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.     

Contribution to the mathematical theory of mass behavior: I. The propagation of single acts

The propagation of a single act in a large population is supposed to depend on some external circumstance and on an “imitation component”, where encounters with individuals who are performing or have already performed the act contribute to the tendency of an individual to perform it. The “tendency” to perform is supposed to be measured by the average frequency of stimuli, randomly distributed in time, impinging on the individual. The deduced equation is a relation between the fraction of the population who have performed the act and time, provided the time course of the “external circumstance” and the way in which the imitation component contributes are known. Several special cases are studied, in particular, cases without the imitation component, cases with imitation only, and various mixed cases. Examples are given of social situations in which such factors may operate and general suggestions are made for the systematization of observations and/or experiments to test the assumptions of the theory.     

Response time and threshold of a random net

The response time of a random net is defined as the expected time (measured in the number of synaptic delays) required for the excitation in the net (measured by the fraction of neurons firing per unit time) to reach a certain level. The response time is calculated in terms of the net parameters as a function of the intensity of the outside stimulation. Two principal types of cases are studied, 1) an instantaneous initial stimulation, and 2) continuously applied stimulation. It is shown that for a certain type of net where the required level of excitation is small, the response time-intensity equation reduces to the one derived on the basis of the “one-factor” theory applied to a neural connection. More general assumptions, however, give different types of equations. The concept of the “net threshold” is defined, and its calculation indicated. The net threshold for instantaneous stimulation is, in general, greater than that for continuous stimulation. The results are discussed with reference to existing theories of reaction times.     

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.     

Closed-loop and open-loop control of posture and movement during human trunk bending

Closed-loop (CL) and open-loop (OL) types of motor control during human forward upper trunk bending are investigated. A two-joint (hip and ankle) biomechanical model of the human body is used. The analysis is performed in terms of the movements along eigenvectors of the motion equation (“eigenmovements” or “natural synergies”). Two analyzed natural synergies are called “H-synergy” (Hip) and “A-synergy” (Ankle) according to the dominant joint in each of these synergies. Parameters of CL control were estimated using a sudden support platform displacement applied during the movement execution. The CL gain in the H-synergy increased and in the A-synergy decreased during the movement as compared with the quiet standing. The analysis of the time course of OL control signal suggests that the H-synergy (responsible for the prime movement, i.e. bending per se) is controlled according to the EP theory whereas for the associated A-synergy (responsible for posture adjustment, i.e. equilibrium maintenance) muscle forces and gravity forces are balanced for any its final amplitude and therefore the EP theory is not applicable to its control.     

Chain processes and their biophysical applications: Part I. General theory

A mathematical theory applicable to the biological effects of radiations as chain processes is developed. The theory may be interpreted substantially as a “hit theory” involving the concepts of “sensitive volume” or “target area”. The variability of the sensitivity of the organism to the radiation and its capacity of recovery between single hits is taken into account. It is shown that in a continuous irradiation of a biological aggregate in which the effect of each single hit cannot be observed, recovery and variation of sensitivity are formally equivalent to each other so that a discrimination between these two phenomena is possible only by discontinuous irradiation or by using different radiation intensities. Methods for the calculation of the “number of hits” and for the determination of the kinetics of the processes from “survival curves” or similar experimental data are given. The relation between the recovery and the Bunsen-Roscoe law is discussed. The case in which the injury of the organism is dependent on the destruction of more than one “sensitive volume” is also considered.     

On the Forkker-Planck equation in the stochastic theory of mortality: I

This paper, consisting of two parts, gives all the mathematical details that were omitted in a previous work by G. A. Sacher and E. Trucco (“The Stochastic Theory of Mortality.”Ann. N. Y. Acad. Sci.,96, 985–1007, cited here as ST). We assume that the reader is familiar with ST, where the stochastic theory of mortality, originally proposed by Sacher, is discussed at length. We recall that the basic model presented there refers to an ensemble of particles performing Brownian motion in one dimension, with the added constraint of two absorbing barriers. These two points, collectively, are designated as the “lethal bound.” Part I (section 1 to 4) deals with the special case in which the two absorbing barriers are symmetrically located at a finite distance from the origin. The solution of the Fokker-Planck equation is obtained from the theory of eigenvalue problems. Quite generally, the eigenfunctions functions belong to the family of Kummer's confluent hypergeometric functions, but the symmetry condition imposed here results in considerable simplification and makes it possible to estimate the first few eigenvalues by a graphical procedure. In section 3 we show how perturbation theory can be applied in the limiting case of “weak homeostasis,” and section 4 deals with the opposite extreme of “strong homeostasis.” A rigorous proof is given for the result corresponding to equation (28) of ST (asymptotic or quasi-static approximation for the “force of mortality”). This work was performed under the auspices of the U.S. Atomic Energy Commission.     

