Contributions to the theory of imitative behavior

The imitation effects in a social group depend both on the size of the group and on the distribution of a certain psychobiological quantity ϕ which measures the tendency of an individual towards a given behavior. The distribution function of ϕ determines the ratio μ of the individuals in the society who adopt a given behavior. When the size of the social group is not too large, the actual distribution of ϕ will deviate from the most probable one, and therefore communities of the same size and having the same parameters may have different values of μ. Approximate equations are developed which give the probability of a given μ for a group of a given size. Possible effects of interactions of communities of different sizes are briefly discussed. A generalization of the theory of imitative behavior to any number of mutually exclusive behaviors is given, and its possible sociological implications are discussed.     

A suggestion for a new approach to the mathematical theory of imitative behavior

The theory of imitative behavior as developed hitherto by the author was based on the assumption that each individual has a natural preference for one of the two mutually exclusive behaviors. The endogenous fluctuations in the central nervous system then result in the individual’s exhibiting the two behaviors alternately with a relative frequency determined by the natural preference. Imitation shifts the natural preference towards one or the other of the two mutually exclusive behaviors. In the present approach it is suggested that the relative frequency of the two mutually exclusive behaviors exhibited alternately is determined by maximizing the “satisfaction function” of the individual, that is by hedonistic factors rather than by purely random fluctuations. Corresponding equations are developed. It is shown that in certain cases, even when the imitation effect is absent, a sort of “pseudoimitation” may occur. Another situation leads, in the case of two individuals only, to a complete “division of labor” between them, with respect to the two behaviors. Each one exhibits only one behavior. After that imitation is introduced explicitly by assuming that imitation by one individual or another increases the satisfaction function of the imitating individual. Results thus obtained show similarities to the results of the old theory.     

On imitative behavior

Previous studies of effects of imitation on individuals in a population, in which the tendencies ϕ towards one or another of two mutually exclusive behaviors are distributed, are amplified by considering the distribution, not of ϕ directly, but of the excitations ɛ₀₁ and ɛ₀₂ of the two centers which mediate each of the two behaviors. It is shown how the distribution of ϕ is derived from those of ɛ₀₁ and ɛ₀₂. It is found that when both tendencies ɛ₀₁ and ɛ₀₂ are weak, the choice of one of the two behaviors not only is originally determined by pure chance, but that it is impossible to effect a change of the behaviour of a large population from one adopted behavior to a possible opposite one, by inhibiting the tendency towards the first behavior. Such a change by inhibition is possible only when the tendencies toward both mutually exclusive behaviors are sufficiently strong. A possible application to the persistency of irrelevant established behavior patterns, such as handshakes, is suggested.     

Mathematical biological of social behavior: II

A group of individuals is considered in which each individual has tendencies to exhibit one or another of two mutually exclusive behaviors. Neurobiophysically this may be described in terms of Landahl's reciprocally inhibited parallel reaction chains. The spontaneous excitations ε₁ and ε₂ at the central connections of each chain are a measure of the “natural” tendency of the individual toward one or the other of the two behaviors. According to equations derived by H. D. Landahl, the probability of one or the other behavior is determined by the difference ε₁ − ε₂. A population of individuals is considered in which ε₁ − ε₂ is distributed in some continuous way, and therefore in which the probability of a given behavior is distributed continuously between 0 and 1. The effect of other individuals exhibiting a given behavior is to increase the corresponding ε of the individual. Thus behavior of others affects the probability for a given behavior of each individual. It is shown that the equations describing the behavior of the population on the basis of this neurobiophysical picture reduce in the first approximation to the differential equations which were postulated by the author in his previous work on social behavior.     

A field theory of neural nets: I: Derivation of field equations

A model is described in which neural activity is represented by a field quantity ϕ, with the neurons as the sources of ϕ. It is shown that, with certain physically realistic assumptions, ϕ satisfies a moderately nonlinear differential equation. It is also found that this equation is isotropic and of second order if and only if the neuronal connectivity has a dependence on distance,p, of the formp ⁻¹ e ^−1/2βp .     

