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  • Articles  (134,304)
  • Springer  (111,811)
  • Wiley  (22,493)
  • 2020-2020
  • 1970-1974  (134,304)
  • 1945-1949
  • 1974  (46,309)
  • 1973  (44,549)
  • 1972  (43,446)
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  • 2020-2020
  • 1970-1974  (134,304)
  • 1945-1949
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  • 1
    Electronic Resource
    Electronic Resource
    Obituary of Nicolas Rashevsky (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. i 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02477019
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  • 2
    Electronic Resource
    Electronic Resource
    On the stochastic theory of compartments: I. A single-compartment system (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 53-63 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A stochastic model is developed for a compartment with a single time-dependent input, and generalized to include inputs from several sources. With the number of particles of a given molecular species in the compartment as the random variable, the mean, variance and third central moment of this variable are calculated from its generating function, and compared with previous results. The behavior of the calculated moments is discussed, and the possibility of applying the model to chemical and biological systems is considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02477024
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  • 3
    Electronic Resource
    Electronic Resource
    A remark on sensory deprivation from the point of view of organismic sets (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 439-441 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It is shown that from the definition of organismic sets (Rashevsky,Organismic Sets. Some Reflections on the Nature of Life and Society, Holland, Michigan, Mathematical Biology, Inc. and Grosse Pointe, Michigan, J. M. Richards Laboratory) a complete sensory deprivation of an organismic set of ordern=2 should result in malfunctioning of the set. A generalization to higher order sets is suggested.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476706
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  • 4
    Electronic Resource
    Electronic Resource
    Optimal branching structure of the vascular tree (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 431-438 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Optimality in branching structure of the vascular tree was studied. Analysis on its physiological roles as the duct system for blood supply to the capillaries predicted that the vascular tree should be constructed with minimum volume under restriction of determinant pressure, flow and location at the origin and the terminals. Mathematical derivations of this conditional extremum problem yielded some equations expressing the relations between the radii of the branches and their branching angles, which provided numerical solutions for branching points of bi- and poli-terminal minimum volume trees. Comparison of the peritoneal vascular tree in a dog with the minimum volume one computed under the same restrictive conditions showed good agreement in their branching structure.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476705
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  • 5
    Electronic Resource
    Electronic Resource
    First order kinetics in a distributed model (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 443-456 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In order to determine the kinetics of passage of a substance through an organ containing a tangle of vessels, we study the response of a tube to various inflows (perfusion, brief injection, ...). The introduction of the catabolic terms and of the spatial dependence between bulk concentration and surface concentration allows one to account for the difference of arteriovenous concentrations observed experimentally for many metabolites. The relationships between the physico-chemical parameters of the organ and the operational parameters of the model demonstrate the importance of the transit time through the considered vessels. If one considers the different pathways as independent, the introduction of the transit time distribution for an inert substance enables one to compute the response of the organ analytically or by recurrence, using convolution. The parameters of the model can be obtained by the moments method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476707
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  • 6
    Electronic Resource
    Electronic Resource
    Stochastic simulation of the gas diffusion in the air phase of the human lung (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 457-466 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The solution of the diffusion equation in the gas phase of the human lung is very difficult because of the structure of the bronchial tree. It is shown by means of physical arguments, how one can reduce the diffusion equation to a simple one-dimensional form. The solution is then obtained by a stochastic simulation, which is easily realized on a digital computer.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476708
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  • 7
    Electronic Resource
    Electronic Resource
    A new mathematical model of the respiratory center (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 467-481 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A new mathematical model of the oscillatory behavior of the respiratory center has been developed based upon published records of neuronal activity during respiration in the pons and medulla. In contrast with a previous model, four, rather than two, networks are assumed to interact in the respiratory center so as to produce the respiratory oscillation. A mathematical description of this interaction, in the form of a set of four first-order, nonlinear, coupled differential equations, is derived; the behavior of the solutions of this system is studied qualitatively, and expressions for the durations of the inspiratory and expiratory phases are obtained in terms of some parameters. It is found that central and chemical influences drive the medullar neurons to a position somewhere between saturation and full cutoff, and the pontine neurons deeply into cutoff. The control of the duration of the different phases by these chemical and central means is discussed. In order to effect a decrease in the magnitude of the various times, the neurons have to be driven towards operating points of higher central facilitation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476709
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  • 8
    Electronic Resource
    Electronic Resource
    A mathematical model for calcium homeostasis (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 483-502 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In vivo control of calcium is analysed under the assumption that hormonal influences via plasma levels of parathormone and calcitonin are of prime (but not absolutely dominating) importance. A brief review concerning the physiological significance of body calcium and the mode of action of these two hormones is presented as an introduction to the basic philosophy of the study. A theoretical quasi-linear lumped-parameter model is developed to describe variations in ionic calcium, parathormone and calcitonin plasma concentrations to specific input stimuli. Formal evaluation of the system response requires the determination of ten constants, together with quantitation of ingested calcium entry into the plasma compartment which isindependent of hormonal influences. Values for various parameters are deduced from published data and experimental procedures are outlined to facilitate determination of the remaining unknowns. It is suggested that the proposed model should prove useful for investigations concerning general hormonal actions on calcium homeostatic mechanisms in both normal and diseased states, with particular reference to calcitonin.
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    URL: http://dx.doi.org/10.1007/BF02476710
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  • 9
    Electronic Resource
    Electronic Resource
    The struggle for life: III. A predator-prey chain (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 521-532 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Ann species predator-prey chain is analyzed to determine what oscillations occur in population sizes. It is found that only the populations of the first and second species in the chain must necessarily oscillate around the point of equilibrium if they do not come to equilibrium. The other species may or may not oscillate.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476712
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  • 10
    Electronic Resource
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    Mathematical models for exchange of substances in regional vascular beds— Measurement of the rates within vivo counters (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 503-520 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Sangren and Sheppard developed a mathematical model for first-order processes taking place in the regional circulation, applicable—for example—to tracer studies of potassium transport. It permits calculation of specific activity at any point along a “tube of flow” or in the cuff of tissue surrounding it as a function of time following a spike injection of tracer. In efforts to relate to the exchange a rate curves obtained within vivo counters pointed at the region of interest, we developed a compartment-system model of the process. In investigating the properties of the Sangren and Sheppard model integrated over an entire circulatory bed, as thein vivo counter would see it, we found that when the distribution of transit times of the “tubes of flow” can be approximated by an exponential sum, the solution reduces to that of the compartment system model. This results in an important simplification in the calculation, and insight into the assumptions underlying the two different models. A curve-fitting computer program for the compartment model has been written and applied to double-isotope studies of potassium transport in the hind leg of the dog.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476711
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  • 11
    Electronic Resource
    Electronic Resource
    A numerical experiment on the determination of kinetic rate constants in the presence of diffusion (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 547-558 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Two chemicals,A andB, are allowed to diffuse together and a reaction described by $$A + B\mathop \rightleftharpoons \limits_{K_{ - 1} }^{K_1 } C$$ is allowed to proceed. This system is described mathematically by a system of partial differential equations. A numerical procedure is presented to find the rate constants ofK 1 andK −1. A systematic analysis of the effects of errors is also presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476714
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  • 12
    Electronic Resource
    Electronic Resource
    Observations on cardiac energetics: I. Theoretical introduction (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 533-546 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Equations are developed to describe the energy expenditure of the human heart. As well as the external potential and kinetic energy terms, general consideration is given to other possible avenues of energy consumption. Emphasis is placed upon using mathematical variables which are readily available for experimental verification. The errors involved in assuming that mean values for the physiological parameters give reasonable estimations for the external mechanical performance are examined, and a theoretical estimation for the discrepancy in the kinetic component is presented. Logical extension of the mathematical derivation leads to a determination of cardiac external mechanical efficiency and clearly demonstrates the significance of the ventricular pressure-volume loop in this context. Finally, experimental procedures are suggested to clarify further some of the conclusions reached through the theoretical analysis.
