The smallest value of the axon density for which ‘ignition’ can occur in a random net

As shown by A. Rapoport (1952), when a very brief stimulation or “instantaneous input” is applied to a random net, the subsequent events are determined by the parameters of the net as follows: If the axon densitya is sufficiently large and the fraction γ of the neurons initially stimulated exceeds a certain value γ₁ (theover-all threshold of the net for instantaneous stimulation), excitation will spread through the net until a steady state is reached in which a fraction γ₂ ⩾ γ₁ of the neurons is firing (“ignition phenomenon”). If γ < γ₁ the activity in the net dies out. However, if the axon density is too small, the activity will ultimately die out, no matter how large the fraction of initially stimulated neurons. Thus there exists a limiting valueA of the axon density below which the net cannot “ignite”. ThisA is a function ofh, theindividual threshold of the neurons constituting the net (we assume hereh≥2, since forh=1 the situation is essentially different). Geometrically γ₁ and γ₂ are determined as the two intersection points of a straight line with a sigmoid curve. Whena<A the two curves do not intersect and fora=A they are tangent. In this paper the “tangency case” is investigated and the general features of the functionA(h) are determined. It is shown thatA increases monotonically withh (as one would expect). For all values ofh>1 we haveA(h)>h, but the fractionA(h)/h and the derivativedA(h)/dh approach unity ash increases. An analytical expression of the functionA(h) valid for very large values ofh is derived.     

“Ignition” phenomena in random nets

The spread of excitation in a “random net” is investigated. It is shown that if the thresholds of individual neurons in the net are equal to unity, a positive steady state of excitation will be reached equal to γ, which previously had been computed as the weak connectivity of the net. If, however, the individual thresholds are greater than unity, either no positive steady state exists, or two such states depending on the magnitude of the axone density. In the latter case the smaller of the two steady states is unstable and hence resembles an “ignition point” of the net. If the initial stimulation (assumed instantaneous) exceeds the “ignition point,” the excitation of the net eventually assumes the greater steady state. Possible connections between this model and the phenomenon of the “preset” response are discussed.     

Contribution to the theory of random and biased nets

The probabilistic theory of random and biased nets is further developed by the “tracing” method treated previously. A number of biases expected to be operating in nets, particularly in sociograms, is described. Distribution of closed chain lengths is derived for random nets and for nets with a simple “reflexive” bias. The “island model” bias is treated for the case of two islands and a single axon tracing, resulting in a pair of linear difference equations with two indices. The reflexive bias is extended to multiple-axon tracing by an approximate method resulting in a modification of the random net recursion formula. Results previously obtained are compared with empirical findings and attempts are made to account for observed discrepancies.     

On the mathematical theory of rumor spread

The applicability of the theory of random nets to the theory of rumor spread is shown. In particular the “weak connectivity” of the net appears as the saturation fraction of “knowers” in a thoroughly mixed population through which a message diffuses where each knower tells the message to a finite average number of individuals. Further it is shown how the time course equation of rumor spread, where time is measured by the number of “removes” from the starters, can be translated into an ordinary continuous time course equation if the distribution of the telling intervals is known.     

Contribution to the mathematical theory of contagion and spread of information: I. Spread through a thoroughly mixed population

An equation is derived from the spread of a “state” by contact through a thoroughly mixed population, in which the probability of transmission depends both on the over-all duration of the process and on the time an individual has been in the “state.” Cases in which this probability is a function of only one or the other of the two “times” are worked out. It is shown that in the case of dependence on “private time” alone the asymptotic value of the fraction of the population effected is the same as that derived by the random net approach.     

Identification of Barley Varieties Tolerant to Cadmium Toxicity

Two successive hydroponic experiments were carried out to identify barley varieties tolerant to Cd toxicity via examining Soil–Plant Analyses Development (SPAD) value, plant height, leaves and tillers per plant, root number and volume, and biomass accumulation. The results showed that SPAD values (chlorophyll meter readings), plant height, leaf number, root number and volume, and biomass accumulation of shoot/root were significantly reduced in the plants grown in 20 μM Cd nutrient solution compared with control, and the uptake and translocation of Zn, Mn, and Cu was also strictly hindered. Furthermore, there was a highly significant difference in the reduction in these growth parameters among varieties, and varieties “Weisuobuzhi” and “Jipi 1” showed the least reduction both in the two experiments, suggesting their high tolerance to Cd toxicity, while “Dong 17” and “Suyinmai 2” with the greatest reduction and the toxicity symptoms appeared rapidly and severely, denoting as Cd-sensitive varieties. Significant variety difference in Cd concentration was also found, with Weisuobuzhi containing the highest and Jipi 1 the lowest Cd concentration in shoots.     

