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  • Springer  (237,472)
  • American Physical Society  (29,290)
  • American Geophysical Union  (8,827)
  • 1965-1969  (171,391)
  • 1960-1964  (104,198)
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  • 1
    Electronic Resource
    Electronic Resource
    Relationship between ion distribution and membrane potential during a steady state (1960)
    Springer
    Bulletin of mathematical biology 22 (1960), S. 323-349 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Equations were derived showing the relationship between the membrane potential and the quantities which influence it under steady state conditions. Essentially, the membrane potential is caused by the valence and concentration of the non-permeating ions. The permeating ions can modify the membrane potential by altering the relative concentration of the non-permeating ions with respect to the concentration of the permeating ions. For muscle, the sodium cations act as the non-permeating ions in the extracellular environment by the maintenance of some type of active metabolic process and large anions act as the non-permeating ions in the intracellular environment. Both of these non-permeating ions contribute about equally to the maintenance of the resting membrane potential. When the active metabolic process for sodium extrusion breaks down or when acids are added, the membrane potential should decrease. Water should enter the cell when the sodium metabolic process is diminished; water should leave the cell when acids are added. When acid is added, it is expected that the cations potassium and sodium will leave the cell with little or no shift of the chloride ions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476718
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  • 2
    Electronic Resource
    Electronic Resource
    Life, information theory, probability, and physics (1960)
    Springer
    Bulletin of mathematical biology 22 (1960), S. 351-364 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A purely information-theoretical approach to the problem of self-replication of elementary living units implies that pure chance is the determining factor in the formation of the first living unit. The probability of such a spontaneous formation can be calculated from the minimum amount of information which an organism must possess in order to replicate itself. An estimation of this amount of information is made here by two different methods. First by a “paper and pencil experiment” which indicates the minimum amount of information needed on a printed page in order that with given tools the page could be reproduced. Second—by an analytical consideration of some hypothetical molecular mechanisms. A general method for handling such problems is suggested. On the basis of estimated information contents it is shown that under most favorable conditions the probability of a spontaneous generation by pure chance during the lifetime of the earth is vanishingly small. It is concluded that dynamic factors, which may reduce tremendously the information content, must play a role in the genesis of life on earth.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476719
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  • 3
    Electronic Resource
    Electronic Resource
    Some remarks on molecular biology (1960)
    Springer
    Bulletin of mathematical biology 22 (1960), S. 365-370 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The binding energy of a very long molecular chain, composed of different classes of molecules, depends in general on the order of the molecules. It is shown that under very general conditions there exists for a givenbrutto chemical composition of a chain, a class of chains which is characterized by a total binding energy which is equal to the total binding energy of any other prescribed chain of different composition within the limits of unsharpness of the energy level. This establishes a criterion formapping of a class of configurations of long chain molecules on another class. To the extent that a mapping constitutes a generalized code those results contribute to the theory of molecular codes. Applying to our results the results of a previous paper (1959,Bull. Math. Biophysics,21, 309–326), we arrive at the conclusion that the self-replication of a living molecule may be the property not of a particular structure but of classes of structures.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476720
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  • 4
    Electronic Resource
    Electronic Resource
    On the utilization of food by planktophage fishes (1960)
    Springer
    Bulletin of mathematical biology 22 (1960), S. 371-389 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Making some plausible assumptions about the over-all mechanism of food catching and consumption by fishes and evaluating in the light of those assumptions some available experimental data, it is possible to calculate from those data the variation of several important factors with the concentration of food. The factors considered are: total rate of metabolism, total diurnal energy expenditure in the process of feeding, average number of hours per day during which the fish feeds, average length of path traveled by a fish per day, and the so-called “energetic coefficient of growth.” A possible relation with the work of N. Rashevsky (Bull. Math. Biophysics,20, 299–308, 1959) is discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476721
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  • 5
    Electronic Resource
    Electronic Resource
    Symposium on Mathematical Theories of Biological Phenomena (1960)
    Springer
    Bulletin of mathematical biology 22 (1960), S. 425-425 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476724
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  • 6
    Electronic Resource
    Electronic Resource
    On the theory of indicator-dilution methods under varying blood-flow conditions (1960)
    Springer
    Bulletin of mathematical biology 22 (1960), S. 417-424 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The theory of measurement of flow and volume by indicator dilution techniques is given in conditions of time-variable flow rates. It is shown that the usual Hamilton (1932,Am. J. Physiol.,99, 534–551) methods can be misleading if the flow changes at a rate of close to that of the transport function.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476723
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  • 7
    Electronic Resource
    Electronic Resource
    An hypothesis of freese and the DNA-protein coding problem (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 305-318 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Freese’s Hypothesis states that a single specific alteration in the sequence of nucleotides of an information-bearing DNA molecule results in a specific mutational effect. Within the framework of the DNA-protein coding problem developed elsewhere, and assuming the quasi-ergodicity of the general coding process, it is shown that Freese’s Hypothesis allows us to derive expressions for the length of the smallest mutable DNA molecule and to obtain a bound for the maximal number of allelic molecules of fixed length. To illustrate these ideas, calculations are carried out on appropriate data from bacternophage and man, and the results are shown to differ by a factor of 10 (modulo the rather crude approximations used). It is further shown that, if ρ(N) and ϱ(N) are respectively the number of information-bearing words of lengthN in a given code and the number of words of lengthN, then the number lim ρ(N)/ϱ(N) depends sensitively on the parameter ∈ which specifiesN→∞ the given code. The implications of this result for the spontaneous aggregation of a sufficient number of information-bearing words to characterize an organism are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476743
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  • 8
    Electronic Resource
    Electronic Resource
    Post-and predoctoral fellowships in mathematical biology (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 319-319 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476744
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  • 9
    Electronic Resource
    Electronic Resource
    The perfect learner (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 321-335 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract As a “base line” of memorization performance, the behavior of a “perfect learner” is considered. He is characterized by a perfect memory and by the ability to choose the best search procedure in problems where the correct response from a given repertoire is to be found to each of several stimuli under the condition of “right” and “wroing” promptings by the experimenter. Expected learning curves are derived for the case of disjoint response repertoires associated with the stimuli under cyclic and random presentation of the stimuli and for the case of a single response repertoire (a one-to-one matching problem) under cyclic presentation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476490
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  • 10
    Electronic Resource
    Electronic Resource
    A method for the computation of aortic distensibility in the living human patient and its use for the determination of the aortic effects of aging, drugs, and exercise (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 337-353 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Detailed equations are given for the computation of aortic distensibility in the intact living human patient from measurements of systolic and diastolic arterial pressures, heart rate and cardiac output. From these equations, the aortic characteristics of a large series of normal men of different ages were computed. Comparing these results with measurements on excised aortas, a more pronounced trend toward increasing aortic stiffness with increasing age is evident in living than in dead aortas. Nor-epinephrine and exercise apparently cause the living aortas to constrict but to become more distensible. The same change occurs after 30 minutes of high spinal anesthesia. The ganglionic blocking agents hexamethonium, pentamethonium, and tetraethylammonium usually cause the living aorta to become stiffer, presumably due to dilatation. The aortas of patients with pulmonary disease usually react in different fashion to exercise or tetraethylammonium. The increased aortic distensibility that occurs with the hypertension induced by nor-epinephrine or exercise acts as a compensatory mechanism by decreasing systolic pressure.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476491
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  • 11
    Electronic Resource
    Electronic Resource
    Dimensional analysis in mathematical biology I. General discussion (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 355-376 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Dimensional analysis is discussed from the viewpoint of its basic group properties and shown to be an algebraic Abelian group that is useful for analysis of physical measurements. The application of the method to various types of equations and the formulation of previously unclassified dimensions are discussed. Functional dimensional analysis is applied to the problems of cell size and biomass proliferation; future applications are also noted. A number of dimensionless terms have been formulated for cellular physiochemical phenomena. They apparently represent the first systematic study of biological dimensionless numbers recorded in the literature. A dimensionless proliferation law is suggested. A brief analysis of the physical dimensionality associated with information measures is carried out. Entropy and “information” are shown to be completely different in their dimensional meaning; other informational measures of possible interest in biology are proposed. The dimensional coding and computor analysis of biomathematical equations is suggested.
