Some considerations on relational equilibria

A previous study (Bull. Math. Biophysics,31, 417–427, 1969) on the definitions of stability of equilibria in organismic sets determined byQ relations is continued. An attempt is made to bring this definition into a form as similar as possible to that used in physical systems determined byF-relations. With examples taken from physics, biology and sociology, it is shown that a definition of equilibria forQ-relational systems similar to the definitions used in physics can be obtained, provided the concept of stable or unstable structures of a system determined byQ-relations is considered in a probabilistic manner. This offers an illustration of “fuzzy categories,” a notion introduced by I. Bąianu and M. Marinescu (Bull. Math. Biophysics,30, 625–635, 1968), in their paper on organismic supercategories, which is designed to provide a mathematical formalism for Rashevsky's theory of Organismic Sets (Bull. Math. Biophysics,29, 389–393, 1967;30, 163–174, 1968;31, 159–198, 1969). A suggestion is made for a method of mapping the abstract discrete space ofQ-relations on a continuum of variables ofF-relations. Problems of polymorphism and metamorphosis, both in biological and social organisms, are discussed in the light of the theory.     

Sporadic sustained nonperiodic oscillations in the activities of organismic sets

In combining the author's theories of organismic sets (Rashevsky,Bull. Math. Biophysics,31, 159–198, 1969a) and Robert Rosen's theory of (M, R)-systems (Bull. Math. Biophysics,20, 245–265, 1958), a conclusion is reached that the number of either normal or pathological phenomena in organismic sets may occur. Those phenomena are characterized by occurring spontaneously once in a while but are not exactly periodic. Some epilepsies are an example of such pathological phenomena in the brain.     

A note on the development of organismic sets

It is suggested that the development of organismic sets is governed not by the maximalization of the integral survival value, as suggested previously (Bull. Math. Biophysics,28, 283–308, 1966;29, 139–152, 1967;30, 163–174, 1968), but by maximizing the number of new relations which appear as an organismic set develops.     

Organismic supercategores: II. On multistable systems

The representation of biological systems in terms of organismic supercategories, introduced in previous papers (Bull. Math. Biophysics,30, 625–636;31, 59–70) is further discussed. To state more clearly this representation some new definitions are introduced. Also, some necessary changes in axiomatics are made. The conclusion is reached that any organismic supercategory has at least one superpushout, and this expresses the fact that biological systems are multistable. This way a connection between some results of Rashevsky’s theory of organismic sets and our results becomes obvious.     

Organismic sets and biological epimorphism

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.     

Organismic supercategories and qualitative dynamics of systems

The representation of biological systems by means of organismic supercategories, developed in previous papers (Bull. Math. Biophysics,30, 625–636;31, 59–71;32, 539–561), is further discussed. The different approaches to relational biology, developed by Rashevsky, Rosen and by Băianu and Marinescu, are compared with Qualitative Dynamics of Systems which was initiated by Henri Poincaré (1881). On the basis of this comparison some concrete result concerning dynamics of genetic system, development, fertilization, regeneration, analogies, and oncogenesis are derived.     

Contributions to relational biology

The principle of biotopological mapping (Rashevsky, 1954,Bull. Math. Biophysics,16, 317–48) is given a generalized formulation, as the principle of relational epimorphism in biology. The connection between this principle and Robert Rosen’s representation of organisms by means of categories (1958,Bull. Math. Biophysics,20, 317–41) is studied. Rosen’s theory of (M,R)-systems, (1958,Bull. Math. Biophysics,20, 245–60) is generalized by dropping the assumption that only terminalM _i components are sending inputs into theR _i components. It is shown that, if the primordial organism is an (M,R)-system, then the higher organisms, obtained by a construction well discussed previously (1958,Bull. Math. Biophysics,20, 71–93), are also (M,R)-systems. Several theorems about such derived (M,R)-systems are demonstrated. It is shown that Rosen’s concept of an organism as a set of mappings throws light on phenomena of synesthesia and also leads to the conclusion that Gestalt phenomena must occur not only in the fields of visual and auditory perception but in perceptions of any modality.     

On relations between sets: III

The theory of relations between sets, proposed and outlined in previous publications (Bull. Math. Biophysics,23, 233–235, 1961;28, 117–124, 1966;28, 309–313, 1966), is tentatively expanded and generalized with a view to biological applications.     

A set-theoretical approach to biology

In a preceding paper (Bull. Math. Biophysics 20, 71–93, 1958) the principle of biotopological mapping was formulated in terms of a continuous mapping of an abstract space, made from the set of biological properties which characterize the organism, by an appropriate definition of neighborhoods. In this paper it is shown that we may consider directly the mappings of the different sets of properties which characterize different organisms without taking recourse to abstract spaces. All the verificable conclusions made in the preceding paper remain valid. A serious difficulty mentioned previously is, however, avoided and the possibility of more general predictions is established.     

