A logical calculus of the ideas immanent in nervous activity

Because of the “all-or-none” character of nervous activity, neural events and the relations among them can be treated by means of propositional logic. It is found that the behavior of every net can be described in these terms, with the addition of more complicated logical means for nets containing circles; and that for any logical expression satisfying certain conditions, one can find a net behaving in the fashion it describes. It is shown that many particular choices among possible neurophysiological assumptions are equivalent, in the sense that for every net behaving under one assumption, there exists another net which behaves under the other and gives the same results, although perhaps not in the same time. Various applications of the calculus are discussed.     

Logical identification of all steady states: The concept of feedback loop characteristic states

Biological regulatory systems can be described in terms of non-linear differential equations or in logical terms (using an “infinitely non-linear” approximation). Until recently, only part of the steady states of a system could be identified on logical grounds. The reason was that steady states frequently have one or more variable located on a threshold (see below); those steady states were not detected because so far no logical status was assigned to threshold values. This is why we introduced logical scales with values 0,¹θ, 1²θ, 2, ..., in which¹θ,²θ, ... are the logical values assigned to the successive thresholds of the scale. We thus have, in addition to the regular logical states,singular states in which one or more variables is located on a threshold. This permits identifyingall the steady states on logical grounds. It was noticed that each feedback loop (or reunion of disjointed loops) can be characterized by a logical state located at the thresholds at which the variables of the loop operate. This led to the concept ofloop-characteristic state, which, as we will see, enormously simplifies the analysis.The core of this paper is a formal demonstration that among the singular states of a system, only loop-characteristic states can be steady. Reciprocally, given a loop-characteristic state, there are parameter values for which this state is steady; in this case, the loop is effective (i.e. it generates multistationarity if it is a positive loop, homeostasis if it is a negative loop). This not only results in the above-mentioned radical simplification of the identification of the steady states, but in an entirely new view of the relation between feedback loops and steady states.     

The inconsistencies of present-day mathematical-physical methods pertaining to current generators in volume conductors used in electrocardiography

The present-day practices of electrocardiography and vectorardiography are based upon the theory that the surface potential differences can be assumed to be due to a single dipole inside the body. It is shown in this paper that a dipole cannot account for all the surface potentials due to realistic current generators, and hence the determination of the current generator from surface potential measurements based upon such a theory will lead to inconsistent representations of the heart for one and the same subject. To demonstrate this point two eccentric dipoles of different strengths and locations representing two muscle fibers are taken to be the current generator in a homogeneous spherical conductor. The exact surface potentials are then expressed by means of the “interior sphere theorem” of the authors. With these expressions the magnitude, direction, and location of the resultant dipole are determined by the method of D. Gabor and C. V. Nelson (J. App. Physics,25, 413–16, 1954). The surface potentials due to this resultant dipole are again exactly expressed by means of the “interior sphere theorem” and compared with those due to the eccentric dipoles assumed. It can be seen that the differences can be considerable. It is suggested that the multipole model of the authors (Bull. Math. Biophysics,20, 203–16, 1958) be used as a more accurate and the only unique representation of the heart. This investigation was supported by the National Heart Institute under a research grant H-2263(c).     

Variation in the outcome of population interactions: bifurcations and catastrophes

 The nature of the association between two species may vary depending on population abundances, age or size of individuals, or environmental conditions. Interactions may switch between beneficial and detrimental depending on the net balance of costs and benefits involved for each species. We study the repercussion of the ecological setting on the outcomes of conditional or variable interactions by means of a model that incorporates density-dependent interaction coefficients; that is, interaction α-functions. These characterize the responsiveness and sensitivity of the association to changes in partner's abundance, and can take positive and negative values. Variable outcomes – and transitions between them – are categorized as homeo- or allo-environmental, that is, occurring under the same ecological setting, or not, respectively. Bifurcation analyses show that these dynamics are moulded by ecological factors that are: intrinsic to the nature of the association (concerning the sensitivity of the interaction), and extrinsic to the association itself (the quality of the environment referred to each species alone). The influence of these factors may be conflicting; consequently, the dynamics involve catastrophic events. In a facultative variable association, stable coexistence is expected when environmental conditions are adverse; otherwise, the exclusion of one species is the likely outcome. Remarkable situations as the switching of victim-exploiter roles illustrate the theoretical perspective. Received: 15 December 2001 / Revised version: 18 November 2002 / Published online: 28 February 2003 Key words or phrases: Variable population interactions – Conditional interactions – Costs and benefits – Density dependent interaction coefficient – Hysteresis – Symbiosis – Mutualism – Parasitism     