Is loss-free counting under statistical control?

A new formula for the statistical uncertainty of “loss-free counting” (LFC) is presented. Its validity is demonstrated by comparing with experimental data obtained with a HPGe γ-ray spectrometer. Also, computer simulation data of nuclear counting with different types of count loss (pileup rejection, extending and nonextending dead time) are in agreement with the predicted counting uncertainty. The proposed formula for LFC uncertainty is applicable to spectrometers with a classical semi-Gaussian pulse-shaping amplifier as well as with a gated-integrator amplifier. Hence, achieving statistical control seems to be a feasible goal.     

Modeling of the actomyosin ATPase activity

In the rapid “quench” kientics of myosin, the “initial phosphate burst” is the excess inorganic phosphate that is produced during the early time-course of ATP hydrolysis by myosin subfragment-1 (S-1) or HMM. In general, the existence of a Pi burst implies a rapid (i.e., generally an order of magnitude faster than the steady-state hydrolysis rate) lysis of the phospho-anhydride bond within the ATP molecule, followed by one or more slower steps that are rate limiting for the process. Thus, the presence of a Pi burst can provide an important clue to the mechanism of the reaction. However, in the case of actomyosin, this clue as long been the subject of controversy and misunderstanding. To measure the (initial) Pi burst, myosin S-1 (or HMM) is rapidly mixed with ATP and then the mixture is acid quenched after a specific time period. The medium produced contains free Pi generated from hydrolysis of the ATP. The quantitative measure of the phosphate generated in this way has always been significantly greater than that expected by steady-state “release” of Pi alone, and it is that very difference between this measured Pi after the quench and that amount of Pi expected to be released by steady-state considerations in that same time period that has been referred to as the “initial Pi burst”. Recent investigations of the kinetics of Pi release have used an entirely new method that directly measures the release of Pi from the enzyme-product complex. These studies have made reference to the properties of the “initial Pi burst” in the presence of actin, as well as to a new kinetic entity: the “burst of Pi release”, and have been often vague concerning the true nature of the initial Pi burst, as well as the properties of Pi release as predicted by the current models of the actin activation of the myosin ATPase activity. The purpose of the current article is to correct this oversight, to discuss the “burst” in some detail, and to display the kinetics predicted by the current models for the actin activation of myosin. Furthermore, predictions for the kinetics of the new “burst of Pi release” are discussed in terms of its ability to discriminate between the two current competing models for actin activation of the myosin ATPase activity.     

Requirements for maintaining the embryonic state of avian tendon cells in culture

Summary Primary avian tendon cells (PAT) maintain their embryonic state when cultured in medium F-12 with very low serum (0.2%) and ascorbate (50 μg per ml); that is, they retain the potential for devoting 20–30% of their total protein synthesis to collagen. However, if the cells are left at a confluent cell density or are derived from confluent cultures, this potential is irreversibly decreased. This effect, along with poor medium formulations, probably accounts for the “dedifferentiation” process that occurs when fibroblasts are cultured. In contrast, PAT cells kept at subconfluent cell densities retain the ability to synthesize high levels of collagen. The one limitation in obtaining long-term cultures of high collagen-producing tendon cells in the inability of serum at low concentrations to remain a potent mitogen after a few subcultures. The quantitative loss of function has long been considered to be a cell culture artifact; however, we propose that this drop in collagen synthesis is a reflection of the developmental programing of these cells. In separate series of experiments using organ cultures, we show that tendon tissue from the embryo makes over 30% collagen, whereas, “young” tendons make 18% and “older” tendons from the adult make less than 1%. Therefore, a quantitative drop in collagen synthesis would be expected if normal development were to occur in culture. Our data are consistent with the idea that cultures of embryonic tendon cells are triggered to mature by a mechanism that correlates with high cell density. This investigation was supported in part by National Science Foundation Grant PCM 77-14982; in part by the Division of Biomedical and Environmental Research of the Department of Energy under contract W-7405-ENG-48; and by a National Institutes of Health Fellowship IF32 CA 05807-01, from the National Cancer Institute to R. I. S.     

Can inverse density dependence at small spatial scales produce dynamic instability in animal populations?