Mathematical theory of motivation interactions of two individuals: I

The behavior of two individuals is considered as consisting of an increase or a decrease of productive output. Motivation for increase is the derivative of a “satisfaction function”. This is an algebraic sum of the well-known Thurstone satisfaction curve and another essentially negative quantity, which is a product of a “reluctance parameter” and the “effort”. Each individual attempts to maximize his own total satisfaction. The resulting behavior is examined under a variety of conditions; namely, 1) equal sharing of produced without prescribed sharing of effort; 2) various contracts prescribing the sharing of effort; 3) situations in which one individual is more aware of the underlying motivations than the other. It is these latter situations which under the simplest assumptions of equal sharing, without prescribed sharing of effort, lead to parasitism, i.e. total cessation of effort on the part of one individual. This happens when one individual becomes aware of the other'sautomatic adjustment of his effort so as to bring about a total optimum output, which is a constant. Parasitism is prevented by various forms of contracts in which either the effort necessary for the total optimum output is shared according to a prescribed ratio or the effort of one individual is fixeda priori as a function of the effort of the other. In the latter case the respective efforts become a function of a single variable, and each of the satisfaction functions is maximized by a particular value or values of this variable. In general, these critical values do not coincide for the two satisfaction functions. The problem of finding forms of contract which will result in identical critical (maximizing) values of the variable for both satisfaction functions leads to a functional equation.     

A problem in the mathematical biophysics of interaction of two or more individuals which may be of interest in mathematical sociology

The behavior of an individual may be discussed in terms of a “satisfaction function”. An individual may be considered to always behave in such a way as to make his satisfaction function a maximum. The interaction of two individuals may consist in a cooperation in the production of any kind of objects of satisfaction. Those objects may be either material goods or anything else. The satisfaction of each individual is determined by his share in the total output as well as by the effort he makes. It is shown that for a prescribed method of sharing a behavior in which each individual attempts to maximize the total satisfaction of both individuals results in a greater output than a behavior in which each individual attempts to maximize his own satisfaction.     

Studies in mathematical biosociology of imitative behavior: I. Effects of income distribution

In connection with previous studies (Rashevsky,Mathematical Biology of Social Behavior, chap. xii), a situation is investigated in which the two mutually exclusive possible behaviors of a society consist of the desire to keep the present socioeconomical situation and the desire to change it inany way. The psychophysiological tendency ϕ towards either of the behaviors is considered to be proportional to the difference between the actual incomei of the individual and his needsi′. Assuming that the distribution functionN ₁(i′) of the needs is a given characteristic of the population, it is shown that the distribution functionN(ϕ) of ϕ in the society can be derived fromN ₁(i′) and from the distributionN ₂(i) of the incomesi. A particular case is worked out as an example. Conditions of stability of a socioeconomic structure are studied in their dependence on the income distribution.     

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.     

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.     

The occurrence of protein kinase C ϕ and λ isoforms in retina of different species

The localization and immunochemical identification of the novel protein kinase C ϕ (nPKC ϕ) and the atypical protein kinase C λ (aPKC λ) isoforms in retinas of different species were analyzed by immunohistochemistry and SDS-PAGE/Western blotting. nPKC ϕ immunoreactivity is associated with bipolar cells of mammalian (rabbit, rat and guinea pig) retinas but not the non-mammalian goldfish retina which has a lower concentration of nPKC ϕ. However, SDS-PAGE and Western blotting data indicate the antigen recognized by the nPKC ϕ monoclonal antibody in the retina is of a lower molecular weight than that expected for nPKC ϕ. This would suggest nPKC ϕ is more susceptible to degradation/breakdown than other PKC isoforms found in the retina or that the nPKC ϕ antibody may be recognizing an unknown retinal antigen. A comparison of nPKC ϕ and nPKC ϕ is present in the developing retina at an earlier stage than cPKC α. The typical ‘transport’ of cPKC α toward axonal terminals by phorbol-12,13-dibutyrate does not occur for nPKC ϕ yet both are translocated from the cytosolic to membrane compartments. The inner plexiform layer and the inner nuclear layer (putative horizontal cells) of all species examined (rabbit, rat, guinea pig and goldfish) exhibited positive immunoreactivity for aPKC λ as confirmed by SDS-PAGE/Western blotting. Special issue dedicated to Dr. Kinya Kuriyama.     

A note on imitative behavior and information

The equations for imitative behavior developed previously indicate that the imitation effect increases with the numberN ₀ of individuals in the social group. In this note it is pointed out that the above conclusion holds only for not too large values ofN ₀. The above conclusion is based on the tacit assumption that each individual is completely informed about the behavior adopted by every other individual. If, however,N ₀ becomes very large, the information per individual decreases. As a result of this, the effects of imitation either increase withN ₀ less rapidly or actually become independent ofN ₀.     

