements, which were initially found with the matched beads. Applying this new method on the cell migration yields satisfying results. Based on the particle tracking, the displacement field obtained by this new image processing technique has clear physical meaning. More importantly, this new method completes the matching of the displacement using the features of the displacement field, thus avoiding the direct matching with the image gray values for the relatively large strain of the substrate around the cell. Accordingly, it greatly decreases mismatching, making data checking unnecessary.     

Evidence of a Hopf bifurcation in frog hair cells

The membrane potential of hair cells in the low-frequency hearing organ of the bullfrog, the amphibian papilla, sinusoidally oscillates at small amplitude in the absence of acoustical input. We stimulate the cell with a series of periodic currents close to this natural frequency and observe that its current-to-voltage transfer function is compressively nonlinear, having a large gain for small stimuli and a smaller gain for larger currents. Along with the spontaneous oscillation, this implies that the cell is poised close to a dynamical instability such as a Hopf bifurcation, because distant from the instability the transfer function becomes linear. The cell's frequency selectivity is enhanced for small stimuli. Simulations show that the cell's membrane capacitance is effectively reduced due to a current gain provided by this dynamical instability. We propose that the Hopf resonance is widely used by transducer cells on the sensory periphery to achieve small-signal amplification.     

On the structure of liver ribosomes

The morphology of liver ribosomes and their subparticles, large and small, has been investigated. Analysis of the images has been carried out by successive selection of models and by X-raying them under conditions simulating negative staining. The relation between the main views has been checked by tilting the specimens in an electron microscope through ± 30 °.The small subparticle consists of an elongated body, to one of the ends of which a short “head” is attached. A model has been proposed, whose projections on rotation with respect to the longitudinal axis would satisfy all observable types of images. According to the proposed model, the “head” is tilted with respect to the elongated portion. The length of the subparticle is 230 Å. The dimensions of the elongated portion in the transverse direction are 110 Å × 75 to 80 Å.The large subparticles in lateral view resemble short “rods” 220 to 240 Å long and about 70 to 95 Å wide. At a distance of about 60 Å from the left end of the particles a projection (60 Å in length) is seen, on the inner side of which a depression, or “channel”, filled with the contrasting substance is always observed. Next to this depression a second projection is located, whose height is about 30 Å. The channel is either a cavity in the body of the large subparticle or a part of the RNA without protein. The length of the channel is about 80 Å, the width is about 50 to 60 Å. The left end of the particles is characterized by two sharpened portions; as a result, a cavity that shows an obtuse angle profile makes its appearance. The opposite end of the particles is cut off at an angle of 45 °. In another view, the subparticles appear to be almost rectangular in shape; they are characterized by dimensions of 150 Å × 220 to 240 Å. It is likely that the large projection is displaced sideways with respect to the longitudinal axis of the particles. The asymmetry associated with this displacement gives rise to preferred arrangements of the subparticles on the supporting film. An analysis has been made of the most typical images of monomeric ribosomes, on the basis of which a suggestion is made about mutual orientation of subparticles in a monomer.     

Simulation of Lamellar Phase Transitions in Block Copolymers and Surfactants

Abstract

Simulations of the lamellar phase transitions of symmetric amphiphilic chains are carried out on a cubic lattice, with the amphiphilic chain length N varied from 6 to 48 lattice sites, corresponding to lengths ranging from surfactants to short block copolymers. We find that the effective interaction energy parameter χN (which incorporates the effect of added solvant) at which the transition from the lamellar ordered state to the disordered state occurs is roughly equal to 18-21. While this result is consistent with an extrapolation of the Fredrickson-Helfand weak-segragation theory to N values in the range of the simulations, the amplitude of the sinusoidal compositional wave in the ordered state near the transition is large for all N studied, in disagreement with the weak segregation theories. Thus, for values of N up to 48, the transition occurs in a “moderate,” rather than weak-segregation regime. Near the disordering transition, fluctuating “bridge” or “hole” defects in the lamellae spontaneously appear; with heating these proliferate and lead to the disordering transition. These fluctuating bridges might help explain anomalous diffusion and rheological behavior observed near the disordering transition. We also find that in the ordered state near the transition, the orientational order parameter, which is proportional to the intrinsic birefringence, falls rapidly with increasing N, roughly as 1.5 N^-2.