Logic in a Dynamic Brain

The ability of the human brain to carry out logical reasoning can be interpreted, in general, as a by-product of adaptive capacities of complex neural networks. Thus, we seek to base abstract logical operations in the general properties of neural networks designed as learning modules. We show that logical operations executable by McCulloch–Pitts binary networks can also be programmed in analog neural networks built with associative memory modules that process inputs as logical gates. These modules can interact among themselves to generate dynamical systems that extend the repertoire of logical operations. We demonstrate how the operations of the exclusive-OR or the implication appear as outputs of these interacting modules. In particular, we provide a model of the exclusive-OR that succeeds in evaluating an odd number of options (the exclusive-OR of classical logic fails in his case), thus paving the way for a more reasonable biological model of this important logical operator. We propose that a brain trained to compute can associate a complex logical operation to an orderly structured but temporary contingent episode by establishing a codified association among memory modules. This explanation offers an interpretation of complex logical processes (eventually learned) as associations of contingent events in memorized episodes. We suggest, as an example, a cognitive model that describes these “logical episodes”.     

A canonical form for neural nets without circles

By “neural net” will be meant “neural net without circles.” Every neural net effects a transformation from inputs (i.e., firing patterns of the input neurons) to outputs (firing patterns of the output neurons). Two neural nets will be calledequivalent if they effect the same transformation from inputs to outputs. A canonical form is found for neural nets with respect to equivalence; i.e., a class of neural nets is defined, no two of which are equivalent, and which contains a neural net equivalent to any given neural net. This research was supported by the U.S. Air Force under Contract AF 49(638)-414 monitored by the Air Force Office of Scientific Research.     

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.     

Micropropagation of <Emphasis Type="BoldItalic">Bambusa balcooa</Emphasis> Roxb. through axillary shoot proliferation

Bambusa balcooa is one of the most commercially important bamboo species. Regeneration of this species by sexual means is impossible because no seeds are set after flowering. Vegetative propagation is hindered due to bulky propagules, low rooting ability of the culm and branch cuttings, and seasonal specificity. This makes in vitro-based methods of regeneration important. This paper describes an efficient micropropagation protocol for multiplication of B. balcooa from nodal explants. Nodal segments were surface sterilized with 0.1% mercuric chloride for 10 min, and cultured on Murashige and Skoog (MS) medium supplemented with 4.4 μM 6-benzylaminopurine (BAP), 2.32 μM kinetin (Kn), and gelled with 0.2% w/v gelrite. Eighty-five percent of explants could be established in vitro with 90% of these achieving bud break. In vitro-formed shoots were successfully multiplied in MS liquid medium supplemented with 6.6 μM BAP, 2.32 μM Kn, 2.5% v/v coconut water, and 100 mg l⁻¹ myo-inositol. Subculturing shoots every 3 wk yielded a consistent proliferation rate of 4.11-fold without decline in vigor. Shoot clusters, containing 5 to 8 shoots, were rooted with 87.5% success in 1/2 MS supplemented with 5.71 μM indole-3-acetic acid (IAA), 4.9 μM indole-3-butyric acid (IBA), and 5.37 μM naphthaleneacetic acid (NAA) within 3 wk. Plants regenerated in this manner were acclimatized in the greenhouse and under a shade net with 88% success.     

Some bio-sociological aspects of the mathematical theory of communication

In the first part of the paper a general discussion of the transmission of information through neural chains is given in terms of the Shannon-Weaver theory. It is pointed out that with the all-or-none law a single chain of neurons connected in series transmits one bit of information per signal. A set ofN independent parallel chains transmitsN bits per signal. If, however, the chains are interconnected, the amount of information is reduced. At the same time, however, the degree of coordination of the final neuromuscular reaction is increased. A relation between the maximum possible speed of a reaction and its degree of coordination is derived, and possible applications to spoken language are suggested. A general quantitative discussion of the relation between amount of information and amount of knowledge which an individual may obtain when confronted with the external world is made and a possible connection with new trends in logical thinking is pointed out. In the second part transmission of information through “social chains” is discussed under certain special assumptions. An expression for the “social channel noise” in terms of the length of the channel is derived. Finally an expression is given for the amount of information transmitted from one individual to another in a social group of uniform density as a function of the physical distance between the two individuals.     

The evolution of “egalitarian” and “despotic” social systems among macaques

Recent studies of captive macaques have revealed considerable inter-species differences in dominance styles among females. In “egalitarian” species such as stumptail (Macaca arctoides) or tonkean macaques (M. tonkeana), social interactions are more symmetrical and less kin-biased than in “despotic” species such as Japanese (M. fuscata) or rhesus macaques (M. mulatta). Field observations of moor macaques (M. maurus), close relatives of tonkean macaques, suggest that tolerance during feeding characterizes their egalitarian dominance style in the natural habitat. Although it has been proposed that communal defense against other groups may be the main selective force in the evolution of egalitarian dominance style among females, few field data support this prediction. A game theory analysis showed that both an “egalitarian” strategy and a “despotic” strategy are possible evolutionarily stable strategies (ESS) under certain conditions. The difference in dominance styles might reflect the difference in ESS. This means that an egalitarian dominance style can emerge without strong between-group contest competition. A phylogenetic comparison among macaques suggests that despotic dominance styles very likely evolved from egalitarian dominance styles. In the future, primate socioecological studies should pay more attention to the evolutionary history of each species.     