A note on topological biology

In line with a recent suggestion by the author (Bull. Math. Biophysics,20, 267–73, September, 1958) that not only does the organism as a whole map on the primordial, but that each organ can also be thus mapped, it is shown that the previously introduced abstract spaces, which represent an organism, contain subspaces which map continuously on the space of the primordial. Several theorems about those subspaces are proven.     

On relations between sets: II

The notion of relations between sets, defined in a previous publication (Bull. Math. Biophysics,23, 233–235, 1961) is generalized and some biological examples are given. A generalization ton-ary relation is suggested.     

Topological biology: A note on Rashevsky's transformation T.

In the bio-topological transformation between graphs denoted by (T ⁽¹⁾ X) N. Rashevsky (Bull. Math. Biophysics,18, 173–88, 1956) considers the number of fundamental sets which (a) have only one specialized point as source (and no other sources), (b) have no points in common (are “disjoined”); he proves that this number is an invariant of the transformation. In this note we show that Rashevsky's Theorem can be extended as follows:The number of fundamental sets of the first category is an invariant of the transformation. We must, however, count the subsidiary points of the transformed graph as specialized points. We recall that fundamental sets of the first category are those whose sources consist of specialized points only (Trucco,Bull. Math. Biophysics,18, 65–85, 1956). But in this modified version of the Theorem the fundamental sets may have more than one source and need not be disjoined.     

Outline of a unified approach to physics,biology and sociology

The theory of organismic sets, introduced by N. Rashevsky (Bulletin of Mathematical Biophysics,29, 139–152, 1967;30, 163–174, 1968), is developed further. As has been pointed out, a society is a set of individuals plus the products of their activities, which result in their interactions. A multicellular organism is a set of cells plus the products of their activities, while a unicellular organism is a set of genes plus the products of their activities. It is now pointed out that a physical system is a set of elementary particles plus the product of their activities, such as transitions from one energy level to another. Therefore physical, biological and sociological phenomena can be considered from a unified set-theoretical point of view. The notion of a “world set” is introduced. It consists of the union of physical and of organismic sets. In physical sets the formation of different structure is governed preponderantly by analytical functions, which are special type of relations. In organismic sets, which represent biological organisms and societies, the formation of various structures is governed preponderantly by requirements that some relations, which are not functions, be satisfied. This is called the postulate of relational forces. Inasmuch as every function is a relation (F-relation) but not every relation is a function (Q-relation), it has been shown previously (Rashevsky,Bulletin of Mathematical Biophysics,29, 643–648, 1967) that the physical forces are only a special kind of relational force and that, therefore, the postulate of relational forces applies equally to physics, biology and sociology. By developing the earlier theory of organismic sets, we deduce the following conclusions: 1) A cell in which the genes are completely specialized, as is implied by the “one gene—one enzyme” principle, cannot be formed spontaneously. 2) By introducing the notion of organismic sets of different orders so that the elements of an organismic set of ordern are themselves organismic sets of order (n−1), we prove that in multicellular organisms no cell can be specialized completely; it performs, in addition to its special functions, also a number of others performed by other cells. 3) A differentiated multicellular organism cannot form spontaneously. It can only develop from simpler, less differentiated organisms. The same holds about societies. Highly specialized contemporary societies cannot appear spontaneously; they gradually develop from primitive, non-specialized societies. 4) In a multicellular organism a specialization of a cell is practically irreversible. 5) Every organismic set of ordern>1, that is, a multicellular organism as well as a society, is mortal. Civilizations die, and others may come in their place. 6) Barring special inhibitory conditions, all organisms multiply. 7) In cells there must exist specially-regulatory genes besides the so-called structural genes. 8) In basically identically-built organisms, but which are built from different material (proteins), a substitution of a part of one organism for the homologous part of another impairs the normal functioning (protein specificity of different species). 9) Even unicellular organisms show sexual differentiation and polarization. 10) Symbiotic and parasitic phenomena are included in the theory of organismic sets. Finally some general speculations are made in regard to the possibility of discovering laws of physics by pure mathematical reasoning, something in which Einstein has expressed explicit faith. From the above theory, such a thing appears to be possible. Also the idea of Poincaré, that the laws of physics as we perceive them are largely due to our psychobiological structure, is discussed.     