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    URL: http://dx.doi.org/10.1007/BF02476492
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  • 12
    Electronic Resource
    Electronic Resource
    A minimal principle in biomechanics (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 377-391 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Expenditure of energy under several simultaneous forms (mechanical, chemical, etc.) is associated with all muscular activity. The energy is directly related to what is commonly called exertion or effort. This paper defines “muscular effort” quantitatively in terms of some of the elements of the dynamics of the human (and animal) body. It postulates that in all likelihood the individual will, consciously or otherwise, determine his motion (or his posture, if at rest) in such a manner as to reduce his total muscular effort to a minimum consistent with imposed conditions, or “constraints”. The principle, formulated in mathematical terms, is sufficient to ascribe to the moments at all body joints—a matter generally of free choice on the part of the individual—their most likely magnitudes. It therefore renders the equations of human (and animal) motion determinate within this context. The paper also describes briefly an iteration method for the solution of these equations, once they have been made determinate. A simple illustrative application of the principle is included.
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    URL: http://dx.doi.org/10.1007/BF02476493
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  • 13
    Electronic Resource
    Electronic Resource
    On the role of chemical systems in the microphysical aspects of primary genetic mechanisms (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 393-403 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It is pointed out that two fundamentally different views of primary genetic processes occur in the literature which are frequently confused. The first is a true communication-theoretic view, which regards the genetic apparatus as containing a real information-source and a transducer which converts that information to useful form. The second view is generally expressed as a template scheme based on the Watson-Crick model; it is shown that in this model there is actually no such thing as genetic information in a communication-theoretic sense. Both views are then discussed on the basis of microphysical principles developed in previous work of the author (Bull. Math. Biophysics,22, 227–255, 1960) in an attempt to find which approach is in closer accord with the biological facts. It is shown that, if the communication-theoretic view is correct, then the information-bearing object must act as a “catalyst,” but it is pointed out that the type of catalysis involved must be of a fundamentally different nature than that occurring in familiar enzyme-catalyzed reactions. On the basis of general considerations of irreversible changes in microphysical measuring systems, it is shown that any type of template must suffer a gradual and irreversible denaturation, which seems to make it unlikely that a template could play a primary role in fundamental genetic processes.
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    URL: http://dx.doi.org/10.1007/BF02476494
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  • 14
    Electronic Resource
    Electronic Resource
    On imitative behavior. II. Time course of change from one behavior to another (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 405-411 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The theory developed in previous papers and based on distribution curves of definite form is generalized to any form of unimodel distributions. The time course of the change from one behavior to another is discussed and a general theorem about the time course is established.
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    URL: http://dx.doi.org/10.1007/BF02476495
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  • 15
    Electronic Resource
    Electronic Resource
    Abstract mathematical molecular biology (1961)
    Springer
    Bulletin of mathematical biology 23 (1961), S. 417-417 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476497
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  • 16
    Electronic Resource
    Electronic Resource
    Training program in computer use for biomedical scientists (1963)
    Springer
    Bulletin of mathematical biology 25 (1963), S. 471-471 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476568
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  • 17
    Electronic Resource
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    Reflexes et centres analyseurs dans les microstructures isotropes (1963)
    Springer
    Bulletin of mathematical biology 25 (1963), S. 421-469 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Résumé Nous appliquons le modèle de neurone introduit dans un article antérieur à l’étude d’une microstructure isotrope. La stabilité de cette microstructure implique l’existence d’une régulation d’activité que le principe de construction adéquate permet de définir entièrement. Nous aboutissons à une conception stratifiée du cerveau. Un réseau de neurones spécialisés exercerait, grâce à certains médiateurs chimiques, une action diffuse qui modulerait les propriétés du réseau localisé classique. Les lois de Pavlov peuvent être retrouvées à partir des propriétés de la microstructure et de celles de la régulation. La microstructure isotrope peut également fonctionner comme analyseur. Un certain nombre de temps caractéristiques apparaissent alors, qui semblent jouer un grand rôle en psychologie.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476567
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  • 18
    Electronic Resource
    Electronic Resource
    Tissue blood flow and the sympathomimetic amines (1964)
    Springer
    Bulletin of mathematical biology 26 (1964), S. 1-7 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In the arteries, blood flow and blood pressure are pulsatile in nature (Roston, 1962a; Roston 1962b). The patterns of blood movement and mural distension in the arteries are important because they may be associated with life-threatening degenerative changes in the arterial walls. As the vascular channels narrow, the pulsation decreases. At the level of the capillaries, almost no pulsation exists (Best and Taylor, 1961). The tissues are affected by the direct flow in the capillaries and not by the pulsation in the arteries. Thus, such quantities as pulse pressure, systolic pressure, and diastolic pressure which characterize blood movement in the arteries are not important as far as the tissues are concerned. Rather, the average pressure and flow in the capillaries are the quantities significant for tissue blood flow. The present study analyzes the local blood circulation in a typical tissue. Logical extension of this analysis results in insights into the physiological behavior of the circulation which integrate a considerable body of experimental data.
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    URL: http://dx.doi.org/10.1007/BF02476615
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  • 19
    Electronic Resource
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    A multidimensional birth process and its application to some problems in the dynamics of biological populations (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 333-345 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper is a sequel to a paper by the author entitled “Restricted Transition Probabilities and Their Applications to Some Problems in the Dynamics of Biological Populations” (Bull. Math. Biophysics, 1966,28, 315–331). The paper is divided into two parts. In part one some aspects of the maximum size attained by the population during a finite time interval are studied for the case the stochastic process underlying the evolution of the population is a birth process. Two interesting by-products emerge from the study presented in part one; namely a combinatorial method of finding solutions to the Kolmogorov differential equations in special cases, and secondly, a set of criteria for the optimum allocation of genotypes in the host population of a host-pathogen system. The optimum allocation of genotypes in the host population is a problem of practical importance in controlling plant pathogens. In part two the theory of restricted transition probabilities developed in the companion paper is applied in finding the distribution of the time to the appearance of the first mutation for the case of a two dimensional birth process. The distribution of the time to the appearance of the first mutation is of importance in understanding the role mutation plays in the evolution of a population, particularly in the pathogen population of a host-pathogen system.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476814
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  • 20
    Electronic Resource
    Electronic Resource
    The motion of the left ventricle. I (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 355-362 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The complex arrangement of the muscle fibers in the ventricular wall and the nonsymmetric contraction and expansion of the ventricle preclude the writing of a differential equation of motion for the ventricle as a whole. We can, however, describe the motion of the ventricle by describing the motion of the dimensional parameters length and diameter; the radius, circumference, cross-sectional area, and volume following naturally from these. The ventricle is assumed to be an ellipsoid of revolution and the dimensional parameters to be periodic functions of time. Each of the parameters is expressed as a Fourier series.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476816
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  • 21
    Electronic Resource
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    Comparaison des modeles de neurone de Hodgkin-Huxley et de Nelson (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 347-354 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Le modèle de Nelson peut-être considéré comme une approximation du modèle de Hodgkin-Huxley. Moins précis, il est plus maniable. Le modèle de Nelson peut également être considéré comme une généralisation du modèle de Hodgkin-Huxley. En effet, il introduit des liaisons synaptiques localisées ou diffusantes, et un processus de facilitation. Le mécanisme des liaisons synaptiques ne se traduit pas facilement dans le langage de Hodgkin-Huxley. Par contre, le processus de facilitation s'interprète facilement. Nelson's model can be taken as an approximation of Hodgkin-Huxley's model. Its precision is lesser, but it is more usable. Nelson's model can also be taken as a generalization of Hodgkin-Huxley's one; for it introduces localized or diffusing synaptic connexions and a facilitating process. The mechanism of synaptic connexions cannot be easily translated into Hodgkin-Huxley's language. On the contrary, the facilitating process is easily interpreted.
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    URL: http://dx.doi.org/10.1007/BF02476815
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  • 22
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    The theory of the left ventricular aneurysm (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 363-370 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A spherical model for the human left ventricle with two different types of aneurysms, circular and rectangular-square, is proposed and meaningful relations are derived between the parameters of the aneurysms and ventricle. Such ventricular parameters as stroke volume, end-diastolic volume, and end-systolic volume are given normal human values to compute values for end-systolic radius and percentage shortening of muscle for various sized circular and rectangular-square aneurysms.