Connectivity of random nets

The weak connectivity γ of a random net is defined and computed by an approximation method as a function ofa, the axone density. It is shown that γ rises rapidly witha, attaining 0.8 of its asymptotic value (unity) fora=2, where the number of neurons in the net is arbitrarily large. The significance of this parameter is interpreted also in terms of the maximum expected spread of an epidemic under certain conditions.     

Short-term soil inorganic N pulse after experimental fire alters invasive and native annual plant production in a Mojave Desert shrubland

Post-fire changes in desert vegetation patterns are known, but the mechanisms are poorly understood. Theory suggests that pulse dynamics of resource availability confer advantages to invasive annual species, and that pulse timing can influence survival and competition among species. Precipitation patterns in the American Southwest are predicted to shift toward a drier climate, potentially altering post-fire resource availability and consequent vegetation dynamics. We quantified post-fire inorganic N dynamics and determined how annual plants respond to soil inorganic nitrogen variability following experimental fires in a Mojave Desert shrub community. Soil inorganic N, soil net N mineralization, and production of annual plants were measured beneath shrubs and in interspaces during 6 months following fire. Soil inorganic N pools in burned plots were up to 1 g m⁻² greater than unburned plots for several weeks and increased under shrubs (0.5–1.0 g m⁻²) more than interspaces (0.1–0.2 g m⁻²). Soil NO₃ ⁻−N (nitrate−N) increased more and persisted longer than soil NH₄ ⁺−N (ammonium−N). Laboratory incubations simulating low soil moisture conditions, and consistent with field moisture during the study, suggest that soil net ammonification and net nitrification were low and mostly unaffected by shrub canopy or burning. After late season rains, and where soil inorganic N pools were elevated after fire, productivity of the predominant invasive Schismus spp. increased and native annuals declined. Results suggest that increased N availability following wildfire can favor invasive annuals over natives. Whether the short-term success of invasive species following fire will direct long-term species composition changes remains to be seen, yet predicted changes in precipitation variability will likely interact with N cycling to affect invasive annual plant dominance following wildfire.     

The smallest value of the axon density for which ‘ignition’ can occur in a random net

As shown by A. Rapoport (1952), when a very brief stimulation or “instantaneous input” is applied to a random net, the subsequent events are determined by the parameters of the net as follows: If the axon densitya is sufficiently large and the fraction γ of the neurons initially stimulated exceeds a certain value γ₁ (theover-all threshold of the net for instantaneous stimulation), excitation will spread through the net until a steady state is reached in which a fraction γ₂ ⩾ γ₁ of the neurons is firing (“ignition phenomenon”). If γ < γ₁ the activity in the net dies out. However, if the axon density is too small, the activity will ultimately die out, no matter how large the fraction of initially stimulated neurons. Thus there exists a limiting valueA of the axon density below which the net cannot “ignite”. ThisA is a function ofh, theindividual threshold of the neurons constituting the net (we assume hereh≥2, since forh=1 the situation is essentially different). Geometrically γ₁ and γ₂ are determined as the two intersection points of a straight line with a sigmoid curve. Whena<A the two curves do not intersect and fora=A they are tangent. In this paper the “tangency case” is investigated and the general features of the functionA(h) are determined. It is shown thatA increases monotonically withh (as one would expect). For all values ofh>1 we haveA(h)>h, but the fractionA(h)/h and the derivativedA(h)/dh approach unity ash increases. An analytical expression of the functionA(h) valid for very large values ofh is derived.     

Multisequence comparisons in protein coding genes

A very powerful method for detecting functional constraints operative in biological macromolecules is presented. This method entails performing a base permanence analysis of protein coding genes at each codon position simultaneously in different species. It calculates the degree of permanence of subregions of the gene by dividing it into segments,c codons long, counting how many sites remain unchanged in each segment among all species compared. By comparing the base permanence among several sequences with the expectations based on a stochastic evolutionary process, gene regions showing different degrees of conservation can be selected. This means that wherever the permanence deviates significantly from the expected value generated by the simulation, the corresponding regions are considered “constrained” or “hypervariable”. The constrained regions are of two types: α and β. The α regions result from constraints at the amino acid level, whereas the β regions are those probably involved in “control” processing. The method has been applied to mitochondrial genes coding for subunit 6 of the ATPase and subunit 1 of the cytochrome oxidase in four mammalian species: human, rat, mouse, and cow. In the two mitochondrial genes a few regions that are highly conserved in all codon positions have been identified. Among these regions a sequence, common to both genes, that is complementary to a strongly conserved region of 12S rRNA has been found. This method can also be of great help in studying molecular evolution mechanisms.     

