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  • Springer  (57,968)
  • American Physical Society  (6,888)
  • American Geophysical Union  (1,814)
  • 1970-1974
  • 1965-1969  (66,670)
  • 1967  (38,015)
  • 1965  (28,655)
Collection
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  • 1970-1974
  • 1965-1969  (66,670)
Year
  • 1
    Electronic Resource
    Electronic Resource
    Bimolecular systems: I. Linear systems of complexes (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 1-16 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02476958
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  • 2
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    DNA histograms of irradiated cells: The case of prophase-trapping (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 17-32 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02476959
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  • 3
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    A note on the mathematical biology of automobile driving (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 187-188 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02476971
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  • 4
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    Mathematical biology of automobile driving (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 181-186 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476970
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  • 5
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    Angles of branching and diameters of branches in the human bronchial tree (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 245-259 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02476898
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  • 6
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    Finite linnaean structures (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 191-206 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02476893
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  • 7
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    Abstract power structures (APS) and related bumping orders (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 207-216 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02476894
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  • 8
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    Stability analysis of a mathematical neuron model (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 217-226 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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  • 9
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    A neural net model for masking phenomena (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 227-232 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02476896
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  • 10
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    The struggle for life: I. Two species (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 377-388 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02476909
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  • 11
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    A note on Rashevsky’s postulate of relational forces (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 403-404 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    URL: http://dx.doi.org/10.1007/BF02476912
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  • 12
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    Organismic sets and biological epimorphism (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 389-393 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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  • 13
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    Out-of-print volumes and back issues of the Bulletin of Mathematical Biophysics (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 407-407 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    URL: http://dx.doi.org/10.1007/BF02476914
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  • 14
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    Model of urethra under uniform circumferential tension (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 409-409 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    URL: http://dx.doi.org/10.1007/BF02476915
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  • 15
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    Bimolecular systems: II. Nonlinear systems of complexes (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 605-613 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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  • 16
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    Bimolecular systems: III. Catalytic systems (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 615-623 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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  • 17
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    A theory of cell hydration governed by adsorption of water on cell proteins rather than by osmotic pressure (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 583-596 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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  • 18
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    Considerations of quasi-stationarity in electrophysiological systems (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 657-664 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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  • 19
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    The dead space in a compartmental lung model (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 711-718 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02476922
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  • 20
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    A charge transfer process in the visual pigments (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 677-690 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 841-862 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02476932
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  • 22
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    A suggestion for a new approach to the mathematical theory of imitative behavior (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 863-877 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    Compartmental systems and generating functions (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 49-63 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Compartmental systems can be represented by direct graphs in which each node corresponds to a generating function and each arm to a transfer generating function. A homomorphism is established between a compartmental system and this representation, in analogy with that obtained through the use of the Laplace transformation. From the values obtained experimentally in a given compartment, through the solution of a difference equation, the generating function for the corresponding node can be calculated and the graph of the system can be built up within the degrees of freedom of the model. From the graph it is possible to calculate the transfer generating function between any two connected nodes, the mean permanence time in a given node, the mean transit time between two nodes, and their precursor-successor order.
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    On the competitive exclusion principle (1965)
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    Bulletin of mathematical biology 27 (1965), S. 85-89 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The Competitive Exclusion Principle, formulated by V. Volterra (Memorie del R. Comitato Talassografico Italiano,131, 1–142, 1927) for a number of species competing for a common ecological niche, is extended to a number of species competing for many ecological niches.
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    Periodic and aperiodic phases of circulatory mixing (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 65-70 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A modification is presented of an earlier theory of the mixing of dye following injection into the circulation. Approximate theoretical relations are given for the concentration of dye in the right heart and in the aorta following right atrial injection. It is shown that when the probability distribution of transit times around the circulation has a prolonged tail, mixing waves are now inscribed about a quasi-exponential relation. Later in time the relation levels off to a uniform asymptotic concentration corresponding to an equilibrium volume of dilution.
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    Cooperative specific adsorption of ions at charged sites in an electric field (1965)
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    Bulletin of mathematical biology 27 (1965), S. 91-104 
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    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract The adsorption of two cations at the anionic sites of a polymer (e.g., such as a protein) in an electric fields is discussed, taking into account cooperative interaction of the cations mediated through the backbone of the polymer. The calculation of the grand partition function explicitly considers the vacant negative sites of the polymer. As in the case without cooperative interaction, the problem reduces to the determination of the largest eigenvalue of asymmetric matrices. The weights of the different neighbor configurations are determined. Approximate formulae for the grand partition function and for those weights are derived. The formal analogy of these cooperative phenomena and those occurring in quantum (bio)chemistry is pointed out exemplifying an earlier suggestion about the basis of quantum biology.
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    A proposed mechanism for the sol-gel transformation (1965)
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    Bulletin of mathematical biology 27 (1965), S. 105-112 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The transformation from gel to sol in cell cytoplasm is treated as the transition from a lattice of macromolecules linked by Ca++ ions to a random distribution of the macromolecules. The transition is a cooperative process, whose probability is expressed in terms of the theory of runs. The process is related to cell metabolism by the assumption that available Ca++ concentration is regulated by metabolically produced endogenous chelating agents.
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    Kinetic criteria of solid state mechanisms in biology (1965)
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    Bulletin of mathematical biology 27 (1965), S. 113-118 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Kinetic criteria for solid state physical mechanisms of electron and ion transport in biological systems are summarized, and the mechanisms are discussed. A reaction which is rate-limited by electron or ion transport across a particle or membrane in accord with Ohm's law will show first order kinetics, with an hyperbolic relationship between rate constant and the sum of substrate plus product. Larger initial substrate concentrations produce smaller rate constants, thus giving the appearance of substrate inhibition. Examples are cytochrome oxidase and peroxidase, and pyruvate carboxylase. Ohmic transport mechanisms may be caused by electron conduction or superconduction through protein, by electron conduction through water, or by conduction of ions through membranes. A reaction which is rate-limited by charge transport across an activation energy barrier at an interface in accord with a logarithmic voltage-current law will show reaction kinetics conforming to the Elovich equation, and will have the appearance of a pair of simultaneous first order processes. Examples include decay of photogenerated free radicals in eye melanin particles and in photosynthetic particles of bacteria, and sodium and potassium ion transport across cell surfaces. The logarithmic voltage-current law may be regarded as an empirical relationship describing behavior of interfaces, justified by extensive experimental data on many types of interfaces, or it may be derived theoretically for individual cases from statistical mechanical and/or solid state physical considerations.
