A contribution to the mathematical biology of social behavior: Riots by oppressed groups

The author’s theory of imitation or mass behavior (N. Rashevsky:Mathematical Biology of Social Behavior, chapter xii, revised edition. Chicago: The University of Chicago Press, 1959; also Rashevsky:Looking at History through Mathematics, Cambridge, Mass.: The MIT Pres, 1968), when society chooses one of two mutually exclusive behaviors, is applied to the interaction of two social groups, an oppressor group and an oppressed one. Using crude approximations, conditions are derived as to when the oppressed group will revolt or riot, when the revolt will be suppressed, and when the oppressors will completely give in and oppression will end. Even in the simple approximation used, the situation depends on 14 parameters showing that a simplistic view on riots such as mere strong punishments is utterly inadequate. It is also shown that situations may exist in which revolution-like changes from one type of behavior of a society to another cannot be prevented by any measures.     

A contribution to the search of general mathematical principles in biology

A somewhat different approach to the principle of biotopological mapping, discussed in previous publications, is given. The organism is considered as a set of properties, each of which is in its turn a set of numerous subproperties which are logically included in the corresponding properties. Topology is introduced by an appropriate definition of neighborhoods, and four postulates are stated which concern the mapping of the spaces corresponding to higher organisms on those of lower ones. A number of conclusions are drawn from the postulates. Some of them correspond to well-known facts. For example, in man and some higher organisms appropriate emotional stimuli should produce gastrointestinal or cardiovascular disturbances; or some microorganisms should produce substances harmful to other microorganisms (antibiotics). Some other conclusions are still awaiting verification. One of them is, for example, that there must exist unicellular organisms which produce antibodies to appropriate antigens.     

A contribution to the mathematical biology of the rates of historical development

The paper is a second step toward a biomathematical theory of the rates of spread of new nonconformist ideas or behaviors in a society. It is intended as a preliminary and purely theoretical study of a very oversimplified case. An equation which determines the distribution function of the tendencies toward conformist and nonconformist behaviors is set up under a number of oversimplified assumptions, and a solution by successive approximations is indicated. The expression for the first approximation is given, and an estimate of the order of magnitude of the rates of changes is made. In conclusion an outline is given for further improvement of the theory.     

Mathematical biology of social behavior: III

A society composed of individuals each of whom can perform one of two mutually exclusive activitiesR ₁ andR ₂ is considered. The tendency toward the performance of those activities is measured by the intensities ε₁ and ε₂ of excitation of two corresponding neural centers, which cross-inhibit each other. It follows from the theory developed by H. D. Landahl that an individual with ε₁ − ε₂ = 0, that is one who has no preference for either one of the two activities, will on the average performR ₁ andR ₂ with equal probability. As ε₁ − ε₂ increases, the probabilityP ₁ ofR ₁ increases, tending to 1. As ε₂ − ε₁ increases, the probabilityP ₂ ofR ₂ increases, tending to 1. We haveP ₁+P ₂=1. The effect of imitation is now studied. The total number of individuals in the society which exhibits an activityR ₁ at a given time is considered as constituting a stimulus which increases ε₁. Similarly, the total number of individuals which exhibits activityR ₂ at a given time constitutes a stimulus which increases ε₂. Using the standard equations of the mathematical biophysics of the central nervous system, equations are established which govern the behavior of such a society and the following conclusions are reached. It the quantity ε₁ − ε₂ is distributed in the society in such a way that the distribution function is symmetric with respect to ε₁ − ε₂ = 0, then on the average one-half of the population exhibitsR ₁, the other halfR ₂. This social configuration may, however, be unstable. The slightest accidental excess of individuals exhibiting, for example,R ₁, may bring it into a stable configuration, in which most individuals exhibitR ₁, and only a smaller fraction exhibitR ₂. A slight initial deviation in favor ofR ₂ brings it into a stable configuration, in which most individuals exhibitR ₂. Thus in this case there may be two stable configurations. If the population is in one of those stable configurations, and the distribution function of ε₁ − ε₂ is made asymmetric, favoring the other activity, the population will pass into a stable configuration, in which that other activity is predominant, if the asymmetry of the distribution exceeds a threshold value. By making some drastic simplifications the equations derived here may be reduced to a form which waspostulated by the author previously in his mathematical theory of human relations.     

A contribution to the mathematical biology of automobile driving: II. Passing as a case of psychophysical discrimination

The decision to pass or not to pass in view of an oncoming car is considered as a case of comparative judgment in which it is to be decided whether the time it will take to pass safely is greater or less than the time it will take to collide with the oncoming car. H. D. Landahl's well-known theory of psychophysical discrimination is used, and it is assumed that the “distracting stimuli” considered previously (Rashevsky, 1959,Bull. Math. Biophysics,21, 375–85) tend to increase the standard deviation of Landahl's fluctuation function. Effects of the “distracting stimuli” on the threshold of the neuroelements in Landahl's circuit are also considered. On this basis an expression is derived which gives the probability of a collision accident in passing as a function of the “distracting stimuli.”     

