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Recent demonstration by the author has shown that the fundamental equations of the mathematical biophysics of the central nervous system can be considered as describing the behavior of very large numbers of neurons, of which each one follows discontinuous laws, such as discussed by W. S. McCulloch and W. Pitts. In that light some of the old problems are discussed. The comparative merits of the “microscopic” and “macroscopic” approaches are discussed for the problem of the point to point correspondence between the retina and the cortex, with the number of connecting fibers much less than the number of cells. Some aspects of discrimination of intensities are also discussed. Finally, a few generalizations of the McCulloch-Pitts treatment are suggested, and a nervous network is constructed which illustrates some aspects of the perception of numbers.

A theory of the random change of the shape of an ameba, produced by protusion and retraction of pseudopods is developed on the assumption that a local disturbance, which reduces the surface tension, travels along the surface of the ameba. Quantitative relations suggesting experimental verification are derived for the average size and average duration of a pseudopod.

The idea was suggested by the author previously (Bull. Math. Biophysics,22, 257–262, 1962) that the keeping of the car close to the center of the lane is a problem of psychophysical discrimination between two conflicting stimuli, namely a stimulus to turn away from the left, resp. right edge of the lane. This is elaborated in the present paper. The effects of discrimination threshold and of the endogenous fluctuations which result in erroneous judgments are discussed. In order that driving should be possible at all, a relation, derived in this paper, must hold between the threshold of discriminationh, the sensitivity coefficientb of the driver to changes in the distance between the car and the edge of the lane, and the width of the lane. General expressions are derived which characterize the stochastic nature of the tracking curve.

The approximate equation is derived for the rate of constriction of a dividing cell, describing the phenomenon from its early stages. The equation previously derived by G. Young for the case when the constriction has already considerably progressed is obtained as a limiting case.

V. S. Ivlev [Experimental Ecology of Nutrition of Fishes, 1955, Moscow (in Russia)] has shown that the food uptake by fishes during a fixed interval of time is an exponential function of the concentration of food. Ivlev's equation is derived here, and it is shown that it can hold only for non-stationary conditions, such as prevailed in Ivlev's experiments. For a stationary state, the rate of food uptake should tend asymptotically to a limiting value as the concentration increases, but the variation is not exponential. Different other aspects of the problem are investigated, and definite new experimental procedures suggested. The implications of Ivlev's findings on the effect of non-uniformity of food distribution upon the rate of food consumption are studied from a mathematical point of view. The conclusion is reached that whereas a fish does not, in the process of eating, move directly to an individual food particle which it perceives, it does move more or less directly to large aggregates of particles, if the latter are distributed nonuniformly.

Two somewhat different approaches to topological biology have been developed in recent years. The latest, set-theoretical approach leads rather immediately to a number of conclusions which are verified experimentally. It also predicts a number of new biological relations. The older, graph-theoretical approach does not lead directly to those conclusions but suggests a number of combinatorial relations between number of organs and number of cell types. Such suggestions are absent from the set-theoretical approach. It is shown that the above-mentioned combinatorial relations are independent of the graph-theoretical method proper and can be introduced into the set-theoretical approach through the addition of an independent postulate.

In imitative behavior, as studied previously by N. Rashevsky (Mathematical Biology of Sociol Behavior, Chapter XIII, The University of Chicago Press, 1950), the reason for the majority of a society to accept a particular behavior is based on purely voluntary action (band-wagon effect). In the present paper effects of coercion of the majority by a small minority group which poses the means for coercion, are studied. More general types of equations are thus obtained and threshold effects found, which bear a resemblance to some such effects studied previously.

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.

Developing further a recent suggestion of H. D. Landahl, equations are derived which give the probability for an individual to judge two stimulus patterns as similar or dissimilar. The possibility of experimental verification of those equations is discussed. Next, a mechanism is described which provided for abstraction by responding only to “essential” components of a stimulus pattern. Equations, verifiable in principle experimentally, are derived. Finally, a mechanism is suggested for logical inferences, and equations are derived which give the probability of making an error in a reasoning consisting of a chain of syllogisms, as well as the probability of being unable to complete the chain of reasoning at all.