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    Analysis of some effects of aortic smooth muscle onin vivo aortic pressure curves (1965)
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    Bulletin of mathematical biology 27 (1965), S. 119-130 
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    Notes: Abstract When aortic pressure curves were predicted previously on the basis of a newly developed model of visco-elastic properties of the aorta, it was necessary to use published viscoelastic constants. These were usually obtained from longitudinal strips of blood vessels long removed from the animal, and therefore probably containing deteriorated smooth muscle. The predicted curves had the same form as actual tracings, substantiating the analysis somewhat, but the pressure levels were low. These low levels, if due to inadequate visco-elastic constants, could be attributed to the use of longitudinal rather than circumferential segments as well as to the use of segments with deteriorated muscle. The present analysis uses data collected by the author testing circumferential viscoelastic properties of fourteen different aortic regions in a way suggested by the author's model of an aortic wall. Moreover, the constants were measured on segments containing muscle relaxed by EDTA solutions and on similar segments containing muscle contracted by neosynephrine. These visco-elastic constants were used in the author's nonlinear differential equation of motion of the aortic wallin vivo to predictin vivo pressure curves. The predicted curves were low in any given aortic region if relaxed constants were used, but at normal levels with contracted constants. In fact, pressure curves predicted using constants obtained from aortic segments containing contracted muscle resembled actual tracings in form and pressure levels. Even the observed variations in the form of the systolic pressure curve down the aorta were predicted by this analysis.
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    Biological similarity relative to allometric quantities (1965)
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    Bulletin of mathematical biology 27 (1965), S. 131-133 
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    Notes: Abstract It is an empirical finding that an allometric quantity with dimensional exponents α, β and γ relative to mass, length, and time, respectively, has a value for its allometric exponentb satisfying the relation $$\tfrac{1}{3}(3\alpha + \beta + {\gamma \mathord{\left/ {\vphantom {\gamma 2}} \right. \kern-\nulldelimiterspace} 2}) \leqslant b \leqslant \tfrac{1}{3}(3\alpha + \beta + \gamma ).$$ A theoretical derivation is given of this double inequality using only the fact of constant density and the plausible assumption that metabolic rate is a dominant allometric quantity.
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    The entropy of a continuous distribution (1965)
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    Bulletin of mathematical biology 27 (1965), S. 135-143 
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    Notes: Abstract C. Shannon's definition (Bell System Technical Journal,27, 379–423, 1948) of the entropy of a continuous distribution is dimensionally incorrect and does not have the same significance as the corresponding definition in the discrete case. A new definition is proposed: this modified entropy is more like the entropy of a discrete distribution in one way, in another more like Shannon's “transmission rate.” The ideas are illustrated by reference to Wright's study of the hereditary influence on the coat pattern of the guinea pig.
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    Theorems on random evolution (1965)
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    Bulletin of mathematical biology 27 (1965), S. 145-150 
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    Notes: Abstract In the following paper, a possible mode of evolution is described which differs from the traditional modes in not being selective in the Darwinian sense.
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    Information theory and genetics (1965)
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    Bulletin of mathematical biology 27 (1965), S. 177-181 
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    Notes: Abstract A description of the kinds of systems susceptible to information theoretical analysis is given. By means of an example, certain common fallacies in the application of communication theory to biology are illustrated. The entropy-information analogy is discussed. *** DIRECT SUPPORT *** A01E2109 00008
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    A note on equivalent mathematical models (1965)
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    Bulletin of mathematical biology 27 (1965), S. 161-175 
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    Notes: Abstract A mathematical model of a process contains parameters supposedly characterizing the system which manifests the process. If the parameters are statistically distributed in a population of such systems, the process manifested by the entire population will in general be described by a different mathematical model. Thus a choice is always at hand between two or more mathematical models, depending on which parameters (if any) are assumed to be distributed and, if so, how. Examples of such alternative interpretations are given for mathematical models of some behavioral processes.
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    Superimposed random coding of stimulus-response connections (1965)
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    Bulletin of mathematical biology 27 (1965), S. 191-202 
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    Notes: Abstract The problem of economically linking a large number of stimuli with a large number of potential responses is considered to resemble a problem of efficient retrieval of documents (the responses) on the basis of their characterization by descriptors (the stimuli to which the responses are appropriate). In this retrieval problem, a method whereby the codes for descriptors are random positions in a coding field, and whereby codes for all applicable descriptors are superimposed in the same field, seems to be the simplest way of avoiding serious difficulties of retrieval. After a review of this method, the possibility is considered that very simple neural mechanisms could embody the essential features of the method. The aim of the discussion is to learn whether very simple structures and patterns of reinforcement would be adequate to carry out useful information processing in the brain, and to show some conceivable functions of simple neural networks which the experimenter might keep in mind. The discussion also shows how the structure of a simple “perceptron”-like network is suggested by the requirements of a retrieval task.
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    Studies on the electroretinogram. II. Model of an a-wave regenerator mechanism (1965)
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    Bulletin of mathematical biology 27 (1965), S. 223-233 
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    Notes: Abstract A model is proposed to relate the regeneration of the ERGa-wave after partial light adaptation to the level of the light adaptation. The model assumes that thea-wave amplitude is a function of some reactive substance associated with ana-wave generator. The maximuma-wave amplitude occurs when the eye is fully dark adapted, and thea-wave generator initiator concentration is at a maximum. Thea-wave generator initiator concentration can be decreased by interacting with a product of the rhodopsin-light energy reaction, and increased by removal of this inhibitor. The removal of the inhibitor depends upon the isomerization of the all-trans-retinene to the 11-cis form. An excess of inhibitory material overa-wave generator initiator would cause a delay in the appearance of thea-wave until the excess inhibitory material is removed. This delay is a linear function of the logarithm of the adapting energy. The agreement of this model with the experimental ERG data is very good.
