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  • Springer  (220,733)
  • American Physical Society  (30,292)
  • 1965-1969  (165,841)
  • 1955-1959  (71,343)
  • 1945-1949  (13,841)
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  • 101
    Electronic Resource
    Electronic Resource
    Studies on the electroretinogram. II. Model of an a-wave regenerator mechanism (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 223-233 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A model is proposed to relate the regeneration of the ERGa-wave after partial light adaptation to the level of the light adaptation. The model assumes that thea-wave amplitude is a function of some reactive substance associated with ana-wave generator. The maximuma-wave amplitude occurs when the eye is fully dark adapted, and thea-wave generator initiator concentration is at a maximum. Thea-wave generator initiator concentration can be decreased by interacting with a product of the rhodopsin-light energy reaction, and increased by removal of this inhibitor. The removal of the inhibitor depends upon the isomerization of the all-trans-retinene to the 11-cis form. An excess of inhibitory material overa-wave generator initiator would cause a delay in the appearance of thea-wave until the excess inhibitory material is removed. This delay is a linear function of the logarithm of the adapting energy. The agreement of this model with the experimental ERG data is very good.
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    URL: http://dx.doi.org/10.1007/BF02477279
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  • 102
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    On the quantitative relationship between survival rate of larvae and their food supply (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 215-222 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The survival rate of fishes in their earlier stages of development and the influencing factors present one of the most fundamental problems of fish population dynamics. After I. Hjort's (Cons. L.'explor. Ner.,20, 3–228, 1914) work, there have been many investigators in this field and there is no doubt about the very important role of ova and larvae mortality in the fate of a given fish generation. Less clear are the ideas concerning factors determining the high mortality of fishes in their earlier stages of development; especially the factor of food supply of larvae during the period of transition to exogenic nutrition. The value of this factor has been estimated differently from different points of view. For example, R. J. H. Beverton and S. J. Holt (On the Dynamics of Exploited Fish Population, 1957) have given to the food supply factor its deserved importance. On the other hand, T. V. Dekhnik (Trudy Sevastopolskoi Biologicheskoi Stantsii,13, 216–244, 1960;Ibid.,14, 222–243, 1961) has proved in her investigations that at least for pelagic larvae of Black Sea fishes there is an excessive amount of food, and that therefore food cannot play an important role in larva survival. Not wanting to stop to review the literature of the problem (see Dekhnik,Trudy Sevastopolskoi Biologicheskoi Stantsii,13, 216–244, 1960), we will only remark that the problem as a whole needs further investigation. Not only new data are needed, but also methods for following up analysis have to be worked out.
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    URL: http://dx.doi.org/10.1007/BF02477278
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  • 103
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    Learning fluctuations and hysteresis phenomena (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 253-259 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The investigation described here is anexperimental one which brings to light some new facts and confirms others already reported. They partly concern the hysteresis phenomena handled by N. Rashevsky (Mathematical Biophysics, 1960) and partly provide a point of departure for future biophysical research to be undertaken by biomathematicians.
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    URL: http://dx.doi.org/10.1007/BF02477281
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  • 104
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    An analysis of aortic pressure curves taking into account visco-elastic properties of the aorta and variations in peripheral resistance (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 27-52 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The aortic pressure curve necessarily reveals the mechanical properties of the aorta and peripheral resistance as well as of the dynamics of blood flow. The present study uses a reasonable model of visco-elastic properties of the aorta, a reasonable form for variations in peripheral resistance and blood flow to predict an aortic pressure tracing. Numerical values of constants measured experimentally were available in the published literature. These were used in the nonlinear differential equations of motion of the system under analysis. The equations yielded to piece-wise solution, giving the aortic circumference and the aortic pressure as functions of time. The form of both curves resembles clinical tracings, but numerical values of circumference were higher and of pressure lower thanin vivo. The discrepancies between predicted and clinical curves may reveal certain inadequacies in published measurements on visco-elastic constants. These measurements have been made on longitudinal rather than circumferential strips often containing dead rather than living muscle. The discrepancies, therefore, indicate specific gaps in our knowledge of aortic behaviorin vitro. The suggested model of the system aided in the design of experiments which could supply data necessary to substantiate or to revise the model.
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    URL: http://dx.doi.org/10.1007/BF02476466
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  • 105
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    The venous circulation (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 373-377 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Some aspects of the circulation through the veins remain unexplained. The pressure gradient which ordinarily exists across a large vein, for example, is much greater than that necessary to maintain the same flow through a rigid tube of identical diameter (Brecher, 1956; Starling and Evans, 1962). During inspiration, blood flow through the thoracic portion of the inferior vena cava increases markedly, while that through the distal abdominal portion does not change. Furthermore, an active source of pressure drop in the chest is necessary to maintain venous flow. For the open chest the pressure drop occurs mainly during ventricular contraction, while in the closed chest it is produced chiefly by inspiration. The present study indicates that the high distensibility of the veins accounts in significant degree for the behavior characteristic of the venous circulation.
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    URL: http://dx.doi.org/10.1007/BF02476842
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  • 106
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    A logical model for growth and differentiation in multi-celled organisms (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 379-387 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper is an attempt to provide a logical model for the process of growth and differentiation in a multi-cellular organism. More specifically it is intended to show how genetic information relating to macroscopic structure and coded in the form of a logical tree could be progressively embodied in the organism as it develops by repeated division from a single cell. The aim is to establish biological analogies rather than mathematical interest, and reproduction, adaption, and the coordinating action of hormones are discussed within the general logical framework.
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    URL: http://dx.doi.org/10.1007/BF02476843
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  • 107
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    A mathematical model of the urethra (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 407-415 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Models having the form of surfaces of revolution may be used to represent the urethra under pre-voiding pressure. From such models are derived formulas for calculating muscle tension from the shape of a urethragram.
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    URL: http://dx.doi.org/10.1007/BF02476845
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  • 108
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    Estimation of secretory rates of hormones from the specific activities of metabolites which have multiple secreted precursors (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 389-406 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The calculation of rates of entry of material into an open system of multiple pools in the steady state from the specific activities of end products, which may be derived from several pools, is described. This analysis may be applied to estimate the rates of secretion of steroid hormones from the specific activities of urinary metabolites which may have various hormones as common precursors. In a previous publication (Gurpideet al., 1963) formulae have been presented by which secretory rates could be calculated after a single injection of the tracers assuming that each of the urinary metabolites was uniquely derived from one of the pools in the system. In the present article similar formulae were derived without this assumption. Consequently, it is shown that, under certain circumstances, non-uniquely derived metabolites can be used to estimate secretory rates, and that it may be unnecessary to consider the pathways of conversion of the hormones to the metabolites or the sites where these conversion occur.
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    URL: http://dx.doi.org/10.1007/BF02476844
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  • 109
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    The use of two radiotracers for transfer-rate determinations in closed two compartment systems (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 431-434 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The sensitivity and “specificity” of measurements for the determination of transferates are enhanced by the use of an additional radiotracer, serving to trace the unlabelled substance. This method presents advantages mostly in systems outside their steady state but only exeptionally in steady state systems.
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    URL: http://dx.doi.org/10.1007/BF02476847
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  • 110
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    Analysis of tracer experiments V: Integral equations of perturbation-tracer analysis (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 417-429 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract An integral equation approach to perturbation-tracer analysis in steady-state multicompartment systems is formulated. The theory is developed for δ function perturbation and tracer inputs and extended to the case of continuous small perturbations and continuous tracer inputs. It is shown that the first order dependence of the initial entry function can then be expressed by means of an integral equation: $$B_1 (t) = \int_{t_2 = - \infty }^\infty {\int_{t_1 = - \infty }^\infty {P(t_1 )T(t_2 )B_1 (t - t_2 ,t_1 - t_2 )dt_1 dt_2 } } $$ whereB 1(t) is the first order initial entry function for the tracer material,P(t1) the perturbation function.T(t 2) is the tracer input function, andB 1(t−t 2 ,t 1 −t 2 ) is a continuous function of two variables characterizing the first order perturbation-tracer response of the system.