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    URL: http://dx.doi.org/10.1007/BF02476713
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  • 13
    Electronic Resource
    Electronic Resource
    Organismic sets. Sex in animal and human societies (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 559-563 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The question is discussed as to the reason why some animal societies, such as bees or ants, are sexually differentiated, that is, onlysome of its members are exhibiting reproducing activities. It is indicated that human society may be on its way to such a sexual differentiation which may eventually come.
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    URL: http://dx.doi.org/10.1007/BF02476715
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  • 14
    Electronic Resource
    Electronic Resource
    Some biochemical threshold mechanisms (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 565-565 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476716
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  • 15
    Electronic Resource
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    Announcement (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 567-567 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476717
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  • 16
    Electronic Resource
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    The structure of DNA as deduced from fiber X-ray crystallography (1973)
    Springer
    Bulletin of mathematical biology 35 (1973), S. 301-311 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract X-ray diffraction patterns obtained experimentally for fibers, together with their chemical structures, can be analyzed theoretically in terms of an integral equation. The partially unknown electron density function can be solved by iteration. This mathematical technique has been applied with success to study the secondary structures of DNA fibers.
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    URL: http://dx.doi.org/10.1007/BF02458338
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  • 17
    Electronic Resource
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    Information Processing in the Visual Systems of Arthropods (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 339-340 
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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/BF02461333
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  • 18
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    Optimality in the control of environmental systems (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 341-345 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract For an environmental system described by a system of nonlinear first-order differential equations, the problem of achieving specified terminal conditions in a given time with a minimum expenditure of resources is considered. The initial conditions and the minimum value are found numerically in a particular example.
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    URL: http://dx.doi.org/10.1007/BF02464612
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  • 19
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    Kinetic approach to neural systems: I (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 535-544 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A kinetic model of neural systems is introduced and discussed with statistical mechanics techniques. It is assumed that, for a macroscopic description of the model, it suffices to consider only the distribution for the velocity and position of the impulses, and the distribution for the excitation and position of the neurons, at any timet. Making use of Boltzmann's method for the study of a dilute gas, coupled differential equations for the rate of change with time of the distributions have been constructed.
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  • 20
    Electronic Resource
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    Flow of a newtonian fluid through a permeable tube: The application to the proximal renal tubule (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 457-476 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Creeping flow of a Newtonian fluid through a rigid permeable tube is considered and the transmural seepage is assumed to obey Darcy's law. Closed-form solutions for the pressure and velocity fields are presented and equations describing the axial variation of the mean cross-sectional pressure, the axial volumetric flow and the transmural fluid flux are derived. Approximate solutions for small seepage rates are given and are applied to the flow in the proximal renal tubule. Probable values for the epithelium permeability and the intraluminal hydrostatic pressure drop are obtained.
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    URL: http://dx.doi.org/10.1007/BF02463260
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  • 21
    Electronic Resource
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    Disseminated storage of information with sequential coding (1973)
    Springer
    Bulletin of mathematical biology 35 (1973), S. 663-688 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The paper demonstrates that it is possible to construct memory models where the information inserted is stored in disseminated form, using sequential coding, the changes in the units forming the models being determined by their geometrical connections and by the incoming stream of information. The models are shown to have large storage capacity and their efficiency can be made insensitive to loss of or damage to a large fraction of their units. The satisfactory verification by computer simulation of the analysis and results described in the present paper will be the subject of a future paper.
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    URL: http://dx.doi.org/10.1007/BF02458369
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  • 22
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    On knowledge and unpleasantness (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 605-605 
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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/BF02463273
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  • 23
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    The Kerner linearization method (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 67-76 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We examine in detail Edward Kerner’s method for linearizing the equations of enzyme kinetics. Our main result is the determination of canonical forms for systems which can be linearized by the method. This is done both in general and in the special cases of two and three dimensions where complete results are obtained. The practical problem of identifying linearizable systems is also considered and computable necessary criteria are presented.
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    URL: http://dx.doi.org/10.1007/BF02461192
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  • 24
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    Continuous determination of the volume of a space in a non steady state condition (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 59-66 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A general method for determination of the volume of a space in a non steady state condition, in case diffusion might be significant, is developed. Instantaneous mixing of indicators with native fluid is assumed in this first stage of investigation. Theoretical expressions are obtained for the volume of the space and the diffusion coefficient as a function of time. An analysis of feasibility of the method is also included.
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  • 25
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    Passive transfer of low-molecular nonelectrolytes across deformable semi-permeable membranes—II: Dynamics of a single muscle fiber swelling and shrinking and related changes of theT-system tubule form (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 379-401 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The theory of transfer of low-molecular nonelectrolytes across deformable semipermeable membranes of large curvature developed in Part I (Rubinstein, 1974) is used to describe the dynamics of swelling and shrinking of a muscle fiber at the influx and efflux of low-molecular nonelectrolytes. A large set of computations showed that the theory explains the experiments described in the literature.
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  • 26
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    Heat transfer in perfused biological tissue—I: General theory (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 403-415 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Green's function for heat and matter transport is calculated for an infinite medium in which a convection field v(r,t) makes a contribution to the total heat and matter current. It is given by a uniformly and absolutely convergent series in which every term is calculated from the preceding one merely by integration. The solution procedure is interpreted physically and illustrated by a simple problem in which v(r,t)=const. in space and time. Since the solution contains no intrinsic spatial symmetry, it can serve as a starting point for a theory of heat and mass transport in perfused biological tissue.
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  • 27
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    A long wavelength solution for a microorganism swimming in a channel (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 435-444 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The hydrodynamics of a microorganism swimming in a channel is investigated. The microorganism is modeled as a two-dimensional sheet swimming at low Reynolds numbers between two rigid walls. The wavelengths of the propulsive waves passing down the sheet are assummed to be very large compared to the channel spacing, but the amplitude of the propulsive waves is arbitrary. Explicit analytical solutions for the propulsive velocity and the rate of energy dissipated in terms of the wave amplitude, channel spacing, wave number, and wave speeds are given.
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  • 28
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    Mathematical models of conception and birth (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 455-456 
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    ISSN: 1522-9602
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  • 29
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    Why are there so many species? (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 477-488 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It is shown from the statistical-mechanical overview of Volterra's ecological model how to reckon the fluctuations of collective variables such as the total population of a genus: and that these fluctuations are much decreased (or that the collective populationsteadiness is enhanced) as the speciation is increased. (A niching of species in time, or phase-niching, is entailed here.) Secondly, it is shown how Preston's log-normal distribution describing the species-abundance relationship, as well as a generalization of such distributions, come forth simply and naturally from the statistical-Volterra-dynamics.