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    On the quantitative relationship between survival rate of larvae and their food supply (1965)
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    Bulletin of mathematical biology 27 (1965), S. 215-222 
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    Notes: Abstract The survival rate of fishes in their earlier stages of development and the influencing factors present one of the most fundamental problems of fish population dynamics. After I. Hjort's (Cons. L.'explor. Ner.,20, 3–228, 1914) work, there have been many investigators in this field and there is no doubt about the very important role of ova and larvae mortality in the fate of a given fish generation. Less clear are the ideas concerning factors determining the high mortality of fishes in their earlier stages of development; especially the factor of food supply of larvae during the period of transition to exogenic nutrition. The value of this factor has been estimated differently from different points of view. For example, R. J. H. Beverton and S. J. Holt (On the Dynamics of Exploited Fish Population, 1957) have given to the food supply factor its deserved importance. On the other hand, T. V. Dekhnik (Trudy Sevastopolskoi Biologicheskoi Stantsii,13, 216–244, 1960;Ibid.,14, 222–243, 1961) has proved in her investigations that at least for pelagic larvae of Black Sea fishes there is an excessive amount of food, and that therefore food cannot play an important role in larva survival. Not wanting to stop to review the literature of the problem (see Dekhnik,Trudy Sevastopolskoi Biologicheskoi Stantsii,13, 216–244, 1960), we will only remark that the problem as a whole needs further investigation. Not only new data are needed, but also methods for following up analysis have to be worked out.
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    Learning fluctuations and hysteresis phenomena (1965)
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    Bulletin of mathematical biology 27 (1965), S. 253-259 
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    Notes: Abstract The investigation described here is anexperimental one which brings to light some new facts and confirms others already reported. They partly concern the hysteresis phenomena handled by N. Rashevsky (Mathematical Biophysics, 1960) and partly provide a point of departure for future biophysical research to be undertaken by biomathematicians.
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    An analysis of aortic pressure curves taking into account visco-elastic properties of the aorta and variations in peripheral resistance (1965)
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    Bulletin of mathematical biology 27 (1965), S. 27-52 
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    Notes: Abstract The aortic pressure curve necessarily reveals the mechanical properties of the aorta and peripheral resistance as well as of the dynamics of blood flow. The present study uses a reasonable model of visco-elastic properties of the aorta, a reasonable form for variations in peripheral resistance and blood flow to predict an aortic pressure tracing. Numerical values of constants measured experimentally were available in the published literature. These were used in the nonlinear differential equations of motion of the system under analysis. The equations yielded to piece-wise solution, giving the aortic circumference and the aortic pressure as functions of time. The form of both curves resembles clinical tracings, but numerical values of circumference were higher and of pressure lower thanin vivo. The discrepancies between predicted and clinical curves may reveal certain inadequacies in published measurements on visco-elastic constants. These measurements have been made on longitudinal rather than circumferential strips often containing dead rather than living muscle. The discrepancies, therefore, indicate specific gaps in our knowledge of aortic behaviorin vitro. The suggested model of the system aided in the design of experiments which could supply data necessary to substantiate or to revise the model.
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    The venous circulation (1965)
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    Bulletin of mathematical biology 27 (1965), S. 373-377 
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    Notes: Abstract Some aspects of the circulation through the veins remain unexplained. The pressure gradient which ordinarily exists across a large vein, for example, is much greater than that necessary to maintain the same flow through a rigid tube of identical diameter (Brecher, 1956; Starling and Evans, 1962). During inspiration, blood flow through the thoracic portion of the inferior vena cava increases markedly, while that through the distal abdominal portion does not change. Furthermore, an active source of pressure drop in the chest is necessary to maintain venous flow. For the open chest the pressure drop occurs mainly during ventricular contraction, while in the closed chest it is produced chiefly by inspiration. The present study indicates that the high distensibility of the veins accounts in significant degree for the behavior characteristic of the venous circulation.
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    A logical model for growth and differentiation in multi-celled organisms (1965)
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    Bulletin of mathematical biology 27 (1965), S. 379-387 
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    Notes: Abstract This paper is an attempt to provide a logical model for the process of growth and differentiation in a multi-cellular organism. More specifically it is intended to show how genetic information relating to macroscopic structure and coded in the form of a logical tree could be progressively embodied in the organism as it develops by repeated division from a single cell. The aim is to establish biological analogies rather than mathematical interest, and reproduction, adaption, and the coordinating action of hormones are discussed within the general logical framework.
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    A mathematical model of the urethra (1965)
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    Bulletin of mathematical biology 27 (1965), S. 407-415 
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    Notes: Abstract Models having the form of surfaces of revolution may be used to represent the urethra under pre-voiding pressure. From such models are derived formulas for calculating muscle tension from the shape of a urethragram.
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    Estimation of secretory rates of hormones from the specific activities of metabolites which have multiple secreted precursors (1965)
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    Bulletin of mathematical biology 27 (1965), S. 389-406 
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    Notes: Abstract The calculation of rates of entry of material into an open system of multiple pools in the steady state from the specific activities of end products, which may be derived from several pools, is described. This analysis may be applied to estimate the rates of secretion of steroid hormones from the specific activities of urinary metabolites which may have various hormones as common precursors. In a previous publication (Gurpideet al., 1963) formulae have been presented by which secretory rates could be calculated after a single injection of the tracers assuming that each of the urinary metabolites was uniquely derived from one of the pools in the system. In the present article similar formulae were derived without this assumption. Consequently, it is shown that, under certain circumstances, non-uniquely derived metabolites can be used to estimate secretory rates, and that it may be unnecessary to consider the pathways of conversion of the hormones to the metabolites or the sites where these conversion occur.
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    The use of two radiotracers for transfer-rate determinations in closed two compartment systems (1965)
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    Bulletin of mathematical biology 27 (1965), S. 431-434 
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    Notes: Abstract The sensitivity and “specificity” of measurements for the determination of transferates are enhanced by the use of an additional radiotracer, serving to trace the unlabelled substance. This method presents advantages mostly in systems outside their steady state but only exeptionally in steady state systems.