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    URL: http://dx.doi.org/10.1007/BF02476817
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  • 23
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    A note on crime control (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 375-378 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The Volterra theory of two competing populations is extended to the contemporary social problem of crime control. Domains of stability for the time dependence of the numbers in the criminal and enforcement groups are exposed by a numerical example. Both augmentation and reduction of enforcement can produce a stable system. Average values of the ratio of members in each group show great sensitivity to the control policies adopted by the remaining sector of the total population.
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    URL: http://dx.doi.org/10.1007/BF02476819
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  • 24
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    Interactions in enzymatic reactions (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 379-390 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The paper deals with interactions of substances via an enzymatic reaction (Bull. Math. Biophysics,25, 141–154, 1963). The substances are the activators, inhibitors and/or substrates of the reaction. Due to the bimolecularity of the processes in the reaction, the quantitative relation between the steady state amount of complexes and the amounts of the substances assumes a typical form. In multiple enzymatic reactions this form is more complicated, though basically similar. Because the substances may influence the steady state amounts of the complexes in opposite directions, the compensation and blocking effects are the properties of enzymatic reactions. The substances with the same direction of influence may potentiate each other. In the enzymatic reaction here considered, the potentiation is always non-negative.
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    URL: http://dx.doi.org/10.1007/BF02476820
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  • 25
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    Decay of cell synchronization: Solutions of the cell-growth equation (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 391-409 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Growth-rate functions in analytic form have been obtained for cell cultures in which the doubling times follow the Gaussian and Poisson distributions. The growth-rate functions are calculated by using Laplace transforms to solve an integral equation previously presented. Oscillatory solutions result if a substantial fraction of the cells in a culture are synchronized to divide at some particular time. The synchrony and, hence, the oscillatory character of the growth-rate function eventually disappear because of the non-zero variance of the doubling-time distribution. If their variances are sufficiently small, the Gaussian and Poisson doubling-time distributions lead to growth-rate functions that become identical in the limit of large time.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476821
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  • 26
    Electronic Resource
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    On the derivation of a mean growth equation for cell cultures (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 411-416 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract IfN(t) is the expected number of cells in a culture at timet, $$\dot N(t)$$ the corresponding time derivative, andf(t−τ)dt the probability that a cell of aget−τ at timet will divide in the succeeding time intervaldt, then according to Hirsch and Engelberg (this issue) there obtains the integral equation $$\dot N(t) = 2\int_{ - \infty }^t {f(t - \tau )\dot N(\tau )d\tau }$$ for describing the dynamics of the cell population. It is the purpose of this note to give two alternative derivations of this equation, one based on the age density equation of Von Foerster, and the other based on a generalized form of the Harris-Bellman equation describing the first moment of an age dependent, branching process. In addition, a probability model is posed from which the Von Foerster equation and, hence, the Hirsch-Engelberg equation readily follows.
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    URL: http://dx.doi.org/10.1007/BF02476822
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  • 27
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    A mathematical theory of enamel solubility and the onset of dental caries: I. The kinetics of dissolution of powdered enamel in acid buffer (1966)
    Springer
    Bulletin of mathematical biology 28 (1966), S. 417-432 
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    Notes: Abstract A model of the dissolution kinetics of powdered enamel is developed based on the kinetic rate termq, $$q = K'H - k'C \cdot P_1$$ , whereH=[H +],C=[Ca ++] andP 1=[HPO 4 = ]. The differential equations describing the rate of mineral dissolution (and the linearly related rate of appearance of calcium and phosphate in solution) have been derived and solved for three basic cases: (1) when thepH of the solution and surface area of the enamel are considered constant, (2) when thepH is assumed constant, but the reduction in surface area during dissolution is considered, and (3) when the rise ofpH resulting from the buffering effect of the dissolved enamel is considered along with the change in surface area. Analytical solutions have been obtained for cases (1) and (2), while a numerical solution has been found for case (3). Data from a study on enamel dissolution are presented that agree with the theory of case (3), and it is noted that apH rise as large as 0.5 can occur, as has been shown elsewhere in the literature.
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    Reward and punishment in learning with possible reference to punishment for crime (1966)
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    Bulletin of mathematical biology 28 (1966), S. 477-481 
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    Notes: Abstract On the basis of Landahl's theory of two-choice learning it is shown that application of punishment for wrong responses, without giving award for correct ones, does not lead to complete learning, no matter how many trials are used. If initially a “wrong response” was learned, then an attempt to inhibit it by punishment alone will in a class of cases lead only to a 50% suppression of that wrong response. Possible connection with the problem of effectiveness of punishment as a deterrent for crime is mentioned.
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    The use of two radiotracers for transfer-rate determinations in closed two compartment systems (1966)
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    Bulletin of mathematical biology 28 (1966), S. 483-483 
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    Cooperative specific adsorption of iona at charged sites in an electric field (1966)
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    Bulletin of mathematical biology 28 (1966), S. 483-483 
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    Operations research around the world meetings (1966)
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    Bulletin of mathematical biology 28 (1966), S. 485-485 
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    Dimensional anlaysis and the equations of electrophoresis (1966)
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    Bulletin of mathematical biology 28 (1966), S. 501-510 
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    Notes: Abstract A set of characteristic parameters is given for electrophoresis accompanied by diffusion, followed by a method of simplification of the transport equation. The concept of electrophoretic similarity is introduced in connection with the presentation of solutions and the final section contains some dimensional considerations of the potential equation.
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    Categories of (ℓ, ℛ)-systems (1966)
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    Bulletin of mathematical biology 28 (1966), S. 511-517 
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    Notes: Abstract We show that when we represent (ℓ, ℛ)-systems with fixed genome as automata (sequential machines), we get automata with output-dependent states. This yields a short proof that ((ℓ, ℛ)-systems from a subcategory of automata—and with more homomorphisms than previously exhibited. We show how ((ℓ, ℛ)-systems with variable genetic structure may be represented as automata and use this embedding to set up a larger subcategory of the category of automata. An analogy with dynamical systems is briefly discussed. This paper presents a formal exploration and extension of some of the ideas presented by Rosen (Bull. Math. Biophyss,26, 103–111, 1964;28, 141–148;28 149–151). We refer the reader to these papers, and references cited therein, for a discussion of the relevance of this material to relational biology.
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    The origin of lift forces in fluttering flight (1966)
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    Bulletin of mathematical biology 28 (1966), S. 487-500 
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    Notes: Abstract A two-dimensional nonlinear integro-differential equation with time-varying coefficients describing the behavior of the fluttering wing-body systems typical of natural flight mechanisms has been deduced from the Navier-Stokes equation which generalizes local pressure and velocity distributions in the externally oscillating air field. The resulting equation for the wing forces is combined with an analogous expression for the forces of gravitation and acceleration associated with the body. The air acceleration force, not previously considered in bio-physical models of insect and bird flight, is shown to arise from a formal analysis of unsteady or time-varying contributions to the velocity field, while the square form of the conventional steady state aerodynamic forces is derived from the intertial terms in the Navier-Stokes equation with the aid of the approximations of Newtonian impact theory. Previous calculations (Houghton, 1964) have indicated that the contribution to gravitational stability of air acceleration and aerodynamic life are roughly in the ratio of 3:1.
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    Evoked potentials arising from neural population elements excited at different times on a warped surface (1966)
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    Bulletin of mathematical biology 28 (1966), S. 519-536 
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    Notes: Abstract Certain types of cortical electrical events are non-propagated so that the associated electric fields must have standing wave characteristics. However, cortical electric events typically are generated by neurone populations which cannot be activated simultaneously on impulse driving. Hence the sum of the standing wave fields due to asynchronous activation of adjoining regions of cortical neurones must give the appearance of a traveling wave. Analysis of cortical waveforms is further complicated by curvature in cortical surfaces. A model is presented that shows the effects of curvature and time lag in activation on the form of the potential at points in space around a laminar array of elements simulating a population of cortical neurones. The results are compared with waveforms evoked by single-shock stimulation of the prepyriform cortex in cats.