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  • 30
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    The myocardium as a composite material: Analysis (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 173-204 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract As a further attempt to determine the stresses and strains of the individual myocardial fibers, the heart muscle is considered as an orthotropic material. A theory is presented which leads to the expression of the equilibrium conditions for the left ventricle in the form of three simultaneous differential equations. Solution of these equations would give the changes in shape of the left ventricle throughout the cardiac cycle, and, in addition, the stresses and strains of the individual myocardial fibers. It is pointed out, however, that meaningful solutions of the equations cannot be obtained at the present time because of difficulties in experimental determination of certain parameters.
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  • 31
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    A note on epilepsies and similar disorders (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 213-222 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper compares two previously published neural models for epilepsies (Bull. Math. Biophysics,33, 539–553, 1971;34, 71–78, 1972). The second model is developed in more detail and an attempt is made to bring it more in line with established neurological findings. The question of classification of some epilepsies is briefly discussed.
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  • 32
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    Suggestions for a mathematical model of a pathological neuron with spontaneous repetitive discharges (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 223-230 
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    ISSN: 1522-9602
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    Notes: Abstract A formal mathematical model is proposed for a spontaneously repetitively firing neuron. It is based on the assumption that an excitatory and inhibitory substance, possibly different from those involved in synaptic transmissions, is formed in the soma of everynormal neuron. Furthermore, the decay of the substances is ascribed to their combination with some other substances, present in healthy individuals. A generalized two factor system of differential equations is used. It is shown that when the normally present substances are absent, possibly due to genetic defects so that the decay constants become zero, the equations lead to undamped sinusoidal solutions of the difference between excitatory and inhibitory factors, thus producing a trulyspontaneous repetitive discharge, in the absence of external currents or other stimulation. It is suggested that convulsants may act by destroying the substances present in healthy individuals. It is further suggested that by administering to epileptics those substances, which are present in normal healthy persons, perhaps by using brain extracts fromhealthy higher animals which sometimes suffer from epilepsy, an actual cure rather than symptomatic treatment by anticonvulsants may be obtained.
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    Structure and kinetic behavior of linear multicompartment systems (1972)
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    Bulletin of mathematical biology 34 (1972), S. 277-291 
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    Notes: Abstract The general kinetic behavior of a multicompartment system is shown to depend upon certain general structural features, including its connectivity, whether it is open, and whether it contains cyclic pathways. Structural influences are clarified by putting the system matrix in a certain form. For systems not strongly connected, a distinction is drawn between partially and completely open systems. Necessary and sufficient conditions are given for non-singularity of the system matrix and for asymptotic stability of the system. Sufficient conditions are given for non-overshooting and monotonic transitions. A system is demonstrated whose solution may contain a prolonged series of damped oscillations; but the oscillations are very slow and small; and it seems unlikely that oscillations could be detected experimentally in any biological system.
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    A dynamical approach to chemical kinetics: Mass-action laws as generalized Riccati equations (1972)
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    Bulletin of mathematical biology 34 (1972), S. 243-275 
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    Notes: Abstract It is shown how the fundamental laws of chemical kinetics for either open or closed systems with an arbitrarily large number of reactants can be represented as a system of Riccati-like differential equations. Through the use of a concise tensor notation, it is shown when and how the differential system is exactly reducible to linear form, a reduction without approximation that parallels the well-known similar reduction of a single simle Riccati equation. An example is worked out to show how open kinetics can lead to oscillatory chemical concentrations of the Change-Higgins type. The biologically central problem of great chemical speciation is discussed from the viewpoint of Gibbs ensemble theory within the linearized kinetics and, approximately, within the starting nonlinear kinetics where it is shown roughly how to estimate, from an overall temperature-like parameter characterizing the whole system, mean chemical levels and mean frequencies of oscillation, and where a gross oscillation of the total mass is estimated in terms of an anharmonic oscillator whose general structure is fixed from the structure of the chemical kinetic laws.
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    Note on oscillations in a closed chain of compartments with linear, unidirectional transport (1972)
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    Bulletin of mathematical biology 34 (1972), S. 293-296 
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    Notes: Abstract A closed chain of compartments in which there is unidirectional transport between adjacent members can exhibit damped oscillations. For a system ofn equivalent compartments, the value ofn which gives the greatest difference between the first maximum and first minimum isn=11, the difference being 1.57%. The greatest difference between the first maximum value and the steady state value is 4% and is obtained whenn=25. The results are illustrated graphically forn equal to 5, 10, 25 and 100.
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    Pourquoi deux sexes? (1972)
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    Bulletin of mathematical biology 34 (1972), S. 297-304 
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    Notes: Abstract This paper is a concrete approach to the problem of the number of the sexes. We try to imagine—on the example of three sexes—the mechanisms which would have to accompany a reproduction with several sexes. We have limited our study to the monohybridism, dihybridism and determinism of the sex.
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    Cilia transport (1972)
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    Bulletin of mathematical biology 34 (1972), S. 325-335 
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    Notes: Abstract An analysis of the effect of cilia on fluid transport in tubules is presented. The applicability of the results for the flow rates observed in the ductus efferentes of the male tract is discussed.
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    Quantitative comparison of dipole models for steady-state currents in excitable membranes (1972)
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    Bulletin of mathematical biology 34 (1972), S. 305-324 
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    Notes: Abstract The dipole models for steady-state currents in excitable membranes of Arndt, Bond and Roper and of Hamel and Zimmerman are compared by fitting the equations to the data of Gilbert and Ehrenstein. The more complex Hammel and Zimmerman model does not fit the data as well as does the simpler Arndt, Bond and Reper model. When fitting the data, the Hammel and Zimmerman current equation reduces to the Arndt, Bond and Roper current equation because of the values assumed by the parameters. An interpretation is given for the parameter values obtained with the Arndt, Bond and Roper model.
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    On the decomposition of a dynamical system into non-interacting subsystems (1972)
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    Bulletin of mathematical biology 34 (1972), S. 337-341 
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    Notes: Abstract It is shown that, under rather general conditions, it is possible to formally decompose the dynamics of ann-dimensional dynamical system into a number of non-interacting subsystems. It is shown that these decompositions are in general not simply related to the kinds of observational procedures in terms of which the original state variables of the system are defined. Some consequences of this construction for reductionism in biology are discussed.
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    Self-reproduction and serial message transfer: Two related problems (1972)
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    Bulletin of mathematical biology 34 (1972), S. 343-353 
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    Notes: Abstract For a certain class of physical machines, termed “structure-determined,” the problem of self-reproduction can be reduced to the problem of serial message reproduction. Serial message reproduction however presupposes a sort of “open system” constraint. This leads to the principle of pseudo, or exogenously standardized, respectively, self-reproduction. It seems to be consistent with both chemical and biological self-reproduction. It thus may reflect a general principle of biological design. The proposed principle is a physico chemical analog to Robert Rosen's abstract relational self-reproduction constraint.
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    Active muscle fiber as an equivalent cardiac current generator (1972)
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    Bulletin of mathematical biology 34 (1972), S. 355-377 
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    Notes: Abstract Equations are derived describing potentials due to an active muscle fiber in an infinite medium in terms of two surface integrals—one of the propagated action potential and the other of the membrane current density, both integrals being taken over the surface of the muscle. These equations are incorporated into an equivalent cardiac current generator in which the left ventricle (i.e. the current source) is represented by a three-dimensional wedge and the thorax (i.e. the volume conductor), by a homogeneous circular cylinder. Since this current generator expresses the body surface potentials in terms of the membrane current density and the membrane potential at any point on the surface of the electrically active muscle fiber, the calculated ECG can be correlated with theactual sources within the heart. This equivalent cardiac generator possesses many of the physical and physiological properties of cardiac muscle. The equations were evaluated numerically on a digital computer. The results indicate that equivalent cardiac current generators of this type can yield clinically significant results and that further research is necessary to investigate their properties fully.
