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  • Springer  (267,999)
  • Public Library of Science (PLoS)
  • 1970-1974  (177,357)
  • 1960-1964  (90,642)
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  • 1
    Electronic Resource
    Electronic Resource
    Contributions to the theory of organismic sets, II: Aggressiveness and submissiveness (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 49-54 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476664
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  • 2
    Electronic Resource
    Electronic Resource
    Contribution to the theory of organismic sets, III (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 55-66 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In line with previous studies on organismic sets, the division of all organismic sets intogeneral autotrophic and heterotrophic is introduced. The first produce their food themselves from some external source of energy, which in general may be an energy of any kind. The others use other organismic sets as the source of their food and energy. On earth we know only one kind of generalgeneral autotrophic organismic sets, namely, the autotrophic plants which use solar radiation as their source of energy and for production of their own food. It is shown why autotrophic animals do not exist on earth except as microorganisms like, e.g.,Euglena. A rigorous proof of the previously derived theorem that in an organismic set of ordern〉1 no element can be completely specialized is given. It requires the introduction of new postulates. Finally, in considering the organic world as a whole, the notion of organismic sets ofmixed order is introduced.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476665
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  • 3
    Electronic Resource
    Electronic Resource
    “A stochastic solution of the 3-body problem in chemical kinetics. The termolecular stochastic process I” (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 67-81 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It appears to be axiomatic that termolecular and higher order reactions occur relatively rarely. The basis for this judgment seems to lie in the supposition that successful 3-Body collisions of 3 interactive species of molecules cannot occur frequently enought to account for chemical or biochemical transformation. In order to provide a more complete mathematical framework than now exists for examining this hypothesis the probability of effective termolecular “δ-collisions” as a function of time is derived. This amounts to adding to the class of reactions for which stochastic models are now available the termolecular reaction. In common with the unimolecular and bimolecular cases this process is seen to satisfy the criterion of consistency-in-the-mean with respect to deterministic formulations. It is planned next to use the termolecular process and the lower order processes in computer-assistedin numero experimental studies aimed at comparing alternative mechanisms of reaction.
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    URL: http://dx.doi.org/10.1007/BF02476666
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  • 4
    Electronic Resource
    Electronic Resource
    Maximum likelihood estimation for stochastic first order reactions (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 83-96 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Small sample properties of the maximum likelihood estimator for the rate constant of a stochastic first order reaction are investigated. The approximate bias and variance of the maximum likelihood estimator are derived and tabulated. If observations of the system are made at timesiτ,i=1, 2, ...,N; τ〉0, the observational spacing τ which minimizes the approximate variance of the maximum likelihood estimator is found. The non-applicability of large sample theory to confidence interval derivation is demonstrated by examination of the relative likelihood. Bartlett’s method is employed to derive approximate confidence limits, and is illustrated by using simulated kinetic runs.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476667
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  • 5
    Electronic Resource
    Electronic Resource
    Organismic supercategories and qualitative dynamics of systems (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 339-354 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476778
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  • 6
    Electronic Resource
    Electronic Resource
    Some realizations of (M, R)-systems and their interpretation (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 303-319 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Some years ago (Rosen 1958a, b; 1959) we described a class of metaphorical, relational paradigms for cellular activity which we termed (M, R)-systems. A sizable amount of subsequent work, to be itemized below, has been devoted to an exploration of some of the properties of these systems. The main purpose of the present paper is to put this class of paradigms into a general system-theoretic perspective, with a particular view to appraising the relation between the type of system description embodied in the (M, R)-system and other kinds of physical and mathematical descriptions of cellular systems. Thus, the principal aim is to establish the relationships and connections between the global relational formalism embodied in the (M, R)-systems and the empirical descriptions which still represent the bulk of our biological knowledge.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476776
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  • 7
    Electronic Resource
    Electronic Resource
    Relational forces and structures produced by them (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 321-338 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract 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.
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    URL: http://dx.doi.org/10.1007/BF02476777
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  • 8
    Electronic Resource
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    Probabilistic and multiple regression modeling of the biologist’s decision processes and its implications for medical diagnosis (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 439-449 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Current psychological research into the inference (diagnostic) process is briefly reviewed, using as a vehicle an investigation of the prediction of the probability of success of hypothetical applicants to a graduate program in biology. Brunswik’s lens model and multiple regression analysis are used, as is a Bayesian approach. Four judges’ (biologists’) predictions are analyzed. Some general conclusions about inference, drawn from the current data in psychology, are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476785
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  • 9
    Electronic Resource
    Electronic Resource
    Anin numero study of glucose-insulin interaction (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 451-462 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A mathematical model has been developed to simulate the glucose-insulin interaction following a glucose load such as occurs in an IVGTT. This model differs from earlier models in that the insulin response to glucose loading is a recurring all or none threshold response. The model has been simulated on a digital computer using the digital analog simulation language CSMP.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476786
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  • 10
    Electronic Resource
    Electronic Resource
    Wave propagation in bone media (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 463-479 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The composite nature of bone dictates the use of a model for bone which is transversely isotropic. We solve the associated sets of partial differential equations governing the dynamic elastic behavoor of a two-layered cylindrical-shaped bone. The solution is analyzed for long, short, and intermediate length waves. The special case of compact bone is treated for long and short wave lengths and a numerical example is worked out to determine the wave speeds (for short wave lengths) given a set of elastic constants, determined by ultrasonic methods, and the bone density, wave frequency, and radius.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476787
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  • 11
    Electronic Resource
    Electronic Resource
    Fourth International Biophysics Congress (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 481-481 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    URL: http://dx.doi.org/10.1007/BF02476788
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  • 12
    Electronic Resource
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    Obituary of Nicolas Rashevsky (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. i 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02477019
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  • 13
    Electronic Resource
    Electronic Resource
    On the stochastic theory of compartments: I. A single-compartment system (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 53-63 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A stochastic model is developed for a compartment with a single time-dependent input, and generalized to include inputs from several sources. With the number of particles of a given molecular species in the compartment as the random variable, the mean, variance and third central moment of this variable are calculated from its generating function, and compared with previous results. The behavior of the calculated moments is discussed, and the possibility of applying the model to chemical and biological systems is considered.
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    URL: http://dx.doi.org/10.1007/BF02477024
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  • 14
    Electronic Resource
    Electronic Resource
    Relationship between ion distribution and membrane potential during a steady state (1960)
    Springer
    Bulletin of mathematical biology 22 (1960), S. 323-349 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Equations were derived showing the relationship between the membrane potential and the quantities which influence it under steady state conditions. Essentially, the membrane potential is caused by the valence and concentration of the non-permeating ions. The permeating ions can modify the membrane potential by altering the relative concentration of the non-permeating ions with respect to the concentration of the permeating ions. For muscle, the sodium cations act as the non-permeating ions in the extracellular environment by the maintenance of some type of active metabolic process and large anions act as the non-permeating ions in the intracellular environment. Both of these non-permeating ions contribute about equally to the maintenance of the resting membrane potential. When the active metabolic process for sodium extrusion breaks down or when acids are added, the membrane potential should decrease. Water should enter the cell when the sodium metabolic process is diminished; water should leave the cell when acids are added. When acid is added, it is expected that the cations potassium and sodium will leave the cell with little or no shift of the chloride ions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476718
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  • 15
    Electronic Resource
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    Life, information theory, probability, and physics (1960)
    Springer
    Bulletin of mathematical biology 22 (1960), S. 351-364 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A purely information-theoretical approach to the problem of self-replication of elementary living units implies that pure chance is the determining factor in the formation of the first living unit. The probability of such a spontaneous formation can be calculated from the minimum amount of information which an organism must possess in order to replicate itself. An estimation of this amount of information is made here by two different methods. First by a “paper and pencil experiment” which indicates the minimum amount of information needed on a printed page in order that with given tools the page could be reproduced. Second—by an analytical consideration of some hypothetical molecular mechanisms. A general method for handling such problems is suggested. On the basis of estimated information contents it is shown that under most favorable conditions the probability of a spontaneous generation by pure chance during the lifetime of the earth is vanishingly small. It is concluded that dynamic factors, which may reduce tremendously the information content, must play a role in the genesis of life on earth.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476719
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  • 16
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    Some remarks on molecular biology (1960)
    Springer
    Bulletin of mathematical biology 22 (1960), S. 365-370 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract 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. Applying to our results the results of a previous paper (1959,Bull. Math. Biophysics,21, 309–326), we arrive at the conclusion that the self-replication of a living molecule may be the property not of a particular structure but of classes of structures.