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    A continuity equation for cell populations (1966)
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    Bulletin of mathematical biology 28 (1966), S. 545-554 
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    Notes: Abstract A continuity equation for cell-number density in a population of cells is derived, and a system of equations for eliminating parameters between the general solution and the initial distribution obtained.
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    The behavior of the weakening and failing myocardium (1966)
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    Bulletin of mathematical biology 28 (1966), S. 537-544 
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    Notes: Abstract Use of an electrical model of the left ventricle of the heart and the arterial system permits analysis of the changes which take place as the capacity of the myocardium for generation of force decreases. The model is simple in structure, and its construction and practical testing would not be difficult. It demonstrates that, as the heart muscle weakens, the peak of intracardiac force occurs later in systole, and the difference between the intracardiac pressure and the aortic pressure in the second half of systole is much greater than for the normal heart. The feedback mechanisms which are proposed to affect myocardial contractility would affect this compensation for cardiac weakening. Indices to categorize the behavior of the normal, compensated though weakened, and decompensated myocardium are proposed.
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    The frequency distribution of chromosome constituents in cells with chromosome-number dispersion (1966)
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    Bulletin of mathematical biology 28 (1966), S. 555-566 
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    Notes: Abstract The frequency distribution in a population of cells of the quantityCD (defined as the amount of some chromosomal substance in a cell which containsC chromosomes) is calculated using the distribution in the population of the amount per chromosome,D, and the distribution of chromosome number,C.
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    Calculation of rate of synthesis of chromosomal DNA in cells with chromosome-number dispersion (1966)
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    Bulletin of mathematical biology 28 (1966), S. 567-574 
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    Notes: Abstract The rate of chromosomal DNA synthesis in an exponentially growing population of cells having chromosome-number dispersion is calculated using DNA histogram data, chromosome-number distribution data, and the assumptions that the synthesis rate is constant and DNA double exactly.
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    Estimation of rate of synthesis of chromosomal DNA in ehrlich ascites tumor cells (1966)
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    Bulletin of mathematical biology 28 (1966), S. 575-584 
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    Notes: Abstract An estimate is made of the rate coefficient for linear DNA synthesis with exact doubling in an exponentially growing population of Ehrlich ascites tumor cells having chromosome-number dispersion. Comparison of calculated and experimental results suggest that the assumptions used in the calculation are tenable, but further experimental evidence is needed to prove this.
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    A sociological approach to biology (1966)
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    Bulletin of mathematical biology 28 (1966), S. 655-661 
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    Notes: Abstract The paper develops further some suggestions made previously (Bulletin of Mathematical Biophysics,28, 283–308, 1966) that certain biological phenomena may be more easily interpreted from a “sociological” point of view by considering the organism as a social aggregate of cells and a cell as a social aggregate of genes. In this light the problems of origin of life on earth, of aging, and of parasitism and symbiosis are discussed. The notion of social aggregates of different orders is introduced.
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    Theorems on random evolution (1966)
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    Bulletin of mathematical biology 28 (1966), S. 663-663 
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    A theoretical and experimental study of the mechanical behavior of the cornea with application to the measurement of intraocular pressure (1966)
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    Bulletin of mathematical biology 28 (1966), S. 585-643 
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    Notes: Abstract A theoretical and experimental study was made of the mechanical behavior of the cornea. The theoretical analysis included an analytical solution for the symmetrical constraint of a thin, shallow, spherical shell by a rigid indenter. The experimental study investigated the rheology of the cornea with particular emphasis on its compliance with the requirements of the Boltzmann Superposition Principle. Representative results of tests on twenty enucleated hog eyes and two human eyes have been reported. The corneas of the human and hog eyes behaved as linear viscoelastic solids; the human eyes differed from the hog eyes in having a long term creep component. Several eyes were tested at the site of procurement, six to seven minutes after the animal's death, and it was established that creep is not an artifact due to aging or enucleation. The analytical and experimental results were combined to study some instruments used to detect the level of pressure in the eye. The theoretical analysis predicted that a type of elastic instability occurs during the process of flattening a small portion of the cornea; this is discussed with reference to the Goldmann and Mackay-Marg tonometers. The role of corneal creep was considered with reference to the response of the Schiøtz indentation tonometer during the time dependent process known as tonography.
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    Further studies on the shape of the tracking curve of an automobile and its dependence on biological parameters of the driver (1966)
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    Bulletin of mathematical biology 28 (1966), S. 645-654 
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    Notes: Abstract Following previous studies, differential equations are established which determine the variation of the stimulus towards a corrective turn of the steering wheel and its effect on the excitation of the centers in the brain which results in the production of the corrective turn. The equations are derived under the highly oversimplified assumption that all excitation thresholds are so small that they can be neglected. Under these assumptions it is found that the tracking curve of a car is a sinusoid with negative damping, that is, with an ever increasing amplitude. Driving under these assumptions is imposible since the car will always eventually jump off the road. The possible effects of the threshold as well as stimuli towards corrective turns other than the distance from the edge of the lane are very briefly discussed. In spite of the negative results of the paper, its interest lies in the circumstance that with the complication of the model, we find that driving depends not only on the reaction times as the only “purely biological” parameter, but on three other neurobiophysical constants. In a subsequent paper (Rashevsky, 1967) it is shown how the introduction of one or more purely biological parameters of the driver makes a stable driving regime possible.
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    The motion of thhe left ventriele: I (1966)
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    Bulletin of mathematical biology 28 (1966), S. 663-663 
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    Bimolecular systems: I. Linear systems of complexes (1967)
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    Bulletin of mathematical biology 29 (1967), S. 1-16 
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    Notes: Abstract A vast number of biologically important processes are based upon bimolecular systems. In these systems intermediate complexes are formed. Bimolecular systems in which no complex-complex interactions occur are called linear systems of complexes. A definition and some characteristic properties of these systems are given here. There may exist a contradiction of Onsager's principle of detailed balancing in these systems; however, no principal differences are found between the steady state behavior of an open system and that of a closed system. It is shown that the steady state behavior of a linear system of complexes of arbitrary complexity has some similarities with the steady state behavior of a simple bimolecular system, e.g., Michaelis-Menten enzymatic reaction. Multiplicity of action of the substances participating in biomolecular processes may produce some qualitative differences in the steady state behavior of the system.
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    DNA histograms of irradiated cells: The case of prophase-trapping (1967)
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    Bulletin of mathematical biology 29 (1967), S. 17-32 
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    Notes: Abstract A time-dependent DNA histogram is calculated for an irradiated population of cells under the limiting assumption that the cells cannot pass through prophase due to the effects of the radiation. The population is assumed to increase exponentially prior to irradiation, but after irradiation to neither gain nor lose cells. Chromosome-number dispersion is taken into account in the calculation. The qualitative behavior of the calculated and experimental histograms are in reasonable agreement. The quantitative agreement between the two is relatively good at short post-irradiation times but is poor at long post-irradiation times (say, greater than half the doubling time). This suggests that recovery phenomena cannot be neglected at long post-irradiation times.
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    A note on the mathematical biology of automobile driving (1967)
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    Bulletin of mathematical biology 29 (1967), S. 187-188 
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    Notes: Abstract It is pointed out that the three different stimuli for a corrective turn, namely the distance from the edge of the lane, the rate of approach to the edge, and the angle between the direction of the car and the direction of the lane (Bull. Math. Biophysics,28, 645–654, 1966,29, 181–186, 1967) may act all three simultaneously. It is found that in that case the tracking curve of the car is stable below a critical speed and becomes unstable above it.
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    Mathematical biology of automobile driving (1967)
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    Bulletin of mathematical biology 29 (1967), S. 181-186 
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    Notes: Abstract Continuing a previous study (Bull. Math. Biophysics, 28, 645–654, 1966), the biophysical mechanism of a corrective turn is investigated for the case where the stimulus for the corrective turn is produced not only by the perception of the nearness of an edge of the lane, but also by the rate of approach of the car towards the edge. In that case it is found that the tracking curve of the car may consist of a series of damped sinusoids and safe driving would be possible at any speed if it were not for the endogenous fluctuation in the driver's central nervous system. If the effect of the rate of approach increases sufficiently rapidly as the distance to the edge of the lane decreases, then a stable undamped oscillating tracking curve is possible. The case is also studied where the driver makes a corrective turn in response to a direct perception of the angle between the direction of the lane and the longitudinal axis of the car.