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    Analytical solutions for transient thermal behavior in biological systems (1972)
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    Bulletin of mathematical biology 34 (1972), S. 413-418 
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    Notes: Abstract Analytical solutions are presented for transient heat conduction in biological media. General boundary conditions and internal sources varied in both spatial and time variables are considered, thus, solutions for many special cases can be obtained with ease from the general solutions presented in this analysis.
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    Simultaneous diffusion and convection in single breath lung washout (1972)
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    Bulletin of mathematical biology 34 (1972), S. 393-412 
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    Notes: Abstract Two mathematical models of pulmonary single breath gas washout (one analytic, one numerical) are developed and their predictions compared with experimental data on human subjects. Weibel's 23 generation symmetric anatomical model is used as a guide to bronchial tree geometry. Experimental plots of nitrogen concentration versus volume expired, dead space versus breath holding time, and dead space versus tidal volume are compared with plots predicted by the models. Agreement is good. A plot of nitrogen concentration in the airways as predicted by the numerical model at different times during inhalation and exhalation of a single breath of oxygen is shown. Model predictions for changes in dead space with changes in washout gas and expiratory flow rate are discussed. Use of the analytic model for obtaining average values of the path length from mouth to alveoli in a given subject is discussed. To the extent of their agreement with experiment, the models provide a sound physical basis for the correlation of airway structure and function.
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    Announcement (1972)
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    Bulletin of mathematical biology 34 (1972), S. 429-429 
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    Generalizations of the Roginsky-Zeldovich (or Elovich) equation for charge transport across biological surfaces (1972)
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    Bulletin of mathematical biology 34 (1972), S. 419-427 
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    Notes: Abstract The Roginsky-Zeldovich (or Elovich) equation, which is −dx/dt=m exp (nx) (x=substrate concentration,t=time,m andn=constants), describes the kinetics of various biological electron and ion transport processes, and has been derived from the concept of charge transport across an activation energy barrier at an interface between dissimilar phases, driven by a difference in redox or ion potentials, with the simplifying assumptions that charge carrier concentration is constant, backward current across the interface is zero, and diffusion of substrate is fast. If charge carrier concentration is proportional to substrate concentration, then the kinetic equation is −dx/dt=mx exp (nx). If backward current is not zero, then −dx/dt=m 1 exp (n 1x) −m 2 exp (n 2 x), wherem 1,m 2,n 1 andn 2 are constants. Kinetic equations for interfacial charge transport in the presence of a significant substrate diffusion potential are also derived.
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    A note on knowledge and unpleasantness (1973)
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    Bulletin of mathematical biology 35 (1973), S. 313-317 
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    Notes: Abstract Various philosophers have repeatedly denounced knowledge as a source of unpleasantness, thereby criticizing education. This paper presents a set theoretical approach to knowledge and education and tries to explain how they could lead occasionally to a feeling of unpleasantness.
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    An analysis of countercurrent exchange with emphasis on renal function (1973)
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    Bulletin of mathematical biology 35 (1973), S. 275-286 
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    Notes: Abstract An analysis of countercurrent exchange in a U-tube is presented for a single-solute, constant-volume flow rate system with spatially varying source fluxes and permeabilities. Analytical solutions are given for the steady-state equations and numerical solutions for the unsteady-state equations. The solutions indicate that an external source of solute delivered to the stream flowing away from the U-tube bend can be distributed by the exchanger so that the concentration in both limbs increases toward the bend. In particular, there exist source fluxes whose magnitude decreases monotonically toward the bend for which the maximum solute concentration occurs at the bend. The point at which a concentration maximum occurs is governed principally by the solute permeability of the barrier separating the two limbs and by the volume flow rate through the exchanger. The system dynamics depend strongly on the relative cross-sectional areas of the two limbs or, equivalently, on the flow velocities within them. The model is used as a basis for discussion of various functional aspects of the renal vasa recta system.
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    Mathematical analysis of circulatory mixing process (1973)
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    Bulletin of mathematical biology 35 (1973), S. 287-300 
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    Notes: Abstract The circulatory mixing process was analyzed as the time course of the dispersion of indicator after its injection into the heart. In simplified models, which had one or two lumped mixing chambers and circulatory pathways connected with them, it was suggested that the extent of dispersion could be evaluated by the variance of indicator distribution in the total circulating blood when the circulation time distributions between the chambers and the concentration curves in the chambers were known. The method of determining the circulation time distributions through the pulmonary, systemic and total circulations was derived and the actual distributions were obtained in dogs by indicator dilution techniques. With the use of these distributions, the time course of the circulatory mixing process was numerically calculated. The results showed that there was considerable difference in velocities of the process between the case of the right heart injection and the left heart injection of the indicator.
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    Reactions chimiques couplees considerees en tant que filtres passe-bande (1973)
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    Bulletin of mathematical biology 35 (1973), S. 319-337 
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    Notes: Abstract An investigation is made as to whether or not the existence of a band-pass filter function, analogous to that in electronics, can be proved from the fundamental laws of chemical kinetics. The problem is important for better understanding of the preference of certain biological rhythms to others. It is shown with simple examples that such behavior is possible for a number of systems of coupled chemical reactions far enough from the thermodynamic equilibrium. It is of interest to generalize this behavior since it could conceivably play a role in the transmission of “usable information” in biology.
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    Organismic sets. Scientific creativity, its spread and applications (1973)
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    Bulletin of mathematical biology 35 (1973), S. 339-344 
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    Notes: Abstract The phenomenon of mental creativity is considered from the standpoint of the theory of organismic sets, developed by the author in a series of previous publications. It is shown how the differences in creativity between different individuals may be interpreted on this basis, and why extreme creativity is rare. A parallel interpretation for facility in observation is given, and it is shown why facility in creativity and observation is much rarer than either individual facility. A further conclusion is drawn regarding the deducibility of the laws of nature by purely logical means.
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    Quantum theory of time-varying stimulation in nerve (1973)
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    Bulletin of mathematical biology 35 (1973), S. 359-374 
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    Notes: Abstract The equation for the quantum transitions (spontaneous and stimulated) of membrane dipoles is solved for the various forms of time-varying stimulation in nerve. From the condition of ever-increasing dipole population in the upper state, the threshold for excitation is determined in each case. The results obtained are in agreement with the established facts. The optimum frequency for stimulation is given asv 0=0.0615/T 2 whereT 2 is the dipole relaxation time. The feature of the theory is that the mathematical formulation is based upon a physical mechanism and the results can thus provide some understanding in the observed phenomena.
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    On gompertz curves (1973)
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    Bulletin of mathematical biology 35 (1973), S. 415-415 
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    Reorientation of membrane dipoles during nerve excitation (1973)
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    Bulletin of mathematical biology 35 (1973), S. 417-418 
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    Announcement (1973)
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    Bulletin of mathematical biology 35 (1973), S. 419-419 
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    Interfilament forces in striated muscle (1973)
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    Bulletin of mathematical biology 35 (1973), S. 565-575 
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    Notes: Abstract Evaluation of the Van der Waals energy per filament suggests that molecular dispersion forces should not be very important in determining the stability of the myofilament lattice in resting muscle. In order to explain the lattice stability and other important properties of the striated muscle, it is suggested that a balance between electrostatic forces and forces developed by some interfibrillar structures is mainly responsible.