The binding energy of a very long molecular chain, composed of different classes of molecules, depends in general on the order of the molecules. It is shown that under very general conditions there exists for a givenbrutto chemical composition of a chain, a class of chains which is characterized by a total binding energy which is equal to the total binding energy of any other prescribed chain of different composition within the limits of unsharpness of the energy level. This establishes a criterion formapping of a class of configurations of long chain molecules on another class. To the extent that a mapping constitutes a generalized code those results contribute to the theory of molecular codes.

In continuation of previous work (Rashevsky,Some Medical Aspects of Mathematical Biology, Springfield, Ill.: Charles C. Thomas, 1964, Chap. 23 and Appendix 14), the study of the effects of the physical parameters of the cells of endocrine glands on the onset of sustained periodical oscillations in the interaction between the anterior pituitary and the thyroid hormones is generalized to include the possible effect of the intercellular fluid and of the degree of vascularization. Some conclusions of the previous study remain valid although some modifications must be made. A decreased relative volume of the intercellular fluid and an increased vascularization favor the conditions for sustained oscillations. The permeability of the cells and the permeability of the capillaries appear explicitly in the expressions which show the conditions for sustained periodicities.

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.

It is shown that the principle of biological epimorphism (Rashevsky,Mathematical Principles in Biology and Their Applications, Springfield, Ill.: Charles Thomas, 1960) is contained in the theory of organismic sets (Bull. Math. Biophysics,29, 139–152, 1967) if an additional postulate not directly connected to mappings is made.

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.

Some general considerations are given regarding the effects of temporal sequences of stimuli in a neuronic network, which consists of a set of parallel chains of excitatory fibers with cross-connections made of inhibitory fibers. It is shown that, in general, the excitation produced by any individual stimulus of the series is a function of the order and duration of the previous stimuli, and that the effect of each stimulus thus depends on the whole temporal pattern considered.

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.

An attempt is made to develop a general theory of the structure of an extremity of a quadruped or biped, based on its mechanical function. The problem is essentially one of dynamics of a chain of linked levers. The equations of motions of such systems are practically insoluble exactly at the present time. A very crude approximation method is therefore outlined. It is shown that, theoretically, the number of links in the chain forming the extremity may be determined by the optimum use of the total available force of the flexors and extensors. Discussion of the formula derived from this point of view shows, however, that actually this is not the determining factor. Other theoretical possibilities are discussed. While no application can be made at present, yet in principle it is shown how mechanical consideration can throw light on the anatomical structure of the extremity.

In a previous paper, in which a possible mathematical approach to history was outlined, it was shown that urbanization plays an important part in the propagation of new ideas. The rate of such propagation influences the rate of historical developments. The present paper deals in more detail with possible mechanisms of formation of earliest cities. Equations are derived which give the limiting size of such cities and their rate of growth. Of particular importance for the spread of new ideas is the spread of information. The latter largely depends on the fraction of individuals who travel between city and country. Expressions for this quantity are derived. An approach is outlined to the mathematical study of the earliest social classes, which may have been formed as a result of military, religious, or economic stratifications.

A neural model based on a generalization of a model proposed in 1938 in the first edition of the author'sMathematical Biophysics (Chicago: Univ. of Chicago Press) is described. It possesses the property that due to some endogenous or exogenous stimulus, which may be of random nature, a pathway may suddenly begin to fire spontaneously. This spontaneous firing may either gradually spread over other pathways and eventually cease, or it may remain localized within one or a few pathways and then cease. Which of the two types of events occurs depends on the values of a number of parameters. The case of spreading reminds one of Jacksonian epilepsy.