A Mathematical Model for Evolution and SETI

Darwinian evolution theory may be regarded as a part of SETI theory in that the factor f_l in the Drake equation represents the fraction of planets suitable for life on which life actually arose. In this paper we firstly provide a statistical generalization of the Drake equation where the factor f_l is shown to follow the lognormal probability distribution. This lognormal distribution is a consequence of the Central Limit Theorem (CLT) of Statistics, stating that the product of a number of independent random variables whose probability densities are unknown and independent of each other approached the lognormal distribution when the number of factors increased to infinity. In addition we show that the exponential growth of the number of species typical of Darwinian Evolution may be regarded as the geometric locus of the peaks of a one-parameter family of lognormal distributions (b-lognormals) constrained between the time axis and the exponential growth curve. Finally, since each b-lognormal distribution in the family may in turn be regarded as the product of a large number (actually “an infinity”) of independent lognormal probability distributions, the mathematical way is paved to further cast Darwinian Evolution into a mathematical theory in agreement with both its typical exponential growth in the number of living species and the Statistical Drake Equation.     

A note on the equations of conditioned reflex

An alternative method is suggested for integrating a certain differential equation associated with a conditioning process, where the stimulus is presented in the form of a “square wave,” i.e., is of constant intensity during an interval of time followed by no stimulus during the next interval, etc. A solution is also given where the stimulus is a rectified sine wave.     

“Theory” in Theory and Practice

A central obstacle to accepting evolution, both among students and the general public, is the idea that evolution is “just a theory,” where “theory” is understood in a pejorative sense as something conjectural or speculative. Although scientists and textbooks constantly explain that the scientific use of “theory” is quite different, the pejorative use continues to cause confusion, in part because of its deep roots in a popular, Baconian, understanding of science. A constructivist approach, whereby students are helped to examine the adequacy of their preconceptions about “theory” for themselves and to revise or replace them appropriately, is recommended.     

“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.     

Non-linear excitation theory: Non-accommodative,sub-threshold effects

The standard two factor excitation theories should be called “preexcitation” theories since they apply only to those events occurring just up to excitation. A true phenomenological excitation theory which describes thewhole excitation cycle must involve non-linear equations. The nature of these non-linearities is suggested by B. Katz's subthreshold response data. From this data is constructed a “local phenomenological characteristic” which is analogous to the current-voltage characteristic of a non-linear electrical or mechanical system capable of displaying relaxation oscillations. Excitation by constant currents is shown to occur where the slope of the characteristic changes sign. The variation of the time constant of excitation with degree of response, explained by W. A. H. Rushton in terms of a liminal length, is described here in purely formal terms. The theory as presented explicity treats only those events in the excitation cycle up to and a little beyond excitation; the complete excitation cycle (including recovery and repetition) is mentioned as being amenable to mathematical treatment by an extension of the present theory.     

Self-reproduction and serial message transfer: Two related problems

For a certain class of physical machines, termed “structure-determined,” the problem of self-reproduction can be reduced to the problem of serial message reproduction. Serial message reproduction however presupposes a sort of “open system” constraint. This leads to the principle of pseudo, or exogenously standardized, respectively, self-reproduction. It seems to be consistent with both chemical and biological self-reproduction. It thus may reflect a general principle of biological design. The proposed principle is a physico chemical analog to Robert Rosen's abstract relational self-reproduction constraint.     

A reinterpretation of the mathematical biophysics of the central nervous system in the light of neurophysiological findings

The fundamental equations for the interaction between neurons used in mathematical biophysics seem at first incompatible with the actual neurophysiological findings on the synaptic transmission. It is shown, however, that those equations may be readily interpreted in terms of accepted neurophysiological views. What has been termed “synapse” in mathematical biophysics must be regarded as a complicated network of internuncial neurons. It is shown that, under rather conditions, the number of those interneurons willstatistically vary with time according to the differential equation postulated for the excitatory and inhibitory factors. The latter are thus interpreted as the number of excitatory and inhibitory interneurons.     