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    Angles of branching and diameters of branches in the human bronchial tree (1967)
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    Bulletin of mathematical biology 29 (1967), S. 245-259 
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    Notes: Abstract The principle of minimal work requires that the conducting airways of the human lung should have a maximum radius for minimal resistance to gas flow. At the same time there is a requirement that the airways should have a minimal volume for economy of space. These two opposing requirements have been investigated mathematically, and a method for calculating the angle of branching which produces minimal volume has been derived. The relationship of the radii of the parent and daughter branches to produce minimal resistance has been similarly defined. By measurement of a bronchial cast from a human lung the extent to which the predicted optimum structure is realized in practice has been shown. The change in structure associated with change of function at the transition from conducting airway to diffusion zone has been demonstrated.
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    Finite linnaean structures (1967)
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    Bulletin of mathematical biology 29 (1967), S. 191-206 
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    Notes: Abstract This paper considers a class of set-theoretical entities, calledn-rank Linnaean structures, which are intended as abstract models of the taxonomic classificatory systems of biology. In the first part, devoted to formalism, finite Linnaean structures are discussed in complete generality; but, in addition, eight distinct subclasses are noted and some of the properties of their elements are explored. In the second part, concerned with applications, it is shown that taxonomic systems may be recast in the form of finite Linnaean structures, and an effort is made to show that some undesirable features of earlier models are avoided without artificiality and without abandoning extensional mathematics.
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    Abstract power structures (APS) and related bumping orders (1967)
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    Bulletin of mathematical biology 29 (1967), S. 207-216 
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    Notes: Abstract Generalizing an idea of M. Richardson (Fundamentals of Mathematics, New York: Macmillan Co., 1958), an APS on a given populationP is a non-empty collection of non-empty subsets ofP such that ifA is in the collection andA⊆B, thenB is in the collection. From a structure of this kind a partial ordering ofP, called therelated bumping order, is derived. The question is raised as to what kinds of partial orderings can be so obtained. For structures determined by voting weights of the members of the population, a complete characterization of all possible bumping orders is obtained.
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    Stability analysis of a mathematical neuron model (1967)
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    Bulletin of mathematical biology 29 (1967), S. 217-226 
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    Notes: Abstract The “second method” of Liapunov is used to perform a stability analysis of a mathematical model of the neuron. This analysis is based on the hypothesis that the firing of the neuron coincides with a temporary state of instability of the system, and that the initiation of all-or-none process depends on the magnitude of membrane depolarization and its first time derivative. It is found that the stability (and hence the possibility of a second firing) is restored approximately when the rate of membrane repolarization is at a maximum. This result predicts that the duration of the period of absolute refractoriness in neurons would be about 75 per cent of the spike duration, and thus shorter than the value usually obtained from experimental measurements.
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    A neural net model for masking phenomena (1967)
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    Bulletin of mathematical biology 29 (1967), S. 227-232 
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    Notes: Abstract Some aspects of masking phenomena are considered in terms of the simplest possible model of two-factor neural elements. The effect of a number of variables can be accounted for, but the introduction of an internuncial element results in a masking function which need not be symmetric about zero delay interval. As an illustration, the results for a special case are compared with available data. In general, such a model results in a masking function which depends on the intensity, area, and duration of the stimuli, as well as on the temporal and spatial separation between them.
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    The struggle for life: I. Two species (1967)
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    Bulletin of mathematical biology 29 (1967), S. 377-388 
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    Notes: Abstract The general equations are discussed describing two species in competition or in symbiosis or feeding one on the other.
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    A note on Rashevsky’s postulate of relational forces (1967)
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    Bulletin of mathematical biology 29 (1967), S. 403-404 
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    Organismic sets and biological epimorphism (1967)
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    Bulletin of mathematical biology 29 (1967), S. 389-393 
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    Notes: Abstract It is shown that the principle of biological epimorphism (Rashevsky,Mathematical Principles in Biology and Their Applications, Springfield, Ill.: Charles Thomas, 1960) is contained in the theory of organismic sets (Bull. Math. Biophysics,29, 139–152, 1967) if an additional postulate not directly connected to mappings is made.
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    Out-of-print volumes and back issues of the Bulletin of Mathematical Biophysics (1967)
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    Bulletin of mathematical biology 29 (1967), S. 407-407 
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    Model of urethra under uniform circumferential tension (1967)
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    Bulletin of mathematical biology 29 (1967), S. 409-409 
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    Bimolecular systems: II. Nonlinear systems of complexes (1967)
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    Bulletin of mathematical biology 29 (1967), S. 605-613 
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    Notes: Abstract This paper deals with bimolecular systems in which also complex-complex interactions occur. Because of the complexity of the problem, an approximation in a form of coupled linear systems of complexes (Bull. Math. Biophysics,29, 1–16, 1967) is considered. Two types of couplings, serial and parallel, are studied. In the serial coupling the nonlinear system of complexes has the same behavior as its subsystems. An entity, initial sensitivity, has interesting properties: in serial coupling it is at most equal to the product and in parallel coupling, at most equal to the sum of partial initial sensitivities.
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    Bimolecular systems: III. Catalytic systems (1967)
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    Bulletin of mathematical biology 29 (1967), S. 615-623 
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    Notes: Abstract Amplification effect in the catalytic bimolecular systems is a consequence of the kinetic characteristic of the catalyst. Two types of the coefficient of amplification are defined. The applicability of these definitions is given by the type of the bimolecular system. In a simple example it is shown that the concept of amplification is meaningful in these systems. Furthermore, two rules, analogous to those for a coupling of amplifiers, are derived for the two basic modes of coupling of catalytic systems. Thus, in biological systems the catalytic reactions may be regarded as biologically effective amplifiers.
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    A theory of cell hydration governed by adsorption of water on cell proteins rather than by osmotic pressure (1967)
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    Bulletin of mathematical biology 29 (1967), S. 583-596 
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    Notes: Abstract It is postulated that cell hydration is governed by adsorption of water on cell proteins in accord with the Bradley adsorption isotherm, and that the action of a solute in the surrounding solution is to lower the vapor pressure of the solution so that cell water adsorption is decreased by moving down the Bradley isotherm. From these concepts, it is derived that cell volume (V) should be related to solute concentration (x) by the equationV=−E log10 x+F whereE andF are constants which are independent of type of solute. For a non-adsorbed solute this agrees well with experimental data. For solutes which are adsorbed by cell proteins, a correction in the above equation may be necessary at higher solute concentrations, which is shown to be compatible with various experimental data. The types of experiments which are generally used to support the osmotic pressure theory of cell hydration agree equally well with the adsorption theory. The virtue of the adsorption theory is that, unlike the osmotic pressure theory of cell swelling, it is compatible with permeability of the cell membrane to solutes, which has been experimentally observed for various solutes.
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    Considerations of quasi-stationarity in electrophysiological systems (1967)
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    Bulletin of mathematical biology 29 (1967), S. 657-664 
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    Notes: Abstract Conditions under which a time varying electromagnetic field problem (such as arises in electrophysiology, electrocardiography, etc.) can be reduced to the conventional quasistatic problem are summarized. These conditions are discussed for typical physiological parameters.
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    The dead space in a compartmental lung model (1967)
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    Bulletin of mathematical biology 29 (1967), S. 711-718 
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    Notes: Abstract A compartmental lung model with any number of synchronously filling and emptying functional chambers and a common dead space or conducting region is considered. It is shown that the model gives rise to an output, in an open circuit washout determination, which is a weighted sum of exponentials. From estimates of these weights and exponential components, estimates of the model parameters can be recovered. Relations giving the unique correspondence between the output parameters and the model parameters are derived and the existence and uniqueness of solutions established.
