Contribution to the probabilistic theory of neural nets: IV. Various models for inhibition

Input-output formulas are derived for a neuron upon which converge single axones of two other neurons, which are subjected to a Poisson shower, where a number of different assumptions are made concerning the mechanism of inhibition. In one assumption so-called “bilateral pre-inhibition” is considered. That is to say, both neuronsN ₁ andN ₂ may exciteN ₃, but if the stimulus of one of them follows within a certain interval σ of the other, the second stimulus is not effective. This model is essentially no different from that involving two excitatory neurons acting upon a neuron having a refractory period. Another mechanism considered involves so-called “pre-and-post” inhibition, in which if two stimuli fromN ₁ andN ₂ fall within σ,both are ineffective. This case being mathematically much more involved than the preceding, an approximation method is used for deriving the input-output formula. Previous papers of this series are denoted by I, II, and III in this paper.     

Physics,biology and sociology: II. Suggestion for a synthesis

In continuation of previous studies (Bull. Math. Biophysics,28, 283–308; 655–661, 1966;29, 139–152, 1967) it is shown that the difference between the “metric” aspects of physics and the “relational” aspects of biological and social sciences disappear by accepting the broader definition of “relation”, such as that given in mathematics and logic. A conceptual superstructure then becomes possible from which all three branches of knowledge may be derived, though none of them can be derived from the others.     

Abstract mathematical molecular biology: II some relational consequences of the “one gene-one enzyme” hypothesis

Following the program outlined in a previous paper (Bull. Math. Biophysics,23, 237–260, 1961), a further abstract study is made of some simple relational systems which possess some properties of living organisms. It is shown that the “one gene-one enzyme” hypothesis leads to the conclusion that either all genes are built of the same chemical building blocks, or that at least all genes have a number of building blocks in common. A consistent relational application of the “one gene-one enzyme” hypothesis leads moreover to the conclusion that replication is not an inherent property of a gene. Rather there must be a set of enzymes which “copy” the genes. The number of enzymes in this set must be less than the number of genes and therefore the activity of those “copying” enzymes cannot be absolutely specific.     

Basidiocarp development inCoprinus macrorhizus

The dikaryon #5026 (A2B2)+#5132 (A7B7) of the basidiomyceteCoprinus macrorhizus was treated with ultraviolet light (UV),N-methyl-N′-nitro-N-nitrosoguanidine (NG), and 5-bromouracil (BU) in order to induced developmental variations in fruiting. Among a total of 11,383 isolates, 742 were homokaryons. Fruiting test was made with the remaining 10,641 dikaryotic isolates, of which 1,594 showed abnormality in basidiocarp development. Both UV and NG were very efficient in inducing such variations, but BU was not. The induced variations were classified into seven basic types as follows: 1) “knotless”, in which hyphal knots, the first sign of fruiting, do not differentiate; 2) “primordiumless,” in which hyphal knots are formed, but no further development occurs; 3) “maturationles”, in which primordia are formed, but the maturation of primordia into adult fruit bodies does not occur; 4) “elongationless”, in which the elongation of stipe is blocked, and a basidiocarp with a very short stipe is produced; 5) “expansionless,” in which pilei do not expand normally; 6) “sporeless,” in which the formation of basidiospores is blocked and albinic pilei bearing none or only a small amount of spores are produced; and 7) “autolysisless,” in which the autolysis of pilei does not appear to occur. It has been noticed that the four steps of maturation, i.e. stipe elongation, pileus expansion, spore formation, and perhaps pileus autolysis, can proceed independently. Compound types of variations such as “elongation-expansionless,” “elongation-sporeless,” “expansion-sporeless,” and “elongation-expansion-sporeless” were also induced. UV treatment induced maturationless at the highest rate, while NG treatment sporeless. Contributions from the Division of Plant Morphology, Department of Biology, Faculty of Science, Okayama University. No. 117.     

Further studies of some aspects of a neurobiophysical model of schizophrenia

The neurobiophysical model of schizophrenia discussed previously (Bull. Math. Biophysics,26, 167–185, 1964;27, 21–26, 1965) is generalized further, to include catatonic and stuporous states. It is concluded that the development of schizophrenia will proceed through different stages of catatonic and non-catatonic states, depending on parameters which characterize on one hand the general inhibition of the individual, on the other hand what may be called his “stability.” Suggestions for possible clinical verifications of the conclusions are made.     

On the reversibility of environmentally induced alterations in abstract biological systems

The environmentally induced alterations in structure of (M, ℜ) which were described previously (R. Rosen,Bull. Math. Biophysics,23, 165–171, 1961) are examined from the standpoint of determining under what circumstances they can be reversed by further environmental interactions. For simplicity we consider only the case of (M, ℜ)-systems possessing one “metabolic” and one “genetic” component. In the case of environmentally induced alteration of the “metabolic” component alone, a necessary and sufficient condition is given for the reversibility of the alteration. In the case of alteration of the “genetic” component, the situation becomes more complex; several partial results are given, but a full analysis is not available at this time. Some possible biological implications of this analysis are discussed. This research was supported by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command, under Contract no. AF-49(638)-917 and Grant no. AF-AFOSR-9-63.     