After giving a brief review of the theory of organismic sets (Bull. Math. Biophysics,29, 139–152, 1967;31, 159–198, 1969), in which the concept of relational forces, introduced earlier (Bull. Math. Biophysics,28, 283–308, 1966a) plays a fundamental role, the author discusses examples of possible different structures produced by relational forces. For biological organisms the different structures found theoretically are in general agreement with observation. For societies, which are also organismic sets as discussed in the above references, the structures can be described only in an abstract space, the nature of which is discussed. Different isomorphisms between anatomical structures, as described in ordinary Euclidean space, and the sociological structures described in an abstract space are noted, as should be expected from the theory of organismic sets.

The theory of organismic sets, developed in previous papers (Bull. Math. Biophysics,29, 139–152; 389–393; 643–647) is further generalized. To conform better with some biological and sociological facts the basic definitions are made more general. The conclusion is reached that every organismic setSo is in general the union of three disjoined subsetsSo1,So2 andSo3. Of these the subsetSo1, called the “core” is equivalent to an organismic set defined in previous publications. Its functioning is essential for the functioning ofSo. The subsetsSo2 andSo3, taken alone, are not organismic sets. The first of them is responsible for such biological or sociological functions which are not necessary for the “immediate” survival ofSo but which are important for adaptation to changing environment and are therefore essential for a “long range survival.” The second one,So3, is responsible for biological or social functions which are irrelevant for the survival ofSo. Biological and sociological examples ofSo2 andSo3 are given. In addition to the fundamental theorem established in the first of the above mentioned papers, three new conclusions are derived. One is that in organismic sets of order higher than zero not all elements are specialized. The second is that every organismic set of order higher than zero is mortal. The third is that with increasing specialization the intensities of some activities in some elements ofSo are reduced. Again the biological and sociological examples are given.

The paper outlines a further extension of a previous theory. It discusses perception of straight lines, perception of length, perception of intersections, disconnected contours; effects of associations; breaking up of a pattern into parts; some suggestions for experimental procedure.

The paper develops further some suggestions made previously (Bulletin of Mathematical Biophysics,28, 283–308, 1966) that certain biological phenomena may be more easily interpreted from a “sociological” point of view by considering the organism as a social aggregate of cells and a cell as a social aggregate of genes. In this light the problems of origin of life on earth, of aging, and of parasitism and symbiosis are discussed. The notion of social aggregates of different orders is introduced.

It is pointed out that the approximation method for diffusion problems, developed by N. Rashevsky in 1937 and successfully used since then by many authors, was in a sense a precursor of relational biology. The connection between the approximation method, relational biology, and the theory of organismic sets, developed in a series of recent papers by N. Rashevsky, is discussed. A number of conclusions known to hold experimentally, are then derived from relational considerations and some of them are applied to organismic sets.

Assuming that the initiation of cancer growth is due to a purely accidental fluctuation of some physicochemical condition, it is possible to derive the equation for the cancer incidence as a function of age. The final result depends on whether it is assumed that a single accidental fluctuation in a cell is sufficient to produce cancer, or that a finite number,k, of repetitions of the accidental fluctuation must occur in the same cell. In principle it is possible to determine from observed incidence curves the numberk. Actually, however, this cannot be done at present, because the difference of the theoretical curves for differentk's is rather small, and the available cancer statistics are not accurate enough to show such differences.

It is pointed out that the property of an organism of regenerating certain lost parts is a subproperty of the general property of reproduction. The principle of adequate design imposes an upper limit upon the total number of subproperties which an organism can possess. Therefore with increasing complexity of an organism in some direction, which by definition is the same as an increase of the total number of subproperties, the number of subproperties needed for regeneration must decrease.