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    Analysis of tracer experiments V: Integral equations of perturbation-tracer analysis (1965)
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    Bulletin of mathematical biology 27 (1965), S. 417-429 
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    Notes: Abstract An integral equation approach to perturbation-tracer analysis in steady-state multicompartment systems is formulated. The theory is developed for δ function perturbation and tracer inputs and extended to the case of continuous small perturbations and continuous tracer inputs. It is shown that the first order dependence of the initial entry function can then be expressed by means of an integral equation: $$B_1 (t) = \int_{t_2 = - \infty }^\infty {\int_{t_1 = - \infty }^\infty {P(t_1 )T(t_2 )B_1 (t - t_2 ,t_1 - t_2 )dt_1 dt_2 } } $$ whereB 1(t) is the first order initial entry function for the tracer material,P(t1) the perturbation function.T(t 2) is the tracer input function, andB 1(t−t 2 ,t 1 −t 2 ) is a continuous function of two variables characterizing the first order perturbation-tracer response of the system.
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    Correlation “waves” in brain-like structures (1965)
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    Bulletin of mathematical biology 27 (1965), S. 435-447 
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    Notes: Abstract A correspondence is established between a tangible model of brain structure (and function) and a system of observer-observed interactions. The observed quantities are “stimuli” in the form of signal amplitude distributions in a mass of neuron-like units; the observer is a set of neurons (not circumscribed in a local region) in which a distributed parameter mirrors the stimulus history of the set, i.e., represents a “memory”. Utilizing the theory of the Perceptron, a contemporary brain model, it is demonstrated that large systems composed of many observer-observed interactions exhibit quantum mechanical behavior on a “macroscopic” scale. This behavior entails wave-like phenomena and the need of applying the superposition mechanics to system information content calculations.
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    Mathematical models for cellular systems. The von foerster equation. Part II (1965)
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    Bulletin of mathematical biology 27 (1965), S. 449-471 
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    Notes: Abstract This is the continuation of Part I, which was published in the September, 1965, issue of theBulletin. The birth rate, α(t), is now assumed to be a linear functional of the age density,n. This gives a simple model of self-replenishing stem cell compartments, and leads to a necessary condition for the existence of a steady state. Some examples are presented to illustrate the formalism. They include: (a) An equivivant population with life spanD and no losses from death or migration. The total number of cells is multiplied by 2 in each time intervalD. As a special case, frequently realized in practice, the population may be increasing exponentially with time (“log-phase” of growth). (b) A compartment with “random” emigration of cells and gamma distribution of life spans. (c) An oversimplified version of L. G. Lajtha’s model describing stem cell kinetics. In section IV a simple case in which the loss function depends explicitly onn is discussed very briefly.
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    Mathematical models of nonuniform intrapulmonary gas distribution (1965)
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    Bulletin of mathematical biology 27 (1965), S. 473-476 
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    Notes: Summary Mathematical models of nonuniform gas distribution in the lungs which assume a two-chambered lung to be ventilated through a third chamber, i.e. a common dead space, have led to diverging results. A breath-by-breath analysis of such a system results in a two-exponential solution whereas a continuous ventilation analysis gives a three-exponential solution. This is caused by the different assumptions made in the two models about the composition of dead space gas. In the breath-by-breath analysis one assumes that theN 2 content of the dead space is zero at the end of inspiration. In the continuous ventilation model one assumes that theN 2 content in the dead space is unknown at all instants during the breathing cycle. No physical significance should be attached to any chamber in this type of analysis. The continuous ventilation model provides a more general solution than the cyclical ventilation model, because the former treats the common dead spaces as an independent unknown.
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    On the electrocardiographic field equation (1965)
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    Bulletin of mathematical biology 27 (1965), S. 493-495 
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    The representation of organisms in terms of predicates (1965)
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    Bulletin of mathematical biology 27 (1965), S. 477-491 
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    Notes: Abstract The different approaches to relational biology developed by N. Rashevsky and R. Rosen consider essentially binary relations between various components of biological functions of the organism. Actually an organism is represented by a set of differentn-ary relations. The present paper is an attempt to outline a possible approach to this more realistic situation. Inasmuch asn-ary relation is ann-place predicate, it is attempted to describe the basic known properties of an organism in terms ofn-place predicates, in which the variables represent the different “components” of the organism. Some possible forms of such predicates are discussed and some general properties of systems of such predicates are studied. It is shown that if the organism is described by predicates of the type discussed here, statements can be derived about the conditions “of reestablishability” of different components. Conclusions similar to those obtained previously by R. Rosen are reached now on a very different basis. A description of the process of cell differentiation in multicellular organisms in terms of predicates studied here is briefly outlined. A comparison of similarities and differences between the approach and Rosen’s description of organisms in terms of the theory of categories is made.
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    Temporal organization in cells (1965)
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    Bulletin of mathematical biology 27 (1965), S. 497-500 
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    29th National Meeting of the Operations Research Society of America (1965)
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    Bulletin of mathematical biology 27 (1965), S. 503-503 
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    Harold A. White March 4, 1933–July 27, 1965 Obituary notice (1965)
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    Bulletin of mathematical biology 27 (1965), S. 501-502 
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    A model for neuron firing with exponential decay of potential resulting in diffusion equations for probability density (1967)
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    Bulletin of mathematical biology 29 (1967), S. 233-243 
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    Notes: Abstract A neuron is assumed to receive synaptic input of both excitatory and inhibitory natures from a large number of neighboring neurons; it is also assumed that a large number of such impulses are required to raise the neuron’s transmembrane potential to its threshold potential, at which it “fires” or “spikes”. The model is similar to one of Gerstein and Mandelbrot, except that in the absence of input an exponential decay of potential toward a resting level is introduced. Computational methods of determining the spike timeinterval distribution are discussed, along with the inverse problem of estimating the parameters of the system from observed spike time-interval data.
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    Pulse velocities in an orthotropic elastic tube (1967)
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    Bulletin of mathematical biology 29 (1967), S. 311-318 
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    Notes: Abstract Moens-Korteweg relations are developed for the velocity of wave propagation through an orthotropic elastic tube based on the three dimensional equations of elasticity. Numerical examples are presented for the femoral artery of a dog and several other orthotropic materials. These results are compared with those obtained from the equations of motion for an orthotropic elastic medium.