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    URL: http://dx.doi.org/10.1007/BF02476720
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  • 17
    Electronic Resource
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    On the utilization of food by planktophage fishes (1960)
    Springer
    Bulletin of mathematical biology 22 (1960), S. 371-389 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Making some plausible assumptions about the over-all mechanism of food catching and consumption by fishes and evaluating in the light of those assumptions some available experimental data, it is possible to calculate from those data the variation of several important factors with the concentration of food. The factors considered are: total rate of metabolism, total diurnal energy expenditure in the process of feeding, average number of hours per day during which the fish feeds, average length of path traveled by a fish per day, and the so-called “energetic coefficient of growth.” A possible relation with the work of N. Rashevsky (Bull. Math. Biophysics,20, 299–308, 1959) is discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476721
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  • 18
    Electronic Resource
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    Symposium on Mathematical Theories of Biological Phenomena (1960)
    Springer
    Bulletin of mathematical biology 22 (1960), S. 425-425 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    URL: http://dx.doi.org/10.1007/BF02476724
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  • 19
    Electronic Resource
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    On the theory of indicator-dilution methods under varying blood-flow conditions (1960)
    Springer
    Bulletin of mathematical biology 22 (1960), S. 417-424 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The theory of measurement of flow and volume by indicator dilution techniques is given in conditions of time-variable flow rates. It is shown that the usual Hamilton (1932,Am. J. Physiol.,99, 534–551) methods can be misleading if the flow changes at a rate of close to that of the transport function.
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    URL: http://dx.doi.org/10.1007/BF02476723
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  • 20
    Electronic Resource
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    A remark on sensory deprivation from the point of view of organismic sets (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 439-441 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It is shown that from the definition of organismic sets (Rashevsky,Organismic Sets. Some Reflections on the Nature of Life and Society, Holland, Michigan, Mathematical Biology, Inc. and Grosse Pointe, Michigan, J. M. Richards Laboratory) a complete sensory deprivation of an organismic set of ordern=2 should result in malfunctioning of the set. A generalization to higher order sets is suggested.
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    URL: http://dx.doi.org/10.1007/BF02476706
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  • 21
    Electronic Resource
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    Optimal branching structure of the vascular tree (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 431-438 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Optimality in branching structure of the vascular tree was studied. Analysis on its physiological roles as the duct system for blood supply to the capillaries predicted that the vascular tree should be constructed with minimum volume under restriction of determinant pressure, flow and location at the origin and the terminals. Mathematical derivations of this conditional extremum problem yielded some equations expressing the relations between the radii of the branches and their branching angles, which provided numerical solutions for branching points of bi- and poli-terminal minimum volume trees. Comparison of the peritoneal vascular tree in a dog with the minimum volume one computed under the same restrictive conditions showed good agreement in their branching structure.
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    URL: http://dx.doi.org/10.1007/BF02476705
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  • 22
    Electronic Resource
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    First order kinetics in a distributed model (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 443-456 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In order to determine the kinetics of passage of a substance through an organ containing a tangle of vessels, we study the response of a tube to various inflows (perfusion, brief injection, ...). The introduction of the catabolic terms and of the spatial dependence between bulk concentration and surface concentration allows one to account for the difference of arteriovenous concentrations observed experimentally for many metabolites. The relationships between the physico-chemical parameters of the organ and the operational parameters of the model demonstrate the importance of the transit time through the considered vessels. If one considers the different pathways as independent, the introduction of the transit time distribution for an inert substance enables one to compute the response of the organ analytically or by recurrence, using convolution. The parameters of the model can be obtained by the moments method.
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    URL: http://dx.doi.org/10.1007/BF02476707
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  • 23
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    Stochastic simulation of the gas diffusion in the air phase of the human lung (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 457-466 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The solution of the diffusion equation in the gas phase of the human lung is very difficult because of the structure of the bronchial tree. It is shown by means of physical arguments, how one can reduce the diffusion equation to a simple one-dimensional form. The solution is then obtained by a stochastic simulation, which is easily realized on a digital computer.
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  • 24
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    A new mathematical model of the respiratory center (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 467-481 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A new mathematical model of the oscillatory behavior of the respiratory center has been developed based upon published records of neuronal activity during respiration in the pons and medulla. In contrast with a previous model, four, rather than two, networks are assumed to interact in the respiratory center so as to produce the respiratory oscillation. A mathematical description of this interaction, in the form of a set of four first-order, nonlinear, coupled differential equations, is derived; the behavior of the solutions of this system is studied qualitatively, and expressions for the durations of the inspiratory and expiratory phases are obtained in terms of some parameters. It is found that central and chemical influences drive the medullar neurons to a position somewhere between saturation and full cutoff, and the pontine neurons deeply into cutoff. The control of the duration of the different phases by these chemical and central means is discussed. In order to effect a decrease in the magnitude of the various times, the neurons have to be driven towards operating points of higher central facilitation.
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    URL: http://dx.doi.org/10.1007/BF02476709
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  • 25
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    A mathematical model for calcium homeostasis (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 483-502 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In vivo control of calcium is analysed under the assumption that hormonal influences via plasma levels of parathormone and calcitonin are of prime (but not absolutely dominating) importance. A brief review concerning the physiological significance of body calcium and the mode of action of these two hormones is presented as an introduction to the basic philosophy of the study. A theoretical quasi-linear lumped-parameter model is developed to describe variations in ionic calcium, parathormone and calcitonin plasma concentrations to specific input stimuli. Formal evaluation of the system response requires the determination of ten constants, together with quantitation of ingested calcium entry into the plasma compartment which isindependent of hormonal influences. Values for various parameters are deduced from published data and experimental procedures are outlined to facilitate determination of the remaining unknowns. It is suggested that the proposed model should prove useful for investigations concerning general hormonal actions on calcium homeostatic mechanisms in both normal and diseased states, with particular reference to calcitonin.
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  • 26
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    The struggle for life: III. A predator-prey chain (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 521-532 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Ann species predator-prey chain is analyzed to determine what oscillations occur in population sizes. It is found that only the populations of the first and second species in the chain must necessarily oscillate around the point of equilibrium if they do not come to equilibrium. The other species may or may not oscillate.