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    A method for simultaneous determination of (I) the blood flows to the organs and (II) the cardiac output (1973)
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    Bulletin of mathematical biology 35 (1973), S. 549-563 
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    Notes: Abstract A method is presented for the simultaneous determination of (i) the blood flow to the organs and (ii) the cardiac output. Part I of the paper deals with the analysis of ann compartment (organ) vascular system model. The data, employed in the analysis, consists of continuous monitoring of the amounts of indicatorM i in the organs (or compartments). An analysis for determination of the cardiac output and the absolute flows to the organs is presented. Since it is difficult to isolate certain organ systems and measure the amounts of indicator in them exclusively, a more realistic model of then compartment vascular system is presented in Part II. Herein, the analysis has accounted for the finite transit time, of the indicator, from one organ system to another. Further, estimation theory is employed to make estimates of blood flow to different organs by taking note of (i) the measurement errors due to the detectors' monitoring (for an organ system) some combination ofM i 's instead of theM i for the particulari th organ and (ii) noise uncertainties introduced by the measuring instruments.
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    Particle distribution for dilute suspension in flow (1973)
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    Bulletin of mathematical biology 35 (1973), S. 577-589 
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    Notes: Abstract Analytic expressions for the velocity profile and particle distribution of a dilute suspension in flow were obtained as functions of radial distance. Einstein's linear viscosity model and the hypothesis of “minimum energy dissipation” were used. The methods of variational calculus were applied during the mathematical development. A parabolic velocity profile, which is a modified form of that for Hagen-Poiseuille flow, and a uniform particle distribution were obtained. An attempt is made to explain the results in light of some of the widely held theories on suspension flow and the rather severe limitations of Einstein's viscosity model. A suggestion for future work is made for improving the results of the present.
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    A semiclassical theory for nerve excitation by a low intensity electromagnetic field (1973)
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    Bulletin of mathematical biology 35 (1973), S. 591-605 
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    Notes: Abstract A possible mechanism for microwave-neuron interaction, when the nerve is irradiated by a thermally insignificant electromagnetic field, is described. The radiation field is treated classically, but the atomic system which interacts with this field is treated quantum mechanically using the density matrix approach. Attention is given to both homogeneous and inhomogeneous broadening effects, and the degrading influence of inhomogeneous broadening upon the neural membrane's ability to interact with the electromagnetic field is shown.
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    The epileptic neuron: I (1973)
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    Bulletin of mathematical biology 35 (1973), S. 709-714 
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    Notes: Abstract Mathematical models of neurons are studied which exhibit spontaneous and repetitive firings in the absence of normally occurring substances. The activity of such neurons could result in the symptoms of epilepsy. The identification of such substances would be important in the treatment of the disease as well as in the understanding of their mode of action. The importance of the geometrical and physical parameters for the generation of spontaneous repetitive discharges is brought out.
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    The epileptic neuron: II (1974)
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    Bulletin of mathematical biology 36 (1974), S. 1-16 
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    Notes: Abstract Conditions under which a neuron can maintain repetitive discharges are studied with the view that some light may be shed on the problem of epilepsy. It is shown that if there are excitatory and inhibitory substances as well as binding substances in volved, then for repetitive discharges to occur, the formation of the substances must be intracellular. The geometric physical characteristics of the system are also determining factors. The role of a fatigue factor is considered in an attempt to understand the frequency and duration of mild and severe epileptic attacks.
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    On the number of phenotype structures in genetic systems (1974)
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    Bulletin of mathematical biology 36 (1974), S. 325-338 
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    Notes: Abstract Phenotype structures in genetic systems are carefully defined in an abstract setting so that a considerable amount of enumerative theory can be brought to bear on the problem of enumerating them. Recent results can be used to simplify the computations, and a natural correspondence is suggested which changes the problem of finding the number of phenotype structures to the problem of determining the numbers of certain graphs with colored points.
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    The interaction of diffusion and perfusion in homogeneous tissue (1974)
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    Bulletin of mathematical biology 36 (1974), S. 505-526 
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    Notes: Abstract A mathematical model is proposed to examine the interaction between blood perfusion and gas diffusion in the uptake of inert gases in tissue. The standard Haldane perfusion model is contrasted with the Hills radial bulk diffusion model in a variety of homogeneous tissue types used in decompression theory. It is the intention of the present analysis to fix ideas on the role of diffusion, perfusion and axial concentration and quantitative studies are given and seem to show that Haldane's perfusion theory is at best a poor approximation even at asymptotic times. It is shown that a strong interaction exists between diffusion and perfusion in muscle tissue and neither approach adequately describes the actual uptake half-time of an inert gas.
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    Necessary and sufficient condition for a model insect population to go to extinction (1974)
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    Bulletin of mathematical biology 36 (1974), S. 527-533 
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    Notes: Abstract A necessary and sufficient condition on the parameters for a model population to become extinct is presented. The mathematical model describes an insect population with overlapping generations where the females are polyandrous and the males are subject to autosterilization. The relationship between the values of the parameters of the model and the time to extinction is illustrated.
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    D'Arcy Thompson's theorems and the shape of the molluscan shell (1974)
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    Bulletin of mathematical biology 36 (1974), S. 567-575 
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    Notes: Abstract The paper deals with properties of mathematical models of biological systems. It is shown that under suitable conditions the number of parameters employed in the model depends only on the biological system and not on the particular model chosen. As an application the mathematical description of the formation of molluscan shells is discussed.
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    Molecular control systems: I—Temporal organization in cells revisited (1974)
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    Bulletin of mathematical biology 36 (1974), S. 595-599 
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    Notes: Abstract An analysis of Goodwin's fundamental nonlinear oscillator in the light of the requirement that concentrations must be intrinsically non-negative reveals that most of the phase plane is “forbidden”, and that there is one distinguished closed curve that has some of the properties of a limit cycle.
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    Note on oscillations of turbidity in a precipitation reaction system of thyroxine (1974)
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    Bulletin of mathematical biology 36 (1974), S. 601-604 
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    Notes: Abstract A model for a precipitation reaction system is presented in which all steps are reversible monomolecular or bimolecular reactions. Under certain conditions, quasi-stable oscillations in turbidity can occur, as has been observed in the precipitation of thyroxine.
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    Stabilité structurelle et morphogénèse (1974)
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    Bulletin of mathematical biology 36 (1974), S. 611-613 
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    Stochastic processes and applications in biology and medicine. Part 1—Theory. Part 2—Models (1974)
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    Bulletin of mathematical biology 36 (1974), S. 607-610 
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    The determination of cell-cycle parameters from measurements of the fraction of labeled mitoses (1972)
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    Bulletin of mathematical biology 34 (1972), S. 1-12 
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    Notes: Abstract A method is described for estimating cell-cycle parameters from experimental fraction-of-labeled-mitoses measurements. The method is closely related to that of J. C. Barrett (1970) but is based on the analysis of Brockwell and Trucco (1970) which takes into account population growth in the calculation of theoreticalFLM-functions. Several sets of experimental data are analyzed, among them the data for the Marshall tumor considered by Barrett. It is found that population growth has a small but nevertheless detectable effect on the estimates of the cell parameters.