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    An investigation of basic assumption in enzyme kinetics using results of the geometric theory of differential equations (1967)
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    Bulletin of mathematical biology 29 (1967), S. 335-341 
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    Notes: Abstract Up to the present time, the following property of the product component in the reversible one substrate-one intermediate-one product enzymic mechanism has been taken only as anassumption, viz., during the course of the reaction, the time-rate of change of product concentration is never negative and the product concentration never exceeds its equilibrium value. Applying the methods of the geometric theory of ordinary differential equations it is shown that this result follows as a direct deduction from the differential equations governing the mechanism together with the initial conditions. Further, the nature of the equilibrium point as a stable node is established.
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    Analysis of tracer experiments: VIII. Integro-differential equation treatment of partly accessible, partly injectable multicompartment systems (1967)
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    Bulletin of mathematical biology 29 (1967), S. 319-333 
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    Notes: Abstract An integro-differential equation treatment of multi-compartment systems is developed which permits formal analysis of the incomplete data which is available from partly accessible, partly injectable systems. New transport functions are defined which can be obtained directly from the experimental data. These functions serve to characterize the communication and topology between different accessible compartments and also the reentrant contributions from inaccessible sites. The method gives solutions consistent with those of the differential equation approach when the system is uniformly contiguous and accessible, more complete solutions than those of the integral equation approach when all measured compartments are injectable, and in addition provides complete or partial solutions for certain otherwise analytically intractable systems. Detailed numerical illustrations of the method are given.
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    General two factor models (1967)
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    Bulletin of mathematical biology 29 (1967), S. 405-405 
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    Mathematical theory of the possible role of intercellular fluid and of vascularization on physiological periodicities (1967)
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    Bulletin of mathematical biology 29 (1967), S. 395-401 
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    Notes: Abstract In continuation of previous work (Rashevsky,Some Medical Aspects of Mathematical Biology, Springfield, Ill.: Charles C. Thomas, 1964, Chap. 23 and Appendix 14), the study of the effects of the physical parameters of the cells of endocrine glands on the onset of sustained periodical oscillations in the interaction between the anterior pituitary and the thyroid hormones is generalized to include the possible effect of the intercellular fluid and of the degree of vascularization. Some conclusions of the previous study remain valid although some modifications must be made. A decreased relative volume of the intercellular fluid and an increased vascularization favor the conditions for sustained oscillations. The permeability of the cells and the permeability of the capillaries appear explicitly in the expressions which show the conditions for sustained periodicities.
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    Equilibrium conditions in polysomic inheritance for a panmictic population (1967)
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    Bulletin of mathematical biology 29 (1967), S. 437-449 
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    Notes: Abstract A general model of a large 2m-ploid breeding population, withr loci ands h alleles at the h th locus is considered. It is assumed that the population is bisexual, non-overlapping and breeds by random mating. The genotypic structure of the population is presented as a bilinear form in the gametic output vectors where the genotype distribution is in the matrix form. Using the concept of the segregation distribution, the genotype proportions in the (n+1)st generation are given. An equilibrium condition for random chromosome segregation is obtained in terms of gene frequencies.
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    The blood flow of the brain (1967)
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    Bulletin of mathematical biology 29 (1967), S. 541-548 
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    Notes: Abstract The rigidity of the skull and the inertial characteristics and incompressibility of its contents cause the elastic cerebral arteries and veins to act over brief periods of time like rigid tubes of relatively small diameter. Poiseuille's law is applicable to their behavior. The use of this law, in combination with the fact that, during brief intervals, the total volume of the cerebral arteries and veins remains constant, permits derivation of a mathematical expression for the average arterial flow in terms of an average arterial radius. The differentiated equation has five positive roots which represent maxima and minima of the average flow in terms of the average arterial radius. The theoretical results have physiological implications and potential clinical usefulness, which are discussed.
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    Erythrocyte distribution in arterial blood flow: I. Basic equations and anisotropic effects (1967)
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    Bulletin of mathematical biology 29 (1967), S. 549-563 
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    Notes: Abstract Equations are derived for the flow of an anisotropic fluid in a tube. It is argued that these provide a model for arterial blood flow. Particular attention is paid to the effect of radial differences in hematocrit. Sequels to this paper (Bull. Math. Biophysics,29, 565–574; forthcoming, 1967) will respectively demonstrate possible wall-directed forces on the erythrocyte and enlarge on the physiological consequence of hematocrit variations. The present article develops the basic equations and explores the possible role of anisotropic effects in blood flow.
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    A re-evaluation of X-ray diffraction patterns of DNA (1967)
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    Bulletin of mathematical biology 29 (1967), S. 575-581 
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    Notes: Abstract Concentric circular lines are present on X-ray diffraction patterns of DNA. They cannot be accounted for by the double helix model. It is suggested that they may indicate the presence of double helical side chains.
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    A model for circadian clocks (1967)
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    Bulletin of mathematical biology 29 (1967), S. 781-791 
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    Notes: Abstract An extension of an earlier model simulating the effects of light on the drosophila eclosion rhythm is presented. The effects of variable light intensity are described. This allows not only the simulation of certain experiments not covered by the earlier model, but also it permits an extension of the model to other organisms. By changing only its sensitivity to light the model simulates the phase response curves of certain mammals as well as Aschoff’srule.
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    A note on the derivation of the stevens power law (1967)
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    Bulletin of mathematical biology 29 (1967), S. 827-829 
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    Notes: Abstract Previous derivations of the Stevens Power Law relating loudness to intensity assume, in addition to level invariance, that the relation must be analytic or at least differentiable. This last condition is replaced here by the weaker one of requiring only continuity.
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    Steady-state diffusion through a finite pore into an infinite reservoir: An exact solution (1965)
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    Bulletin of mathematical biology 27 (1965), S. 57-65 
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    Notes: Abstract An outline is given of an analysis that leads to an exact solution for the problem of steady-state diffusion through a finite thick pore into an infinite region surrounding the mouth of the pore. From this exact formula a simple expression for the flux is derived. This expression approximates the flux with a relative error of less than 3.42 per cent independently of the ratiol/a wherel is the length of the pore anda its radius. If desired, more accurate expressions for the flux can be obtained from the exact solution.