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    URL: http://dx.doi.org/10.1007/BF02476712
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  • 27
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    Mathematical models for exchange of substances in regional vascular beds— Measurement of the rates within vivo counters (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 503-520 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Sangren and Sheppard developed a mathematical model for first-order processes taking place in the regional circulation, applicable—for example—to tracer studies of potassium transport. It permits calculation of specific activity at any point along a “tube of flow” or in the cuff of tissue surrounding it as a function of time following a spike injection of tracer. In efforts to relate to the exchange a rate curves obtained within vivo counters pointed at the region of interest, we developed a compartment-system model of the process. In investigating the properties of the Sangren and Sheppard model integrated over an entire circulatory bed, as thein vivo counter would see it, we found that when the distribution of transit times of the “tubes of flow” can be approximated by an exponential sum, the solution reduces to that of the compartment system model. This results in an important simplification in the calculation, and insight into the assumptions underlying the two different models. A curve-fitting computer program for the compartment model has been written and applied to double-isotope studies of potassium transport in the hind leg of the dog.
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    URL: http://dx.doi.org/10.1007/BF02476711
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  • 28
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    A numerical experiment on the determination of kinetic rate constants in the presence of diffusion (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 547-558 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Two chemicals,A andB, are allowed to diffuse together and a reaction described by $$A + B\mathop \rightleftharpoons \limits_{K_{ - 1} }^{K_1 } C$$ is allowed to proceed. This system is described mathematically by a system of partial differential equations. A numerical procedure is presented to find the rate constants ofK 1 andK −1. A systematic analysis of the effects of errors is also presented.
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    URL: http://dx.doi.org/10.1007/BF02476714
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  • 29
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    Observations on cardiac energetics: I. Theoretical introduction (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 533-546 
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    Notes: Abstract Equations are developed to describe the energy expenditure of the human heart. As well as the external potential and kinetic energy terms, general consideration is given to other possible avenues of energy consumption. Emphasis is placed upon using mathematical variables which are readily available for experimental verification. The errors involved in assuming that mean values for the physiological parameters give reasonable estimations for the external mechanical performance are examined, and a theoretical estimation for the discrepancy in the kinetic component is presented. Logical extension of the mathematical derivation leads to a determination of cardiac external mechanical efficiency and clearly demonstrates the significance of the ventricular pressure-volume loop in this context. Finally, experimental procedures are suggested to clarify further some of the conclusions reached through the theoretical analysis.
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    Organismic sets. Sex in animal and human societies (1972)
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    Bulletin of mathematical biology 34 (1972), S. 559-563 
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    Notes: Abstract The question is discussed as to the reason why some animal societies, such as bees or ants, are sexually differentiated, that is, onlysome of its members are exhibiting reproducing activities. It is indicated that human society may be on its way to such a sexual differentiation which may eventually come.
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    Some biochemical threshold mechanisms (1972)
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    Bulletin of mathematical biology 34 (1972), S. 565-565 
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    Announcement (1972)
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    Bulletin of mathematical biology 34 (1972), S. 567-567 
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    The structure of DNA as deduced from fiber X-ray crystallography (1973)
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    Bulletin of mathematical biology 35 (1973), S. 301-311 
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    Notes: Abstract X-ray diffraction patterns obtained experimentally for fibers, together with their chemical structures, can be analyzed theoretically in terms of an integral equation. The partially unknown electron density function can be solved by iteration. This mathematical technique has been applied with success to study the secondary structures of DNA fibers.
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    An hypothesis of freese and the DNA-protein coding problem (1961)
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    Bulletin of mathematical biology 23 (1961), S. 305-318 
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    Notes: Abstract Freese’s Hypothesis states that a single specific alteration in the sequence of nucleotides of an information-bearing DNA molecule results in a specific mutational effect. Within the framework of the DNA-protein coding problem developed elsewhere, and assuming the quasi-ergodicity of the general coding process, it is shown that Freese’s Hypothesis allows us to derive expressions for the length of the smallest mutable DNA molecule and to obtain a bound for the maximal number of allelic molecules of fixed length. To illustrate these ideas, calculations are carried out on appropriate data from bacternophage and man, and the results are shown to differ by a factor of 10 (modulo the rather crude approximations used). It is further shown that, if ρ(N) and ϱ(N) are respectively the number of information-bearing words of lengthN in a given code and the number of words of lengthN, then the number lim ρ(N)/ϱ(N) depends sensitively on the parameter ∈ which specifiesN→∞ the given code. The implications of this result for the spontaneous aggregation of a sufficient number of information-bearing words to characterize an organism are discussed.
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    Post-and predoctoral fellowships in mathematical biology (1961)
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    Bulletin of mathematical biology 23 (1961), S. 319-319 
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    The perfect learner (1961)
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    Bulletin of mathematical biology 23 (1961), S. 321-335 
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    Notes: Abstract As a “base line” of memorization performance, the behavior of a “perfect learner” is considered. He is characterized by a perfect memory and by the ability to choose the best search procedure in problems where the correct response from a given repertoire is to be found to each of several stimuli under the condition of “right” and “wroing” promptings by the experimenter. Expected learning curves are derived for the case of disjoint response repertoires associated with the stimuli under cyclic and random presentation of the stimuli and for the case of a single response repertoire (a one-to-one matching problem) under cyclic presentation.
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    A method for the computation of aortic distensibility in the living human patient and its use for the determination of the aortic effects of aging, drugs, and exercise (1961)
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    Bulletin of mathematical biology 23 (1961), S. 337-353 
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    Notes: Abstract Detailed equations are given for the computation of aortic distensibility in the intact living human patient from measurements of systolic and diastolic arterial pressures, heart rate and cardiac output. From these equations, the aortic characteristics of a large series of normal men of different ages were computed. Comparing these results with measurements on excised aortas, a more pronounced trend toward increasing aortic stiffness with increasing age is evident in living than in dead aortas. Nor-epinephrine and exercise apparently cause the living aortas to constrict but to become more distensible. The same change occurs after 30 minutes of high spinal anesthesia. The ganglionic blocking agents hexamethonium, pentamethonium, and tetraethylammonium usually cause the living aorta to become stiffer, presumably due to dilatation. The aortas of patients with pulmonary disease usually react in different fashion to exercise or tetraethylammonium. The increased aortic distensibility that occurs with the hypertension induced by nor-epinephrine or exercise acts as a compensatory mechanism by decreasing systolic pressure.
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    Dimensional analysis in mathematical biology I. General discussion (1961)
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    Bulletin of mathematical biology 23 (1961), S. 355-376 
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    Notes: Abstract Dimensional analysis is discussed from the viewpoint of its basic group properties and shown to be an algebraic Abelian group that is useful for analysis of physical measurements. The application of the method to various types of equations and the formulation of previously unclassified dimensions are discussed. Functional dimensional analysis is applied to the problems of cell size and biomass proliferation; future applications are also noted. A number of dimensionless terms have been formulated for cellular physiochemical phenomena. They apparently represent the first systematic study of biological dimensionless numbers recorded in the literature. A dimensionless proliferation law is suggested. A brief analysis of the physical dimensionality associated with information measures is carried out. Entropy and “information” are shown to be completely different in their dimensional meaning; other informational measures of possible interest in biology are proposed. The dimensional coding and computor analysis of biomathematical equations is suggested.
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    A minimal principle in biomechanics (1961)
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    Bulletin of mathematical biology 23 (1961), S. 377-391 
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    Notes: Abstract Expenditure of energy under several simultaneous forms (mechanical, chemical, etc.) is associated with all muscular activity. The energy is directly related to what is commonly called exertion or effort. This paper defines “muscular effort” quantitatively in terms of some of the elements of the dynamics of the human (and animal) body. It postulates that in all likelihood the individual will, consciously or otherwise, determine his motion (or his posture, if at rest) in such a manner as to reduce his total muscular effort to a minimum consistent with imposed conditions, or “constraints”. The principle, formulated in mathematical terms, is sufficient to ascribe to the moments at all body joints—a matter generally of free choice on the part of the individual—their most likely magnitudes. It therefore renders the equations of human (and animal) motion determinate within this context. The paper also describes briefly an iteration method for the solution of these equations, once they have been made determinate. A simple illustrative application of the principle is included.