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    Transfer times across the human body (1972)
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    Bulletin of mathematical biology 34 (1972), S. 33-44 
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    Notes: Abstract The radioactivity disappearance curves of glucose-6-14C albumin-I131 after a single injection of tracer into a human subject have been determined in detail, particularly at early time intervals. The curves, expressed as sums of exponentials, have been analyzed as the infinite sum of convolutions of single passage time densities. The resultant transfer time distribution of a single circulatory pass allows examination of all delays in the system no matter how long they take. The structural detail evident by this means and the long mean time of a single pass of glucose (〉5 min) supports the thesis that factors other than rapid and uniform diffusion play a role in the extravascular movements of glucose molecules.
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    A bisexual multitype branching process with applications in population genetics (1972)
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    Bulletin of mathematical biology 34 (1972), S. 13-31 
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    Notes: Abstract A bisexual multiple branching process is studied. Consider a population with respect to three genotypes in both the female and male populations and let $$X(n) = \left\langle {X_1 (n), X_2 (n), X_3 (n)} \right\rangle and Y(n) = \left\langle {Y_1 (n), Y_2 (n), Y_3 (n)} \right\rangle$$ be random vectors giving the number of females and males (respectively) of each genotype in generationn. The mating of females and males is accommodated in the model withZ ij (n) representing the number of females of theith genotype mated with a male of thejth genotype in generationn. The mating system is such that a female may be mated to only one male but a male may be mated with more than one female. By arranging the nine random variablesZ ij (n),i, j=1, 2, 3, in a 1×9, vectorZ(n) it is shown that under certain conditions there is a positive constant ϱ such that when ϱ〉1 the vectorsZ n /ρn,X n /ρn andY n /ρn converge almost surely asn→∞ to random vectors with fixed directions. The paper is divided into four sections. In section 1 the model is described in detail and its potential applications to population genetics are discussed. In section 2, the generating function of the transition probabilities of theZ-process are derived. Section3 is devoted to the study of the limiting behavior of the first and second moments of theZ-process, and in section4 the results of section3 are utilized to study the behavior of the random vectorsZ(n),X(n) andY(n) asn→∞.
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    Theory of initial current density in membrane voltage-clamp experiments (1972)
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    Bulletin of mathematical biology 34 (1972), S. 45-52 
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    Notes: Abstract Herein we show that the voltage-clamp current density at zero time calculated from electrodiffusion equations is linear in the clamping voltage for a simple membrane (no charge structure) and for a membrae with fixed charges. Such membranes are nonexcitable. Excitable membranes can be represented by a homogeneous membrane with dipole layers at the surface. In this case the initial current density will be linear in the clamping voltage if a critical field for a dipole layer reorientation is not passed through in changing from holding to clamping potential. Otherwise, deviation from nonlinearity may occur. This is in agreement with experimental data for the squid giant axon.
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    A note on sporadic nonperiodic oscillations in the activity of the endocrine system and on the endocrine—Nervous system feedback (1972)
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    Bulletin of mathematical biology 34 (1972), S. 65-69 
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    Notes: Abstract By generalizing a previous paper on periodicities in the endocrine system (Bull. Math. Biophysics,30, 735–749, 1968), it is shown that nonperiodic, sporadic oscillations in the system are also possible. A procedure of describing the feedback mechanism between the endocrine system and the central nervous system is suggested. It is shown that the combined system: endocrine—CNS, also may show sporadic fluctuations.
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    Wall shear in pulsatile flow (1972)
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    Bulletin of mathematical biology 34 (1972), S. 79-86 
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    Notes: Abstract This paper deals with a mathematical attempt to determine the wall shear during normal flow of blood in the ascending and the descending thoracic aorta. A simple model is used, but the results obtained are in agreement with published experimental results for the descending thoracic aorta. It is suggested that the degree of fluctuation in the pressure gradient at a given station is the major factor in determining the level of wall shear at that point.
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    A neural circuit which exhibits some features of epileptic attacks (1972)
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    Bulletin of mathematical biology 34 (1972), S. 71-78 
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    Notes: Abstract A neural model based on a generalization of a model proposed in 1938 in the first edition of the author'sMathematical Biophysics (Chicago: Univ. of Chicago Press) is described. It possesses the property that due to some endogenous or exogenous stimulus, which may be of random nature, a pathway may suddenly begin to fire spontaneously. This spontaneous firing may either gradually spread over other pathways and eventually cease, or it may remain localized within one or a few pathways and then cease. Which of the two types of events occurs depends on the values of a number of parameters. The case of spreading reminds one of Jacksonian epilepsy.
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    Diffusion exchange between a membrane-bounded sphere and its surrounding (1972)
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    Bulletin of mathematical biology 34 (1972), S. 93-102 
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    Notes: Abstract The diffusion equation is solved for a membrane-bounded sphere situated in an infinite medium with different diffusion properties. The formal solution is obtained through Laplace transformation in the time variable. It is not possible to find a closed form solution in terms of analytical functions, and therefore a numerical inversion technique is applied to obtain the final solution. The application on a biological problem is discussed.
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    Determination of reaction rate constants in the mammillary system (1972)
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    Bulletin of mathematical biology 34 (1972), S. 87-92 
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    Notes: Abstract It is shown that the individual rate constants can be determined for the composite chemical system: $$A + B_i \rightleftarrows C_i ; i = 1...N$$ with only measurements of the unbound species,A(t), required. The dissociation rate constants can be determined by direct analysis of a single steady state tracer study. The association constants then follow from the analysis of stable equilibrium determinations reported earlier (Hart, 1965). An approximate solution when tracer methods are in-applicable is also given.
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    Some biochemical threshold mechanisms (1972)
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    Bulletin of mathematical biology 34 (1972), S. 103-112 
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    Notes: Abstract Certain arrangements of enzymatic (bimolecular) subsystems lead to characteristic threshold-type response. Two simple cases of such systems are studied here in terms of steady state behavior and explicit relationships between system and curve parameters. It is found that the curvature of the threshold curve is directly related to the equivalent Michaelis constant and, in the case of saturated threshold curve, the slope of the curve at the idealized threshold is limited by the ratio of saturation to threshold. This slope may be appreciably increased up to a stepwise response at the threshold if a multisubstrate complex of the enzyme is the only species which affects the enzyme mediated transport.
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    Theory of neural mediation of intraocular dynamics (1972)
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    Bulletin of mathematical biology 34 (1972), S. 113-148 
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    Notes: Abstract Many experimental studies have indicated that the intraocular pressure is subject to mediation by adrenergic mechanisms affecting both the rate of formation of the aqueous humor and the resistance of the pathway through which the aqueous humor flows out of the eye. Thus, for example, the role of adrenergic drugs in glaucoma therapy is well known. How the mediation is accomplished has not been clarified in detail. Several possible mechanisms have been suggested, and all may indeed be involved. The present study is concerned with the basis and mathematical formulation of one of them and the consequences with respect to aqueous dynamics. The analysis leads to expressions for the aqueous outflow resistance and the formation rate, as well as other quantities of interest. The theoretical behavior is shown to compare favorably with the results of infusion studies and various other experiments, and to provide a unified picture of much of the pressure-flow behavior of both the living and the dead eye.