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    The single file model for the diffusion of ions through a membrane (1965)
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    Bulletin of mathematical biology 27 (1965), S. 79-86 
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    Notes: Abstract The model proposed by A. L. Hodgkin and R. D. Keynes (Jour. of Physiol.,128, 61–88, 1955) for the diffusion of potassium through the nerve membrane is extended to cover an arbitrary number of species of ions with charges not necessarily the same. One type of interference is also investigated.
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    On the theory of the cardiovascular system (1965)
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    Bulletin of mathematical biology 27 (1965), S. 71-83 
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    Notes: Abstract Most theoretical studies of the circulation have focussed on the transmission line properties of arteries. Only a small number of papers have dealt with the circulation as a closed (lumped) system with two pumps connected by the lesser and greater circulation (Beneken, inCirculatory Analog Computers, No. Holland Publ. Co., Amsterdam, 1963; Defares,et al., inCirculatory Analog Computers, No. Holland Publ. Co., Amsterdam, 1963; Grodins,Quart. Rev. of Biology,34, 93, 1959; Guyton,Cardiac Output and its Regulation, Saunders Publ. Co., New York, 1963). F. W. Cope's recent studies in this journal (Bull. Math. Biophysics,22, 19, 1960;23, 337, 1961;24, 137, 1962) deal with essentially the same questions, although here the circuit is not “closed”. We have attempted to extend the analysis of the areflex (closed) circulation. The complete study is reported elsewhere (Defares,et al., Acta Physiol, et Parmac. Neerl., 1963).
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    Equations of heat distribution within the body (1965)
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    Bulletin of mathematical biology 27 (1965), S. 67-78 
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    Notes: Abstract A vector integral equation describing heat distribution within the body has been derived. The factors considered are heat conduction, forced convection via the circulatory system, environmental exchange, metabolic heat production, and change in heat content. The vector partial differential equation and alternative forms incorporating boundary conditions were also developed. A difference equation based on a first-order approximation to the fundamental equations was derived to form the basis of a model for heat distribution within the body. It has been shown that factors involving conduction and convection must be considered independently unless the temperature of the blood flowing from a region of the body is equal to the average temperature of the tissue in that region. If this relation between tissue and blood temperature does exist, only a single temperature from each eleeent is needed to describe the heat distribution. In this latter case, models which ascribe all heat transfer to “equivalent” conduction or to convection can give valid predictions.
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    Determination of equilibrium constants and maximum binding capacities in complexIn vitro systems: I. The mammillary system (1965)
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    Bulletin of mathematical biology 27 (1965), S. 87-98 
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    Notes: Abstract It is shown that in a system containingn types of mutually noninteracting binding sites, the association constants are then roots of annth order polynomial while the maximum binding capacities can be evaluated by solving a set ofn simultaneous linear equations. Thenth order polynomial and the system ofn linear equations are defined in terms of 2n intermediate coefficients, the coefficients being themselves evaluated by substituting 2n sets of appropriate experimental data into an auxiliary system of 2n linear equations. The existence and uniqueness of the solutions are established.
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    Southern Regional Graduate Summer Session in Statistics at Oklahoma State University (1965)
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    Bulletin of mathematical biology 27 (1965), S. 111-111 
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    Important announcement to our contributors and readers (1965)
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    Bulletin of mathematical biology 27 (1965), S. 113-113 
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    A theory of ion transport across cell surfaces by a process analogous to electron transport across liquid-solid interfaces (1965)
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    Bulletin of mathematical biology 27 (1965), S. 99-109 
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    Notes: Abstract A kinetic theory of ion transport across cell surfaces has been developed in a form analogous to the kinetic theory of electron transport across solid-liquid interfaces of biological particles. The ionic theory is based on the observation that, at least in one instance, the voltage-current behavior for ion conduction across a cell surface is describable by the Tafel equation, in analogy to the conduction of electrons across solid-liquid interfaces. The theory predicts that the kinetics of ion transport across cell surfaces should conform to the Elovich rate equation, which is shown to be true for various experimental data.
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    Letter to the editor (1965)
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    Bulletin of mathematical biology 27 (1965), S. 114-114 
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    Mathematical development of a physical model of some visco-elastic properties of the aorta (1965)
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    Bulletin of mathematical biology 27 (1965), S. 115-115 
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    A biographical sketch of Nicolas Rashevsky (1965)
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    Bulletin of mathematical biology 27 (1965), S. 3-4 
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    A letter read to professor Rashevsky at a dinner in his honor December 11, 1964 (1965)
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    Bulletin of mathematical biology 27 (1965), S. 5-10 
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    Some comments on re-estalishability (1965)
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    Bulletin of mathematical biology 27 (1965), S. 11-14 
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    Notes: Abstract The present note consists of two separate but related parts. In the first, a new graphtheoretic proof is presented that an (ℳ,R)-system must always contain a nonreestablishable component. The second considers some questions concerning the relation between re-establishability and the time-lag structure in (ℳ,R)-systems. It is supposed that the reader is familiar with the terminology of the author's previous work on (ℳ,R)-systems, particularly R. Rosen,Bull. Math. Biophysics,20, 245–260, 1958.
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    Model studies of blood-glucose regulation (1965)
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    Bulletin of mathematical biology 27 (1965), S. 21-37 
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    Notes: Abstract A simplified, linearized model of the system regulating blood-glucose concentrations is reviewed. This model, which predicts a damped sine wave response to an oral glucose load, lumps the large number of kinetic parameters into a much smaller number which can, at least in part, characterize the human glucose regulatory system. The predictions based on the model are compared with measurements of blood-glucose and blood-insulin concentrations during the oral glucose-tolerance test. Various other conditions are simulated and their implications are discussed in terms of the mathematical model used.
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    On compartmentalization (1965)
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    Bulletin of mathematical biology 27 (1965), S. 15-19 
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    Notes: Abstract The notion of a compartment is discussed in terms of the Markovian process. From the stochastic matrix (the elements of which are state transition probabilities between different states of a particle of a chemical element), one may find a (generally) nonstochastic matrix; the elements of this second matrix are probabilities that, starting from some initial state, the particle will reach another seleced state (W. Feller, 1962,An Introduction to Probability Theory). Forming equivalence classes of states it can be shown that the equivalence classes based on an equivalence relation, which holds for the elements of the above-mentioned nonstochastic matrix, are essential for the notion of a compartment. From this procedure it is also obvious that a rigorous definition of a physically realizable compartment is impossible. Some conclusions on the practical use of compartmental analysis are drawn.