All else being equal, inversely density-dependent (IDD) mortality destabilizes population dynamics. However, stability has not been investigated for cases in which multiple types of density dependence act simultaneously. To determine whether IDD mortality can destabilize populations that are otherwise regulated by directly density-dependent (DDD) mortality, I used scale transition approximations to model populations with IDD mortality at smaller “aggregation” scales and DDD mortality at larger “landscape” scales, a pattern observed in some reef fish and insect populations. I evaluated dynamic stability for a range of demographic parameter values, including the degree of compensation in DDD mortality and the degree of spatial aggregation, which together determine the relative importance of DDD and IDD processes. When aggregation-scale survival was a monotonically increasing function of density (a “dilution” effect), dynamics were stable except for extremely high levels of aggregation combined with either undercompensatory landscape-scale density dependence or certain values of adult fecundity. When aggregation-scale survival was a unimodal function of density (representing both “dilution” and predator “detection” effects), instability occurred with lower levels of aggregation and also depended on the values of fecundity, survivorship, detection effect, and DDD compensation parameters. These results suggest that only in extreme circumstances will IDD mortality destabilize dynamics when DDD mortality is also present, so IDD processes may not affect the stability of many populations in which they are observed. Model results were evaluated in the context of reef fish, but a similar framework may be appropriate for a diverse range of species that experience opposing patterns of density dependence across spatial scales.     

Unique Determination of Some Homoplasies at Hybridization Events

Phylogenetic relationships may be represented by rooted acyclic directed graphs in which each vertex, corresponding to a taxon, possesses a genome. Assume the characters are all binary. A homoplasy occurs if a particular character changes its state more than once in the graph. A vertex is “regular” if it has only one parent and “hybrid” if it has more than one parent. A “regular path” is a directed path such that all vertices after the first are regular. Assume that the network is given and that the genomes are known for all leaves and for the root. Assume that all homoplasies occur only at hybrid vertices and each character has at most one homoplasy. Assume that from each vertex there is a regular path leading to a leaf. In this idealized setting, with other mild assumptions, it is proved that the genome at each vertex is uniquely determined. Hence, for each character the vertex at which a homoplasy occurs in the character is uniquely determined. Without the assumption on regular paths, an example shows that the genomes and homoplasies need not be uniquely determined.     

Experimental translocation of juvenile water voles in a Scottish lowland metapopulation

Population density affects dispersal success because residents can hinder or facilitate immigration into a new site, via a “social fence effect” or “social attraction” (or “conspecific attraction”), respectively. These mechanisms can affect the dynamics of fragmented populations and the success of translocations. However, information on the settlement behaviour of dispersers is rare. We conducted a manipulative field experiment using wild water voles, which exist in metapopulations along waterways in Scotland. We translocated 17 young of dispersal age into either an occupied site or a vacant site containing good habitat, which had recently become extinct due to a feral predator (American mink) moving through. We monitored the movements of translocated voles using radio telemetry. Translocated voles were less likely to settle in occupied sites with higher densities of residents, suggesting a possible social fence effect at high density. There was evidence of a social attraction mechanism, because voles never remained at new sites unless another individual arrived soon after translocation, and they were more likely to settle in occupied or colonised sites than vacant ones. Voles remained in the transient phase of dispersal for many days, and often followed a “stepping stone” trajectory, stopping for several days at successive sites. We suggest that trajectories followed by dispersing water voles, the time scale and long dispersal distances found in this species are conducive to locating conspecifics at low density and colonising vacant habitat. These results are encouraging for prospects of metapopulation persistence and future translocation success.     

Contribution to the probabilistic theory of neural nets: IV. Various models for inhibition

Input-output formulas are derived for a neuron upon which converge single axones of two other neurons, which are subjected to a Poisson shower, where a number of different assumptions are made concerning the mechanism of inhibition. In one assumption so-called “bilateral pre-inhibition” is considered. That is to say, both neuronsN ₁ andN ₂ may exciteN ₃, but if the stimulus of one of them follows within a certain interval σ of the other, the second stimulus is not effective. This model is essentially no different from that involving two excitatory neurons acting upon a neuron having a refractory period. Another mechanism considered involves so-called “pre-and-post” inhibition, in which if two stimuli fromN ₁ andN ₂ fall within σ,both are ineffective. This case being mathematically much more involved than the preceding, an approximation method is used for deriving the input-output formula. Previous papers of this series are denoted by I, II, and III in this paper.     

Study of the formation of a fast electron beam in a coaxial gas diode in the unlimited emission model

The formation of a fast electron beam in a coaxial gas diode is simulated using the OOPIC Pro particle-in-cell code under the assumption of unlimited cathode emissivity. It is shown that fast electrons are generated near the cathode. The so-called “contracting-capacitor” mechanism of electron acceleration does not operate in the parameter range under study, because the boundary of the dense plasma is spread out due to preionization of the electrode gap by fast electrons.     