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    URL: http://dx.doi.org/10.1007/BF02476846
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  • 111
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    Correlation “waves” in brain-like structures (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 435-447 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A correspondence is established between a tangible model of brain structure (and function) and a system of observer-observed interactions. The observed quantities are “stimuli” in the form of signal amplitude distributions in a mass of neuron-like units; the observer is a set of neurons (not circumscribed in a local region) in which a distributed parameter mirrors the stimulus history of the set, i.e., represents a “memory”. Utilizing the theory of the Perceptron, a contemporary brain model, it is demonstrated that large systems composed of many observer-observed interactions exhibit quantum mechanical behavior on a “macroscopic” scale. This behavior entails wave-like phenomena and the need of applying the superposition mechanics to system information content calculations.
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    URL: http://dx.doi.org/10.1007/BF02476848
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  • 112
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    Mathematical models for cellular systems. The von foerster equation. Part II (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 449-471 
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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, 1965, issue of theBulletin. The birth rate, α(t), is now assumed to be a linear functional of the age density,n. This gives a simple model of self-replenishing stem cell compartments, and leads to a necessary condition for the existence of a steady state. Some examples are presented to illustrate the formalism. They include: (a) An equivivant population with life spanD and no losses from death or migration. The total number of cells is multiplied by 2 in each time intervalD. As a special case, frequently realized in practice, the population may be increasing exponentially with time (“log-phase” of growth). (b) A compartment with “random” emigration of cells and gamma distribution of life spans. (c) An oversimplified version of L. G. Lajtha’s model describing stem cell kinetics. In section IV a simple case in which the loss function depends explicitly onn is discussed very briefly.
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    URL: http://dx.doi.org/10.1007/BF02476849
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  • 113
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    Mathematical models of nonuniform intrapulmonary gas distribution (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 473-476 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary Mathematical models of nonuniform gas distribution in the lungs which assume a two-chambered lung to be ventilated through a third chamber, i.e. a common dead space, have led to diverging results. A breath-by-breath analysis of such a system results in a two-exponential solution whereas a continuous ventilation analysis gives a three-exponential solution. This is caused by the different assumptions made in the two models about the composition of dead space gas. In the breath-by-breath analysis one assumes that theN 2 content of the dead space is zero at the end of inspiration. In the continuous ventilation model one assumes that theN 2 content in the dead space is unknown at all instants during the breathing cycle. No physical significance should be attached to any chamber in this type of analysis. The continuous ventilation model provides a more general solution than the cyclical ventilation model, because the former treats the common dead spaces as an independent unknown.
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  • 114
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    On the electrocardiographic field equation (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 493-495 
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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/BF02476852
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  • 115
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    The representation of organisms in terms of predicates (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 477-491 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The different approaches to relational biology developed by N. Rashevsky and R. Rosen consider essentially binary relations between various components of biological functions of the organism. Actually an organism is represented by a set of differentn-ary relations. The present paper is an attempt to outline a possible approach to this more realistic situation. Inasmuch asn-ary relation is ann-place predicate, it is attempted to describe the basic known properties of an organism in terms ofn-place predicates, in which the variables represent the different “components” of the organism. Some possible forms of such predicates are discussed and some general properties of systems of such predicates are studied. It is shown that if the organism is described by predicates of the type discussed here, statements can be derived about the conditions “of reestablishability” of different components. Conclusions similar to those obtained previously by R. Rosen are reached now on a very different basis. A description of the process of cell differentiation in multicellular organisms in terms of predicates studied here is briefly outlined. A comparison of similarities and differences between the approach and Rosen’s description of organisms in terms of the theory of categories is made.
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  • 116
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    Temporal organization in cells (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 497-500 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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  • 117
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    29th National Meeting of the Operations Research Society of America (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 503-503 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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  • 118
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    Harold A. White March 4, 1933–July 27, 1965 Obituary notice (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 501-502 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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  • 119
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    Etude des premiers etages sensoriels (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 1-10 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Résumé Les premiers étages sensoriels sont étudiés en utilisant notre modèle de neurone et en supposant que les réseaux responsables de la perception sont particulièrement solides, stables, économiques. Nous montrons que les premiers neurones doivent être spontanément périodiques et autorégulés. La nécessité fonctionnelle des premiers étages de la voie visuelle est démontrée. Par analogie, nous étudions la voie auditive.
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  • 120
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    The flow of a viscous liquid in a converging tube (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 11-24 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The problem of the viscous flow of an incompressible Newtonian liquid in a converging tapered tube has been solved in spherical polar coordinates. The method of the solution involves the Stokes' stream function and a technique introduced by Stokes in the study of a sphere oscillating in a fluid. The theory for the flow in a rigid tube includes: (1) the pulsatile flow with both radial and angular velocity components; (2) the steady state flow with both radial and angular velocity components and (3) the very slow steady state flow with only a radial velocity component present. For a tapered elastic tube, the velocity of the propagated pulse wave is determined. The solution given is in terms of the elastic constants of the system and the coordinates for this type of geometry. The pulse velocity is then related to the velocity in an elastic cylindrical tube with the necessary correction terms to account for the tapered tube.
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  • 121
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    Some multi-dimensional branching processes as motivated by a class of problems in mathematical genetics I (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 25-50 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In this paper a class of branching processes applicable to populations reproducing by some asexual means or by a simple selfing system of mating is studied. The paper is divided into three parts. In part one the mathematical model is introduced, part two is a mathematical analysis of the model, and in part three concrete applications and examples are given. Many of the proofs of the theorems in part two are omitted but will appear in a subsequent issue of theBulletin.
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  • 122
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    Design of neural nets with intermittent response and certain other relevant studies (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 51-74 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The application of the earlier results (Pavlidis, T. 1965. “A New Model for Simple Neural Nets and its Application in the Design of a Neural Oscillator.”Bull. Math. Biophysics,27, No. 2, 215–229) to the design of more complex neural nets is attempted. The following cases are considered: 1. Chains of neurons where it is proven that the frequency of the output pulses does not depend on the value of the input as long as it is above a certain threshold. 2. Groups of neurons with backward inhibition which present an intermittent mode of operation. 3. Neural nets with periodic facilitation which permit time sharing of certain components for different functions. 4. A neural net which can detect the sign of the input even if the main receptor is sensitive only to the absolute value of it, is presented. 5. A velocity estimating neural net which in combination with one of the nets with intermittent response provides a model for the smooth eye tracking movements.
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  • 123
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    Academic learning, a variational model (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 75-90 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract By assigning coordinates to the information space comprising all knowledge, rigorous mathematical interpretations can be placed on such terms as academic ability, memory and creativity such that these psychometric concepts can be incorporated into a framework of functional analysis which then permits the optimization of long-term academic learning processes through the location of the teaching trajectories in information space which will maximize the knowledge accumulated in a generalized educational system composed of a complex of subject-pupil-teacher interactions. The concepts of discrete and continuous information spaces are discussed in connection with subject-subject, subjectpupil and pupil-pupil interactions, and the advantages of using variational versus dynamic programming methods of optimizing alternative educational systems are evaluated.
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  • 124
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    On chains of related sets (1966)
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    Bulletin of mathematical biology 28 (1966), S. 103-106 
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    ISSN: 1522-9602
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    Notes: Abstract IfK is a partition of a setK which is partially ordered by the relationR andR is a collection of pairs of sets ofK such that the sets of each pair are related byR in the sense of Rashevsky, thenR is a relation which partially ordersK. Necessary and sufficient conditions thatK be a chain are obtained, and ifK is a chain under these conditions, it is shown thatK is unique.