Mathematical biosociology of beliefs and prejudices

Following a suggestion made previously (Bull. Math. Biophysics,13, 61, 1951), it is assumed that every individual has both a tendency to behavearationally, by accepting everything on faith, and rationally, by subjecting everything to rational analysis. Arational behavior is characterized by various beliefs, prejudices, etc., which are considered to be conditioned reactions, learned by the individual before he completely develops his faculties for rational thinking. The two tendencies are assumed to be due to excitations of two different regions of the central nervous system, and are measured by the intensities ɛ _f and ɛ _r of those excitations. Those intensities are further assumed to increase linearly with time, the increases of the two beginning, in general, at different ages. The rates of increase are considered as normally distributed in the population. The relative frequency of arational and rational behavior is determined by the difference φ=ɛ _f =ɛ _r according to equations 0 developed previously (Bull. Math. Biophysics,11, 255, 1949). It is shown that with the above assumptions the majority of the population, which starts with arational behavior, will, within two or three of generations, either change to rational behavior or continue indefinitely to behave arationally. This will hold as long as imitative factors are present. Expressions for the numbers of individuals who behave rationally and arationally are derived. If the intensity of conditioning toward an arational behavior decreases with increasing size of the rationally behaving minority, or, if the rationally behaving individuals are not influenced by imitation, then a slow secular trend toward rational behavior may be present. An expression is also derived for the fraction of individuals who behave rationally as a function of age. This fraction increases with increase of the age at which the beginning conditioning toward any beliefs or prejudices begins.     

Patterns of orientation behavior in marsh frogs (Rana ridibunda Pall.) of southern populations

The behavior of frogs from southern areas with an arid climate released during the reproductive period between their “home” pond and a less distant river is shown to be independent of weather conditions. The experiments were performed on frogs living near the village of Dosang, Astrakhan oblast. The frogs were released 60 to 150 m from their “home” breeding waterbody and 60 to 80 m from an “alien” waterbody. Four experiments were performed on 27 individuals. The movements of the frogs were traced by the method of “tracking by a thread.” The results revealed no preference in the frogs to return to their own breeding ponds, rather than to the nearby river. This behavior is peculiar to frogs of southern populations.     

The flexible grouping and behavioral character of a flock of Suffolk ewes (Ovis aries)

The spatial patterning and grouping behavior of domestic sheep (Ovis aries) under ranch management was studied. Activity was highly synchronized among flock members and was often initiated by several individuals, but no specific individual was always responsible. In the pasture, they were not randomly dispersed, but stayed together, and individuals did not develop specific favorite areas. They usually split into several groups. The size and composition of the group was very flexible and changed often. The group did not have fixed members, although there seemed to be certain partiality for an individual's association. In certain situations, the sheep sometimes formed a large flock composed of most members. They showed 2 types of flock formation: splitting into small groups, and assembling into one large group. They changed immediately from one formation type to another, without any particular individual's initiation. They seemed to have preference for association partners in a flock, but this was not found to be a determinant factor of subgroup composition. Furthermore a large group is not formed from the association of other stable subgroups. They seemed to be attracted by co-existence or the “flock” itself, rather than association with preferred individuals.     

Mate preferences in female canaries (Serinus canaria) within a breeding season

Divorce and remating in birds can be described as strategies used to enhance reproductive success. Mate switching often occurs because pairs failed to brood at least one chick during the previous breeding season. In the present study, we evaluated the influence of reproductive success on female preferences in domesticated canaries (Serinus canaria). For that purpose, females previously paired and having reared young were placed in a choice test situation: They were allowed to choose between their previous mate and a familiar male (a male neighbor during the breeding period). During these choice tests, females tended to stay near their previous mate longer than near a male neighbor when their reproductive success was “good” (at least two chicks). On the other hand, females with “poor” reproductive success (one chick) did not show a preference for their previous mate. Furthermore, in the present study, we observed that during choice tests males reacted to the presence of their previous mate in a particular way, by gathering nest material. This behavior was more scarcely observed in neighbor males which, on the contrary, sang significantly more than previous mates did.     

Mathematical theory of motivation interactions of two individuals: II

The behavior of two individuals, consisting of effort which results in output, is considered to be determined by a satisfaction function which depends on remuneration (receiving part of the output) and on the effort expended. The total output of the two individuals is not additive, that is, together they produce in general more than separately. Each individual behaves in a way which he considers will maximize his satisfaction function. Conditions are deduced for a certain relative equilibrium and for the stability of this equilibrium, i.e., conditions under which it will not “pay” the individual to decrease his efforts. In the absence of such conditions “exploitation” occurs which may or may not lead to total parasitism. Some forms of the inverse problem are considered, where the form of behavior is given and forms of the satisfaction function are deduced which lead to it.     