If two individuals work on the production of the same object of satisfaction and if their cooperative efforts result in an increased overall productivity, then the following statement holds: if each individual tries to maximize his own satisfaction, each individual will have less of the object of satisfaction than when each tries to maximize the sum of the satisfactions of both individuals. This theorem was demonstrated previously for the case of a rather special choice of the satisfaction function. It is now shown that it holds for a much wider class of such functions. In addition to previously studied cases of methods of sharing the object of satisfaction, several new cases are investigated. An attempt is made to give a mathematical formulation for a society in which individuals contribute according to their ability and share according to their needs. It is shown that in order for such a description to be meaningful, certain requirements must be imposed on the satisfaction function. Even with those requirements fulfilled, the possibility of such a society requires the fulfillment of an inequality between certain functionals.

This paper is a continuation of some discussions published previously (Bull. Math. Biophysics,16, 317–347, 1954). A different example of a primordial graph is suggested and discussed. Then six different types of geometric transformations, similar in some respects to the one discussed in the previous paper are proposed. Another set of seven different transformations which emphasize the central role of the circulatory system is described. A theorem about the “lumping” of several points of a graph into one point is proved and possible biological aspects of it are discussed.

The theory of organismic sets (Bull. Math. Biophysics,31, 159–189, 1969) is applied to the theory of leadership in human society. The ability of making decisions, required for leadership, is a product of the activities of the cells of the cerebral cortex, which are elements of the subsetS 02 of the organismic set “man” (loc. cit.). Products of the activities of the elements of an organismic set do not need to be of a material nature. Such things as thoughts, feelings, attitudes, etc., are also products of the activitiesa 1 of the elements. An individual can makeall necessary decisions for adaptation in a changing environment, when his subsetS 02 contains as a proper subset a set {a 12 ′ ∼ ⊂S 02 of activities. It is shown that such individuals are rare. If none exist, then the one who possesses a subset {a 12 * ∼ ⊂ {a 12 ′ ∼ of higher cardinalityc m than any other individual, will be the leader. The possibility is discussed that fromN individualsN′ 〈N possess subsets {a 12 * ∼ ⊂ {a 12 ′ ∼ all of the same cardinalityc m but differing in the type of their elements, thus resulting in several leaders. It is then discussed what determines which of theN −N′ individuals will choose a particular oneN′ individuals as leader. Cooperation and competition between leaders is discussed.

It is suggested that the effect of sudden temperature changes upon the organism may be due to the variation of two factors, whose rates of changes are functions of the temperature. It is assumed that the variations of the two factors follow differential equations which are formally identical with those of the two-factor theory of excitation, and in which the temperature formally plays the role of the stimulus. By proper choice of the parameters the two factors are made equal for any constant temperature. Assuming that the greater the absolute value of the difference of the two factors, the greater is the instability of the organism, and the greater is its susceptibility to disease, relations between the suddenness of temperature changes and disease incidence are derived. The incidence of weather-affected disease for a given period of time is found to be a linear function of the mean hourly temperature variation during that period, a relation which can be in principle verified.

Following a previous study by A. Weinberg, the author investigates periodical diffusion phenomena produced in a spherical cell by a simple coupled set of chemical reactions. The general solution even for a spherical cell does not possess spherical symmetry. It is found that periodic oscillations are possible with a frequency spectrum determined by a set of “eigenvalues”. However, these oscillations are all damped even if the system of coupled reactions which is responsible for them has non-damped solutions. Therefore, although very complex and highly asymmetrical configurations of concentrations may be thus produced in the cell, none of those configurations, except some possible centrally symmetric ones, is lasting.

Let two or more individuals each possess different quantities of two objects of satisfaction. Under certain conditions they may agree to exchange part of the objects if this leads to an increase of each one's satisfaction. The equations which govern this process have been derived by G. E. Evans (1930) for the case of two individuals. A different proof of these equations is given here and the equations are generalized to the case of more than two individuals.