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    On the use of the Von Foerster equation for the solution and generalization of a problem studied by S. A. Tyler and R. Baserga (1965)
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    Bulletin of mathematical biology 27 (1965), S. 39-48 
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    Notes: Abstract In a population of cells labeled with a single injection of tritiated thymidine at timet=0, it is assumed that a constant fraction, 1−z, of the cells which are potentially able to divide fail to do so, and that the cells which do divide all have identical generation time,D. Death and emigration of cells are neglected. In mitosis, the partitioning of label among the two daughter cells is supposed to follow the binomial probability law. Using the formalism developed by H. Von Foerster the fraction of labeled cells in the total population is computed as a function oft, the time after injection of label. Ift is an integral multiple ofD the results coincide with those of S. A. Tyler and R. Baserga.
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    Development of structure in a society with a dominance relation when new members are added successively (1965)
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    Bulletin of mathematical biology 27 (1965), S. 151-160 
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    Notes: Abstract A society with a dominance relation is considered to be built up by starting with a small society and adding new members in succession. As each member is added he engages in contests with each of the older members to determine the dominance relation between them. The probability that the older member dominates is considered to depend on the size of the society and linearly on the older members score. A recurrence relation for the hierarchy index is derived. The approach of the society to a hierarchical structure is considered for various special cases of this probability. Reasonable assumptions concerning this dominance probability are shown to lead to structures close to the hierarchy. If the new member dominates all the older ones below a certain rank, and is dominated by all those above this rank, then the hierarchy will persist if it is the initial structure, or the structure will tend to hierarchy as the size increases, if it is not the initial structure.
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    Theory of hemoglobin facilitated oxygen transport (1965)
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    Bulletin of mathematical biology 27 (1965), S. 183-190 
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    Notes: Abstract The transport of oxygen in a hemoglobin-saturated medium is theoretically investigated using classical transport theory. It is found that all the chemical complexes can be expressed as a single function of oxygen pressure. A potential difference together with apH shift is predicted to occur across the medium.
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    Notes on magnetic actions upon the nervous system (1965)
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    Bulletin of mathematical biology 27 (1965), S. 203-214 
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    Notes: Abstract Several physical effects (magnetomotive force on ions, magnetic induction of electrical field, magnetic changes of inductance) are quantitatively analyzed in an attempt to attain an insight on how externally applied static magnetic fields influence the activity of the neuron and the Nervous System as a whole or in part. The possible magnetic action on shifting excited zones of the axon appears as most promising for prediction and interpretation of measurable effects. Magnetic fields may modify nervous functions by multiplication and addition of very small biophysical effects.
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    Molecular set theory: II. An aspect of the biomathematical theory of sets (1965)
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    Bulletin of mathematical biology 27 (1965), S. 235-251 
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    Notes: Abstract In an earlier paper (Molecular Set Theory: I.Bull. Math. Biophysics,22, 285–307, 1960) the author proposed a “Molecular Set Theory” as a formal mathematical meta-theoretic system for representing complex reactions not only of biological interest, but also of general chemical interest. The present paper is a refinement and extension of the earlier work along more formal algebraic lines. For example the beginnings of an algebra of molecular transformations is presented. It also emphasizes that this development, together with the genetical set theory of Woodger's and Rashevsky's set-theoretic contributions to Relational Biology, points to the existence of a biomathematical theory of sets which is not deducible from the general mathematical, abstract theory of sets.
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    Man/machine interaction in adaptively controlled experimental conditions (1965)
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    Bulletin of mathematical biology 27 (1965), S. 261-273 
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    Notes: Abstract Systems in which a human subject interacts with an adaptive control mechanism through display and response facilities are examined. A cybernetic model is discussed, together with supporting experimental data.
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    Computer simulation of stem-cell kinetics (1965)
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    Bulletin of mathematical biology 27 (1965), S. 275-290 
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    Notes: Abstract Computer simulation of the stem-cell system has been motivated by a desire to provide a device which may assist the experimenter in his development of complex research strategies in the rapidly developing investigative fields that relate to erythropoiesis and granulopoiesis. A simulation program, written in FORTRAN II for the IBM-7094, is being developed with the requirements of flexibility and broad applicability in mind. The biological model for which it originally was developed differs from those previously advanced by visualizing differentiation into the erythrocytic and granulocytic series as occurring during the post-mitotic phase of a stem cell's growth, the cell's susceptibility to erythropoietic and granulopoietic stimuli varying as its biochemical development unfolds. One might test this model by attempting to demonstrate an asymmetrical interference between erythropoietic and granulopoietic challenges to the stem-cell system. A method for establishing an initial stable configuration of the model is presented. The simulation of the introduction of exogenous erythropoietic stimuli is described. And there is a brief description of the feedback mechanism employed to stabilize the size of the stem-cell population.
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    A note on visual integrative mechanisms (1965)
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    Bulletin of mathematical biology 27 (1965), S. 305-310 
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    Notes: Abstract In a system as complex and as effective as the eye, the cooperative interaction of different mechanisms may be taken as axiomatic. With this as a starting point, various visual phenomena are considered, such as short term memory, eye movements, and flicker fusion. Simple data on mean values lend support to the proposition that the spatial and temporal characteristics of these phenomena are matched with one another. The significance of this for a mechanism of vision is discussed.
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    Matrix algebra of a generalized linear stochastic model. Nonconfluent and general confluent cases (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 311-315 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Matrix algebra is the natural tool for the study of linear stochastic models with many parameters. Complete solutions are given for the nonconfluent and the general confluent cases. It is shown that the axiomatics of a generalized linear stochastic model are naturally described within the framework of the linear algebra in an Euclidean space.