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    Eighth International Congress of Electroencephalography and Clinical Neurophysiology (1972)
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    Bulletin of mathematical biology 34 (1972), S. 149-150 
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    A fit to nerve membrane rectification curves with a double-dipole-layer membrane model (1972)
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    Bulletin of mathematical biology 34 (1972), S. 151-172 
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    Notes: Abstract Following Wei's suggestion that nerve stimulation and conduction properties are due to dipole layers at the two membrane surfaces (Wei, 1969), we have done steady-state electro-diffusion calculations in the constant field approximation for a simple double-dipole-layer model. We are thereby able to quantitatively fit the recent potassium iso-osmotic rectification curves of Gilbert and Ehrenstein for the squid giant axon membrane. For the squid axon membrane in a natural ion environment, only the outside dipole layer is present in the fit to the data.
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    Geometrical parameters of the individual myocardial fibers (1972)
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    Bulletin of mathematical biology 34 (1972), S. 205-211 
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    Notes: Abstract In this paper the left ventricle of the heart was considered as a shell of varying thickness. The first and second fundamental forms of the middle surface, of the shell, as well as Euler's theorem were used for deriving expressions giving the length and curvature of the individual myocardial fibers. Recent anatomical studies have shown that the myocardial fibers in the middle of the left ventricular wall follow a course nearly parallel to the horizontal plane (Streeteret al., 1969). In previous papers (Voukydis 1969, 1970) a mathematical description of the curvature and length of the individual myocardial fibers was presented. Unfortunately, both the curvature and the length formulas contained the cotangent of the fiber helix angle, which approaches infinity as the fiber assumes a course parallel to the horizontal plane. Consequently, these two formulas cannot be used for fibers nearly parallel to the horizontal plane. The present paper will give an alternative way for calculating the length and curvature of the individual myocardial fibers, based on the fundamental forms of surface and on Euler's theorem.
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    Some remarks on the central nervous system (1972)
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    Bulletin of mathematical biology 34 (1972), S. 231-242 
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    Notes: Abstract It is pointed out that the successes obtained in the mathematical biology of the central nervous system are based mostly on a number of more or less complicated neuronic circuit models, each inventedad hoc for the purpose of explaining a given phenomenon. The individual models remain disconnected from each other, however, and the unity of the CNS is not apparent. (Rashevsky,Mathematical Biophysics, 3rd Edition, Vol. II, 1960. New York, Dover Publications, Inc.) Some “field theories” of the CNS, as for example that of Griffith (Bull. Math. Biophysics,25, 111–120, 1963;27, 187–195, 1965), give more expression to this unity but lose in the explanation of specific phenomena. The present paper starts with the picture thatevery neuron in the brain isdirectly or indirectly affected to some extent byevery other neuron. This leads to a system of equations with a very large number of variables. Such a system can be replaced in the limiting case by an integral equation of the first kind. At least two specific results can be obtained with this approach and suggestions for further improvement are made.
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    Fiber stress profiles in the left ventricle of the heart during diastole: Effects of distension and hypertrophy (1972)
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    Bulletin of mathematical biology 34 (1972), S. 379-392 
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    Notes: Abstract Pressure-volume and volume-dimensions relationships, obtained from excised dog left ventricles were used for calculating the stresses acting along the longitudinal axis of the individual myocardial fibers. The calculations were based on a set of empirical and theoretical equations. The pressure-volume relationship as well as the volume-dimensions relationships for the excised left ventricle were expressed in the form of empirical equations; the fiber orientation was written as a function of the fiber location within the left ventricular wall; finally, the fiber stress was determined by means of theoretically derived formulas. Simultaneous solutions for the fibers of a meridian cut through the left ventricular myocardial shell were obtained by means of a digital computer and presented in the form of diagrams. The results showed that at low degrees of distension of the left ventricle there are two zones of higher stresses at the equatorial area, one near the epicardium and one near the endocardium. As the distension proceeds under the effect of progressively increasing intraventricular pressure, these two zones become less well defined, whereas a new zone of higher stresses appears near the apex. At high degrees of distension, the ventricle assumes a more spherical shape and the equatorial zones of higher stresses are replaced by zones of lower stresses. Increase in the myocardial mass results in appearance of the equatorial lower stress zones at lower degrees of distension.
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    A note on open linear systems (1974)
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    Bulletin of mathematical biology 36 (1974), S. 97-99 
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    Notes: Abstract A theorem is given which states a necessary and sufficient condition for the specific activity to be uniform throughout an open compartmental system in the steady state.
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    Introduction to Mathematics for Life Scientists (1974)
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    Bulletin of mathematical biology 36 (1974), S. 101-102 
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    A dynamic model of an environmental system withn interacting components andp degrees of freedom (1974)
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    Bulletin of mathematical biology 36 (1974), S. 91-96 
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    Notes: Abstract The nonlinear differential equations of growth proposed by Volterra (1931) are used as the basis for a dynamic model of ann-element system withp specified, terminal conditions andp missing initial conditions. The resulting two-point boundary-value problem can be solved, ifp is not large, by the shooting method used previously by Huddleston (1967). A numerical example withn=4 andp=2 is solved on a digital computer, and some results are presented.
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    Use of a heat transfer analogy for a mathematical model of respiratory tract deposition (1974)
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    Bulletin of mathematical biology 36 (1974), S. 105-116 
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    Notes: Abstract Mathematical models predicting the aerosol deposition in the respiratory tract are reviewed. Data in the literature indicated not only that the air flow in the trachea and major bronchi may not be laminar, but also that the entrance effect of the tube or airway has not been considered. A new approach to a mathematical model of respiratory tract deposition, based on the analogy of the heat and mass transfer, is discussed.
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    Isotherms forr species adsorbed at linear polymers with nearest neighbor cooperative interaction (1974)
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    Bulletin of mathematical biology 36 (1974), S. 117-126 
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    Notes: Abstract The equation of the cooperative specific isotherm is derived for the general case ofr species interacting cooperatively at nearest neighboring sites. Explicit expressions for the special casesr=2–5 are given.
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    Pseudo-sound generation at atherosclerotic constrictions in arteries (1974)
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    Bulletin of mathematical biology 36 (1974), S. 143-155 
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    Notes: Abstract Sounds and murmurs have long been employed to qualitatively diagnose cardiovascular disease. However, quantitative diagnosis has been hindered by the lack of understanding of the sound generation and transmission mechanisms. Clinical phonoangiographic studies have shown that simple assumptions about low frequency sound transmission through tissue surrounding an artery are inadequate for obtaining meaningful quantitative diagnosis. Therefore, a theory is developed which relates internal turbulent flow in constricted peripheral arteries to the sound observed at the surface of the skin by means of assumptions of similarity and local axial homogeneity of the internal turbulence. It is found that the spectrum of pressure at the wall of the artery is related to the spectrum of the pressure at the surface of the skin by a filtering factor approximately proportional to ω-2. This arises not because of frequency dependent volumetric absorption in the surrounding medium, as with ultrasound, but because of the manner in which stochastic signals add when observed.
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    Measuring a mitotic oscillator: The ARC discontinuity (1974)
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    Bulletin of mathematical biology 36 (1974), S. 171-182 
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    Notes: Abstract Mitosis occurs synchronously in up to 108 nuclei in the syncytial plasmodium ofPhysarum polycephalum. Any two phases of the mitotic cycle may be mixed by fusing plasmodial pairs. A topological property of the synchronized phase of the fused pair as a function of parental phases, the arc discontinuity, characterizes the underlying oscillator, and indicates mitosis is controlled by a moderate relaxation oscillator which rotates more rapidly near its singularity than its limit cycle. A model oscillator is briefly described.