Two definitions of stability of equilibria

In classical physics the stability of an equilibrium requires that any, even infinitesimal, displacement from the configuration of equilibrium results in forces which tend to restore the original equilibrium configuration. In case of several stable equilibrium configurations, the height of the threshold, which must be exceeded by the deviarion from the stable equilibrium in order to bring the configuration into another stable equilibrium is taken as a measure of stability of the first configuration. In quantum mechanics, and in the recent work of I. Baianu, S. Comorosan and M. Marinescu (Bull. Math. Biophysics,30, 625–635, 1968;31, 59–70, 1969;32, 539–561, 1970) on organismic supercategories, preference is given to take, as ameasure of the degree of stability of a configuration, or of a “state”, the length of time during which the system remains in that configuration. It is shown that under rather general conditions the two criteria are equivalent.     

On multipole theory in electrocardiography

This paper continues a comparison of the Taylor series and spherical harmonic forms of multipole representations initiated by Yeh (Bull. Math. Biophysics,24, 197–207, 1962). It is shown that while transformations from Taylor series form into spherical harmonic form is always possible, the inverse cannot be accomplished as suggested by Yeh; corrected transformation equations are given. It is also shown that direct measurement of Taylor coefficients, as outlined in Yeh, Martinek, and de Beaumont (Bull. Math. Biophysics,20, 203–216, 1958), is actually not possible. Accordingly, only the spherical harmonic coefficients can be determined by measurement of surface potentials, as in electrocardiography.     

Molecular set theory: II. An aspect of the biomathematical theory of sets

In an earlier paper (Molecular Set Theory: I.Bull. Math. Biophysics,22, 285–307, 1960) the author proposed a “Molecular Set Theory” as a formal mathematical meta-theoretic system for representing complex reactions not only of biological interest, but also of general chemical interest. The present paper is a refinement and extension of the earlier work along more formal algebraic lines. For example the beginnings of an algebra of molecular transformations is presented. It also emphasizes that this development, together with the genetical set theory of Woodger's and Rashevsky's set-theoretic contributions to Relational Biology, points to the existence of a biomathematical theory of sets which is not deducible from the general mathematical, abstract theory of sets.     

Probability distributions associated with distinct hits on targets

Some probability distributions connected with distinct hits on targets, using two different firing schemes, are developed. It is assumed that any shot has a probabilityp, not necessarily unity, of hitting the target at which it was aimed. The development uses a well-known expression for the probability that exactlyt ofN possible events occur simultaneously. Some of the formulae developed here include as special cases the probabilities derived separately and by more complicated arguments in papers by N. Rashevsky. (Bull. Math. Biophysics,17, 45–50, 1955) and A. Rapoport (Bull. Math. Biophysics,13, 133–38, 1951).     

A neural model for conditioned avoidance learning

A mathematical model for learning of a conditioned avoidance behavior is presented. An identification of the net excitation of a neural model (Rashevsky, N., 1960.Mathematical Biophysics. Vol. II. New York: Dover Publications, Inc.) with the instantaneous probability of response is introduced and its usefulness in discussing block-trial learning performances in the conditioned avoidance situation is outlined for normal and brain-operated animals, using experimental data collected by the author. Later, the model is applied to consecutive trial learning and connection is made with the approach of H. D. Landahl (1964. “An Avoidance Learning Situation. A Neural Net Model.”Bull. Math. Biophysics,26, 83–89; and 1965, “A Neural Net Model for Escape Learning.”Bull. Math. Biophysics,27, Special Edition, 317–328) wherein lie further data with which the model can be compared.     

Analysis of tracer experiments: IV. The kinetics of generalN compartment systems

The methods of C. W. Sheppard and A. S. Householder (Jour. App. Physcis,22, 510–20, 1951), H. D. Landahl (Bull. Math. Biophysics,16, 151–54, 1954) and H. E. Hart (Bull. Math. Biophysics,17, 87–94, 1955;ibid.,19, 61–72, 1957;ibid.,20, 281–87, 1958) are employed in studying the kinetics of generalN compartment systems. It is shown that the nature of the transfer processes occurring in fluid flow systems and the chemical processes occurring in quadratic systems and in catalyzed quadratic systems can in principle be completely determined for all polynomial dependencies. Systems involving three-body and higher-order interactions can be completely solved, however, only if supplementary information is available. Research supported by the Atomic Energy Commission, Contract AT (30-1)-1551.     

A derivation of a rote learning curve from the total uncertainty of a task

The derivation of H. D. Landahl’s learning curve (1941,Bull. Math. Biophysics,3, 71–77) from a single information-theoretical assumption obtained previously (Rapoport, 1956,Bull. Math. Biophysics,18, 317–21) is extended to obtain the entire family of such curves with the number of stimuliM (to each of which one ofN responses is to be associated) as a parameter. No additional assumptions are required. The entire family thus appears as a function of a single free parameter,k, all other parameters being experimentally determined. The theory is compared with a set of experiments involving the learning of artificial languages. An alternative quasi-neurological model leading to the same equation is offered.