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  • 125
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    On multipole theory in electrocardiography (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 161-166 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper continues a comparison of the Taylor series and spherical harmonic forms of multipole representations initiated by Yeh (Bull. Math. Biophysics,24, 197–207, 1962). It is shown that while transformations from Taylor series form into spherical harmonic form is always possible, the inverse cannot be accomplished as suggested by Yeh; corrected transformation equations are given. It is also shown that direct measurement of Taylor coefficients, as outlined in Yeh, Martinek, and de Beaumont (Bull. Math. Biophysics,20, 203–216, 1958), is actually not possible. Accordingly, only the spherical harmonic coefficients can be determined by measurement of surface potentials, as in electrocardiography.
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  • 126
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    Some multi-dimensional branching processes as motivated by a class of problems in mathematical genetics II (1966)
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    Bulletin of mathematical biology 28 (1966), S. 181-190 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper is a continuation of a paper, “Some Multi-Dimensional Branching Processes as Motivated by a Class of Problems in Mathematical Genetics I,” by C. J. Mode, which appeared in a previous issue of theBulletin. Its purpose is two-fold; namely to discuss the mathematical existence of the model and to supply the mathematical proofs of some theorems in section two of the paper mentioned above. This paper should be read in conjunction with the previous paper.
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  • 127
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    A resolution of Rosen’s paradox for self-reproducing automata (1966)
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    Bulletin of mathematical biology 28 (1966), S. 191-194 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract Rosen (Bull. Math Biophysics. 1959) has argued that a self-reproducing automaton of the type originally described by von Neumann is impossible because of a logical paradox inherent in its definition. The paradox is resolved by explicitly allowing errors (mutations) in the system and thus introducing evolution. There is no paradox in an automaton, originating from a slightly different ancestor through mutation. The von Neumann model thus becomes realistic and useful for a discussion of biological phenomena.
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  • 128
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    Ventricular septal defect with bidirectional shunting: Mathematical considerations (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 167-179 
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    ISSN: 1522-9602
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    Notes: Abstract Previously proposed formulae for the quantitative estimation of bidirectional shunts across ventricular septal defects require determination of the oxygen contents of mixed venous, pulmonary artery, pulmonary venous, and aortic blood. Because these formulae do not take into account the mixing of oxygenated with unoxygenated blood within the ventricles, their use must result in underestimation of shunt flows in each direction. A mathematical model for a ventricular defect is examined, in which it is assumed that mixing of blood occurs in each of six sites in the venae cavae or right atrium, right ventricle, pulmonary artery, left atrium, left ventricle, and aorta. A total of fourteen streams of blood can flow from one to another of these mixing sites. As long as complete mixing occurs in the six specified mixing sites, any degree of mixing or non-mixing of the various streams is permitted. From the equations characterizing the model, formulae are derived in which the shunt flow in each direction is expressed in terms of the oxygen contents in the six mixing sites and the fractions of blood which enter the shunt from either side without prior mixing in a ventricular mixing site. The previously reported formulae, which apply when no ventricular mixing is allowed to occur, lead to theoretical minimum values for the shunt flows in each direction. At the opposite extreme where all the shunting blood is required to mix in a ventricle before entering the shunt, formulae for maximum possible shunt flows are also obtained. The absolute values for the left-to-right and right-to-left shunt flows, which must lie somewhere between the theoretical maximum and minimum values, cannot be computed from blood gas data alone.
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    Control of the contractility of the myocardium in a feedback system (1966)
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    Bulletin of mathematical biology 28 (1966), S. 195-205 
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    Notes: Abstract Experimental evidence strongly suggests that the contractility of the intact heart in situ, in contrast to that of striated muscle elsewhere in the body, is controlled in a close-cycle system. Thus, the variation of intraventricular pressure during systole follows a complex pattern, whose relative form remains quite constant regardless of the duration of ejection. By use of the single-chambered model of the cardiovascular system, a mathematical representation of a feasible feedback mechanism is developed. The requirement that the feedback system must satisfy mathematical principles eliminates relationships apparently reasonable from a physiological viewpoint. A clinical application which the mathematical development suggests is that early arterial hypertension may arise from an abnormal feedback mechanism with excessively large cardiac output in the initial portion of systole.
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    On the shape of a DNA molecule (1966)
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    Bulletin of mathematical biology 28 (1966), S. 207-216 
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    Notes: Abstract Due to the lack of direct X-ray evidence for base pairing being the only mechanism for the formation of double helix in a DNA crystal, an alternative explanation is suggested so that the observed DNA loop becomes essential.
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    A population growth model incorporating delayed responses (1966)
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    Bulletin of mathematical biology 28 (1966), S. 219-233 
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    A note on group ring expression of self-fertilized population (1966)
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    Bulletin of mathematical biology 28 (1966), S. 217-218 
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    Notes: Abstract Validity of group ring expression of selfed population is shown for cases in which there are differences in recombination probabilities between two sexual sides of a plant.
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    Analysis of tracer experiments: VII. General multi-compartment systems imbedded in non-homogeneous inaccessible media (1966)
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    Bulletin of mathematical biology 28 (1966), S. 261-282 
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    Notes: Abstract An integral equation analysis of generaln compartment steady state systems imbedded in static media of arbitrary complexity has been developed. A set of initial entry functions can be found which serve to determine a corresponding set of partitioned initial entry functions. The partitioned functions, in turn, can be used to predict the probabilities and time courses of various transport histories and to determine all steady state rates of flow between measured compartments. The method is quite general, being completely applicable, for example, to closed systems, to cyclic systems and to systems in which relatively rapid (but finite) exchange between compartments occurs.
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    A note on relations between sets (1966)
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    Bulletin of mathematical biology 28 (1966), S. 309-313 
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    Notes: Abstract From the definition of a strong and weakn-ary relation betweenn sets, given in a previous paper (Bulletin of Mathematical Biophysics,27, 477–492), it follows that for a given set ofn sets and givenn-ary relationR between them there can exist only one strong relation, but a large number of weak ones. An expression for the total number of possible weak relations is derived and the notion of the degree of weakness of a relation is introduced and discussed.
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    Reconstruction of protein and nucleic acid sequences: IV. The algebra of free monoids and the fragmentation stratagem (1966)
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    Bulletin of mathematical biology 28 (1966), S. 235-260 
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    Notes: Abstract The problem of determining the sequence of a biopolymer from its fragments is stated in mathematical terms. Using concrete properties of a free monoid, certain general classes of biopolymers are shown to be insolvable from fragment data produced by complete digestion where enzymes specific for any possible combination of chemical bonds are employed.
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    Physics, biology, and sociology: A reappraisal (1966)
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    Bulletin of mathematical biology 28 (1966), S. 283-308 
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    Notes: Abstract To the extent that all biological phenomena are perceivable only through their physical manifestations, it may be justified to assume that all biological phenomena will be eventually represented in terms of physics; perhaps not of present day physics, but of some “extended” form of it. However, even if this should be correct, it must be kept in mind that representing individual biological phenomena in terms of physics is not the same as deducing from known physical laws the necessity of biological phenomena. Drawing an analogy from pure mathematics, it is possible that while every biological phenomenon may be represented in terms of physics, yet biological statements represent a class of “undecidable” statements within the framework of physics. Such a conjecture is reinforced by the history of physics itself and illustrated on several examples. The 19th century physicists tried in vain todeduce electromagnetic phenomena from mechanical ones. A similar situation may exist in regard to biological and social sciences. Quite generally, the possibility of representing a class B phenomena in terms of class A phenomena does not imply that the phenomena of class B can be deduced from those of class A. The consequences of the above on the relation between physics, biology, and sociology are studied. A tentative postulational formulation of basic biological principles are given and some consequences are discussed. It is pointed out that not only can the study of biological phenomena throw light on some physical phenomena, but that the study of social phenomena may be of value for the understanding of the structures and functions of living organisms. The possibility of a sort of “socionics” is indicated.
