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  • Springer  (172,932)
  • 1995-1999
  • 1980-1984  (43,712)
  • 1975-1979  (129,220)
  • 1925-1929
  • 1981  (43,712)
  • 1978  (44,825)
  • 1977  (43,877)
  • 1976  (40,518)
Collection
Keywords
Publisher
Years
  • 1995-1999
  • 1980-1984  (43,712)
  • 1975-1979  (129,220)
  • 1925-1929
Year
  • 101
    Electronic Resource
    Electronic Resource
    Bounds on the populations in the Dreitlein-Smoes model of oscillatory kinetic systems (1976)
    Springer
    Bulletin of mathematical biology 38 (1976), S. 623-631 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The model recently proposed by Dreitlein and Smoes for oscillatory kinetic systems is studied. Diffusion of the oscillating species is taken into account, and bounds on the total number of individuals of each species are determined for both two- and three-dimensional finite regions with various boundary conditons applied. It is found that in general the effect of diffusion on the system behavior is to reduce the maximum possible radius of limit cycles. In particular, in some cases global limit cycle behavior is precluded.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02458637
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  • 102
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    Flow limits of Kedem-Katchalsky equations for fluid flux (1976)
    Springer
    Bulletin of mathematical biology 38 (1976), S. 671-677 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The Kedem-Katchalsky equations for fluid flux across membranes may not be adequate for large solvent flows. In particular, for an example of two membranes in series, it is argued that they would predict physically unreasonable behavior. An alternate equation for solute flow is proposed for a simple sieving membrane. For the same example, this equation predicts more physically reasonable results.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02458641
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  • 103
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    On the geometry of space-and time-equivalent catastrophes (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 1-25 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The geometrical nature of the Elementary catastrophes (Thom, 1969) is reviewed. Histories of the movement of catastrophe manifolds and bifurcation sets are presented for some of the space-equivalent unfoldings described by Wassermann (1975). These unfoldings provide descriptions of the variation with time of the stability of stationary states of associated potential energy functions. Identification of these stationary energy states with stationary states of a system therefore provides a description of its behavior with time. Qualitative descriptions of this type are particularly useful when the complexity of a system prevents a detailed quantitative description. Histories of bifurcation set movements suggest different types of system behavior at different space-like coordinates. This type of theory may be a useful model for the processes leading to differentiation of cells and to emergence of adult forms of a biological organism.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02463127
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  • 104
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    Prediction of limit cycles in mathematical models of biological oscillations (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 27-44 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Three conjectures are given which predict the existence of unique stable limit cycle oscillations in a class of piecewise linear (PL) differential equations. The equations are appropriate to model biological or other complex systems in which there are switchlike interactions between the elements of the network. Methods are presented which can be used to develop mathematical models which are conjectured to display stable limit cycle oscillations, from qualitative experimental information about relative phases of activity in the dynamical systems. Several illustrative numerical examples are given, and one experimental example from neurobiology is discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02463128
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  • 105
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    Least energy regulation of the arterial system (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 79-93 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The formulation and results of the Kalman State Regulator Problem are applied to a mathematical model of the arterial system of a dog to obtain an optimal control for blood pressure. The criterion for optimality is minimum energy per cycle.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02463131
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  • 106
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    Pressure-flow relationships in a model for the arterial system (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 95-105 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A mathematical model is developed in state variable form suitable for the study of the control of blood pressure and flow in the mammalian cardiovascular system. The applicability of the model to steady state, both mean and pulsatile, and transient phenomena is demonstrated by the agreement of the results with experimental data. This model was developed to study the neural and renal-endocrine-electrolyte control of cardiovascular functions.
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    URL: http://dx.doi.org/10.1007/BF02463132
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  • 107
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    Conditions for periodic solutions of volterra differential systems (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 133-160 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract As a contribution to the discussion of oscillatory models for interacting species it is shown that two-species Volterra models can never have limit cycles, and a complete enumeration is given of conditions which the parameters of these models must satisfy in order that a part of the phase space be filled with a family of closed curves; sketches of phase portraits are also given. These results complement and correct older results by Bautin and by Coppel on quadratic differential systems. The paper opens with a brief discussion of some more practical aspects of the ecological application of oscillatory models.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02461432
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  • 108
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    The kinetics of diffused substances in some biological systems (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 107-121 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The purpose of this paper is to present a method for the determination of a solution for a coupled system of first order initial boundary-value problems arising from some biological systems. The physical problem is to determine the suspended and the superficial molecular concentrations of a traced substance passing through an organ containing a tangle of vessels, such as the kidney-ureter system. The approach to the problem is by successive approximation which leads to a recursion formula for the determination of the solution as well as error estimates for the approximations. The recursion formula involves only direct integration which indicates a promising possibility in obtaining numerical results by using a computer. In addition to the determination of a solution, some qualitative analysis of the solution is given. This includes the existence of a unique solution, the continuous dependency of the solution on the data, and the stability problem of a steady-state solution.