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    URL: http://dx.doi.org/10.1007/BF02477286
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    Properties of solution of the diffusion-reaction equation (1965)
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    Bulletin of mathematical biology 27 (1965), S. 291-303 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract An arbitrary set of chemical reactions is considered to occur among chemical speciesX i . In a closed uniform reaction system certain linear combinations of the concentrations of theX i are constants. The general construction of all such linear combinations with non-negative coefficients is given in terms of the molecular formulae for theX i . It is shown that to each such linear combination there corresponds another which is a harmonic function when the reactions take place in an open spatially distributed stationary reaction system of arbitrary shape. Under the usual boundary conditions these harmonic functions are constants. With some restrictions upon the diffusion and permeability coefficients these constants are evaluated. This evaluation is the basis for relations between the total concentration of a given chemical group (e.g., the sum of the concentrations of a free molecule, or ion, and its various bound forms) in the reaction system, and in the surrounding medium. The bearing of these relations on apparent active transport is noted and illustrated.
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    URL: http://dx.doi.org/10.1007/BF02477284
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    Analysis of tracer experiments: VI. Determination of partitioned initial entry functions (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 329-332 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It is shown that the partitioned initial entry functions previously introduced in multicompartment analysis can be directly and uniquely determined from the experimental data.
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    URL: http://dx.doi.org/10.1007/BF02477288
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    A neural net model for escape learning (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 317-328 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract An escape learning situation is discussed in terms of a neural model in which a stimulus can result in a conditioned excitement and a specific conditioned response. By using the simplest relations between the strengths of conditioning and the number of reinforcements and by introducing a distribution of fluctuations occurring regularly in time, one can calculate the probabilities of various responses, as well as the various latencies, in successive trials. The results are in moderately satisfactory agreement with the data of R. L. Solomon and L. C. Wynne (Psychol. Monogr.,67, No. 4, 1953). Consequences of the model for various experimental situations are discussed.
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    Effects of the collateral circulation of the heart (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 1-8 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Collateral circulation minimizes the myocardial injury which results from narrowing of a coronary artery. A large collateral circulation has disadvantages, however. It may divert so much of the limited blood flow through the adjacent nonarteriosclerotic coronary artery that the blood supply of the normal muscle supplied by that artery may be inadequate during heavy exercise. In the presence of a large collateral circulation, both the normal and ischemic regions of the heart may be extremely vulnerable to small arteriosclerotic changes narrowing the patent artery near the aorta. The effective increase in flow which results from arteriolar vasodilatation produced by drugs may be much greater in the presence of a small collateral circulation than a large one.
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    URL: http://dx.doi.org/10.1007/BF02476463
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    Simple theory of transfusions (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 9-20 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A transfusion can be hypothesized to be required when a determination factor (D=probability of adverse effects if transfusion not given/adverse effects if transfusion is given) exceeds some predetermined value.D varies between the limits 0 and ∞, and in most clinical situations will be a small number on the order of 20. Since the probabilities contributing to the denominator ofD are essentially independent of each other, they can be summed to obtain the probability of ill effects. A method of handling an exception to this, the incompatibility reactions following multiple transfusions within a short time interval, is pointed out. The probability of adverse effects if a transfusion is not given is more difficult to evaluate; values gathered from the literature are presented, as well as methods for obtaining further data. Two techniques for estimating future transfusion requirements are discussed. One is a correlative procedure, which functions by analyzing similar cases admitted to the hospital. The second procedure is an estimate of stability (homeostasis), based on a parameter introduced by B. C. Patten (Scince,134, 1010–1011, 1961). The dilution of endogenous cells and plasma by transfusions is considered and the consequences of many small transfusions compared with those of few (and larger) transfusions.
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    Some possible quantitative aspects of a neurophysiological model of schizophrenias (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 21-26 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In continuation of a previous paper (Bull. Math. Biophysics,26, 167–185, 1964) simple equations are derived for the rate of development of schizophrenia as a function of some psychobiological parameters of the individual and of an index which characterizes the frequency of traumatic experiences of the individual. A clue to the understanding of why schizophrenia is more likely to develop at an early adult age is thus provided.
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    longitudinal diffusion along the nephron during stop flow (1965)
    Springer
    Bulletin of mathematical biology 27 (1965), S. 53-56 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A mathematical analysis is presented which shows that during stop flow experiments longitudinal diffusion of solute along the nephron is of too small a magnitude to interfere with the interpretation of data.
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    URL: http://dx.doi.org/10.1007/BF02476467
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    Note on the analysis of thymidine labeling in cells with chromosome-number dispersion (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 33-40 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A method of analyzing thymidine labeling in a population of cells is formulated. The formulation establishes a unique relation between a specific set of labeling data and a specific set of cells in the population, viz. that set of cells having a particular chromosome number. The analysis employs a cell-state variable, i.e., a quantity which specifies the progress of a cell through its lifecycle. This variable is defined in terms of the nucleo-protein content and configuration of the chromosomes. The relation mentioned above leads immediately to an expression for the number of cells present at a particular time following labeling which have a given amount of label per cell and a given chromosome number.
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    URL: http://dx.doi.org/10.1007/BF02476960
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    Dose-rate dependence of radiation-induced metaphase delay (1967)
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    Bulletin of mathematical biology 29 (1967), S. 41-56 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract An equation relating radiation-induced metaphase delay to the dose-rate and duration of irradiation is obtained. The equation is derived from a model specifying the effects of radiation on the normal chromosome coiling process. The basic assumptions of the model are (1) that normal coiling proceeds by contractile protein acting on segements of a viscoelastic chromosomal fiber; (2) that radiation causes cross-linking of adjacent chromosomal fibers which hinders the coiling process.
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    URL: http://dx.doi.org/10.1007/BF02476961
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    Bistable mathematical and mechanical models related to the urethra (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 57-65 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Normal micturition is controlled primarily by a neural system. Certain physical effects become evident when neural control is destroyed, and the automatic or autonomous bladder phenomena occur. It is shown in this paper that a physical system simulating the alternating periods of continence and voiding of the automatic bladder may comprise only passive elastic components, and that periodic voiding does not per se imply neural control.
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    URL: http://dx.doi.org/10.1007/BF02476962
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    Further comments on replication in $$\left( {\mathfrak{M},\Re } \right)$$ -systems-systems (1967)
    Springer
    Bulletin of mathematical biology 29 (1967), S. 91-94 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A number of inaccuracies in previous papers are pointed out and amended, and some implications of the correct situation are outlined.
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    URL: http://dx.doi.org/10.1007/BF02476964
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