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.     

“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.     

A statistical approach to the theory of the central nervous system

A “probabilistic” rather than a “deterministic” approach to the theory of neural nets is developed. Neural nets are characterized by certain parameters which give the probability distributions of different kinds of synaptic connections throughout the net. Given a “state” of the net (i.e., the distribution of firing neurons) at a given moment, an equation for the state at the next moment of quantized time is deduced. Certain very special cases involving constant distributions are solved. A necessary condition for a steady state is deduced in terms of an integral equation, in general non-linear.     

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.     

A quantum model for controlled enzymic reactions: II. Controlled biochemical networks

Two control units, the switching and the two factor discriminating net are described. They are derived as a consequence of the enzymic oscillatory behavior induced by substrate “perturbation”. A complex network encompassing long sequences of metabolic reactions is constructed and the organization of cellular metabolic activities in well defined “regimes” and “states” inferred.     

Measuring Nitrogen and Phosphorus Uptake by Intact Roots of Mature Acer saccharum Marsh., Pinus resinosa Ait., and Picea abies (L.) Karst

A common method for measuring uptake by intact roots in situ is the depletion method, wherein intact fine roots are separated from soil and placed in nutrient solution. The difference between initial and final amounts of nutrient in solution is attributed to root uptake. Variations on this method include applying pretreatment solutions, training roots to grow into bags or trays, and varying concentrations of nutrient solution. We tested whether variations in methods affected measured net uptake rates of NH ₄ ⁺ , NO ₃ ⁻ , and PO ₄ ³⁻ . Intact roots of 60 year-old sugar maple (Acer saccharum Marsh.), red pine (Pinus resinosa Ait.), and Norway spruce (Picea abies (L.) Karst.) were given one of four treatments prior to measuring net uptake. “Trained” roots were grown in a sand-soil mixture. “Recovered” roots were excavated and allowed to recover in nutrient solution for two or four days (“two-day recovery” and “four-day recovery”, respectively). “No recovery” roots were excavated and used immediately in experiments. We exposed roots to three concentrations of nutrient solutions to observe the effects of initial nutrient solution concentration. Initial nutrient solution concentration was an important source of variation in measured uptake rates, and N uptake was stimulated by low antecedent concentrations. We found no significant differences in net uptake rates between pretreatments for any of the species studied, indicating that our pretreatments were not effective in improving measurement of uptake. Such pretreatments may not be necessary for measuring net uptake and may not hinder the comparison of rates measured using variations of the depletion method.     

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 Semio-Mathematical Nature of Codes

The relational structure of RNA, DNA, and protein bears an interesting similarity to the determination problem in category theory. In this paper, we present this deep-structure similarity and use it as a springboard for discussing some abstract properties of coding in various systems. These abstract properties, in turn, may shed light on the evolution of the DNA world from a semiotic perspective. According to the perspective adopted in this paper, living systems are not information processing systems but “meaning-making” systems. Therefore, what flows in the genetic system is not “information” but “value.” We define meaning, meaning-making, and value and then use these terms to explain the abstract dynamics of coding, which can illuminate many forms of sign-mediated activities in biosystems.     

A preliminary study of vocal communication in wild long-tailed macaques (Macaca fascicularis). I. Vocal repertoire and call emission

Vocal communication in wild long-tailed macaques (Macaca fascicularis) is described in terms of (1) a preliminary vocal repertoire and the situations in which calls occur in the natural habitat of this species and (2) quantitative measurement of the natural occurrence of calls in the field. Although a number of calls are relatively discrete (e.g., a male loud call), gradation is pronounced for both wide-spectrum (“harsh”) and narrow-spectrum (“clear”) vocal signals. Thirteen general types of harsh calls are identified provisionally as elements of the vocal repertoire. The exact number of discrete clear calls contributing to the vocal repertoire could not be ascertained precisely, but these calls were classified operationally into six broadly acoustically different classes in order to measure natural vocal behavior. Vocalizations tended to occur in temporal “clusters” during sample, periods. Narrow-band clear or “coo” calls were more frequently performed by macaques than wide-band harsh calls. The possible functional implications of the correlated occurrence of multiple vocal signals are discussed.     