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    URL: http://dx.doi.org/10.1007/BF02463133
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  • 109
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    State, security and insurance (1978)
    Springer
    The Geneva risk and insurance review 10 (1978), S. 3-43 
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    ISSN: 1554-9658
    Source: Springer Online Journal Archives 1860-2000
    Topics: Economics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02409805
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  • 110
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    Heat transfer in perfused biological tissue—II. The “macroscopic” temperature distribution in tissue (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 183-199 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In Part I, the general solution of the heat transport equation was given for a medium of constant thermal properties, containing a general convection fieldv(r,t). Starting from this general solution, an average or “macroscopic” temperature distribution is calculated in this paper, since in most physiological problems only an average or “macroscopic” temperature is accessible to the measurement. It is shown that the averaged temperature distribution is influenced alone by the order of the local symmetries of the convection field. These local symmetries can be deduced from anatomical data on the vascular architecture. In order to perform the averaging process, a detailed discussion, and an estimate of the correlation between the microscopic temperature variation and the convection field is carried out. Hereby, great emphasis is put on the experimental consequences of these results. As an illustrative example, the calculation of the macroscopic temperature distribution in a medium traversed by parallel capillaries is outlined.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02461434
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  • 111
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    A note on dipole sources in conducting biological tissues (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 201-209 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The study of current flow fields in biologicaltissue requires finding the potential field from dipole sources such that the normal gradient vanishes at the exterior surface. A convenient way to determine the dipole field is by taking the gradient of the potential field set up by a point source. However, the point source problem is ill-posed when the normal gradient is required to vanish at the outer surface. In the paper the nature of this problem is discussed and several methods for overcoming the difficulty are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02461435
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  • 112
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    A hydrodynamic model of aqueous flow in the posterior chamber of the eye (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 223-235 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract An analysis of the hydrodynamics of aqueous flow in the posterior chamber of the anterior segment of the eye is presented. The viscid, incompressible description of fluid dynamics in a spherical geometry is utilized to reduce the problem to a biharmonic-type equation using a “Stokes Stream Function”. Analogous to Poiseuille flow in a cylindrical pipe, velocity profiles are deduced and an Ohms Law relationship between pressure and flow is derived in terms of the geometry of the assumed model. This result is then incorporated into a synthesized electric circuit analog of flow between ciliary artery and episcleral vein. Applications to open angle and pupilary block glaucoma are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02461437
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  • 113
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    On the nerve impulse equation: The dynamic responses of nerve impulse (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 247-255 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract With the assumption of dipole interaction with the membrane matrix, the dipole barrier under an applied field shows a minimum in its time transient. Kinetic equations governing the migration of ions are presented. Na+ activation, Na+ inactivation and K+ delay are all part of the same mechanism in this model with no other separate assumptions needed. Voltage Clamp equation and action potential equation are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02461439
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  • 114
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    Metabolic model and dosimetric data for copper (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 265-269 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A metabolic model for copper is derived and presented in the context of the International Commission on Radiological Protection (ICRP) dosimetry models for the lungs and gastrointestinal tract.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02461441
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  • 115
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    Analytical conditions for the conservative form of the ecological equations (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 257-264 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The analytical conditions by which a Volterra's general system describingn interacting species can be put in the “conservative” form have been examined. The cases forn=2, 3, 4 have been analyzed in detail and a general condition for any value ofn is deduced. The analytical and biological constraints following by this approach are compared to the conclusions drawn by Leigh on the ground of purely biological considerations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02461440
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  • 116
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    On the stochastic theory of compartments: III. General time-dependent reversible systems (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 237-246 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract General formulation of stochastic single- and multi-compartment reversible systems with time-dependent transitions is made. The correspondence between the stochastic mean and the deterministic value is established in case of time-dependence and it is shown how the consequence of this can be utilized to compute the distribution and the moments of each individual compartment of the system. A two-compartment reversible system previously proposed by Cardenas and Matis (1975a) is analyzed on the basis of the theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02461438
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  • 117
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    Mathematical models of metabolic regulation, symposia biologica hungarica (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 271-272 
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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/BF02461442
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  • 118
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    Optimal control of pests in the presence of predators (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 319-333 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Optimal control theory is applied to the problem of controlling pests by biological and chemical means simultaneously. The net birth rate of the pests is controlled chemically while at the same time predators are allowed to operate. Several numerical examples are included.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02461604