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    Genetic nets and dissipative structures: An algebraic approach (1974)
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    Bulletin of mathematical biology 36 (1974), S. 183-196 
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    Notes: Abstract Genetic nets represent an attempt to model genome structure. Depending on the interaction dynamics assumed, they can constitute highly non-linear chemical systems having multiple steady states; hence their relevance to the theory of dissipative structures. Their typical size and possible complexity makes it difficult to study them by means of customary analytical techniques based on differential equations. We have therefore considered an algebraic approach derived from regarding the nets as finite-state automata. This view has revealed a surprisingly rich algebraic structure which can be used to investigate problems concerned with the relation between biological structure and function. This algebraic structure is described with particular reference to the genetic nets of Tsanev and Sendov.
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    URL: http://dx.doi.org/10.1007/BF02458602
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    Finite fluctuations and multiple steady states far from equilibrium (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 205-213 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The role of finite fluctuations in transitions between nonequilibrium steady states in nonlinear systems is investigated. Attention is focused on a model biochemical system for which the usual deterministic chemical kinetics predicts a far-from-equilibrium region of multiple steady states. A stochastic approach to chemical kinetics is adopted to study explicitly the effect of fluctuations around the coexisting stable states on a predicted hysteresis in the transition between those states. A numerical solution of the stochastic master equation for the system yields results which differ qualitatively from predictions of the purely macroscopic theory. Possible implications of these results are considered, and several important aspects of the computational scheme are discussed in some detail.
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    The global asymptotic stability of prey-predator systems with second-order dissipation (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 215-217 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Models of biological development, evolution and control should take into account that very small numbers of cells or chemicals or individuals eventually grow into stable, large populations. The simplified two-component model used in these studies includes the following: (1) first-order decay; (2) first-order autocatalysis; (3) negative feedback; (4) positive feedback; (5) second-order decay; (6) second-order autocatalysis. A positive definite Lyapunov function is constructed and shown to have a negative definite total derivative. The stationary statex〉0,y〉0, therefore possesses global asymptotic stability. This means that sustained oscillations cannot occur. Another stationary state,x=y=0, is shown to be unstable. This means that infinitesimally small perturbations ofx=y=0 will result in evolution of the variables to the stable stationary state. This result contrasts with that obtained with the Lotka-Volterra model in that small perturbations ofx=y=0 for that model result in sustained, oscillating excursions; the smaller the initial perturbations, the larger these excursions will be. A simulation illustrates that stable populations of 1020 x's andy's can arise from a singlex andy.x grows more or less continuously, buty remains extremely small for 80 per cent of the time interval required for the variables to approach their stable populations.
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    URL: http://dx.doi.org/10.1007/BF02458605
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    Semiconductor theory of ion transport in thin lipid membranes: II. Surface recombination (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 247-264 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Following the theory of surface recombination in semiconductors, we have derived an expression for the rate of ion recombination at the membrane surface. The surface recombination rate is used in the boundary conditions of current flows at the interfaces. Expressions for the ion fluxes are then derived as functions of environmental variables and membrane parameters. Our analysis strongly suggests that the ion flow through a thin lipid membrane consists of two major components: the surface barrier jumping current and the surface recombination current that are controlled decisively by surface barrier height, surface trap density and surface recombination rates. These general formulations are useful not only for the calculation but also for the understanding of ion transport in thin lipid membranes under a variety of experimental conditions. The implications of this theory to biological membranes and its possible extensions are discussed.
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    URL: http://dx.doi.org/10.1007/BF02461327
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    Transport of O2 along a model pathway through the respiratory region of the lung (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 275-303 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A pathway through the system of branching in the respiratory region of the lung is modelled by a circular cylinder, closed at one end, with partitions which define the component respiratory units. In this model the transport of O2 during inspiration, generated by diffusion is compared with that produced by diffusion together with convection and the importance of convection in the respiratory region in promoting oxygen uptake at the alveolar wall is discussed. For this discussion it is only necessary to consider inspiration. The equations are solved numerically for flow rates of 10, 85 and 200 liters/min. O2 uptake at the wall and curves of constant O2 concentration are shown to illustrate the influence of convection. It is found that after a 2 sec inspiration from an O2 tension of 98 mm Hg and a lung volume of 2300 ml, convection is about 12 per cent as important as diffusion at a flow rate of 85 liters/min, whereas at 10 liters/min convection is only about 0.4 per cent as important as diffusion.
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    URL: http://dx.doi.org/10.1007/BF02461329
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    Theoretical relationship between the optimal models of the vascular tree (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 311-323 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Two models of optimal branching structure of the vascular tree are compared. Murray’s minimum work model derived from minimum energy loss due to flow and volume in the duct system is proved to be included as a mathematical group in the authors’ model defined by the minimum volume under determinant pressure, flow and position at the terminals. The problem about heterotypical trees which are identical at the terminal conditions but different in the topological order of branch combinations are discussed, applying the results of analyses on the equivalent duct of uniform terminal pressure trees. It is proved that the minimum work tree has the least energy loss compared with its heterotypical minimum volume trees and is a better model of branching structure of the vascular tree.
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    URL: http://dx.doi.org/10.1007/BF02461331
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    A morphogenetic system matched to a group (1973)
    Springer
    Bulletin of mathematical biology 35 (1973), S. 615-625 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A period of morphogenesis is operationally defined as a system. This period precedes cell differentiation and is axially expressed. Six interacting elements of structure are defined utilizing Spemann and Mangold's heteroplastic transplantation experiments. The system generates thirty-six interactions, given a rule of composition and specific premises. The element set and operation is matched to the symmetric group S3.
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    Asymptotic behavior in a deterministic epidemic model (1973)
    Springer
    Bulletin of mathematical biology 35 (1973), S. 607-614 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The effects of a periodic contact rate and of carriers are considered for a generalization of Bailey's simple epidemic model. In this model it is assumed that individuals become susceptible again as soon as they recover from the infection so that a fixed population can be divided into a class of infectives and a class of susceptibles which vary with time. If the contact rate is periodic, then the number of infectives as time approaches infinity either tends to zero or is asymptotically periodic depending on whether the total population size is less than or greater than a threshold value. The behavior for large time of the number of infectives is determined for three modifications of the model which involve carriers.
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    Electron-phonon (trapped photon) coupling and infrared coaxial transmission line theory of energy transport in mitochondria and nerve (1973)
    Springer
    Bulletin of mathematical biology 35 (1973), S. 627-644 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper develops quantitatively for the kinetics of mitochondrial oxidation the implications of the hypothesis that phosphorylation is accomplished by phonons in the mitochondrial solid. The concept is used that the phonon acts like a trapped photon to produce a reverse photocurrent, which inhibits oxidation, in the absence of phonon dissipation due to phosphorylation or uncoupling. The same conceft in reverse is shown to explain qualitatively the generation or membrane potentials in nerve axon. Infrared optical phonons generated in the above processes are expected to have limited mibility, but it is shown that the structure and geometry of the lipid bilayer membrane are well adapted to the rapid and efficient dissemination of this energy throughout the mitochondrion or nerve axoni n the form of infrared electromagnetic waves by a coaxial transmission line mechanism. The mitochondrion may act like an infrared coaxial resonant cavity.
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