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.     

Correlations between the psychological peculiarities of an individual and the efficacy of a single neurofeedback session (by the EEG characteristics)

Modifications of different EEG rhythms induced by a single neurofeedback session (by the EEG characteristics) directed toward an increase in the ratio of the spectral powers (SPs) of the α vs θ oscillations were compared with the psychological characteristics of the tested subjects (the group included 30 persons). A generally accepted neurofeedback technique was used; the intensity of acoustic white noise served as the feedback signal. EEG potentials were recorded from the C3 and C4 leads. Psychological testing was carried out using Eysenck’s (EPQ), Rusalov’s (OST), and (16 PF) questionnaires. The directions of changes in the SPs of EEG frequency components were found to significantly correlate with some individuality-related peculiarities of the tested subjects. The SP of the δ rhythm correlated with the EPQ scale “neuroticism,” OST scale “social plasticity,” and 16 PF factors H (“parmia”), I (“premsia”), and Q₃ (“self-control of behavior”). The SP of the θ component demonstrated correlations with the OST scales “ergisity,” “plasticity,” and “social temp” and with 16 PF factors M (“autia”), Q₄ (“frustration”), and Q1 (“exvia”). The SP of the α rhythm correlated with 16 PF factors Q₃ (“self-control of behavior”), G (“strength of superEgo”), O (“hypothymia”), L (“protension”), and N (“shrewdness”). The SP of the β rhythm correlated with the OST scale “emotionality,” while that of the γ rhythm showed correlations with the 16 PF indices L (“protension”) and M (“autia”). Changes in the ratio of the α vs θ SPs correlated with the EPQ scale “neuroticism.” Thus, our data confirm the statement that a high individual variability of the results of a single (first in the series) neurofeedback session is to a great extent related to peculiarities of the individual psychological pattern of the subject. Neirofiziologiya/Neurophysiology, Vol. 38, No. 3, pp. 239–247, May–June, 2006.     

Spread of information through a population with socio-structural bias: I. Assumption of transitivity

A previously derived iteration formula for a random net was applied to some data on the spread of information through a population. It was found that if the axon density (the only free parameter in the formula) is determined by the first pair of experimental values, the predicted spread is much more rapid than the observed one. If the successive values of the “apparent axon density” are calculated from the successive experimental values, it is noticed that this quantity at first suffers a sharp drop from an initial high value to its lowest value and then gradually “recovers”. An attempt is made to account for this behavior of the apparent axon density in terms of the “assumption of transitivity”, based on a certain socio-structural bias, namely, that the likely contacts of two individuals who themselves have been in contact are expected to be strongly overlapping. The assumption of transitivity leads to a drop in the apparent axon density from an arbitrary initial value to the vicinity of unity (if the actual axon density is not too small). However, the “recovery” is not accounted for, and thus the predicted spread turns out to beslower than the observed.     

Two types of using of space in the resident common shrews <Emphasis Type="Italic">Sorex araneus</Emphasis> L.

The home range of resident animals is considered as “familiar area” including a “foraging area.” It has been revealed that the activity of an average animal unit in the “foraging area” could be approximated by normal distribution. Estimation of activity distribution in the “familiar area” (beyond the “foraging area”) was impeded by means of marking since it might be difficult to record distant movements, and the method does not provide an essential body of data. In the case of the common shrew Sorex araneus, the “familiar area” was estimated using pitfall as animals evade them in the known areal. The “foraging area” radius of the average shrew was taken to be 30 m (95% of the animal unit activity), the radius of “familiar area” was within the range of 180–240 m. The “foraging area” was expected to provide the animal with vital resources, and the “familiar area” reflects its need for exploratory activity.     