A contribution to the mathematical biophysics of visual aesthetics

In continuation of previous studies, inequalities between different parameters of the brain are derived which determine whether an individual prefers in general visual patterns consisting of a relatively small number of relatively strongly excited elements, or such patterns which consist of a very large number of relatively weakly excited elements. As has been discussed in a previous publication, the first type of pattern is usually represented by artificial human-made designs, whereas the second type of pattern is formed predominantly in natural landscapes and sceneries. Thus the inequalities established in this paper give us the biophysical conditions which determine an individual's preference for either artificial designs or for landscapes and other natural objects.     

A contribution to the mathematical biophysics of cell growth and shapes: II

A theoretical study of the growing nerve cell filopodium is made using the assumptions of volume constancy, cylindrical shape, and substrate track of an earlier paper, but assuming additionally that a retarding force per unit area proportional to the rate of elongation is also acting. Equations of elongation for two different cases are derived.     

A contribution to the mathematical biophycics of visual perception and aesthetics

In continuation of previous studies of the mathematical biophysics of visual perception in relation to the aesthetic evaluation of visual patterns, an expression for the total intensity of excitation in a discriminating center as a function of the intensity of the peripheral stimulus is derived. This expression is applied to the case of aesthetic judgments of similar polygons of different sizes. The theoretical conclusions are tested experimentally by use of standard psychological scaling methods. The theoretical predictions are found to be in agreement with the experimental results.     

A mathematical contribution to the study of orientation of organisms

Organisms orient themselves to a stimulus by two general methods. One method is by directed orientation (taxis); the other is by undirected locomotory reaction (kinesis). An equation, and the methods for finding the necessary parameters of this equation, is derived for the distribution of organisms within a container, with the following limitations: (1) the organisms have no accommodation, (2) they are always active, and (3) the stimulus changes slowly with position. Necessary modifications of the equation are then derived, so that the last two limitations may be eliminated. The equation cannot be solved excatly because of its complexity; hence an approximation method must be used. This method is discussed, an approximate solution is found, and a time constant for equilibrium to be established is derived. Applications tovarious experiments in the literature are then made with fairly satisfactory results. A new interpretation of the theory of klino-kinesis with accommodation is found upon application of the equations developed to experimental work. Further limitations and uses of these equations are then discussed. This work was done while the author was Public Health Service Research Fellow of The National Institute of Mental Halth, Federal Security Agency.     

A contribution to the mathematical biophysics of psychophysical discrimination III

The neural mechanism previously discussed is further generalized. The case is considered in which a random variation is associated with each stimulus. The mechanism is generalized and equations are derived for discriminations between stimuli differing in several modalities. The latter indicates an analysis by the factor method. Suggestions are made in connection with the use of triads and with the problem of a multidimensional psychophysics.     

Ultramicrobacteria: Formation of the concept and contribution of ultramicrobacteria to biology

Ultramicrobacteria (UMB) are species of the domain Bacteria characterized by very small sizes of proliferating cells (less than 0.1 μm³ in volume) and small genomes (3.2 to 0.58 Mb). Some authors use the term nanobacteria as a synonym of UMB. Several tens of UMB species have been isolated from various natural habitats: sea water, soil, silt, Greenland ice sheet, permafrost soils, and intestines of humans and insects. Under laboratory conditions, they are cultivated on different nutrient media. In the second prokaryotic domain, the Archaea, ultrasmall forms (ultramicroarchaea) have also been described, including nanoarchaea (members of the genus Nanoarchaeum) with a cell volume of less than 0.1 μm³. The term nanobacteria is used in the literature also to denote ultrasmall bacterium-like particles occurring in rocks, sands, soils, deep sub-surface layers, meteorites, and clinical samples. The systematic position and the capacity for self-reproduction of these particles are still unclear. The cultured UMB forms are characterized by highly diverse morphology, ultrastructural organization, physiology, biochemistry, and ecology. UMB form three groups according to the type of cell wall structure and the reaction to Gram staining: (1) gram-negative, (2) gram-positive, and (3) cell wall-lacking. Their cells divide by constriction, septation, or budding. The unique processes performed by UMB are dehalorespiration and obligate or facultative epibiotic parasitism. The UMB that synthesize organic compounds in ocean waters with the involvement of proteorhodopsin play a great role in the biosphere. UMB have been found in seven large phylogenetic groups of prokaryotes, where their closest relatives are organisms with larger cells typical of bacteria, which is evidence of the polyphyletic origin of the currently known UMB species and the reductive mode of their evolution.     

The contribution of oxytocin and vasopressin to mammalian social behavior: potential role in autism spectrum disorder

Oxytocin (OT) and arginine-vasopressin (AVP) are 2 peptides that are produced in the brain and released via the pituitary gland to the peripheral blood, where they have diverse physiological functions. In the last 2 decades it has become clear that these peptides also play a central role in the modulation of mammalian social behavior by their actions within the brain. Several lines of evidence suggest their involvement in autism spectrum disorder (ASD), which is known to be associated with impaired social cognition and behavior. Recent clinical trials using OT administration to autistic patients have reported promising results. Here, we aim to describe the main data that suggest a connection between these peptides and ASD. Following a short illustration of several major topics in ASD biology we will (a) briefly describe the oxytocinergic and vasopressinergic systems in the brain, (b) discuss a few compelling cases manifesting the involvement of OT and AVP in mammalian social behavior, (c) describe data supporting the role of these peptides in human social cognition and behavior, and (d) discuss the possibility of the involvement of OT and AVP in ASD etiology, as well as the prospect of using these peptides as a treatment for ASD patients.     