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    Re-establishability in abstract biology (1966)
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    Bulletin of mathematical biology 28 (1966), S. 371-374 
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    Notes: Abstract It is shown that any (ℳ ℛ) has some component which cannot be re-established after it has been inhibited. If there is only one such component, it must be central, that is, its inhibition stops the whole system. These results hold even when it is not assumed that ℳ is connected.
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    Restricted transition probabilities and their applications to some problems in the dynamics of biological populations (1966)
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    Bulletin of mathematical biology 28 (1966), S. 315-331 
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    Notes: Abstract In this paper a theory of a class of restricted transition probabilities is developed and applied to a problem in the dynamics of biological populations under the assumption that the underlying stochastic process is a continuous time parameter Markov chain with stationary transition probabilities. The paper is divided into three parts. Part one contains sufficient background from the theory of Markov processes to define restricted transition probabilities in a rigorous manner. In addition, some basic concepts in the theory of stochastic processes are interpreted from the biological point of view. Part two is concerned with the problem of finding representations for restricted transition probabilities. Finally, in part three the theory of restricted transition probabilities is applied to the problem of finding and analyzing some properties of the distribution function of the maximum size attained by the population in a finite time interval for a rather wide class of Markov processes. Some other applications of restricted transition probabilities to other problems in the dynamics of biological populations are also suggested. These applications will be discussed more fully in a companion paper.
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    A mathematical theory of enamel solubility and the onset of dental caries: II. Some solubility equilibrium considerations of hydroxyapatite (1966)
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    Bulletin of mathematical biology 28 (1966), S. 433-441 
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    Notes: Abstract Equilibrium solubility considerations are presented based on the assumption that equating the kinetic expressionq, developed in part I, to zero can describe the equilibrium or steady state between hydroxyapatite and salt solutions. From this expression is derived Hodge's empirical equilibrium equation,C=KH. Further, a lograithmic transformation of this equation results in an expression that accounts for the equilibrium calcium, phosphorus andpH relation found by Levinskas and Neuman. Finally, it also shows the relation between log (C·P) andpH necessary for typical artificial carious lesions as found by Coolidge, Besic and Jacobs. A discussion of a recent theory of hydroxyapatite solubility of LaMer reveals calculation errors that vitiate his results. It is shown that logK 1 (K 1 is the ratio of the rate constants inq and can serve as a solubility equilibrium constant for hydroxyapatite) varies by only 1.2 units when calculated from three diverse sets of data. This variation is less than that reported by LaMer (when the errors of calculation in that work are corrected) and considerably less than the range of 11 among attempts to calculate a conventionalpK sp , as summarized by Hodge.
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    On mass behavior (1966)
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    Bulletin of mathematical biology 28 (1966), S. 465-475 
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    Notes: Abstract In imitative behavior, as studied previously by N. Rashevsky (Mathematical Biology of Sociol Behavior, Chapter XIII, The University of Chicago Press, 1950), the reason for the majority of a society to accept a particular behavior is based on purely voluntary action (band-wagon effect). In the present paper effects of coercion of the majority by a small minority group which poses the means for coercion, are studied. More general types of equations are thus obtained and threshold effects found, which bear a resemblance to some such effects studied previously.
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    A mathematical theory of enamel solubility and the onset of dental caries: III. Development and computer simulation of a model of caries formation (1966)
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    Bulletin of mathematical biology 28 (1966), S. 443-464 
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    Notes: Abstract Part III attempts to develop a diffusion controlled model of caries in the intact enamel employing the kinetic results of the previous two parts. A model of the enamel as a granular bed with a diffusible organic matrix filling the interstices is considered. The basic equations of diffusion and simultaneous reaction are developed under the assumption that all the reactions are so rapid as compared with the diffusion rate, that they are in a quasi-equilibrium state. The resultant system of seven coupled, non-linear parabolic partial differential equations is of such complexity that only numerical solutions could be attempted. Stability restrictions inherent in the problem dictated the use of the DuFort-Frankel numerical solution for parabolic boundary problems. Numerical solutions giving the concentration of all reactants, the rate of mineral loss, and the enamel porosity were obtained for a variety of boundary conditions. It is found that departure from the equilibrium condition expressed in part II is necessary for the occurrence of an attack on the enamel. The rate and pattern of penetration is then determined primarily by the concentrations of undissociated buffer, and salts, together with the rate of diffusion in the surrounding medium. The possibility of a relatively intact surface layer persisting over a demineralized subsurface region due solely to the composition of the demineralizing medium is noted. Remineralization behavior in portions of the carious lesion occurs in the model under certain boundary conditions.
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    A model for neuron firing with exponential decay of potential resulting in diffusion equations for probability density (1967)
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    Bulletin of mathematical biology 29 (1967), S. 233-243 
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    Notes: Abstract A neuron is assumed to receive synaptic input of both excitatory and inhibitory natures from a large number of neighboring neurons; it is also assumed that a large number of such impulses are required to raise the neuron’s transmembrane potential to its threshold potential, at which it “fires” or “spikes”. The model is similar to one of Gerstein and Mandelbrot, except that in the absence of input an exponential decay of potential toward a resting level is introduced. Computational methods of determining the spike timeinterval distribution are discussed, along with the inverse problem of estimating the parameters of the system from observed spike time-interval data.
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    Pulse velocities in an orthotropic elastic tube (1967)
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    Bulletin of mathematical biology 29 (1967), S. 311-318 
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    Notes: Abstract Moens-Korteweg relations are developed for the velocity of wave propagation through an orthotropic elastic tube based on the three dimensional equations of elasticity. Numerical examples are presented for the femoral artery of a dog and several other orthotropic materials. These results are compared with those obtained from the equations of motion for an orthotropic elastic medium.
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    An investigation of basic assumption in enzyme kinetics using results of the geometric theory of differential equations (1967)
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    Bulletin of mathematical biology 29 (1967), S. 335-341 
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    Notes: Abstract Up to the present time, the following property of the product component in the reversible one substrate-one intermediate-one product enzymic mechanism has been taken only as anassumption, viz., during the course of the reaction, the time-rate of change of product concentration is never negative and the product concentration never exceeds its equilibrium value. Applying the methods of the geometric theory of ordinary differential equations it is shown that this result follows as a direct deduction from the differential equations governing the mechanism together with the initial conditions. Further, the nature of the equilibrium point as a stable node is established.
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    Analysis of tracer experiments: VIII. Integro-differential equation treatment of partly accessible, partly injectable multicompartment systems (1967)
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    Bulletin of mathematical biology 29 (1967), S. 319-333 
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    Notes: Abstract An integro-differential equation treatment of multi-compartment systems is developed which permits formal analysis of the incomplete data which is available from partly accessible, partly injectable systems. New transport functions are defined which can be obtained directly from the experimental data. These functions serve to characterize the communication and topology between different accessible compartments and also the reentrant contributions from inaccessible sites. The method gives solutions consistent with those of the differential equation approach when the system is uniformly contiguous and accessible, more complete solutions than those of the integral equation approach when all measured compartments are injectable, and in addition provides complete or partial solutions for certain otherwise analytically intractable systems. Detailed numerical illustrations of the method are given.