Isolation and characterization of two types of actinophage infectingStreptomyces scabies

Two types of actinophages, ϕS and ϕL, were isolated from soil samples by usingStreptomyces scabies, a potato scab pathogen, as indicator strain. The phages were partially characterized according to their physicochemical properties, plaques and particles morphology, and their host range; this varied from narrow (for ϕS) to wide (for ϕL). The adsorption rate constants of the ϕS and ϕL were 3.44 and 3.18 pL/min, and their burst sizes were 1.61 and 3.75 virions per mL, respectively. One-step growth indicated that ϕS and ϕL have a latent period of 1/2 h followed by a rise period of 1/2 h. The temperate character of these phages was tested in other isolates ofStreptomyces. Four of the phages (ϕSS3, ϕSS12, ϕSS13 and ϕSS17) were identified as temperate phages, since they were able to lysogenize SS3, SS12, SS13 and SS17. ϕSS3, ϕSS12 and ϕSS13 were homoimmune, and they were heteroimmune with respect to ϕSS17. The restriction barriers of lysogenic isolates (SS12, SS13 and SS17) interfered with the blockage of plaque formation by phages (ϕSS12, ϕSS13 or ϕSS17) propagated on them, about 75% of lysogenic isolates had restriction systems. The exposure of the lysogenic isolates (SS12, SS13 and SS17) to UV-irradiation prevented the possible restriction barriers of these isolates so that these barriers could be overcome.     

The root–soil system of Norway spruce subjected to turning moment: resistance as a function of rotation

The reactions of trees to wind, rockfall, and snow and debris flow depend largely on how strong and deformable their anchorage in the soil is. Here, the resistive turning moment M of the root–soil system as a function of the rotation ϕ at the stem base plays the major role. M(ϕ) describes the behavior of the root–soil system when subject to rotational moment, with the maximum M(ϕ) indicating the anchorage strength M _a of the tree. We assessed M(ϕ) of 66 Norway spruce (Picea abies L. Karst) by pulling them over with a winch. These 45- to 170-year-old trees grew at sites of low and high elevation, with a diameter at breast height DBH = 14–69 cm and a height H = 9–42 m. M(ϕ) displayed a strong nonlinear behavior. M _a was reached at a lower ϕ for large trees than for small trees. Thus overhanging tree weight contributed less to M _a for the large trees. Overturning also occurred at a lower ϕ for the large trees. These observations show that the rotational ductility of the root–soil system is higher for small trees. M _a could be described by four monovariate linear regression equations of tree weight, stem weight, stem volume and DBH ² ·H (0.80 < R ² < 0.95), and ϕ at M _a, ϕ _a, by a power law of DBH²·H (R ² = 0.85). We found significantly higher M _a for the low-elevation spruces than for the high-elevation spruces, which were more shallowly anchored, but no significant difference in ϕ _a. The 66 curves of M(ϕ), normalized (_n) by M _a in M-direction and by ϕ _a in ϕ-direction, yielded one characteristic average curve: . Using and the predictions of M _a and ϕ _a, it is shown that M(ϕ) and the curves associated with M(ϕ) can be predicted with a relative standard error ≤25%. The parameterization of M(ϕ) by tree size and weight is novel and provides useful information for predicting with finite-element computer models how trees will react to natural hazards.     

The potential effects of sex,posture and living condition on lateralized behaviors in ring-tailed lemurs (<Emphasis Type="Italic">Lemur catta</Emphasis>)

This research explores the effects of posture, sex, and living condition on hand and side preferences of semi-free-ranging, adult ring-tailed lemurs (Lemur catta) housed at the Duke University Primate Center in Durham, NC. Data were collected on 11 adult individuals (five females and six males) during normal daily activities over a ten-week period from May–July 2001. Variables analyzed in this study include unimanual behaviors (i.e., reach, hold, and limb used to start locomotion) and other potentially lateralized behaviors that do not involve handuse (i.e., whole-body turning and tail position). The data were analyzed to investigate potential individual and population level side biases for each behavior; potential sex biases in side preference for each behavior; and for ‘reach’, potential effects of posture (sitting, tripedal stance, or bipedal stance) on individual hand preferences. Additionally, to investigate potential effects of living condition on lateral biases, the data from this study were compared to data collected on the same individual Lemurs living under more restrictive living conditions during the previous year. Largely, as predicted based on available literature, we found that there was a significant sex difference across all hand-use categories and for whole-body turning, and that posture was a significant factor in the expression of hand preference for reaching. Contrary to previous research, the effect of living condition on lateral preferences was minimal, and no side preferences were found at the population level for any of the behaviors analyzed.