A note on the mathematical biology of automobile driving

It is pointed out that the three different stimuli for a corrective turn, namely the distance from the edge of the lane, the rate of approach to the edge, and the angle between the direction of the car and the direction of the lane (Bull. Math. Biophysics,28, 645–654, 1966,29, 181–186, 1967) may act all three simultaneously. It is found that in that case the tracking curve of the car is stable below a critical speed and becomes unstable above it.     

A branching process, its application in biology: influence of demographic parameters on the social structure in mammal groups

Branching processes are widely used in biology. This theoretical tool is used in cell dynamics, epidemics and population dynamics. In population dynamics, branching processes are mainly used to access extinction probabilities of populations, groups or families, with the Galton-Watson branching process. Many mammal species live in socially-structured groups, and the smallest units of these groups are lineages (or families) of kin-related individuals. In many primate species, these lineages are matrilines, as females remain in their natal groups most of the time, whereas males generally disperse. Lineage parameters, such as numbers of matrilines, size of each matriline and average degree of relatedness, could strongly influence the genetic composition of groups. Evidence indicates that division along matrilines could induce substantial differentiation among fission groups. Here, we develop a novel mathematical model based on the branching process theory describing demographic dynamics of groups. The main result of this model is an explicit analytical expression of the joint distribution of numbers of lineages and sizes of socially-structured groups. We investigated the influence of parameters such as natality and mortality on the outcome of the process, including extinction probability. Finally, we discuss this theoretical result with respect to biological significance.     

The contribution of the Nobel laureates to biochemistry and molecular biology

The Nobel Prize winners works are considered and analyzed. Main attention is paid to the works devoted to biochemistry or problems related to biochemistry. The most prominent biochemical investigations are considered in detail. In some cases it is given the history and chronology of important biochemical discoveries. The survey is devoted due to the 170th anniversary of birth of Alfred Bernhard Nobel--a famous scientist and businessman, the founder of the Nobel Prizes.     

A contribution to the biology ofEurytoma curculionum Mayr (Hym., Eurytomidae)

Zusammenfassung Eurytoma curculionum Mayr wird als Parasit vonApion gemeldet. Der Beschreibung seiner Eistruktur folgen biologische Angaben.      

A history of the study of solid tumour growth: The contribution of mathematical modelling

A miscellany of new strategies, experimental techniques and theoretical approaches are emerging in the ongoing battle against cancer. Nevertheless, as new, ground-breaking discoveries relating to many and diverse areas of cancer research are made, scientists often have recourse to mathematical modelling in order to elucidate and interpret these experimental findings. Indeed, experimentalists and clinicians alike are becoming increasingly aware of the possibilities afforded by mathematical modelling, recognising that current medical techniques and experimental approaches are often unable to distinguish between various possible mechanisms underlying important aspects of tumour development. This short treatise presents a concise history of the study of solid tumour growth, illustrating the development of mathematical approaches from the early decades of the twentieth century to the present time. Most importantly these mathematical investigations are interwoven with the associated experimental work, showing the crucial relationship between experimental and theoretical approaches, which together have moulded our understanding of tumour growth and contributed to current anti-cancer treatments. Thus, a selection of mathematical publications, including the influential theoretical studies by Burton, Greenspan, Liotta et al., McElwain and co-workers, Adam and Maggelakis, and Byrne and co-workers are juxtaposed with the seminal experimental findings of Gray et al. on oxygenation and radio-sensitivity, Folkman on angiogenesis, Dorie et al. on cell migration and a wide variety of other crucial discoveries. In this way the development of this field of research through the interactions of these different approaches is illuminated, demonstrating the origins of our current understanding of the disease.     

Mathematical biology of social behavior: IV. Imitation effects as a function of distance

In previous publications, social groups have been studied in which each individual has a preference for one of two possible mutually exclusive activities. This preference is measured by a quantity ϕ. The value ϕ=0 corresponds to no preference; a preference for one activity is measured by a positive ϕ, the preference for the other by a negative ϕ. The quantity ϕ varies from individual to individual. It has been shown previously that, owing to effects of imitation, even when the average ϕ for the group is zero, one of the two behaviours will be chosen by the majority of the group. Whereas in previous studies the imitation effect was considered as independent of the distance between the imitating and imitated individuals, in the present study the case is considered in which the effect of imitation decreases with the distance between the individuals. It is found that under certain conditions a greater percentage near the center of the area occupied by the group, rather than near the periphery, exhibits the chosen behavior. The possible sociological meaning of this gradient of behavior is discussed.