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    General two factor models (1967)
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    Bulletin of mathematical biology 29 (1967), S. 405-405 
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    Mathematical theory of the possible role of intercellular fluid and of vascularization on physiological periodicities (1967)
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    Bulletin of mathematical biology 29 (1967), S. 395-401 
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    Notes: Abstract In continuation of previous work (Rashevsky,Some Medical Aspects of Mathematical Biology, Springfield, Ill.: Charles C. Thomas, 1964, Chap. 23 and Appendix 14), the study of the effects of the physical parameters of the cells of endocrine glands on the onset of sustained periodical oscillations in the interaction between the anterior pituitary and the thyroid hormones is generalized to include the possible effect of the intercellular fluid and of the degree of vascularization. Some conclusions of the previous study remain valid although some modifications must be made. A decreased relative volume of the intercellular fluid and an increased vascularization favor the conditions for sustained oscillations. The permeability of the cells and the permeability of the capillaries appear explicitly in the expressions which show the conditions for sustained periodicities.
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    Equilibrium conditions in polysomic inheritance for a panmictic population (1967)
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    Bulletin of mathematical biology 29 (1967), S. 437-449 
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    Notes: Abstract A general model of a large 2m-ploid breeding population, withr loci ands h alleles at the h th locus is considered. It is assumed that the population is bisexual, non-overlapping and breeds by random mating. The genotypic structure of the population is presented as a bilinear form in the gametic output vectors where the genotype distribution is in the matrix form. Using the concept of the segregation distribution, the genotype proportions in the (n+1)st generation are given. An equilibrium condition for random chromosome segregation is obtained in terms of gene frequencies.
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    The blood flow of the brain (1967)
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    Bulletin of mathematical biology 29 (1967), S. 541-548 
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    Notes: Abstract The rigidity of the skull and the inertial characteristics and incompressibility of its contents cause the elastic cerebral arteries and veins to act over brief periods of time like rigid tubes of relatively small diameter. Poiseuille's law is applicable to their behavior. The use of this law, in combination with the fact that, during brief intervals, the total volume of the cerebral arteries and veins remains constant, permits derivation of a mathematical expression for the average arterial flow in terms of an average arterial radius. The differentiated equation has five positive roots which represent maxima and minima of the average flow in terms of the average arterial radius. The theoretical results have physiological implications and potential clinical usefulness, which are discussed.
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    Erythrocyte distribution in arterial blood flow: I. Basic equations and anisotropic effects (1967)
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    Bulletin of mathematical biology 29 (1967), S. 549-563 
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    Notes: Abstract Equations are derived for the flow of an anisotropic fluid in a tube. It is argued that these provide a model for arterial blood flow. Particular attention is paid to the effect of radial differences in hematocrit. Sequels to this paper (Bull. Math. Biophysics,29, 565–574; forthcoming, 1967) will respectively demonstrate possible wall-directed forces on the erythrocyte and enlarge on the physiological consequence of hematocrit variations. The present article develops the basic equations and explores the possible role of anisotropic effects in blood flow.
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    A re-evaluation of X-ray diffraction patterns of DNA (1967)
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    Bulletin of mathematical biology 29 (1967), S. 575-581 
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    Notes: Abstract Concentric circular lines are present on X-ray diffraction patterns of DNA. They cannot be accounted for by the double helix model. It is suggested that they may indicate the presence of double helical side chains.
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    A model for circadian clocks (1967)
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    Bulletin of mathematical biology 29 (1967), S. 781-791 
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    Notes: Abstract An extension of an earlier model simulating the effects of light on the drosophila eclosion rhythm is presented. The effects of variable light intensity are described. This allows not only the simulation of certain experiments not covered by the earlier model, but also it permits an extension of the model to other organisms. By changing only its sensitivity to light the model simulates the phase response curves of certain mammals as well as Aschoff’srule.
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    A note on the derivation of the stevens power law (1967)
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    Bulletin of mathematical biology 29 (1967), S. 827-829 
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    Notes: Abstract Previous derivations of the Stevens Power Law relating loudness to intensity assume, in addition to level invariance, that the relation must be analytic or at least differentiable. This last condition is replaced here by the weaker one of requiring only continuity.
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    Theoretical considerations on potentiation in drug interaction (1958)
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    Bulletin of mathematical biology 20 (1958), S. 1-23 
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    Notes: Abstract To account for some of the more important aspects of drug interaction we shall consider a model which can also account for certain general properties of the action of a single drug. A simple model in which there may be enzymatic detoxification of a drug is studied theoretically. The relation between time for appearance of an effect due to the drug and the size of the dose is found to contain the same parameters as the relation between the effectiveness of paired doses and the interval of time between doses. A similar situation holds when the drug is given at a constant rate. When two drugs are administered together, their effect will depend on the manner of interaction, how much of each drug is given, which is given first, and on the interval of time between each administration. A number of plausible types of interaction is considered theoretically in terms of the model, analytical expressions being given for a number of cases. The interaction may be synergistic or antagonistic. In the former case the potentiation may be more than or less than additive depending on the order of delivery and on the time between injections. Methods for the estimation of the parameters from data are discussed.
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    Some extensions of the use of matrices in population theory (1959)
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    Bulletin of mathematical biology 21 (1959), S. 13-17 
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    Notes: Abstract This paper extends Leslie's vector and matrix treatment of populations. A simple matrix is given for species in which adult mortality and fertility are independent of age, but in which the juvenile mortality rate differs from the adult. The population vector can be changed into a population matrix. This should allow equations using functions of the size of the population to be developed. Genetic variables such as sex or other polymorphisms can be introduced, and the notation allows different systems of selection or non-random mating to be specified.
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    Steady-state diffusion through a finite pore into an infinite reservoir: An exact solution (1965)
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    Bulletin of mathematical biology 27 (1965), S. 57-65 
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    Notes: Abstract An outline is given of an analysis that leads to an exact solution for the problem of steady-state diffusion through a finite thick pore into an infinite region surrounding the mouth of the pore. From this exact formula a simple expression for the flux is derived. This expression approximates the flux with a relative error of less than 3.42 per cent independently of the ratiol/a wherel is the length of the pore anda its radius. If desired, more accurate expressions for the flux can be obtained from the exact solution.
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    The single file model for the diffusion of ions through a membrane (1965)
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    Bulletin of mathematical biology 27 (1965), S. 79-86 
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    Notes: Abstract The model proposed by A. L. Hodgkin and R. D. Keynes (Jour. of Physiol.,128, 61–88, 1955) for the diffusion of potassium through the nerve membrane is extended to cover an arbitrary number of species of ions with charges not necessarily the same. One type of interference is also investigated.
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    On the theory of the cardiovascular system (1965)
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    Bulletin of mathematical biology 27 (1965), S. 71-83 
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    Notes: Abstract Most theoretical studies of the circulation have focussed on the transmission line properties of arteries. Only a small number of papers have dealt with the circulation as a closed (lumped) system with two pumps connected by the lesser and greater circulation (Beneken, inCirculatory Analog Computers, No. Holland Publ. Co., Amsterdam, 1963; Defares,et al., inCirculatory Analog Computers, No. Holland Publ. Co., Amsterdam, 1963; Grodins,Quart. Rev. of Biology,34, 93, 1959; Guyton,Cardiac Output and its Regulation, Saunders Publ. Co., New York, 1963). F. W. Cope's recent studies in this journal (Bull. Math. Biophysics,22, 19, 1960;23, 337, 1961;24, 137, 1962) deal with essentially the same questions, although here the circuit is not “closed”. We have attempted to extend the analysis of the areflex (closed) circulation. The complete study is reported elsewhere (Defares,et al., Acta Physiol, et Parmac. Neerl., 1963).