A number of shortcomings of the diffusion drag force theory of cell division are pointed out, especially in regard to the difficulties of interpreting mitotic phenomena. As an alternative a different theory is outlined, based on the emphasis of colloidal phenomena in the nucleus. The nucleus is assumed to contain very long molecules, which are disoriented in the resting stage but which become oriented prior and during the mitosis, forming the mitotic spindle. Such an oriented structure is of the nature of a tactoid. The molecular mechanism of the formation of the spindle is outlined and some equations are derived which suggest possible quantitative verification. The interplay of the molecular forces is assumed also to be responsible for the splitting of the chromosomes. Arrangement of chromosomes in the equatorial plate and their movement to the poles are discussed, and equations describing those phenomena are outlined.

By considering the general energy balance between the energy of muscular contraction in the extremities, kinetic energy imparted to the animal and the energy lost in the lifting of the center of gravity and in the possible muscular viscosity, an expression is derived for the speed of an animal in terms of the constant which determine its shape and structure. The relation discussed previously, namely, that the speed is proportional to the square root of the length of the extremities, turns out to be only a first rough approximation in a rather special case. Reasons are given for the number of joints actually observed in animal extremities.

Some aspects which involve the interaction of the human element and of the machine element in driving are discussed. As an example a simple equation is derived for the maximum safe speed of a car on an empty road. The parameters of the equation are partly of physiological nature, partly of mechanical nature. Another example treats in a similar manner the problem of a car passing another car moving in the same direction.

In a recent paper (Bull. Math. Biophysics, 20, 245, 1958), Robert Rosen applied topological considerations to the study of an organism as a whole. Those considerations have no direct relation to the principle of biotopological mapping. They rather represent a topological model of an organism, especially a model of the repair mechanisms which organisms possess for lost or impaired parts. In this note it is shown that the model introduced by Rosen may possibly be derived from the principle of biotopological mapping plus a proper definition of the primordial. Such a derivation may also provide a clue to a proper biotopological approach to the problem of multiplication of organisms.

Generalising the previously studied neurobiophysical schemes, consisting of excitatory and inhibitory elements, a neural mechanism is discussed, which may be involved in the perception of combinations of musical tones. Equations, giving the total value of central excitation for a combination of any two tones are derived, available observations are discussed in the light of the suggested theory.

The purpose of this paper is to explore further the possibilities of a theory of structure (form) as determined by function. The paper is preliminary in nature and emphasis is placed more on the methodological approach than on immediately verifiable conclusions. An approximate mathematical treatment of diffusion between alveoli and blood is given, taking into account the periodic variations of the concentration of gases in the lung and in the bloodstream due to the periodicity of breathing. Expressions for the blood concentration and the alveolar concentration of oxygen as functions of time are derived in terms of the frequency of breathing and of some structural geometric parameters of the lung. It is then shown what tentative conclusions can be drawn from such considerations in regard to the structure of the lung.

In his theory of psychophysical discrimination H. D. Landahl assumes that because of the symmetry of the neural circuit considered the fluctuation of the difference of excitations at the two parallel connections is the same as if the fluctuations occurred only at either one of the connections. It is shown that if the excitation at both connections fluctuates independently according to a simple exponential probability function, the difference of excitation fluctuates according to a different probability function.

The fact that the growth of every organism, as it develops through cell multiplication from the ovum, eventually ceases, makes plausible the hypothesis that each cell of an organism exerts some kind of inhibitory influence upon the growth and multiplication of every other cell through the medium of some factorj. Each cell, however, is considered to possess an amount of a “growth factor”g, the activity of which is inhibited byj. In the course of development, most cells differentiate. A number of them remains relatively undifferentiated, and those are used for repairs of natural tear and wear as well as for healing of wounds. With age either the number of those cells or their “potency” decreases, which accounts for a general decrease in the speed of wound healing. Due to statistical fluctuations, the amount of theg is not uniformly distributed amongst the cells. Some of them have accidentally a greater amount than others. As the number or potency of the undifferentiated cells decreases with age, so does their inhibitory influence upon other cells. Therefore, those cells which do have an accidental dental excess ofg-factor will begin to grow and multiply, producing neoplasms. Putting the above assumptions into mathematical form, it is possible to derive a relation which gives theprobability of cancer incidence as a function of age. The theoretical relation is compared to the rather inadequate empirical data and found in agreement with them. A relation for the speed of wound healing as a function of age is also derived, and also found in general agreement with the rather scant data. Several other conclusions are discussed and possibilities of their practical applications pointed out.