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    On the role of chemical systems in the microphysical aspects of primary genetic mechanisms (1961)
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    Bulletin of mathematical biology 23 (1961), S. 393-403 
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    Notes: Abstract It is pointed out that two fundamentally different views of primary genetic processes occur in the literature which are frequently confused. The first is a true communication-theoretic view, which regards the genetic apparatus as containing a real information-source and a transducer which converts that information to useful form. The second view is generally expressed as a template scheme based on the Watson-Crick model; it is shown that in this model there is actually no such thing as genetic information in a communication-theoretic sense. Both views are then discussed on the basis of microphysical principles developed in previous work of the author (Bull. Math. Biophysics,22, 227–255, 1960) in an attempt to find which approach is in closer accord with the biological facts. It is shown that, if the communication-theoretic view is correct, then the information-bearing object must act as a “catalyst,” but it is pointed out that the type of catalysis involved must be of a fundamentally different nature than that occurring in familiar enzyme-catalyzed reactions. On the basis of general considerations of irreversible changes in microphysical measuring systems, it is shown that any type of template must suffer a gradual and irreversible denaturation, which seems to make it unlikely that a template could play a primary role in fundamental genetic processes.
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    On imitative behavior. II. Time course of change from one behavior to another (1961)
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    Bulletin of mathematical biology 23 (1961), S. 405-411 
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    Notes: Abstract The theory developed in previous papers and based on distribution curves of definite form is generalized to any form of unimodel distributions. The time course of the change from one behavior to another is discussed and a general theorem about the time course is established.
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    Abstract mathematical molecular biology (1961)
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    Bulletin of mathematical biology 23 (1961), S. 417-417 
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    Training program in computer use for biomedical scientists (1963)
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    Bulletin of mathematical biology 25 (1963), S. 471-471 
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    Reflexes et centres analyseurs dans les microstructures isotropes (1963)
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    Bulletin of mathematical biology 25 (1963), S. 421-469 
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    Notes: Résumé Nous appliquons le modèle de neurone introduit dans un article antérieur à l’étude d’une microstructure isotrope. La stabilité de cette microstructure implique l’existence d’une régulation d’activité que le principe de construction adéquate permet de définir entièrement. Nous aboutissons à une conception stratifiée du cerveau. Un réseau de neurones spécialisés exercerait, grâce à certains médiateurs chimiques, une action diffuse qui modulerait les propriétés du réseau localisé classique. Les lois de Pavlov peuvent être retrouvées à partir des propriétés de la microstructure et de celles de la régulation. La microstructure isotrope peut également fonctionner comme analyseur. Un certain nombre de temps caractéristiques apparaissent alors, qui semblent jouer un grand rôle en psychologie.
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    Tissue blood flow and the sympathomimetic amines (1964)
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    Bulletin of mathematical biology 26 (1964), S. 1-7 
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    Notes: Abstract In the arteries, blood flow and blood pressure are pulsatile in nature (Roston, 1962a; Roston 1962b). The patterns of blood movement and mural distension in the arteries are important because they may be associated with life-threatening degenerative changes in the arterial walls. As the vascular channels narrow, the pulsation decreases. At the level of the capillaries, almost no pulsation exists (Best and Taylor, 1961). The tissues are affected by the direct flow in the capillaries and not by the pulsation in the arteries. Thus, such quantities as pulse pressure, systolic pressure, and diastolic pressure which characterize blood movement in the arteries are not important as far as the tissues are concerned. Rather, the average pressure and flow in the capillaries are the quantities significant for tissue blood flow. The present study analyzes the local blood circulation in a typical tissue. Logical extension of this analysis results in insights into the physiological behavior of the circulation which integrate a considerable body of experimental data.
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    Information Processing in the Visual Systems of Arthropods (1974)
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    Bulletin of mathematical biology 36 (1974), S. 339-340 
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    Optimality in the control of environmental systems (1974)
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    Bulletin of mathematical biology 36 (1974), S. 341-345 
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    Notes: Abstract For an environmental system described by a system of nonlinear first-order differential equations, the problem of achieving specified terminal conditions in a given time with a minimum expenditure of resources is considered. The initial conditions and the minimum value are found numerically in a particular example.
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    Kinetic approach to neural systems: I (1974)
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    Bulletin of mathematical biology 36 (1974), S. 535-544 
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    Notes: Abstract A kinetic model of neural systems is introduced and discussed with statistical mechanics techniques. It is assumed that, for a macroscopic description of the model, it suffices to consider only the distribution for the velocity and position of the impulses, and the distribution for the excitation and position of the neurons, at any timet. Making use of Boltzmann's method for the study of a dilute gas, coupled differential equations for the rate of change with time of the distributions have been constructed.
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    Flow of a newtonian fluid through a permeable tube: The application to the proximal renal tubule (1974)
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    Bulletin of mathematical biology 36 (1974), S. 457-476 
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    Notes: Abstract Creeping flow of a Newtonian fluid through a rigid permeable tube is considered and the transmural seepage is assumed to obey Darcy's law. Closed-form solutions for the pressure and velocity fields are presented and equations describing the axial variation of the mean cross-sectional pressure, the axial volumetric flow and the transmural fluid flux are derived. Approximate solutions for small seepage rates are given and are applied to the flow in the proximal renal tubule. Probable values for the epithelium permeability and the intraluminal hydrostatic pressure drop are obtained.
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    Disseminated storage of information with sequential coding (1973)
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    Bulletin of mathematical biology 35 (1973), S. 663-688 
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    Notes: Abstract The paper demonstrates that it is possible to construct memory models where the information inserted is stored in disseminated form, using sequential coding, the changes in the units forming the models being determined by their geometrical connections and by the incoming stream of information. The models are shown to have large storage capacity and their efficiency can be made insensitive to loss of or damage to a large fraction of their units. The satisfactory verification by computer simulation of the analysis and results described in the present paper will be the subject of a future paper.
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    On knowledge and unpleasantness (1974)
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    Bulletin of mathematical biology 36 (1974), S. 605-605 
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    The Kerner linearization method (1974)
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    Bulletin of mathematical biology 36 (1974), S. 67-76 
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    Notes: Abstract We examine in detail Edward Kerner’s method for linearizing the equations of enzyme kinetics. Our main result is the determination of canonical forms for systems which can be linearized by the method. This is done both in general and in the special cases of two and three dimensions where complete results are obtained. The practical problem of identifying linearizable systems is also considered and computable necessary criteria are presented.
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    Continuous determination of the volume of a space in a non steady state condition (1974)
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    Bulletin of mathematical biology 36 (1974), S. 59-66 
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    Notes: Abstract A general method for determination of the volume of a space in a non steady state condition, in case diffusion might be significant, is developed. Instantaneous mixing of indicators with native fluid is assumed in this first stage of investigation. Theoretical expressions are obtained for the volume of the space and the diffusion coefficient as a function of time. An analysis of feasibility of the method is also included.