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    Equations of heat distribution within the body (1965)
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    Bulletin of mathematical biology 27 (1965), S. 67-78 
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    Notes: Abstract A vector integral equation describing heat distribution within the body has been derived. The factors considered are heat conduction, forced convection via the circulatory system, environmental exchange, metabolic heat production, and change in heat content. The vector partial differential equation and alternative forms incorporating boundary conditions were also developed. A difference equation based on a first-order approximation to the fundamental equations was derived to form the basis of a model for heat distribution within the body. It has been shown that factors involving conduction and convection must be considered independently unless the temperature of the blood flowing from a region of the body is equal to the average temperature of the tissue in that region. If this relation between tissue and blood temperature does exist, only a single temperature from each eleeent is needed to describe the heat distribution. In this latter case, models which ascribe all heat transfer to “equivalent” conduction or to convection can give valid predictions.
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    Determination of equilibrium constants and maximum binding capacities in complexIn vitro systems: I. The mammillary system (1965)
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    Bulletin of mathematical biology 27 (1965), S. 87-98 
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    Notes: Abstract It is shown that in a system containingn types of mutually noninteracting binding sites, the association constants are then roots of annth order polynomial while the maximum binding capacities can be evaluated by solving a set ofn simultaneous linear equations. Thenth order polynomial and the system ofn linear equations are defined in terms of 2n intermediate coefficients, the coefficients being themselves evaluated by substituting 2n sets of appropriate experimental data into an auxiliary system of 2n linear equations. The existence and uniqueness of the solutions are established.
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    Southern Regional Graduate Summer Session in Statistics at Oklahoma State University (1965)
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    Bulletin of mathematical biology 27 (1965), S. 111-111 
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    Important announcement to our contributors and readers (1965)
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    Bulletin of mathematical biology 27 (1965), S. 113-113 
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    A theory of ion transport across cell surfaces by a process analogous to electron transport across liquid-solid interfaces (1965)
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    Bulletin of mathematical biology 27 (1965), S. 99-109 
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    Notes: Abstract A kinetic theory of ion transport across cell surfaces has been developed in a form analogous to the kinetic theory of electron transport across solid-liquid interfaces of biological particles. The ionic theory is based on the observation that, at least in one instance, the voltage-current behavior for ion conduction across a cell surface is describable by the Tafel equation, in analogy to the conduction of electrons across solid-liquid interfaces. The theory predicts that the kinetics of ion transport across cell surfaces should conform to the Elovich rate equation, which is shown to be true for various experimental data.
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    Letter to the editor (1965)
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    Bulletin of mathematical biology 27 (1965), S. 114-114 
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    Mathematical development of a physical model of some visco-elastic properties of the aorta (1965)
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    Bulletin of mathematical biology 27 (1965), S. 115-115 
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    A biographical sketch of Nicolas Rashevsky (1965)
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    Bulletin of mathematical biology 27 (1965), S. 3-4 
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    A letter read to professor Rashevsky at a dinner in his honor December 11, 1964 (1965)
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    Bulletin of mathematical biology 27 (1965), S. 5-10 
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    Some comments on re-estalishability (1965)
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    Bulletin of mathematical biology 27 (1965), S. 11-14 
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    Notes: Abstract The present note consists of two separate but related parts. In the first, a new graphtheoretic proof is presented that an (ℳ,R)-system must always contain a nonreestablishable component. The second considers some questions concerning the relation between re-establishability and the time-lag structure in (ℳ,R)-systems. It is supposed that the reader is familiar with the terminology of the author's previous work on (ℳ,R)-systems, particularly R. Rosen,Bull. Math. Biophysics,20, 245–260, 1958.
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    Model studies of blood-glucose regulation (1965)
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    Bulletin of mathematical biology 27 (1965), S. 21-37 
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    Notes: Abstract A simplified, linearized model of the system regulating blood-glucose concentrations is reviewed. This model, which predicts a damped sine wave response to an oral glucose load, lumps the large number of kinetic parameters into a much smaller number which can, at least in part, characterize the human glucose regulatory system. The predictions based on the model are compared with measurements of blood-glucose and blood-insulin concentrations during the oral glucose-tolerance test. Various other conditions are simulated and their implications are discussed in terms of the mathematical model used.
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    On compartmentalization (1965)
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    Bulletin of mathematical biology 27 (1965), S. 15-19 
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    Notes: Abstract The notion of a compartment is discussed in terms of the Markovian process. From the stochastic matrix (the elements of which are state transition probabilities between different states of a particle of a chemical element), one may find a (generally) nonstochastic matrix; the elements of this second matrix are probabilities that, starting from some initial state, the particle will reach another seleced state (W. Feller, 1962,An Introduction to Probability Theory). Forming equivalence classes of states it can be shown that the equivalence classes based on an equivalence relation, which holds for the elements of the above-mentioned nonstochastic matrix, are essential for the notion of a compartment. From this procedure it is also obvious that a rigorous definition of a physically realizable compartment is impossible. Some conclusions on the practical use of compartmental analysis are drawn.
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    On the use of the Von Foerster equation for the solution and generalization of a problem studied by S. A. Tyler and R. Baserga (1965)
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    Bulletin of mathematical biology 27 (1965), S. 39-48 
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    Notes: Abstract In a population of cells labeled with a single injection of tritiated thymidine at timet=0, it is assumed that a constant fraction, 1−z, of the cells which are potentially able to divide fail to do so, and that the cells which do divide all have identical generation time,D. Death and emigration of cells are neglected. In mitosis, the partitioning of label among the two daughter cells is supposed to follow the binomial probability law. Using the formalism developed by H. Von Foerster the fraction of labeled cells in the total population is computed as a function oft, the time after injection of label. Ift is an integral multiple ofD the results coincide with those of S. A. Tyler and R. Baserga.
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    Development of structure in a society with a dominance relation when new members are added successively (1965)
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    Bulletin of mathematical biology 27 (1965), S. 151-160 
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    Notes: Abstract A society with a dominance relation is considered to be built up by starting with a small society and adding new members in succession. As each member is added he engages in contests with each of the older members to determine the dominance relation between them. The probability that the older member dominates is considered to depend on the size of the society and linearly on the older members score. A recurrence relation for the hierarchy index is derived. The approach of the society to a hierarchical structure is considered for various special cases of this probability. Reasonable assumptions concerning this dominance probability are shown to lead to structures close to the hierarchy. If the new member dominates all the older ones below a certain rank, and is dominated by all those above this rank, then the hierarchy will persist if it is the initial structure, or the structure will tend to hierarchy as the size increases, if it is not the initial structure.
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    Theory of hemoglobin facilitated oxygen transport (1965)
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    Bulletin of mathematical biology 27 (1965), S. 183-190 
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    Notes: Abstract The transport of oxygen in a hemoglobin-saturated medium is theoretically investigated using classical transport theory. It is found that all the chemical complexes can be expressed as a single function of oxygen pressure. A potential difference together with apH shift is predicted to occur across the medium.
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    Notes on magnetic actions upon the nervous system (1965)
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    Bulletin of mathematical biology 27 (1965), S. 203-214 
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    Notes: Abstract Several physical effects (magnetomotive force on ions, magnetic induction of electrical field, magnetic changes of inductance) are quantitatively analyzed in an attempt to attain an insight on how externally applied static magnetic fields influence the activity of the neuron and the Nervous System as a whole or in part. The possible magnetic action on shifting excited zones of the axon appears as most promising for prediction and interpretation of measurable effects. Magnetic fields may modify nervous functions by multiplication and addition of very small biophysical effects.