The twentieth century has witnessed a geometrization of physics, that is, a reduction of the basic concepts of physics to geometric concepts. The topological approach to biology, recently proposed and to some extent developed by the author, is a small step in the direction of geometrization of biology, but is unable to achieve the main purpose of such a geometrization of biology, namely, the reduction to geometric concepts of such purely biological concepts as ingestion, digestion, assimilation, etc. To achieve this purpose we must find geometric structures or spaces, in which different geometric properties stand to each other in the same formal logical relation, as the different concepts of biology stand to each other. If this were possible, then a set of geometric theorems could be “translated” by an appropriate “glossary” into a set of biological laws. While not offering a solution to this problem, the present paper illustrates the possibility of such an approach on several examples. Certain new types of topological spaces are introduced, which are used for illustration purposes only. It is shown, however, how from a theorem about such spaces a verifiable biological prediction could be made, if these spaces were to be taken seriously. A possible application to biology of E. Artin's theory of braids is outlined.

It is shown how the application of the principle of maximum energy transformation, discussed in previous papers, leads to expressions which give the probability of occurrence of a pseudopod of a given size and duration in function of other physical constants of the amoeba.

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.

In a preceding paper (Rashevsky, 1969. “Outline of a Unified Approach to Physics, Biology and Sociology.”Bulletin of Mathematical Biophysics,31, 159–198) certain isomorphisms between biological and social systems on the one hand and physical systems on the other were studied. The notion or relational forces, of which ordinary physical forces are a particular case, was introduced. In the present paper an attempt is made to establish analogies between stable equilibria in physical systems, equilibria due to physical forces, and stable equilibria in biological and social systems which are due to purely relational forces. The notion of relational forces causing multiple equilibria similar to multiple equilibria in some physical systems is studied, and it is outlined how this notion may possibly help the understanding of such phenomena as polymorphism, metamorphosis and the existence of rudimentary organs or rudimentary functions.

A neural mechanism is discussed which explains certain states of anxieties, in which an individual is worried about things which may be possible, but are exceedingly improbable. Certain quantitative relations are derived.Mutatis mutandis similar considerations apply to some states of elations. An application to certain types of paranoia is discussed.

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.

On the basis of previously proposed mathematical models of social behavior, the present note investigates the possibility of the control of behavior remaining permanently in the hands of one class, if this class possesses sufficient means for influencing mass behavior. The conclusion is reached that, with the assumptions made, if the behavior imposed by the controlling class leads to sufficiently strong dissatisfaction, the control will pass to another class, no matter how strong the controlling power of the first.

In the theory of organismic sets (Bull. Math. Biophysics,31, 159–198, 1969) we considered organisms as sets endowed with certain “activities,” the latter’s resulting in a set of “products.” Those products may be of a material nature, like a hormone secreted by a cell, or of a non-material nature, like a feeling or an attitude. In the present paper aggressiveness and submissiveness are considered as such non-material products of the activities of the brain cells. A general description of aggressiveness and submissiveness is given in terms of organismic sets. Cycles in “peck order” are thus naturally explained.

In previous papers cases were considered in which a visual pattern consists of a relatively small number of relatively strongly excited elements. Those cases are of interest in the theory of visual perception and aesthetic rating of artificial man made patterns. By an extension of the theory so as to take into account the finite thresholds of the inhibiting fibers a theory of aesthetic ratings of pattern consisting of a very large number of elements is outlined. This type of theory is applicable to natural patterns, such as landscapes.