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    Passive transfer of low-molecular nonelectrolytes across deformable semi-permeable membranes—II: Dynamics of a single muscle fiber swelling and shrinking and related changes of theT-system tubule form (1974)
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    Bulletin of mathematical biology 36 (1974), S. 379-401 
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    Notes: Abstract The theory of transfer of low-molecular nonelectrolytes across deformable semipermeable membranes of large curvature developed in Part I (Rubinstein, 1974) is used to describe the dynamics of swelling and shrinking of a muscle fiber at the influx and efflux of low-molecular nonelectrolytes. A large set of computations showed that the theory explains the experiments described in the literature.
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    Heat transfer in perfused biological tissue—I: General theory (1974)
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    Bulletin of mathematical biology 36 (1974), S. 403-415 
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    Notes: Abstract Green's function for heat and matter transport is calculated for an infinite medium in which a convection field v(r,t) makes a contribution to the total heat and matter current. It is given by a uniformly and absolutely convergent series in which every term is calculated from the preceding one merely by integration. The solution procedure is interpreted physically and illustrated by a simple problem in which v(r,t)=const. in space and time. Since the solution contains no intrinsic spatial symmetry, it can serve as a starting point for a theory of heat and mass transport in perfused biological tissue.
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    A long wavelength solution for a microorganism swimming in a channel (1974)
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    Bulletin of mathematical biology 36 (1974), S. 435-444 
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    Notes: Abstract The hydrodynamics of a microorganism swimming in a channel is investigated. The microorganism is modeled as a two-dimensional sheet swimming at low Reynolds numbers between two rigid walls. The wavelengths of the propulsive waves passing down the sheet are assummed to be very large compared to the channel spacing, but the amplitude of the propulsive waves is arbitrary. Explicit analytical solutions for the propulsive velocity and the rate of energy dissipated in terms of the wave amplitude, channel spacing, wave number, and wave speeds are given.
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    Mathematical models of conception and birth (1974)
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    Bulletin of mathematical biology 36 (1974), S. 455-456 
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    Why are there so many species? (1974)
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    Bulletin of mathematical biology 36 (1974), S. 477-488 
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    Notes: Abstract It is shown from the statistical-mechanical overview of Volterra's ecological model how to reckon the fluctuations of collective variables such as the total population of a genus: and that these fluctuations are much decreased (or that the collective populationsteadiness is enhanced) as the speciation is increased. (A niching of species in time, or phase-niching, is entailed here.) Secondly, it is shown how Preston's log-normal distribution describing the species-abundance relationship, as well as a generalization of such distributions, come forth simply and naturally from the statistical-Volterra-dynamics.
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    Stochastic models for an enzyme reaction in an open linear system (1971)
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    Bulletin of mathematical biology 33 (1971), S. 97-115 
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    Notes: Abstract A stochastic model is developed for an enzyme reaction in an open linear system. The proposed model assumes that the open system maintains the concentration of substrate and inhibitor at constant levels and that the product molecules are removed from the system by a first order reaction. Stochastic models for several enzyme reactions occurring in this open system are shown to correspond to special cases of theGI/M/∞ queue. Takács’ (1958) results for this queueing system are used to obtain the stochastic properties of the enzyme systems. A specific model we studied assumed completely competitive inhibition in an open system. The stationary distribution for the number of product molecules in the system is obtained. The enzyme reaction which incorporated the “intermediate chain hypothesis” can also be investigated by the queueing theory approach. It is shown that for this open system, if the model which incorporated the intermediate chain hypothesis has the same deterministic properties as the Michaelis-Menten model, then the latter has greater stochastic variation than the former.
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    International Conference on Stochastic Point Processes (1971)
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    Bulletin of mathematical biology 33 (1971), S. 153-153 
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    An integral method for the analysis of blood flow (1971)
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    Bulletin of mathematical biology 33 (1971), S. 117-128 
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    Notes: Abstract The existing methods to solve the problems of pulsatile flow in the cardiovascular system are based on either linear axisymmetric equations or non-linear one-dimensional equations. The solutions thus obtained give only a mediocre comparison with measurements. In this paper, a non-linear axisymmetric theory is proposed. The starting point of the present theory is a third degree polynomial representation of the velocity profile. Integral methods are then applied to obtain the governing equations. To ascertain the accuracy of the theory proposed above, the calculations for a simple case involving pulsatile flow in a long rigid tube were performed. The results are: (a) the average velocities compare very well with exact solutions and (b) the velocity profiles for a given frequency agree very well with exact solutions for flow in small tubes, but tend to differ as tube size is increased.
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    Methode d’objectivation d’organisations temporelles privilégiées d’activités neuronales unitaires (1971)
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    Bulletin of mathematical biology 33 (1971), S. 129-151 
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    Notes: Résumé Le but de ce travail est la mise en évidence d’éventuels “patterns” temporels privilégiés de potentiels d’action neuronaux masqués par la superposition d’une activité aléatoire. Dans la première partie, on propose un modèle susceptible de rendre compte de cette activité aléatoire. Dans la seconde, on expose une méthode d’extraction des patterns privilégiés, compatible avec les paramètres du modèle neuronal proposé. Son algorithme fait notamment intervenir l’estimation de la fonction d’expectation. Cette méthode peut, en fait, ment intervenir l’estimation de la fonction d’expectation. Cette méthode peut, en fait, être appliquée à l’étude de séries temporelles d’événements dans des domaines très divers.
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    International Symposium on Mathematical Biology (1971)
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    Bulletin of mathematical biology 33 (1971), S. 155-156 
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    Molecular set theory. III: The wide sense kinetics of molecular sets (1971)
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    Bulletin of mathematical biology 33 (1971), S. 355-372 
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    Notes: Abstract An effort is made to begin widening the scope of kinetics by merging the concepts and point of view of molecular set theory with the stochastic approach to kinetics, beginning with the simplest unimolecular molecular set transformation. In this spirit the new concept ofmolecular set variable is introduced as the basic unit of kinetics as opposed to simply the traditionalconcentration (or cardinality) unit, connoting that the composition as well as the size of a molecular set are significant dynamic features of a system. The changes in state (or “value”) of the molecular set variable are characterized by a Markovian stochastic process and the relationship between this process and the corresponding unimolecular process for the concentration variable introduced earlier is discussed. The possible role of molecular set theory in terms of the underlying biomathematical structure of relational biology is also considered.
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    Contribution a l’etude de la forme des plantes: discussion d’un modele de ramification (1971)
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    Bulletin of mathematical biology 33 (1971), S. 387-401 
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    Notes: Abstract The problem of the forms of plants and models of branching are discussed using experimental data on the mistletoe. The number of branches by division, the distribution of divisions with regard to the number of branches per division and to the level of division, the geometrical characters of branches according to the level of division and the host, the stability of model are studied. One gives an interpretation of the model of branching as a model of growth.
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    A quantum model for controlled enzymic reactions (1971)
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    Bulletin of mathematical biology 33 (1971), S. 373-386 
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    Notes: Abstract A quantum model for the general enzymic reaction,E+S ⇌ ES → P, is presented, starting with the assumptions that any chemical substanceS, which may be a substrate for a particularE (S)-enzyme is a microphysical system and any enzymeE-molecule, capable of interacting with anS-substrate is a “measuring system” which will “measure” one or more of theS-observables. According to the above assumptions a stochastic model of the reaction is constructed and a computer simulation of the steady state performed. The results thus obtained predicted fluctuations in the enzymic reaction rate, function of the substrate “perturbation”. On an experimental basis it is demonstrated that the irradiation of an enzymic substrate with low energies results in the inducement of a dose-dependent oscillatory behavior in the corresponding enzymic reaction rate. In the reaction type, the oscillations thus induced in theE-activity by the corresponding substrates are out-of-phase, realizing a biochemical discriminating net. Likewise, in an $$S_1 \xleftarrow{{E_1 }}S\xrightarrow{{E_2 }}S_2$$ reaction type, the oscillations induced by the irradiatedS-substrate in the activities of the respective enzyme, realize a biochemical switching net.