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    A charge transfer process in the visual pigments (1967)
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    Bulletin of mathematical biology 29 (1967), S. 677-690 
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    Notes: Abstract A physical model that incorporates all the experimental information on the formation of the visual pigment rhodopsin is presented. The visual pigments consist of a chromophore bound to an appropriate protein. Thus rhodopsin (λm 505 mμ) is formed by a Schiff’s base linkage C19H27CH=NH+-opsin (λm 440 mμ) between 11-cis retinal (λm 380 mμ) and the protein opsin (λm 280 mμ). It is found that there exists a red shift in the spectrum of rhodopsin from the Schiff’s base. The model brings an explanation for this red shift. It is shown that such a shift may be due to a charge transfer process (R. S. Mulliken,J. Am. Chem. Soc.,74, 811–824, 1952) between an electron at the double bond of carbons C11−C12 and an atomic orbital of the sulphur present in cysteine. This provides an explanation of the presence of SH-groups in the protein after the absorption of light. A one-electron approximation is used and the dipole momentμ NV ; hence, the oscillator strengthf of the transitionNV is estimated and compared with the experimentally determined extinction coefficient ∈m by mixing 3.5×10−3 M of 11-cis retinal with 8.3×10−5 M of cysteine at pH ranges 6 through 8. Reasonable agreement is found. Solvent, concentration and temperature dependence are shown also.
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    Optimal models for social change (1967)
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    Bulletin of mathematical biology 29 (1967), S. 841-862 
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    Notes: Abstract By assigning coordinates to the environmental function space comprising all physical and mental stimuli, mathematical interpretations can be based on such terms as adaptability, and reactivity which relate to individuals interacting with their environment within a society. These psychometric concepts are incorporated into a framework of functional analysis, which permits the optimization of social change by maximizing the satisfaction integral through the use of variational or dynamic programming methods in conjunction with some optimal social policy. The approach provides a mathematical connection between psychology and sociology, and further demonstrates that existing forms of government are simulated by differential equations belonging to the same general class. The synthesis of new classes of functional equations describing social progress is visualized as a legitimate objective for abstract mathematical sociology.
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    Note from the editor (1968)
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    Bulletin of mathematical biology 30 (1968), S. 1-1 
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    A suggestion for a new approach to the mathematical theory of imitative behavior (1967)
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    Bulletin of mathematical biology 29 (1967), S. 863-877 
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    Notes: Abstract The theory of imitative behavior as developed hitherto by the author was based on the assumption that each individual has a natural preference for one of the two mutually exclusive behaviors. The endogenous fluctuations in the central nervous system then result in the individual’s exhibiting the two behaviors alternately with a relative frequency determined by the natural preference. Imitation shifts the natural preference towards one or the other of the two mutually exclusive behaviors. In the present approach it is suggested that the relative frequency of the two mutually exclusive behaviors exhibited alternately is determined by maximizing the “satisfaction function” of the individual, that is by hedonistic factors rather than by purely random fluctuations. Corresponding equations are developed. It is shown that in certain cases, even when the imitation effect is absent, a sort of “pseudoimitation” may occur. Another situation leads, in the case of two individuals only, to a complete “division of labor” between them, with respect to the two behaviors. Each one exhibits only one behavior. After that imitation is introduced explicitly by assuming that imitation by one individual or another increases the satisfaction function of the imitating individual. Results thus obtained show similarities to the results of the old theory.
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    Erythrocyte distribution in arterial blood flow: IV. Simplification of the proposed equations (1968)
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    Bulletin of mathematical biology 30 (1968), S. 27-32 
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    Notes: Abstract In a previous paper (Bull. Math. Biophysics,29, 565–574, 1967) the author developed equations to represent velocity and hematocrit profiles in quasi-Poiseuille flow of blood. It was assumed that energy dissipation was minimized and that the viscosity depended on hematocrit and shear rate according to the Casson formula. These equations are simplified considerably, placed in a form more suitable for numerical solution and shown to depend on a single dimensionless parameter. Typicalin vivo values for this parameter are calculated.
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    Éléments d’une théorie générale de la croissance d’une population (1968)
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    Bulletin of mathematical biology 30 (1968), S. 33-46 
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    Notes: Abstract In this paper, discrete models of reproduction are studied. In part one, definitions are given, particularly on order of the reproduction; part two concerns the growth of the population; part three, the phenomena of delay or acceleration; and part four, the consequences of mortality.
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    Neuroendocrine control of thyroid secretion in living systems: A feedback control system model (1968)
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    Bulletin of mathematical biology 30 (1968), S. 3-26 
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    Notes: Abstract A model of the regulation of thyroid hormone in the bloodstream of living systems is formulated and analyzed. The portion of this model defined as theregulator includes components representing the thyroid, anterior pituitary and hypothalamic organs and their intercommunicating channels, that is, the peripheral plasma and hypophysial portal circulations and certain neuro-secretory connections. The loss of hormones from the plasma in the living system associated with physiological mechanisms within the peripheral tissue space and the excretory pathways is represented in the model by a lumpedload on the regulator. The model is reduced to a system of differential equations involving eleven parameters and variables, all of which are identified with certain physiological structures and states. Five of these are currently observable by available laboratory techniques and two others are computable explicity from the equations of the model; the remaining four can be computed in the same way to within a multiplicative constant. Procedires for carrying out ten of these measurements and calculations are suggested. On the basis of the equations and parameters of the model, a discussion of the normal behavior and the response of this system to certain types of disturbances is presented. A systematic effort has been made in the development of this model to include all relevant physiological data and relationships reported in the biological literature. A summary of this literature, reflecting the views and interpretations made by the authors of this paper, is included for completeness and ease of reference.
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    Linear bulk diffusion into heterogeneous tissue (1968)
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    Bulletin of mathematical biology 30 (1968), S. 47-59 
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    Notes: Abstract In this paper an expression is derived which describes the transient overall uptake of an inert solute by a section of tissue excised with parallel faces and placed upon an impermeable base. The approach diverges from the conventional analyses for perfused tissue (Morales and Smith,Bull. Math. Biophysics,6, 125–141, 1944;7, 47–99, 1945) because the extravascular zone is regarded as a heterogeneous diffusion medium. Account for this is taken by regarding tissue as effectively composed of two phases—a continuous (extracellular) phase similar to water, and a dispersed phase comprising cells of irregular profile. In both phases the relevant mode of uptake is taken as bulk diffusion rather than surface permeation, thus emphasizing the influence of the internal geometry of the tissue upon its overall exchange response.
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    The identification of recirculating systems in the frequency domain (1968)
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    Bulletin of mathematical biology 30 (1968), S. 87-104 
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    Notes: Abstract A method for the identification of flow systems by frequency domain analysis has been extended to include systems with recirculation and truncated data curves. Application of the technique to clinical indicator-dilution curves indicates that the method may be useful in the quantitation of intracardiac shunts. A number of numerical examples which demonstrate the accuracy of the method are included.
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    A systems-mathematical interpretation of psychoanalytic theory (1968)
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    Bulletin of mathematical biology 30 (1968), S. 61-86 
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    Notes: Abstract By assigning time-varying coordinates to all environmental stimuli, it has been possible to axiomatize psychoanalytic theory on the five principles of multiple causation, growth-aging influence, genetic influence, historic influence and conscious-unconscious activity. The theorems of summation of response and the inevitability of conscious-unconscious conflict with their corollaries follow directly from the axiomatic foundations, as does the existence of an adaptation-defense mechanism. The interpretation of the defense mechanism in terms of an ego-id feedback system provides the basis for the structural existence of conscious-conscious and unconscious-unconscious conflict.
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    Cellular systems as sequential machines: Stability properties (1968)
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    Bulletin of mathematical biology 30 (1968), S. 117-122 
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    Notes: Abstract In previous studies of (M,R) (Rosen, 1961; Demetrius, 1966), it was assumed that changes in the structure of (M,R) which were induced by environmental alternations occurred without error. Here, the effect of both “genetic” and “metabolic” malfunctions on the behavior of (M,R) is examined and a subclass of these systems whose behavior is invulnerable to such errors is specified.