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    Molecular set theory: II. An aspect of the biomathematical theory of sets (1965)
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    Bulletin of mathematical biology 27 (1965), S. 235-251 
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    Notes: Abstract In an earlier paper (Molecular Set Theory: I.Bull. Math. Biophysics,22, 285–307, 1960) the author proposed a “Molecular Set Theory” as a formal mathematical meta-theoretic system for representing complex reactions not only of biological interest, but also of general chemical interest. The present paper is a refinement and extension of the earlier work along more formal algebraic lines. For example the beginnings of an algebra of molecular transformations is presented. It also emphasizes that this development, together with the genetical set theory of Woodger's and Rashevsky's set-theoretic contributions to Relational Biology, points to the existence of a biomathematical theory of sets which is not deducible from the general mathematical, abstract theory of sets.
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    Man/machine interaction in adaptively controlled experimental conditions (1965)
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    Bulletin of mathematical biology 27 (1965), S. 261-273 
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    Notes: Abstract Systems in which a human subject interacts with an adaptive control mechanism through display and response facilities are examined. A cybernetic model is discussed, together with supporting experimental data.
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    Computer simulation of stem-cell kinetics (1965)
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    Bulletin of mathematical biology 27 (1965), S. 275-290 
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    Notes: Abstract Computer simulation of the stem-cell system has been motivated by a desire to provide a device which may assist the experimenter in his development of complex research strategies in the rapidly developing investigative fields that relate to erythropoiesis and granulopoiesis. A simulation program, written in FORTRAN II for the IBM-7094, is being developed with the requirements of flexibility and broad applicability in mind. The biological model for which it originally was developed differs from those previously advanced by visualizing differentiation into the erythrocytic and granulocytic series as occurring during the post-mitotic phase of a stem cell's growth, the cell's susceptibility to erythropoietic and granulopoietic stimuli varying as its biochemical development unfolds. One might test this model by attempting to demonstrate an asymmetrical interference between erythropoietic and granulopoietic challenges to the stem-cell system. A method for establishing an initial stable configuration of the model is presented. The simulation of the introduction of exogenous erythropoietic stimuli is described. And there is a brief description of the feedback mechanism employed to stabilize the size of the stem-cell population.
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    A note on visual integrative mechanisms (1965)
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    Bulletin of mathematical biology 27 (1965), S. 305-310 
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    Notes: Abstract In a system as complex and as effective as the eye, the cooperative interaction of different mechanisms may be taken as axiomatic. With this as a starting point, various visual phenomena are considered, such as short term memory, eye movements, and flicker fusion. Simple data on mean values lend support to the proposition that the spatial and temporal characteristics of these phenomena are matched with one another. The significance of this for a mechanism of vision is discussed.
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    Matrix algebra of a generalized linear stochastic model. Nonconfluent and general confluent cases (1965)
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    Bulletin of mathematical biology 27 (1965), S. 311-315 
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    Notes: Abstract Matrix algebra is the natural tool for the study of linear stochastic models with many parameters. Complete solutions are given for the nonconfluent and the general confluent cases. It is shown that the axiomatics of a generalized linear stochastic model are naturally described within the framework of the linear algebra in an Euclidean space.
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    Properties of solution of the diffusion-reaction equation (1965)
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    Bulletin of mathematical biology 27 (1965), S. 291-303 
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    Notes: Abstract An arbitrary set of chemical reactions is considered to occur among chemical speciesX i . In a closed uniform reaction system certain linear combinations of the concentrations of theX i are constants. The general construction of all such linear combinations with non-negative coefficients is given in terms of the molecular formulae for theX i . It is shown that to each such linear combination there corresponds another which is a harmonic function when the reactions take place in an open spatially distributed stationary reaction system of arbitrary shape. Under the usual boundary conditions these harmonic functions are constants. With some restrictions upon the diffusion and permeability coefficients these constants are evaluated. This evaluation is the basis for relations between the total concentration of a given chemical group (e.g., the sum of the concentrations of a free molecule, or ion, and its various bound forms) in the reaction system, and in the surrounding medium. The bearing of these relations on apparent active transport is noted and illustrated.
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    Analysis of tracer experiments: VI. Determination of partitioned initial entry functions (1965)
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    Bulletin of mathematical biology 27 (1965), S. 329-332 
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    Notes: Abstract It is shown that the partitioned initial entry functions previously introduced in multicompartment analysis can be directly and uniquely determined from the experimental data.
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    A neural net model for escape learning (1965)
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    Bulletin of mathematical biology 27 (1965), S. 317-328 
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    Notes: Abstract An escape learning situation is discussed in terms of a neural model in which a stimulus can result in a conditioned excitement and a specific conditioned response. By using the simplest relations between the strengths of conditioning and the number of reinforcements and by introducing a distribution of fluctuations occurring regularly in time, one can calculate the probabilities of various responses, as well as the various latencies, in successive trials. The results are in moderately satisfactory agreement with the data of R. L. Solomon and L. C. Wynne (Psychol. Monogr.,67, No. 4, 1953). Consequences of the model for various experimental situations are discussed.
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    Effects of the collateral circulation of the heart (1965)
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    Bulletin of mathematical biology 27 (1965), S. 1-8 
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    Notes: Abstract Collateral circulation minimizes the myocardial injury which results from narrowing of a coronary artery. A large collateral circulation has disadvantages, however. It may divert so much of the limited blood flow through the adjacent nonarteriosclerotic coronary artery that the blood supply of the normal muscle supplied by that artery may be inadequate during heavy exercise. In the presence of a large collateral circulation, both the normal and ischemic regions of the heart may be extremely vulnerable to small arteriosclerotic changes narrowing the patent artery near the aorta. The effective increase in flow which results from arteriolar vasodilatation produced by drugs may be much greater in the presence of a small collateral circulation than a large one.
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    Simple theory of transfusions (1965)
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    Bulletin of mathematical biology 27 (1965), S. 9-20 
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    Notes: Abstract A transfusion can be hypothesized to be required when a determination factor (D=probability of adverse effects if transfusion not given/adverse effects if transfusion is given) exceeds some predetermined value.D varies between the limits 0 and ∞, and in most clinical situations will be a small number on the order of 20. Since the probabilities contributing to the denominator ofD are essentially independent of each other, they can be summed to obtain the probability of ill effects. A method of handling an exception to this, the incompatibility reactions following multiple transfusions within a short time interval, is pointed out. The probability of adverse effects if a transfusion is not given is more difficult to evaluate; values gathered from the literature are presented, as well as methods for obtaining further data. Two techniques for estimating future transfusion requirements are discussed. One is a correlative procedure, which functions by analyzing similar cases admitted to the hospital. The second procedure is an estimate of stability (homeostasis), based on a parameter introduced by B. C. Patten (Scince,134, 1010–1011, 1961). The dilution of endogenous cells and plasma by transfusions is considered and the consequences of many small transfusions compared with those of few (and larger) transfusions.
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    Some possible quantitative aspects of a neurophysiological model of schizophrenias (1965)
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    Bulletin of mathematical biology 27 (1965), S. 21-26 
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    Notes: Abstract In continuation of a previous paper (Bull. Math. Biophysics,26, 167–185, 1964) simple equations are derived for the rate of development of schizophrenia as a function of some psychobiological parameters of the individual and of an index which characterizes the frequency of traumatic experiences of the individual. A clue to the understanding of why schizophrenia is more likely to develop at an early adult age is thus provided.
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    longitudinal diffusion along the nephron during stop flow (1965)
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    Bulletin of mathematical biology 27 (1965), S. 53-56 
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    Notes: Abstract A mathematical analysis is presented which shows that during stop flow experiments longitudinal diffusion of solute along the nephron is of too small a magnitude to interfere with the interpretation of data.