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    The myocardium as a composite material: Analysis (1972)
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    Bulletin of mathematical biology 34 (1972), S. 173-204 
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    Notes: Abstract As a further attempt to determine the stresses and strains of the individual myocardial fibers, the heart muscle is considered as an orthotropic material. A theory is presented which leads to the expression of the equilibrium conditions for the left ventricle in the form of three simultaneous differential equations. Solution of these equations would give the changes in shape of the left ventricle throughout the cardiac cycle, and, in addition, the stresses and strains of the individual myocardial fibers. It is pointed out, however, that meaningful solutions of the equations cannot be obtained at the present time because of difficulties in experimental determination of certain parameters.
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    A note on epilepsies and similar disorders (1972)
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    Bulletin of mathematical biology 34 (1972), S. 213-222 
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    Notes: Abstract This paper compares two previously published neural models for epilepsies (Bull. Math. Biophysics,33, 539–553, 1971;34, 71–78, 1972). The second model is developed in more detail and an attempt is made to bring it more in line with established neurological findings. The question of classification of some epilepsies is briefly discussed.
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    Suggestions for a mathematical model of a pathological neuron with spontaneous repetitive discharges (1972)
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    Bulletin of mathematical biology 34 (1972), S. 223-230 
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    Notes: Abstract A formal mathematical model is proposed for a spontaneously repetitively firing neuron. It is based on the assumption that an excitatory and inhibitory substance, possibly different from those involved in synaptic transmissions, is formed in the soma of everynormal neuron. Furthermore, the decay of the substances is ascribed to their combination with some other substances, present in healthy individuals. A generalized two factor system of differential equations is used. It is shown that when the normally present substances are absent, possibly due to genetic defects so that the decay constants become zero, the equations lead to undamped sinusoidal solutions of the difference between excitatory and inhibitory factors, thus producing a trulyspontaneous repetitive discharge, in the absence of external currents or other stimulation. It is suggested that convulsants may act by destroying the substances present in healthy individuals. It is further suggested that by administering to epileptics those substances, which are present in normal healthy persons, perhaps by using brain extracts fromhealthy higher animals which sometimes suffer from epilepsy, an actual cure rather than symptomatic treatment by anticonvulsants may be obtained.
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    Structure and kinetic behavior of linear multicompartment systems (1972)
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    Bulletin of mathematical biology 34 (1972), S. 277-291 
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    Notes: Abstract The general kinetic behavior of a multicompartment system is shown to depend upon certain general structural features, including its connectivity, whether it is open, and whether it contains cyclic pathways. Structural influences are clarified by putting the system matrix in a certain form. For systems not strongly connected, a distinction is drawn between partially and completely open systems. Necessary and sufficient conditions are given for non-singularity of the system matrix and for asymptotic stability of the system. Sufficient conditions are given for non-overshooting and monotonic transitions. A system is demonstrated whose solution may contain a prolonged series of damped oscillations; but the oscillations are very slow and small; and it seems unlikely that oscillations could be detected experimentally in any biological system.
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    A dynamical approach to chemical kinetics: Mass-action laws as generalized Riccati equations (1972)
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    Bulletin of mathematical biology 34 (1972), S. 243-275 
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    Notes: Abstract It is shown how the fundamental laws of chemical kinetics for either open or closed systems with an arbitrarily large number of reactants can be represented as a system of Riccati-like differential equations. Through the use of a concise tensor notation, it is shown when and how the differential system is exactly reducible to linear form, a reduction without approximation that parallels the well-known similar reduction of a single simle Riccati equation. An example is worked out to show how open kinetics can lead to oscillatory chemical concentrations of the Change-Higgins type. The biologically central problem of great chemical speciation is discussed from the viewpoint of Gibbs ensemble theory within the linearized kinetics and, approximately, within the starting nonlinear kinetics where it is shown roughly how to estimate, from an overall temperature-like parameter characterizing the whole system, mean chemical levels and mean frequencies of oscillation, and where a gross oscillation of the total mass is estimated in terms of an anharmonic oscillator whose general structure is fixed from the structure of the chemical kinetic laws.
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    Note on oscillations in a closed chain of compartments with linear, unidirectional transport (1972)
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    Bulletin of mathematical biology 34 (1972), S. 293-296 
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    Notes: Abstract A closed chain of compartments in which there is unidirectional transport between adjacent members can exhibit damped oscillations. For a system ofn equivalent compartments, the value ofn which gives the greatest difference between the first maximum and first minimum isn=11, the difference being 1.57%. The greatest difference between the first maximum value and the steady state value is 4% and is obtained whenn=25. The results are illustrated graphically forn equal to 5, 10, 25 and 100.
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    Pourquoi deux sexes? (1972)
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    Bulletin of mathematical biology 34 (1972), S. 297-304 
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    Notes: Abstract This paper is a concrete approach to the problem of the number of the sexes. We try to imagine—on the example of three sexes—the mechanisms which would have to accompany a reproduction with several sexes. We have limited our study to the monohybridism, dihybridism and determinism of the sex.
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    Cilia transport (1972)
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    Bulletin of mathematical biology 34 (1972), S. 325-335 
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    Notes: Abstract An analysis of the effect of cilia on fluid transport in tubules is presented. The applicability of the results for the flow rates observed in the ductus efferentes of the male tract is discussed.
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    Quantitative comparison of dipole models for steady-state currents in excitable membranes (1972)
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    Bulletin of mathematical biology 34 (1972), S. 305-324 
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    Notes: Abstract The dipole models for steady-state currents in excitable membranes of Arndt, Bond and Roper and of Hamel and Zimmerman are compared by fitting the equations to the data of Gilbert and Ehrenstein. The more complex Hammel and Zimmerman model does not fit the data as well as does the simpler Arndt, Bond and Reper model. When fitting the data, the Hammel and Zimmerman current equation reduces to the Arndt, Bond and Roper current equation because of the values assumed by the parameters. An interpretation is given for the parameter values obtained with the Arndt, Bond and Roper model.
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    On the decomposition of a dynamical system into non-interacting subsystems (1972)
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    Bulletin of mathematical biology 34 (1972), S. 337-341 
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    Notes: Abstract It is shown that, under rather general conditions, it is possible to formally decompose the dynamics of ann-dimensional dynamical system into a number of non-interacting subsystems. It is shown that these decompositions are in general not simply related to the kinds of observational procedures in terms of which the original state variables of the system are defined. Some consequences of this construction for reductionism in biology are discussed.
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    Self-reproduction and serial message transfer: Two related problems (1972)
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    Bulletin of mathematical biology 34 (1972), S. 343-353 
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    Notes: Abstract For a certain class of physical machines, termed “structure-determined,” the problem of self-reproduction can be reduced to the problem of serial message reproduction. Serial message reproduction however presupposes a sort of “open system” constraint. This leads to the principle of pseudo, or exogenously standardized, respectively, self-reproduction. It seems to be consistent with both chemical and biological self-reproduction. It thus may reflect a general principle of biological design. The proposed principle is a physico chemical analog to Robert Rosen's abstract relational self-reproduction constraint.