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    Cellular systems as graphs (1968)
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    Bulletin of mathematical biology 30 (1968), S. 105-116 
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    Notes: Abstract The definition of an (M,R) is formulated in a way that emphasizes its mathematical properties. Neglecting interactions between the components, it is shown that: (1) An (M,R) contains only one non-reestablishable component. (2) If an (M,R) contains only one non-reestablishable component, then that component is central. Examples are given to illustrate the biological significance of these two results. The notion of “lag-independence” is introduced, and it is shown that if a system possesses only one non-reestablishable component which is “lag-independent” then all components are lag-independent. The concepts of reestablishability, centrality and lag-independence are applied in order to suggest various criteria for optimal organization of (M,R).
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    A theoretical model for the control mechanisms in biochemical reactions (1968)
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    Bulletin of mathematical biology 30 (1968), S. 123-133 
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    Notes: Abstract A mathematical representation for the analysis of control mechanisms in biochemical reactions is presented. First, the theoretical concept of concentration in biological systems is developed. Then a system consisting of two functions λ and τ is constructed as a network of single output automata. The range of λ is taken to be formed by a set of twostates qualitatively different from the “repair function” Φ f of a mappingf: A→B in the stimulated Φ1 and unstimulated state Φ0. Likewise, the range of τ is formed by the set δ={f o ,f 1} wheref 1 means the mappingf in its stimulated state andf o in the unstimulated one. It is demonstrated that the mathematical structure described acts as a control mechanism over thef and Φ f , so that two biochemical components,A→B, are transformed at a controlled rate. Some of the biological applications of this model are briefly examined. The Jacob-Monod model, the enzymatic adaptation phenomenon, and the “rheon unit” hypothesis are discussed within our framework. Eventually, a concrete model for the RNA-polymerase mechanism, based on the above discussion, is presented.
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    A contribution to Rashevsky’s mathematical theory of development (1968)
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    Bulletin of mathematical biology 30 (1968), S. 135-151 
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    Notes: Abstract The application of Rashevsky’s transformationT to a primordial graph yields a set of graphs corresponding to different stages in the development of the organism. However, sinceT is multiple-valued the graphs obtained are not ordered. To obtain an ordering, it is first shown that the set of graphs under consideration is equivalent to a well defined setO (for “organism”) ofn-tuples. A metric is then introduced which is based on a biological consideration discussed by Rashevsky (Bull. Math. Biophysics,16, 317–348, 1954). Since a metric implies an ordering of the setO, with a knowledge of the structure of the primordial, one can obtain the developmental sequence. Unfortunately, at present, the structure of the primordial graph is unknown which makes the direct application of the above principle impossible. Consequently, an indirect approach which makes use of more accessible biological phenomena is discussed as well. The hypothesis thatrate of development decreases exponentially and the implications this has with regard to the metric onO are discussed. It is shown that if the hypothesis is accepted the search for the developmental sequence is narrowed.
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    Mathematical studies of the interaction of respiratory gases with whole blood I. O2 absorption (1960)
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    Bulletin of mathematical biology 22 (1960), S. 391-415 
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    Notes: Abstract Some progress has been made on the problem of the interaction of respiratory gases with whole blood. A practical working model for oxygen absorption in and interaction with whole blood is developed by assuming that oxygen molecules compete with protons for binding sites on the hemoglobin molecule and by invoking the Wyman-Allen (Jour. of Polymer Science,5, 499–518, 1951) hypothesis that two oxygen molecules go on the hemoglobin at one time. Extensive tests of this model against saturation measurements on blood from humans, horses, oxen and sheep are made. Values for the equilibrium constants are calculated and compared. In addition a second working model has been developed in an attempt to explain why O2 saturation measurements when expressed as (100 percent — percent saturation) are an exponential function of oxygen partial pressure. Considerations which make plausible the following expression for saturation, [1−2e −γx/h1/2/(1+(1/20)(β′/h 1/2+h 1/2/β′))] are presented. Herex denotes oxygen tension,h denotes hydrogen ion concentration and β′ and γ are parameters.
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    Mathematical studies of the interaction of respiratory gases with whole blood II. CO2 absorption (1961)
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    Bulletin of mathematical biology 23 (1961), S. 1-14 
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    Notes: Abstract Some progress has been made on the problem of the interaction of respiratory gases with whole blood. A working mathematical model for the O2−CO2 interaction phenomena has been developed from mathematical studies of the data. The Edsall-Wyman (1958) model for CO2 absorption is improved upon in this paper by consolidating it with the O2 absorption model developed in paper I of this set (Bernard, S. R.,Bull. Math. Biophysics,22, 391–415, 1960). This improved model assumed the effect of O2 on CO2 absorption is mediated through the electrical charge possessed by the hemoglobin molecule,i.e., O2 molecules bound to hemoglobin displace protons from the hemoglobin thereby increasing the negative charge on the hemoglobin and at the same time increasing the acidity of the solution. The model is tested against the data.
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    On the mutual influence of individuals in a society (1961)
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    Bulletin of mathematical biology 23 (1961), S. 15-18 
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    Notes: Abstract On the basis of previously proposed mathematical models of social behavior, the present note investigates the possibility of the control of behavior remaining permanently in the hands of one class, if this class possesses sufficient means for influencing mass behavior. The conclusion is reached that, with the assumptions made, if the behavior imposed by the controlling class leads to sufficiently strong dissatisfaction, the control will pass to another class, no matter how strong the controlling power of the first.
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    Contribution to the mathematical biophysics of automobile driving (1961)
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    Bulletin of mathematical biology 23 (1961), S. 19-29 
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    Notes: Abstract Traffic in one direction on a multilane highway is considered, and a general expression for the number of cars which pass a car travelling at a given velocity, as well as the number of cars which the given car passes, is derived for the case when the speeds of different cars are distributed in some arbitrary manner. Closed expressions are derived and discussed for a rectangular distribution. Each passing by another car or of another car is considered as a distracting stimulus which affects the reaction times of the driver. Using previously derived expressions for the safe speed as a function of reaction times, expressions for the safe average speed are derived, in terms of the volume of traffic and of the spread of the distribution of speeds.
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    Glucose metabolism—A “mixed-cycle” system (1961)
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    Bulletin of mathematical biology 23 (1961), S. 99-103 
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    Notes: Abstract Emphasis upon the importance of homeostatic feedback has drawn attention away from the complexity of biological processes. A study of glucose metabolism indicates the importance of open-cycle as well as closed-cycle mechanisms. Besides the glucose-dependent mechanism of insulin secretion, many open-cycle processes involving the liver, adrenal glands and kidneys, play important roles in the variation of blood glucose. In addition, glucose utilization by the tissues is essentially open-cycle in nature.
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    Southern Regional Graduate Summer Session in Statistics at Virginia Polytechnic Institute (1961)
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    Bulletin of mathematical biology 23 (1961), S. 105-106 
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    On the unreliability of judgments of certain social situations by minority groups (1961)
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    Bulletin of mathematical biology 23 (1961), S. 413-416 
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    Notes: Abstract In certain situations like the aftermath of a revolution when discontent rises amongst certain groups of the population, it is frequently observed that the discontented groups are firmly convinced that their point of view is shared by the majority of the population. Yet future events prove that this is far from being the case. This effect is partly attributable to “wishful thinking,” partly to a purely social mechanism. The wishful-thinking effect may be considered as a case of psycho-physical discrimination in which a bias is introduced proportional to the degree of satisfaction anticipated from a given situation. H. D. Landahl's well-known equations can be applied to this case. The social factor is based on the circumstance that an individual associates by preference with such other individuals as have similar opinions. This results in an actual error of estimation of the relative minority or majority because of different frequencies of contact with individuals of the two opposing groups. Both factors may be combined into one equation.
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    Announcements (1961)
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    Bulletin of mathematical biology 23 (1961), S. 421-422 
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    A general target theory of radiobiological action: II (1963)
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    Bulletin of mathematical biology 25 (1963), S. 367-385 
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    Notes: Abstract In Part II we prove some of the more complicated theorems stated and used in Part I. In particular, we derive the distribution functionsD 1,D 2, andD 3, and prove some of their properties under various limiting conditions.