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]     

Suggestions for a mathematical biophysics of some psychoses

We may consider that most of the human behavior is a set of learned responses to certain patterns which recur frequently in the course of human life. Some “abnormal” events or experiences may result in the learning of abnormal responses, and thus in abnormal behavior. The “abnormal” responses may begin to be learned after some of the normal response patterns have been fairly well established. The development of both normal and abnormal behavior may thus be represented by learning curves of the type studied by H. D. Landahl. Applying some of the results of the theory of learning curves and considering that the normal and abnormal reactions may reciprocally inhibit each other, a quantitative theory of some psychoses may be developed. In particular, the effects of shock may be deduced from the assumption that they cause the more recently learned abnormal reactions to be “unlearned” more readily, than the earlier learned “normal” reactions. The effectiveness of shock treatments as a function of the duraction of psychosis is discussed from this point of view.     

Fire regime: history and definition of a key concept in disturbance ecology

“Fire regime” has become, in recent decades, a key concept in many scientific domains. In spite of its wide spread use, the concept still lacks a clear and wide established definition. Many believe that it was first discussed in a famous report on national park management in the United States, and that it may be simply defined as a selection of a few measurable parameters that summarize the fire occurrence patterns in an area. This view has been uncritically perpetuated in the scientific community in the last decades. In this paper we attempt a historical reconstruction of the origin, the evolution and the current meaning of “fire regime” as a concept. Its roots go back to the 19th century in France and to the first half of the 20th century in French African colonies. The “fire regime” concept took time to evolve and pass from French into English usage and thus to the whole scientific community. This coincided with a paradigm shift in the early 1960s in the United States, where a favourable cultural, social and scientific climate led to the natural role of fires as a major disturbance in ecosystem dynamics becoming fully acknowledged. Today the concept of “fire regime” refers to a collection of several fire-related parameters that may be organized, assembled and used in different ways according to the needs of the users. A structure for the most relevant categories of parameters is proposed, aiming to contribute to a unified concept of “fire regime” that can reconcile the physical nature of fire with the socio-ecological context within which it occurs.     

“Little Monkeys on the Grass…” How People for and Against Evolution Fail to Understand the Theory of Evolution

In an informal survey, only five percent of 306 college freshmen students in an introductory biology course provided a correct scientific definition for the theory of evolution. The other respondents provided answers that ranged from “organisms improving themselves” (42 percent) to “monkeys becoming humans” (seven percent). Some of the potential reasons for the lack of understanding of the concept of evolution are explored.     

Sibling rivalry between seeds within a fruit: Some population genetic models

Competition between seeds within a fruit for parental resources is described using one-locus-two-allele models. While a “normal” allele leads to an equitable distribution of resources between seeds (a situation which also corresponds to the parental optimum), the “selfish” allele is assumed to cause the seed carrying it to usurp a higher proportion of the resources. The outcome of competition between “selfish” alleles is also assumed to lead to an asymmetric distribution of resources, the “winner” being chosen randomly. Conditions for the spread of an initially rare selfish allele and the optimal resource allocation corresponding to the evolutionarily stable strategy, derived for species with n-seeded fruits, are in accordance with expectations based on Hamilton’s inclusive fitness criteria. Competition between seeds is seen to be most intense when there are only two seeds, and decreases with increasing number of seeds, suggesting that two-seeded fruits would be rarer than one-seeded or many-seeded ones. Available data from a large number of plant species are consistent with this prediction of the model. Based on a talk given at the Haldane Centenary Symposium held on 6 November 1992 at Ahmedabad as part of the 58th Annual Meeting of the Indian Academy of Sciences.     

Contribution to the probabilistic theory of neural nets: II. Facilitation and threshold phenomena

The output curve of a single neuron with a threshold of response with respect to the frequency of the stimuli is derived. If the stimuli are regularly spaced in time, the output curve has discontinuities. If the threshold and/or refractory period are sufficiently large, the output curve approaches the “all-or-none” curve. In the case of completely randomized stimuli, the output curve is sigmoid. The equation of this curve is derived and some properties are studied. Threshold and “all-or-none” effects can be achieved by “pyramiding” neurons of this type to converge on neurons of higher order.