Some remarks on the mathematical bio-sociology of the relativity of injustice

The author's theory of the adoption of certain types of behavior patterns (Rashevsky, N., 1957, “Contributions to the Theory Initiative Behavior”.Bull. Maths. Biophysics,19, 91–119; 1968,Looking at History through Mathematics, Cambridge, Massachusetts: M.I.T. Press) consisting of elementary behaviors for each of which there is an opposite one and the two are mutually exclusive, is applied to describe the changes in the general type of behavior of a society. The elementary acts of which the whole problem consists may be either overt activities or beliefs or opinions. The general behavior patternsadopted by the society are considered as the “proper” or “just” ones. Any deviation from it in either one or more of the component elementary behaviors is considered as “unjust” and is subject to some punitive action. The total number of possible mutually exclusive behavior patterns is very large but finite. Within this very large range of possible patterns, we find that this notion of justice is relative, because changes from any behavior pattern to any other may occur. It is further shown that the amount of punishment for the deviation from the accepted pattern in order to be effective as well as efficient must be applied in different ways to different individuals even for the same transgression.     

The entropy of a continuous distribution

C. Shannon's definition (Bell System Technical Journal,27, 379–423, 1948) of the entropy of a continuous distribution is dimensionally incorrect and does not have the same significance as the corresponding definition in the discrete case. A new definition is proposed: this modified entropy is more like the entropy of a discrete distribution in one way, in another more like Shannon's “transmission rate.” The ideas are illustrated by reference to Wright's study of the hereditary influence on the coat pattern of the guinea pig.     

Traditional Knowledge and Management of <Emphasis Type="Italic">Feijoa (Acca sellowiana)</Emphasis> in Southern Brazil

Traditional Knowledge and Management of Feijoa (Acca sellowiana) in Southern Brazil. This paper investigates traditional knowledge of the use and management of Acca sellowiana in southern Brazil. Fifty-six informants from three rural communities were assigned to one of four subgroups (“maintainers,” “managers,” “cultivators,” or “users”) based on their responses regarding management and use of A. sellowiana. Traditional knowledge related to use of this species is widespread among rural residents, but traditional knowledge related to management is fragmented depending on whether one uses, manages, or cultivates the species. Knowledge held in rural communities suggests that A. sellowiana could play an expanded role in local economies as well as biodiversity conservation. We suggest that participatory research could stimulate greater local use as well as on-farm conservation of A. sellowiana.     

Heterogeneity of heterochromatin in six species ofCtenomys (Rodentia: Octodontoidea: Ctenomyidae) from Argentina revealed by a combined analysis of C- and RE-banding

Exceptional chromosomal variability makesCtenomys an excellent model for evolutionary cytogenetic analysis. Six species belonging to three evolutionary lineages were studied by means of restriction endonuclease and C-chromosome banding. The resulting banding patterns were used for comparative analysis of heterochromatin distribution on chromosomes. This combined analysis allowed intra- and inter-specific heterochromatin variability to be detected, groups of species belonging to different lineages to be characterized, and phylogenetic relationships hypothesized from other data to be supported. The “ancestral group”,Ctenomys pundti andC. talarum, share three types of heterochromatin, the most abundant of which was also found in C. aff.C. opimus, suggesting that the latter species also belongs to the “ancestral group”. Additionally, within the subspeciesC. t. talarum, putative chromosomal rearrangements distinguishing two of the three chromosomal races were identified. Two species belong to an “eastern lineage”,C. osvaldoreigi andC. rosendopascuali, and share only one type of heterochromatin homogeneously distributed across their karyotypes.C. latro, the only analyzed species from the “chacoan” lineage, showed three types of heterochromatin, one of them being that which characterizes the “eastern lineage”.C. aff.C. opimus, because of its low heterochromatin content, is the most primitive karyotype of the genus yet described. The heterochromatin variability showed by these species, reflecting the evolutionary divergence toward different heterochromatin types, may have diverged since the origin of the genus. Heterochromatin amplification is proposed as a trend withinCtenomys, occurring independently of chromosomal change in diploid numbers.     

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.     