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    On the theory of fixation of isohemagglutinins on the erythrocytes (1963)
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    Bulletin of mathematical biology 25 (1963), S. 387-392 
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    Notes: Abstract The graphical treatment utilized by Marmasse in order to test “Wurmser’s theory of agglutination” has been applied, taking into account all the data available. Contrary to Marmasse’s conclusion, the application of this graphical method is not a valid argument against the theory.
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    On the Fokker-Planck equation in the stochastic theory of mortality: II (1963)
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    Bulletin of mathematical biology 25 (1963), S. 343-366 
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    Notes: Abstract This is the continuation of part I, which was published in the September, 1963, issue ofThe Bulletin. Section 5 treats the special case in which the left absorbing barrier recedes to −∞, leaving essentially only one barrier at a finite distance Λ (〉0) from the origin. The eigenfunctions are now parabolic cylinder functions. The limiting cases Λ→+∞ and Λ→0 are also considered. Though meaningless for practical applications to our problem, they are of interest, mathematically, because the Green’s function for the solution of the Fokker-Planck equation assumes a particularly simple form. In section 6 we study, by means of an example, how the “force of mortality” may vary with time before attaining its final asymptotic value. Section7, still dealing with only one absorbing barrier, shows that our results for “strong homeostasis” are identical with those derived by Chandrasekhar for the escape of particles through a potential barrier in the limiting case of quasi-static flow. Precise conditions are given for the validity of both the quasi-static and the Smoluchowski approximations to the Fokker-Planck equation. Finally, in section 8, a brief mention is made of Gevrey’s method for the solution of parabolic partial differential equations.
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    Studies in stochastic learning theory: I path-dependent linear models for experimenter-subject controlled experiments (1963)
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    Bulletin of mathematical biology 25 (1963), S. 393-419 
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    Notes: Abstract The derivation of learning models relative to choice behavior in experimenter-subject controlled experiments with two outcomes (right or wrong) is considered from the point of view that any such model must satisfy a criterion of optimality. The criterion adopted for investigation, termed optimal asymptotic behavior, is that of the subject asymptotically learning which of the alternatives has the greater probability of being correct. A class of path-dependent linear models is posed as possible candidates. It is shown that no members of this class satisfy the criterion although two of them approach it by making a learning parameter small enough. The possible implications of this are discussed.
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    On the possibilities of error-detecting codes in nucleic acid molecules (1964)
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    Bulletin of mathematical biology 26 (1964), S. 25-29 
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    Notes: Abstract Error-detecting codes have been known to mathematicians and to electrical engineers for over ten years. In general, such codes utilize an additional orparity bit for purposes of detecting errors by the addition of all positive binary bits or “1’s” occurring in any code word. However, since the process of addition is required for such code detection, it is not surprising that these codes have not been applied to the nucleic acid molecule. In 1962, P. I. Hershberg (Trans. I.R.E., CS-10, 280–4, 1962) outlined a categorical constraint which permitted the realization of a class of error-detecting codes which did not require parity bits. This class of codes is applied to the nucleic acid molecule in the present paper.
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    On some topological properties of the systems of compartments (1964)
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    Bulletin of mathematical biology 26 (1964), S. 31-38 
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    Notes: Abstract Compartment systems are often used as models for tracer and drug kinetics. The structure of a compartment system is here analyzed by means of theory of graphs methods. In particular the precursor-successor relationship between any two compartments is classified according to the structure of the graph of the system and to the values of the elements of the matrix associated with it.
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    The equations of growth (1964)
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    Bulletin of mathematical biology 26 (1964), S. 39-43 
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    Notes: Abstract An application of a bifurcation theorem shows the existence of periodic solutions of a system of differential equation used to describe competition between two species. It is then shown that the results are more general than those previously established.
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    An analysis of the 2o and 10o field color-matching functions (1964)
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    Bulletin of mathematical biology 26 (1964), S. 9-24 
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    Notes: Abstract The 2o and 10o field color-matching functions are independent: one specification is not a linear transformation of the other, even after correcting for macular pigment effects. Therefore, the “true” color-matching functions which directly describe the linear responses of the eye must be different for the two field sizes. This means that a given stimulus will, in general, have a different chromaticity depending upon the field size, regardless of the choice of any one colorimetric co-ordinate system for all field sizes. However, in spite of these chromaticity differences, a large uniform field usually appears nearly uniform. Such color uniformity implies that even though chromatic differences occur as a function of retinal position or field size, these differences are small. If this is the case, then the underling “true” color-matching functions determining the observed color-matching functions must be nearly, but not quite, identical. These differences vanish as identity between the sets of color-matching functions is approached. This property suggests a method of calculating the “true” color-matching functions. The “true” color-matching functions must approximate those obtained by minimizing the chromaticity differences between two independent sets of data. This can be done by assuming that the coefficients of transformation should be adjusted so as to produce as nearly identical chromaticities for spectrum stimuli as possible. In this paper, it is also assumed that the “true” color-matching functions have no negative values, as if they were based on actual absorption spectra. This article describes the calculation of the “true” 2o and 10o field color-matching functions satisfying these two conditions. For both field sizes, the maxima of the three functions are near 435, 540, and 585 mμ, after correcting for the filtering effects of the ocular media and macular pigment.
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    A note on the elementary α-perceptron (1964)
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    Bulletin of mathematical biology 26 (1964), S. 45-47 
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    Notes: Abstract In this note the principal convergence theorem (F. Rosenblatt,Principles of Neurodynamics, Spartan Books, Washington D.C., 1962, 111–116) is proved by a new method.
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    On the adsorption of ions at charged sites in an electric field: I (1964)
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    Bulletin of mathematical biology 26 (1964), S. 49-55 
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    Notes: Abstract Considering only nearest neighbor interactions, an expression is obtained for the grand partition function for the adsorption of two kinds of monovalent positive ions at a long chain of one type of monovalent negative fixed sites in an electric field. Expressions are obtained for the fractions of sites which are occupied by each kind of ion as well as of those which are unoccupied as a function of the potential of the electric field.
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    Two remarks on the mathematical biology of automobile driving (1964)
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    Bulletin of mathematical biology 26 (1964), S. 57-61 
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    Notes: Abstract In connection with a series of previous papers by this author (Bulletin of Mathematical Biophysics,21, 299–308, 375–385;22, 257–262, 263–267;23, 19–29;24, 319–325) results obtained by A. Crawford (Economics 5, 417–428) on the effects of irrelevant lights on reaction times toward a given light stimulus are discussed. The conclusions from a previous paper of this author (Bulletin of Mathematical Biophysics,23, 19–29) are elaborated.
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    Analysis of a simple avoidance situation. II. A model (1964)
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    Bulletin of mathematical biology 26 (1964), S. 77-81 
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    Notes: Abstract A mathematical model has been constructed to describe experimental data recorded in a study of a simple avoidance situation. The theoretical description makes use of the concept of the effective number of shocks. The model explains the existence of oscillations encountered in previous experiments.
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    Studies in stochastic learning theory: II. Interpretation of response probabilities (1964)
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    Bulletin of mathematical biology 26 (1964), S. 63-75 
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    Notes: Abstract Response probabilities are interpreted from two points of view. One corresponds to fluctuations in physical parameters suggestive of a neurological basis, and the other corresponds to fluctuations in stimulus sample constitution. The two interpretations are shown to be equivalent under rather general conditions, giving the same type of relation between response and training states. This relation is different from that obtained via the response strength concept used in Part I. As a step toward evaluating the difference in predicted behavior for these different response-training relations, a general functional-difference equation is derived that describes the moments of the corresponding stochastic process in experimenter-subject controlled experiments. As an immediate application, it is used to obtain the continuity condition for the solution of the functional equation treated in Part I, and to justify the differentiability conditions assumed in establishing asymptotic properties of the solution as a function of the reinforcement parameter.
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    An avoidance learning situation. A neural net model (1964)
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    Bulletin of mathematical biology 26 (1964), S. 83-89 
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    Notes: Abstract A simple avoidance situation is considered in terms of a neural net learning model. Data for the control situation can be represented by an expression having three parameters which determine the initial and the steady state activities together with the transient aspects. The introduction of a learning parameter then allows one to calculate satisfactorily the results obtained in the experimental situation in which shock is applied.
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