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    Theory of biological similarities, nondimensional parameters and invariant numbers (1966)
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    Bulletin of mathematical biology 28 (1966), S. 91-102 
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    Notes: Abstract In previous papers (1955–1957) a theory of biological similarity was established, assuming that the limits are the mechanical and the electrodynamical similarity criteria. The range of this theory lies between the coefficient of the time exponent (γ) for mechanical (0.5γ) and electrodynamical (1.0γ) similarities, being the mode 0.93γ. Moreover, for certain functions this restricted theoretical range should be extended to the hydrodynamical similarity criterion (2γ), so that the dimensionless numbers commonly used in Physics (Reynolds, Froude, Weber, etc.) can be included within the total range (0.5–2γ) of biological similarities. From dimensional analysis of physiological, functions it was possible to obtain, by means of dimensional and solution matrices, a group of “nondimensional numbers” by applying Buckingham's Pi-theorem. Nevertheless, only if a single similarity criterion was applied, the residual weight exponent was exactly zero; in all other instances the weight exponent was not zero, due to the existence of a range for biological similarities and to the statistical meaning of exponent (b) of the allometric equations. From the similarity criteria “invariant numbers” can be obtained, by means of which it is possible to establish correlations between numerous morphological and physiological characteristics of a particular system (circulation, respiration, metabolism, etc.).
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    On relations between sets: II (1966)
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    Bulletin of mathematical biology 28 (1966), S. 117-124 
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    Topics: Biology , Mathematics
    Notes: Abstract The notion of relations between sets, defined in a previous publication (Bull. Math. Biophysics,23, 233–235, 1961) is generalized and some biological examples are given. A generalization ton-ary relation is suggested.
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    A matrix approach to the theory of biotopological mapping (1966)
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    Bulletin of mathematical biology 28 (1966), S. 107-116 
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    Topics: Biology , Mathematics
    Notes: Abstract A method is introduced for using matrices to represent the organism-graphs of Rashevsky's theory of biotopological mapping. The representation is made in such a way as to reveal the structure of these graphs. Using insight gained from the consideration of the matrix representations, a theorem is proved concerning the primordial origins of organisms and counterexamples are displayed to show the necessity of the hypotheses of this theorem.
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    Biological effects of magnetic fields (1966)
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    Bulletin of mathematical biology 28 (1966), S. 137-138 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    Bionics Symposium 1966 (1966)
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    Bulletin of mathematical biology 28 (1966), S. 139-139 
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    Further studies of some aspects of a neurobiophysical model of schizophrenia (1966)
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    Bulletin of mathematical biology 28 (1966), S. 125-135 
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    Topics: Biology , Mathematics
    Notes: Abstract The neurobiophysical model of schizophrenia discussed previously (Bull. Math. Biophysics,26, 167–185, 1964;27, 21–26, 1965) is generalized further, to include catatonic and stuporous states. It is concluded that the development of schizophrenia will proceed through different stages of catatonic and non-catatonic states, depending on parameters which characterize on one hand the general inhibition of the individual, on the other hand what may be called his “stability.” Suggestions for possible clinical verifications of the conclusions are made.
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    Abstract biological systems as sequential machines: III. Some algebraic aspects (1966)
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    Bulletin of mathematical biology 28 (1966), S. 141-148 
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    Topics: Biology , Mathematics
    Notes: Abstract Using the relationship between (M,R) and sequential machines developed in previous work, it is shown that the totality of (M,R) which can be formed over a given categoryA itself forms a category in a natural fashion.
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    A note on replication in (M,R)-systems (1966)
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    Bulletin of mathematical biology 28 (1966), S. 149-151 
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    Notes: Abstract The condition which allows the existence of induced replication maps in (M,R)-systems is shown to place strong restrictions on the “richness” of the category from which these systems can be constructed. This condition also admits of a simple biological interpretation, which can be checked empirically, and which may offer insight into the physical and biological realizations of these abstract systems.
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    Abstract biological systems as sequential machines: Behavioral reversibility (1966)
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    Bulletin of mathematical biology 28 (1966), S. 153-160 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Rosen’s identification of abstract biological systems, called (M,R)-systems, with sequential machines is formally characterized. It is then shown that the determination of environmental alterations of (M,R)-systems from a knowledge of the response sequence and the structure of the system, which we call behavioral reversibility, can be interpreted as information-losslessness of sequential machines. Applying this relationship, necessary conditions for behavioral reversibility are derived. It is further shown that, similar to Rosen’s work on structural reversibility, (M,R)-systems are behaviorally reversible only if the number of physically realizable mappings are restricted.
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    Note on the analysis of thymidine labeling in cells with chromosome-number dispersion (1967)
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    Bulletin of mathematical biology 29 (1967), S. 33-40 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A method of analyzing thymidine labeling in a population of cells is formulated. The formulation establishes a unique relation between a specific set of labeling data and a specific set of cells in the population, viz. that set of cells having a particular chromosome number. The analysis employs a cell-state variable, i.e., a quantity which specifies the progress of a cell through its lifecycle. This variable is defined in terms of the nucleo-protein content and configuration of the chromosomes. The relation mentioned above leads immediately to an expression for the number of cells present at a particular time following labeling which have a given amount of label per cell and a given chromosome number.
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    Dose-rate dependence of radiation-induced metaphase delay (1967)
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    Bulletin of mathematical biology 29 (1967), S. 41-56 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract An equation relating radiation-induced metaphase delay to the dose-rate and duration of irradiation is obtained. The equation is derived from a model specifying the effects of radiation on the normal chromosome coiling process. The basic assumptions of the model are (1) that normal coiling proceeds by contractile protein acting on segements of a viscoelastic chromosomal fiber; (2) that radiation causes cross-linking of adjacent chromosomal fibers which hinders the coiling process.
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    Bistable mathematical and mechanical models related to the urethra (1967)
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    Bulletin of mathematical biology 29 (1967), S. 57-65 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Normal micturition is controlled primarily by a neural system. Certain physical effects become evident when neural control is destroyed, and the automatic or autonomous bladder phenomena occur. It is shown in this paper that a physical system simulating the alternating periods of continence and voiding of the automatic bladder may comprise only passive elastic components, and that periodic voiding does not per se imply neural control.
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    Further comments on replication in $$\left( {\mathfrak{M},\Re } \right)$$ -systems-systems (1967)
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    Bulletin of mathematical biology 29 (1967), S. 91-94 
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    Topics: Biology , Mathematics
    Notes: Abstract A number of inaccuracies in previous papers are pointed out and amended, and some implications of the correct situation are outlined.
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    The theory of urine formation in water diuresis with implications for antidiuresis (1967)
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    Bulletin of mathematical biology 29 (1967), S. 67-89 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We investigate a model of the renal medulla in which active NaCl transport is restricted to the thick ascending limb of Henle's loop. The model contains a vas rectum, a loop of Henle, salt, and water. The model generates interstitial osmolality curves consonant with the known functioning of the kidney in water diuresis. Using data from the white rat and the curves generated by the model, one can predict the permeability of the thin limb of Henle's loop to NaCl and the percentage of total renal blood flow entering the inner medulla. In this model interstitial osmolality at the papilla can be about twice plasma osmolality, so that NaCl transport restricted to the outer medulla can contribute significantly to the work required in producing a hypertonic urine. However, the interstitial osmolality monotonically decreases proceeding from the junction of the outer and inner medulla to the papilla, and the maximum interstitial osmolality in the outer medulla is greater than the maximum interstitial osmolality in the inner medulla. Thus we infer that a source of active transport located in the inner medulla is needed to explain the high osmolalities observed in hydropenia. A sketch of an alternative model, a “lineal multiplication mechanism”, for the renal concentrating process is presented in which active transport in the inner medulla is restricted to active salt transport by the collecting duct. The lineal multiplication mechanism makes no use of counter-current multipliers in the inner medulla.
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