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

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

Role of secondary metabolites as defense chemicals against ice-ice disease bacteria in biofouler at carrageenophyte farms

Carrageenophyte farming is an expanding economical activity in North Borneo Island, Malaysia. During routine monitoring of “ice-ice” disease and epiphyte outbreak at commercial farms, it was apparent that culture lines were heavily (60–80%) infested with biofoulers, particularly Acanthophora spp. and Laurencia majuscula. However, only L. majuscula showed dominance and flourished even during “ice-ice” disease outbreak. Presence of chemical defense against seaweed pathogens was investigated in two populations of L. majuscula collected from three major carrageenophyte farms in two districts; (A) Lohok Butun, Selakan Island, and Bum-Bum Island, in Semporna district, and (B) Telutuh, Carrington Reef, and Balambangan Island, in Kudat district. The first population contained elatol (1), and iso-obtusol (2), and, second population contained (Z)-10,15-dibromo-9-hydroxy-chamigra-1,3(15),7(14)-triene (3) and (E)-10-15-dibromo-9-hydroxy-chamigra-1,3(15),7(14)-triene (4), as their antibacterial metabolites. All four metabolites showed highly selective inhibition against “ice-ice” disease bacteria compared to human pathogens at 30 μg disk⁻¹. In addition, seasonal variation of these compounds at two representative farms (Selakan Island [P-1] and Balambangan Island [P-2]) revealed a 120–170% increase in concentration during “ice-ice” disease outbreak. Microscopy of fresh specimens showed the presence of corps en cerise, which is the synthesis and storage site of halogenated metabolites at superficial cortical cells, branch tips, and trichoblasts. This suggests the importance of these metabolites as defense chemicals against “ice-ice” disease bacteria in L. majuscula that grows on seaweed culture lines.     

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 the equilibrium distribution of population in space

Spatial equilibrium distributions of population are derived from the spatial distribution of net rates of reproduction, and from a relationship between migratory flow and gradients of population density and of locational “attractiveness.” Conditions are discussed for which population approaches a uniform spatial density. Under certain conditions a particularly simple statement of the equilibrium conditions is possible in terms of the “potential of population,” a concept introduced by demographers (J. Q. Stewart,Geographical Review,37, 46–85, 1947) to measure the proximity of a point to people. This paper was first written at the Center for Advanced Study in the Behavioral Sciences.     

Applications of matrix algebra to communication nets

A “generic” problem amenable to matrix algebraic treatment is outlined. Several examples are given and one, a communication system, is studied in some detail. A typical structure matrix is used to describe the channels of communication and a “status” matrix is used to describe the distribution of information in the system at any time. A theorem is proved relating the status matrix at any timet to thetth power of the structure matrix. The elements of the communication system are interpreted as individuals who can send messages to each other. For the individuals attempting to solve a “group problem” certain relations are derived between the structure and status matrices and time of solution. The structure of the communication system is permitted to vary with time. A general theorem is proved relating the status matrix to the matrix product of the series of structure matrices representing the changing structure of the system. Some suggestions are made for further generalizations. In particular, it is suggested that so-called “higher order” information transmission can be similarly treated.     

Contributions to topological biology: Some considerations on the primordial graph and on some possible transformations

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.     

Models for reserpine effects on normal and brain-operated animals on a continuous reinforcement bar pressing schedule

Mathematical models are presented for the relationship between changes in the bar pressing rate of albino rats on a continuous reinforcement schedule and alterations of the brain amines norepinephrine and serotonin induced by electrolytic lesions in the hypothalamus and by the drug, reserpine. The discussion utilizes central concepts from some of the more prominent hypotheses on the relation of the amines to behavior and for the potentiation effects of the drug by the lesion (A. Carlsson, 1964. “Functional Significance of Drug-induced Changes in Brain Monoamine Levels.” InBiogenic Amines, eds. H. E. Himwich and W. A. Himwich, New York: Elsevier, 9–27; B. B. Brodie and E. Costa, 1962. “Some Current Views on Brain Monoamines.”Psychopharm. Service Centre Bull.,2, 1–25.) Laboratory data (F. Grabarits and J. Harvey, 1966. “The Effects of Reserpine on Behavior and on Brain Concentrations of Serotonin and Norepinephrine in Control Rats and Rats with Hypothalamic Lesions.”J. Pharm.,153, 403–411) are interpreted, and suggestions for further experimental and theoretical work are made.     

HLA-C “blank” alleles express class I gene products. Biochemical analysis of four different HLA-C “blank” polypeptides

Monoclonal antibodies (mAbs) W6/32, HC10, and 4E were used to precipitate class I antigens from 21 selected individuals with at least oneHLA-C “blank” allele. In 19 of these individuals, characteristicHLA-C banding patterns which could be precipitated by all three HLA class I mAbs were observed on one-dimensional isoelectric focusing gels-obviously the gene products ofHLA-C “blank”. At least four allelic HLA-C “blank” gene products with different isoelectric points could be discerned. All of them segregated withHLA-C “blank” haplotypes in informative families; two of them were associated withHLA-B51, one withHLA-B38, and one withHLA-B18. Reactivity of the HLA-C “blank” heavy chains with mAb W6/32 indicates that they are able to associated with beta-2 microglobulin, and hence are most probably expressed at the cell surface.