Variables influencing one-zero and instantaneous time sampling outcomes

Previous studies have shown that instantaneous time sampling results are more highly correlated with true duration than are one-zero sampling results. One-zero results, however, reflected a composite of both duration and frequency better than did instantaneous results. It is shown in this analysis that these outcomes are the logical result of the definitions of the two sampling modes. When interval lengths meet specific requirements, correct frequency counts can be obtained with either mode and systematic duration error with one-zero sampling can be eliminated by subtracting frequency from the number of intervals scored ‘one.’     

A note on the quantum-theoretic basis of primary genetic activity

In previous work (Bull. Math. Biophysics,23, 393–403, 1960) it was shown that, if primary genetic processes are of an essentially microphysical nature, the objects bearing the primary genetic information must act in a catalytic fashion. At the same time it was pointed out that the kind of catalysis involved in the primary genetic process was fundamentally different, in specific ways, from that occurring, e.g., in enzyme systems. The present work demonstrates that, if the information-bearing objects of the general theory are identified with molecules of DNA, and the primary gene products are considered to be RNA of the “messenger” variety, then the predictions of the general theory can be compared with experimental data from various recently isolated polymerase systems, which appear to “copy” a sequence of nucleotides from DNA into RNAin vitro, and with certainin vivo microbial systems. It is found that these data provide detailed support for the conclusions drawn from the general theory. However, it is emphasized that the identification of the information-bearing objects and primary gene products as DNA and RNA respectively, which allows us to compare the theory with the cited data, is by no means the only identification which can be made; i.e., other interpretations of the general theory are certainly not precluded. 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 Grant No. AF-AFOSR-9-63.     

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.     

A note on the excitation of nerve pathways

LetS denote the intensity of a stimulus, ν the frequency of discharge of a fiber, and ν _T the integral frequency of discharge of a nerve pathway. The relation between ν _T andS can be derived when the relation between ν andS, as well as the distribution function of the fiber thresholds are given. It is investigated under what conditions the functions ν _T (S) and ν(s) will be both of the same exponential form, and it is shown that this can happen only ifS exceeds the highest threshold of the pathway.     

Foraging patterns and kleptoparasitism among three sympatric cormorants (Phalacrocorax spp.) from the Delhi region, North India

Cormorants, described as ‘foot-propelled pursuit divers’, constitute an important component of aquatic food webs and exhibit unique foraging behaviour patterns, which can be properly understood through a comparative study. Since, after a foraging dive they surface to ingest the prey, the intensity of kleptoparasitic attacks on the surface can have a major impact upon the net energetic gain for each individual. Inspite of the fact that cormorants and their habitats are severely threatened in India, their foraging behaviour has not been adequately studied. Such considerations prompted us to undertake field studies on three sympatric cormorants (Phalacrocorax spp.) at 60 different sites in the Delhi region of North India, during 2004–2007. By means of video photography, some key foraging parameters including group size, prey size and patterns of kleptoparasitic attacks were quantified. Along a loose body size gradient, we observed significant differences among the three species with respect to not only their preference for wetland size but also prey size. The frequency of a kleptoparasitic attacks depended upon the group size and foraging behaviour of each species. It was observed that several foraging bouts were abruptly terminated due to human disturbances, mostly at sites lying outside the protected areas. This observation points towards the need to conserve small waterbodies in the countryside, currently threatened by pollution and urbanization, for the benefit of a variety of waterbirds including cormorants.     

Photosynthesis and antioxidative enzyme activities in five Indian mangroves with respect to their adaptability

Five typical mangroves were taken (Bruguiera gymnorrhiza, Excoecaria agallocha, Heritiera fomes, Phoenix paludosa and Xylocarpus granatum) both from Sundarbans (in situ) and grown in a mesophytic environment (ex situ, in the Institute’s premises) for 12–15 years. A comparative account of PAR utilization for maximum photosynthesis, stomatal conductance and production of two antioxidant enzymes (peroxidase and Superoxide dismutase) were done between the in situ and ex situ habitats. The present work revealed that the average net photosynthesis was slightly higher in mangroves from non-saline habitats than that of the native ones. At the same time, stomatal conductances were remarkably reduced under salinity-stressed habitats when compared with those of the mesophytic counterparts, by 25–52%. Salinity imposed increase of antioxidant enzymes was observed. Both the investigated antioxidant enzymes showed considerable increase in saline-grown individuals and proved their efficient scavenging ability to evolve reactive oxygen species (ROS), but these increases were relatively lower in Heritiera and Xylocarpus even though the net photosynthesis was higher. This might be related to their lower adaptability under increased salinity stress than those of the other three species investigated.