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    Active muscle fiber as an equivalent cardiac current generator (1972)
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    Bulletin of mathematical biology 34 (1972), S. 355-377 
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    Notes: Abstract Equations are derived describing potentials due to an active muscle fiber in an infinite medium in terms of two surface integrals—one of the propagated action potential and the other of the membrane current density, both integrals being taken over the surface of the muscle. These equations are incorporated into an equivalent cardiac current generator in which the left ventricle (i.e. the current source) is represented by a three-dimensional wedge and the thorax (i.e. the volume conductor), by a homogeneous circular cylinder. Since this current generator expresses the body surface potentials in terms of the membrane current density and the membrane potential at any point on the surface of the electrically active muscle fiber, the calculated ECG can be correlated with theactual sources within the heart. This equivalent cardiac generator possesses many of the physical and physiological properties of cardiac muscle. The equations were evaluated numerically on a digital computer. The results indicate that equivalent cardiac current generators of this type can yield clinically significant results and that further research is necessary to investigate their properties fully.
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    Analytical solutions for transient thermal behavior in biological systems (1972)
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    Bulletin of mathematical biology 34 (1972), S. 413-418 
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    Notes: Abstract Analytical solutions are presented for transient heat conduction in biological media. General boundary conditions and internal sources varied in both spatial and time variables are considered, thus, solutions for many special cases can be obtained with ease from the general solutions presented in this analysis.
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    Simultaneous diffusion and convection in single breath lung washout (1972)
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    Bulletin of mathematical biology 34 (1972), S. 393-412 
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    Notes: Abstract Two mathematical models of pulmonary single breath gas washout (one analytic, one numerical) are developed and their predictions compared with experimental data on human subjects. Weibel's 23 generation symmetric anatomical model is used as a guide to bronchial tree geometry. Experimental plots of nitrogen concentration versus volume expired, dead space versus breath holding time, and dead space versus tidal volume are compared with plots predicted by the models. Agreement is good. A plot of nitrogen concentration in the airways as predicted by the numerical model at different times during inhalation and exhalation of a single breath of oxygen is shown. Model predictions for changes in dead space with changes in washout gas and expiratory flow rate are discussed. Use of the analytic model for obtaining average values of the path length from mouth to alveoli in a given subject is discussed. To the extent of their agreement with experiment, the models provide a sound physical basis for the correlation of airway structure and function.
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    Announcement (1972)
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    Bulletin of mathematical biology 34 (1972), S. 429-429 
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    Generalizations of the Roginsky-Zeldovich (or Elovich) equation for charge transport across biological surfaces (1972)
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    Bulletin of mathematical biology 34 (1972), S. 419-427 
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    Notes: Abstract The Roginsky-Zeldovich (or Elovich) equation, which is −dx/dt=m exp (nx) (x=substrate concentration,t=time,m andn=constants), describes the kinetics of various biological electron and ion transport processes, and has been derived from the concept of charge transport across an activation energy barrier at an interface between dissimilar phases, driven by a difference in redox or ion potentials, with the simplifying assumptions that charge carrier concentration is constant, backward current across the interface is zero, and diffusion of substrate is fast. If charge carrier concentration is proportional to substrate concentration, then the kinetic equation is −dx/dt=mx exp (nx). If backward current is not zero, then −dx/dt=m 1 exp (n 1x) −m 2 exp (n 2 x), wherem 1,m 2,n 1 andn 2 are constants. Kinetic equations for interfacial charge transport in the presence of a significant substrate diffusion potential are also derived.
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    Mathematical studies of the interaction of respiratory gases with whole blood I. O2 absorption (1960)
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    Bulletin of mathematical biology 22 (1960), S. 391-415 
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    Notes: Abstract Some progress has been made on the problem of the interaction of respiratory gases with whole blood. A practical working model for oxygen absorption in and interaction with whole blood is developed by assuming that oxygen molecules compete with protons for binding sites on the hemoglobin molecule and by invoking the Wyman-Allen (Jour. of Polymer Science,5, 499–518, 1951) hypothesis that two oxygen molecules go on the hemoglobin at one time. Extensive tests of this model against saturation measurements on blood from humans, horses, oxen and sheep are made. Values for the equilibrium constants are calculated and compared. In addition a second working model has been developed in an attempt to explain why O2 saturation measurements when expressed as (100 percent — percent saturation) are an exponential function of oxygen partial pressure. Considerations which make plausible the following expression for saturation, [1−2e −γx/h1/2/(1+(1/20)(β′/h 1/2+h 1/2/β′))] are presented. Herex denotes oxygen tension,h denotes hydrogen ion concentration and β′ and γ are parameters.
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    Mathematical studies of the interaction of respiratory gases with whole blood II. CO2 absorption (1961)
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    Bulletin of mathematical biology 23 (1961), S. 1-14 
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    Notes: Abstract Some progress has been made on the problem of the interaction of respiratory gases with whole blood. A working mathematical model for the O2−CO2 interaction phenomena has been developed from mathematical studies of the data. The Edsall-Wyman (1958) model for CO2 absorption is improved upon in this paper by consolidating it with the O2 absorption model developed in paper I of this set (Bernard, S. R.,Bull. Math. Biophysics,22, 391–415, 1960). This improved model assumed the effect of O2 on CO2 absorption is mediated through the electrical charge possessed by the hemoglobin molecule,i.e., O2 molecules bound to hemoglobin displace protons from the hemoglobin thereby increasing the negative charge on the hemoglobin and at the same time increasing the acidity of the solution. The model is tested against the data.
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    On the mutual influence of individuals in a society (1961)
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    Bulletin of mathematical biology 23 (1961), S. 15-18 
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    Notes: Abstract 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.
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    Contribution to the mathematical biophysics of automobile driving (1961)
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    Bulletin of mathematical biology 23 (1961), S. 19-29 
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    Notes: Abstract Traffic in one direction on a multilane highway is considered, and a general expression for the number of cars which pass a car travelling at a given velocity, as well as the number of cars which the given car passes, is derived for the case when the speeds of different cars are distributed in some arbitrary manner. Closed expressions are derived and discussed for a rectangular distribution. Each passing by another car or of another car is considered as a distracting stimulus which affects the reaction times of the driver. Using previously derived expressions for the safe speed as a function of reaction times, expressions for the safe average speed are derived, in terms of the volume of traffic and of the spread of the distribution of speeds.
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    Glucose metabolism—A “mixed-cycle” system (1961)
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    Bulletin of mathematical biology 23 (1961), S. 99-103 
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    Notes: Abstract Emphasis upon the importance of homeostatic feedback has drawn attention away from the complexity of biological processes. A study of glucose metabolism indicates the importance of open-cycle as well as closed-cycle mechanisms. Besides the glucose-dependent mechanism of insulin secretion, many open-cycle processes involving the liver, adrenal glands and kidneys, play important roles in the variation of blood glucose. In addition, glucose utilization by the tissues is essentially open-cycle in nature.
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    Southern Regional Graduate Summer Session in Statistics at Virginia Polytechnic Institute (1961)
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    Bulletin of mathematical biology 23 (1961), S. 105-106 
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    A note on knowledge and unpleasantness (1973)
    Springer
    Bulletin of mathematical biology 35 (1973), S. 313-317 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Various philosophers have repeatedly denounced knowledge as a source of unpleasantness, thereby criticizing education. This paper presents a set theoretical approach to knowledge and education and tries to explain how they could lead occasionally to a feeling of unpleasantness.