Beyond the binary case in random nets

Experiments on random binary, ternary, etc. (P=2, 3,…, 10) switching nets are reported. Behavioral cycle lengths are examined as functions of output variety,P, input connectance,K, and net size,N. Overall, output variety appears an influential, well-behaved net property. Strong, but well-behaved interactions appear among net variables. In high connectance nets, median cycle length grows approx. asP ^N/2. Other factors constant, one-connected nets show the shortest cycles, and connectance effects appear to converge asymptotically aroundN. Data for cycle length as a function of net size suggest a concavity not compatible with the Kauffman “square root law” (Kauffman, 1969). Evidence of a positive relationship between cycle length and run-in length is found in two-input nets; weaker evidence is obtained that in higher connectance nets this relationship becomes negative in sign. The “modular complexity” ofP>2 nets is examined briefly.     

On the fisher and the cubic-polynomial equations for the propagation of species properties

For precise boundary conditions of biological relevance, it is proved that the steadily propagating plane-wave solution to the Fisher equation requires the unique (eigenvalue) velocity of advance 2(Df)^1/2, whereD is the diffusivity of the mutant species andf is the frequency of selection in favor of the mutant. This rigorous result shows that a so-called “wrong equation”, i.e. one which differs from Fisher's by a term that is seemingly inconsequential for certain initial conditions, cannot be employed readily to obtain approximate solutions to Fisher's, for the two equations will often have qualitatively different manifolds of exact solutions. It is noted that the Fisher equation itself may be inappropriate in certain biological contexts owing to the manifest instability of the lowerconcentration uniform equilibrium state (UES). Depicting the persistence of a mutantdeficient spatial pocket, an exact steady-state solution to the Fisher equation is presented. As an alternative and perhaps more faithful model equation for the propagation of certain species properties through a homogeneous population, we consider a reaction-diffusion equation that features a cubic-polynomial rate expression in the species concentration, with two stable UES and one intermediate unstable UES. This equation admits a remarkably simple exact analytical solution to the steadily propagating plane-wave eigenvalue problem. In the latter solution, the sign of the eigenvelocity is such that the wave propagates to yield the “preferred” stable UES (namely, the one further removed from the unstable intermediate UES) at all spatial points ast→∞. The cubic-polynomial equation also admits an exact steady-state solution for a mutant-deficient or mutant-isolated spatial pocket. Finally, the perpetuating growth of a mutant population from an arbitrary localized initial distribution, a mathematical problem analogous to that for ignition in laminar flame theory, is studied by applying differential inequality analysis, and rigorous sufficient conditions for extinction are derived here.     

A canonical form for neural nets without circles

By “neural net” will be meant “neural net without circles.” Every neural net effects a transformation from inputs (i.e., firing patterns of the input neurons) to outputs (firing patterns of the output neurons). Two neural nets will be calledequivalent if they effect the same transformation from inputs to outputs. A canonical form is found for neural nets with respect to equivalence; i.e., a class of neural nets is defined, no two of which are equivalent, and which contains a neural net equivalent to any given neural net. This research was supported by the U.S. Air Force under Contract AF 49(638)-414 monitored by the Air Force Office of Scientific Research.     

Observations on spontaneous hand use in the common marmoset (Callithrix jacchus)

Four male and four female marmoset monkeys were observed to make a total of 5,600 “right”, “left”, or “both” hand responses over seven categories of spontaneous behaviour. Significant and consistent hand preferences were shown for the majority of monkeys in two of the categories of behaviour, but not in the other five. The paper considers some of the methodological difficulties involved in recording “handedness” in a species such as the common marmoset.     

J D Bernal (1901–1971) in perspective

Conclusion We must conclude that the sub-title of Bernal’s “The Social Function of Science” — “What science does: what science could do” is still the relevant challenge and indicates Bernal’s chief contribution, besides the foundation of molecular biology to our civilization. It is manifest that resources spent on armaments are a monstrous pathological symptom of our social structure. The ancient problem of “what is property” and what may be “owned” and by whom or by what organs of society is awakening.     