Functional trait assembly through ecological and evolutionary time

A classic community assembly hypothesis is that all guilds must be represented before additional species from any given guild enter the community. We conceptually extend this hypothesis to continuous functional traits, refine the hypothesis with an eco-evolutionary model of interaction network community assembly, and compare the resultant continuous trait assembly rule to empirical data. Our extension of the “guild assembly rule” to continuous functional traits was rejected, in part, because the eco-evolutionary model predicted trait assembly to be characterized by the expansion of trait space and trait/species sorting within trait space. Hence, the guild rule may not be broadly applicable. A “revised” assembly rule did, however, emerge from the eco-evolutionary model: as communities assemble, the range in trait values will increase to a maximum and then remain relatively constant irrespective of further changes in species richness. This rule makes the corollary prediction that the trait range will, on average, be a saturating function of species richness. To determine if the assembly rule is at work in natural communities, we compared this corollary prediction to empirical data. Consistent with our assembly rule, trait “space” (broadly defined) commonly saturates with species richness. Our assembly rule may thus represent a general constraint placed on community assembly. In addition, taxonomic scale similarly influences the predicted and empirically observed relationship between trait “space” and richness. Empirical support for the model’s predictions suggests that studying continuous functional traits in the context of eco-evolutionary models is a powerful approach for elucidating general processes of community assembly.     

The Charms of Nature: Darwin on Meaning and Value

I review George Levine’s provocative and highly original book Darwin Loves You. Levine, whose “home discipline” is English Literature, offers a compelling interpretation of Darwin’s works, evaluating their content and Darwin’s prose style to identify a distinctly Darwinian attitude toward nature as a source of meaning and value. Levine believes that Darwin exemplifies the capacity to feel “enchantment” about the natural world, suggesting that, if Darwin’s example were followed, a “Darwinian re-enchantment of the world” would be brought about. This would offer a secular, non-supernatural basis for purpose, meaning, and value. I conclude with a few critical remarks about the scope and cogency of Levine’s proposal.     

Effects of animal density,volume, and the use of 2D/3D recording on behavioral studies of copepods

Studies on the behavior of copepods require both an appropriate experimental design and the means to perform objectively verifiable numerical analysis. Despite the growing number of publications on copepod behavior, it has been difficult to compare these studies. In this study, we studied two species of copepods, Eurytemora affinis and Pseudodiaptomus annandalei, and employed recently developed scaling and non-scaling methodology to investigate the effects of density and volume on the swimming behavior of individual organisms in still water. We also compared the results of two- and three-dimensional projections of the swimming tracks. A combination of scale-dependent and scale-independent analysis was found to characterize a number of behavioral observations very effectively. We discovered that (i) density has no effect except to increase the time spent in the swimming state of “breaking”, (ii) smaller volumes resulted in more complex trajectories, and larger volumes, like density, increased the time spent in the swimming state “breaking”, and (iii) three-dimensional projections gave a more accurate estimation of speed and the time spent cruising. When only a vertical 2D projection was used, “cruising” could be confused with “sinking”. These results indicate that both experimental conditions and the selection of 2D or 3D projection have important implications regarding the study of copepod behavior. The development of standardized procedures with which to compare the observations made in different studies is an issue of particular urgency.     

Outline of a probabilistic approach to animal sociology: II

Under certain assumptions concerning the probabilities of “mutations,” i.e. changes of structure of bird societies, it is shown that the probability distribution for all possible structures of a society ofN individuals approaches a limit independent of the initial probability distribution. A formula for the limiting distribution is derived.     

Computational experience with an algorithm for tetrangle inequality bound smoothing

An important component of computer programs for determining the solution conformation of proteins and other flexible molecules from nuclear magnetic resonance data are the so-called “bound smoothing algorithms”, which compute lower and upper limits on the values of all the interatomic distances from the relatively sparse set which can usually be measured experimentally. To date, the only methods efficient enough for use in large problems take account of only the triangle inequality, but an appreciable improvement in the precision of the limits is possible if the algebraic relations between the distances among each quadruple of atoms are also considered. The goal of this paper is to use a recently improved algorithm for computing these “tetrangle inequality limits” to determine just how much improvement really is possible, given the types of experimental data that are usually available.     

Qualitative analysis of a model generating long potential waves in Ba-treated nerve cells—I. Reduced systems

This study is related to a model describing the behavior of barium-treatedAplysia neurons generating regular burst-plateau patterns. The model is represented by an autonomous dynamical system, defined inR ⁴ and depending on a small parameter. This paper is restricted to the qualitative study of three “reduced systems” deduced from the “complete system”. Part of the study is performed with the use of the qualitative theory of singular perturbations. The predicted behaviors are compared with experimental results.