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    URL: http://dx.doi.org/10.1007/BF02458339
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    An analysis of countercurrent exchange with emphasis on renal function (1973)
    Springer
    Bulletin of mathematical biology 35 (1973), S. 275-286 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract An analysis of countercurrent exchange in a U-tube is presented for a single-solute, constant-volume flow rate system with spatially varying source fluxes and permeabilities. Analytical solutions are given for the steady-state equations and numerical solutions for the unsteady-state equations. The solutions indicate that an external source of solute delivered to the stream flowing away from the U-tube bend can be distributed by the exchanger so that the concentration in both limbs increases toward the bend. In particular, there exist source fluxes whose magnitude decreases monotonically toward the bend for which the maximum solute concentration occurs at the bend. The point at which a concentration maximum occurs is governed principally by the solute permeability of the barrier separating the two limbs and by the volume flow rate through the exchanger. The system dynamics depend strongly on the relative cross-sectional areas of the two limbs or, equivalently, on the flow velocities within them. The model is used as a basis for discussion of various functional aspects of the renal vasa recta system.
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    URL: http://dx.doi.org/10.1007/BF02458336
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  • 91
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    Mathematical analysis of circulatory mixing process (1973)
    Springer
    Bulletin of mathematical biology 35 (1973), S. 287-300 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The circulatory mixing process was analyzed as the time course of the dispersion of indicator after its injection into the heart. In simplified models, which had one or two lumped mixing chambers and circulatory pathways connected with them, it was suggested that the extent of dispersion could be evaluated by the variance of indicator distribution in the total circulating blood when the circulation time distributions between the chambers and the concentration curves in the chambers were known. The method of determining the circulation time distributions through the pulmonary, systemic and total circulations was derived and the actual distributions were obtained in dogs by indicator dilution techniques. With the use of these distributions, the time course of the circulatory mixing process was numerically calculated. The results showed that there was considerable difference in velocities of the process between the case of the right heart injection and the left heart injection of the indicator.
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    URL: http://dx.doi.org/10.1007/BF02458337
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  • 92
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    Reactions chimiques couplees considerees en tant que filtres passe-bande (1973)
    Springer
    Bulletin of mathematical biology 35 (1973), S. 319-337 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract An investigation is made as to whether or not the existence of a band-pass filter function, analogous to that in electronics, can be proved from the fundamental laws of chemical kinetics. The problem is important for better understanding of the preference of certain biological rhythms to others. It is shown with simple examples that such behavior is possible for a number of systems of coupled chemical reactions far enough from the thermodynamic equilibrium. It is of interest to generalize this behavior since it could conceivably play a role in the transmission of “usable information” in biology.
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    URL: http://dx.doi.org/10.1007/BF02458340
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  • 93
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    Organismic sets. Scientific creativity, its spread and applications (1973)
    Springer
    Bulletin of mathematical biology 35 (1973), S. 339-344 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The phenomenon of mental creativity is considered from the standpoint of the theory of organismic sets, developed by the author in a series of previous publications. It is shown how the differences in creativity between different individuals may be interpreted on this basis, and why extreme creativity is rare. A parallel interpretation for facility in observation is given, and it is shown why facility in creativity and observation is much rarer than either individual facility. A further conclusion is drawn regarding the deducibility of the laws of nature by purely logical means.
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    URL: http://dx.doi.org/10.1007/BF02458341
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  • 94
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    Quantum theory of time-varying stimulation in nerve (1973)
    Springer
    Bulletin of mathematical biology 35 (1973), S. 359-374 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The equation for the quantum transitions (spontaneous and stimulated) of membrane dipoles is solved for the various forms of time-varying stimulation in nerve. From the condition of ever-increasing dipole population in the upper state, the threshold for excitation is determined in each case. The results obtained are in agreement with the established facts. The optimum frequency for stimulation is given asv 0=0.0615/T 2 whereT 2 is the dipole relaxation time. The feature of the theory is that the mathematical formulation is based upon a physical mechanism and the results can thus provide some understanding in the observed phenomena.
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    URL: http://dx.doi.org/10.1007/BF02458343
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  • 95
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    On the unreliability of judgments of certain social situations by minority groups (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 413-416 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In certain situations like the aftermath of a revolution when discontent rises amongst certain groups of the population, it is frequently observed that the discontented groups are firmly convinced that their point of view is shared by the majority of the population. Yet future events prove that this is far from being the case. This effect is partly attributable to “wishful thinking,” partly to a purely social mechanism. The wishful-thinking effect may be considered as a case of psycho-physical discrimination in which a bias is introduced proportional to the degree of satisfaction anticipated from a given situation. H. D. Landahl's well-known equations can be applied to this case. The social factor is based on the circumstance that an individual associates by preference with such other individuals as have similar opinions. This results in an actual error of estimation of the relative minority or majority because of different frequencies of contact with individuals of the two opposing groups. Both factors may be combined into one equation.
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    URL: http://dx.doi.org/10.1007/BF02476496
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    Announcements (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 421-422 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476499
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  • 97
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    A general target theory of radiobiological action: II (1963)
    Springer
    Bulletin of mathematical biology 25 (1963), S. 367-385 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In Part II we prove some of the more complicated theorems stated and used in Part I. In particular, we derive the distribution functionsD 1,D 2, andD 3, and prove some of their properties under various limiting conditions.
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  • 98
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    On the theory of fixation of isohemagglutinins on the erythrocytes (1963)
    Springer
    Bulletin of mathematical biology 25 (1963), S. 387-392 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The graphical treatment utilized by Marmasse in order to test “Wurmser’s theory of agglutination” has been applied, taking into account all the data available. Contrary to Marmasse’s conclusion, the application of this graphical method is not a valid argument against the theory.
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    On the Fokker-Planck equation in the stochastic theory of mortality: II (1963)
    Springer
    Bulletin of mathematical biology 25 (1963), S. 343-366 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This is the continuation of part I, which was published in the September, 1963, issue ofThe Bulletin. Section 5 treats the special case in which the left absorbing barrier recedes to −∞, leaving essentially only one barrier at a finite distance Λ (〉0) from the origin. The eigenfunctions are now parabolic cylinder functions. The limiting cases Λ→+∞ and Λ→0 are also considered. Though meaningless for practical applications to our problem, they are of interest, mathematically, because the Green’s function for the solution of the Fokker-Planck equation assumes a particularly simple form. In section 6 we study, by means of an example, how the “force of mortality” may vary with time before attaining its final asymptotic value. Section7, still dealing with only one absorbing barrier, shows that our results for “strong homeostasis” are identical with those derived by Chandrasekhar for the escape of particles through a potential barrier in the limiting case of quasi-static flow. Precise conditions are given for the validity of both the quasi-static and the Smoluchowski approximations to the Fokker-Planck equation. Finally, in section 8, a brief mention is made of Gevrey’s method for the solution of parabolic partial differential equations.
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    URL: http://dx.doi.org/10.1007/BF02476563
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    Studies in stochastic learning theory: I path-dependent linear models for experimenter-subject controlled experiments (1963)
    Springer
    Bulletin of mathematical biology 25 (1963), S. 393-419 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The derivation of learning models relative to choice behavior in experimenter-subject controlled experiments with two outcomes (right or wrong) is considered from the point of view that any such model must satisfy a criterion of optimality. The criterion adopted for investigation, termed optimal asymptotic behavior, is that of the subject asymptotically learning which of the alternatives has the greater probability of being correct. A class of path-dependent linear models is posed as possible candidates. It is shown that no members of this class satisfy the criterion although two of them approach it by making a learning parameter small enough. The possible implications of this are discussed.
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    URL: http://dx.doi.org/10.1007/BF02476566
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