Promotive effect of 5-aminolevulinic acid on chlorophyll,antioxidative enzymes and photosynthesis of Pakchoi (<Emphasis Type="Italic">Brassica campestris</Emphasis> ssp<Emphasis Type="Italic">.</Emphasis><Emphasis Type="Italic">chinensis</Emphasis> var. communis Tsen et Lee)

The objective of this article is to study the effect of 5-aminolevulinic acid (ALA) and enhanced chlorophyll content, antioxidative enzymes and photosynthesis rate by foliar application of ALA. We evaluated three concentrations (control-distilled water, T1-50 mg l⁻¹, T2-150 mg l⁻¹, T3-250 mg l⁻¹) of ALA and seven cultivars, “Sanchidaye” (Sa-1), “Lichuandasuomian” (Li-1), “Aijiaohuang” (Ai-1), “Qingyou” No. 4 (Qi-1), “Aikang” No. 5 (Ak-1), “Hanxiao” (Ha-1) and “Shulv” (Sl-1). “Ak-1” showed strongest response of POD (peroxidase) enzyme activity (0.4 U g⁻¹ min⁻¹) in 250 mg l⁻¹ ALA solution. The highest CAT (catalase) activity (0.8 U g⁻¹ min⁻¹) after administration of 250 mg l⁻¹ ALA was observed in “Li-1”. Meanwhile, highest (1.42 mg l⁻¹) total chlorophyll content was also observed in “Ak-1”, when leaves were treated in 50 mg l⁻¹ ALA, “Li-1” and “Ai-1” showed strongest response of specific activity of superoxide dismutase (SOD) in 50 mg l⁻¹ and 50 mg l⁻¹ ALA. Two hundred and fifty milligram per milliliter of ALA-treatment significantly improved the net photosynthetic rate.     

Source-sink landscape theory and its ecological significance

Exploring the relationships between landscape pattern and ecological processes is the key topic of landscape ecology, for which, a large number of indices as well as landscape pattern analysis model were developed. However, one problem faced by landscape ecologists is that it is hard to link the landscape indices with a specific ecological process. Linking landscape pattern and ecological processes has become a challenge for landscape ecologists. “Source” and “sink” are common concepts used in air pollution research, by which the movement direction and pattern of different pollutants in air can be clearly identified. In fact, for any ecological process, the research can be considered as a balance between the source and the sink in space. Thus, the concepts of “source” and “sink” could be implemented to the research of landscape pattern and ecological processes. In this paper, a theory of sourcesink landscape was proposed, which include: (1) In the research of landscape pattern and ecological process, all landscape types can be divided into two groups, “source” landscape and “sink” landscape. “Source” landscape contributes positively to the ecological process, while “sink” landscape is unhelpful to the ecological process. (2) Both landscapes are recognized with regard to the specific ecological process. “Source” landscape in a target ecological process may change into a “sink” landscape as in another ecological process. Therefore, the ecological process should be determined before “source” or “sink” landscape were defined. (3) The key point to distinguish “source” landscape from “sink” landscape is to quantify the effect of landscape on ecological process. The positive effect is made by “source” landscape, and the negative effect by “sink” landscape. (4) For the same ecological process, the contribution of “source” landscapes may vary, and it is the same to the “sink” landscapes. It is required to determine the weight of each landscape type on ecological processes. (5) The sourcesink principle can be applied to non-point source pollution control, biologic diversity protection, urban heat island effect mitigation, etc. However, the landscape evaluation models need to be calibrated respectively, because different ecological processes correspond with different source-sink landscapes and evaluation models for the different study areas. This theory is helpful to further study landscape pattern and ecological process, and offers a basis for new landscape index design. __________ Translated from Acta Ecologica Sinica, 2006, 26(5): 1444–1449 [译自: 生态学报]