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  • 119
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    Transient behaviour of the single loop solute cycling model of the renal medulla (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 273-300 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Transient solutions are developed for the buildup of a concentration gradient in the single loop solute cycling model of the renal medulla. The “pump” from ascending limb to descending limb is considered in both unsaturated and completely saturated modes of operation. Both analytic solutions and semianalytic solutions obtained from inverting Laplace transforms are considered. The classic representation of concentration buildup by the multiplication process is compared with calculated profiles. The “single effect” is found to vary both in time and space.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02461602
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  • 120
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    The Krylov-Bogoliubov-Mitropolskii method applied to models of population dynamics (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 335-345 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In this study the possibility of applying the asymptotic method of Krylov-Bogoliubov-Mitropolskii to problems of population dynamics is shown. Especially a general Volterra-Gause-Witt type model for prey-predator interaction is investigated. A discussion on the results obtained is given for the general model and for a particular case as well.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02461605
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  • 121
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    A dynamical field theory for dissipative systems: The hierarchical structure of field thermodynamics (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 301-318 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The analytical exploration of hierarchical structures is a central problem in modern biology. In many ways, the most basic tenet of any theory of any biological system is that its behavior lies within the laws of thermodynamics. Thus an investigation into the hierarchical structure of field thermodynamics is in effect an investigation into the underlying structures common to all biological systems. In field thermodynamics, the dynamics is prescribed at the hierarchical level of description concerned with the motions of classes of the various molecular species flowing in the system. Of crucial importance are scale factors which convert potentials into potential energies per mole of the various molecular species. On the other hand, the overall behavior of the system is presented at a higher hierarchical level of description in terms of systemic properties such as charge density or entropy density. Three questions are investigated: (1) how is the transition from the specific to the systemic level of description effected, (2) given the scale factors at the specific level how does one find those at the systemic level, and (3) what relationship can one, find between the scale factors at the two levels of description? Answering these questions necessitates an examination of the nature of the forces at the two levels of the hierarchy of descriptions.
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    URL: http://dx.doi.org/10.1007/BF02461603
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  • 122
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    Reconstitution of the transition matrix from experimental data (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 369-376 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The identification of a linear compartment model, which may describe a chemical or biological process, is a difficult task, since the available data is generally limited. In this paper we propose a method for determining the state transition matrix by minimizing a given quadratic criterion. To solve the resulting matrix equation, an assumption has to be made which constitutes a necessary condition for the identifiability of the model. Moreover when this assumption is satisfied, it is shown that the knowledge of one line or one column of the transition matrix is sufficient to define it completely.
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    URL: http://dx.doi.org/10.1007/BF02461608
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  • 123
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    Analytical solution to isometric mechanogram of hill’s model of striated muscle (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 359-368 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The two-element muscle model considered consists of a contractile element defined by a hyperbolic force-velocity relation connected in series with an “exponential spring”. Differential equations for the isometrically developed force during a tetanic contraction and the corresponding contractile element shortening velocity are derived and their stability is investigated. Analytical solutions to both equations are obtained. Two numerical examples are given, the second chosen to illustrate pressure-induced hypertrophy of a cardiac muscle.
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    URL: http://dx.doi.org/10.1007/BF02461607
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  • 124
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    Lag time in microbe growth as an age-dependent branching process with two phase types (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 347-358 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We study a two-type, age-dependent branching process in which the branching probabilities of one of the types may vary with time. Specifically this modification of the Bellman-Harris process starts with a Type I particle which may either die or change to a Type II particle depending upon a time varying probability. A Type II particle may either die or reproduce with fixed probabilities but may not return to a particle of Type I. In this way the process models the lag phenomenon observed in microbe growth subsequent to transfer to a new culture medium while the organism is adapting to its new environment. We show that if the mean reproduction rate of Type II particles exceeds 1, then the population size grows exponentially. Further the extinction probability for this process is related to that of the Bellman-Harris process. Finally the governing equations are solved for several choices of the growth parameters and the solutions are graphically displayed showing that a wide variety of behavior can be modeled by this process.
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    URL: http://dx.doi.org/10.1007/BF02461606
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  • 125
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    Linear age-dependent population growth with harvesting (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 377-385 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper studies the effect of harvesting a fraction of a population where the population growth is modelled by a linear age-dependent model, the Von Foerster equation. Two harvesting strategies are considered: the first is where a fraction of the population greater than agec is removed, and the second is where a fraction of the population of age greater thanc but less thanc+n is removed. In the case where the death rate and fertility rate are time independent, the effect of harvesting on the stable age distribution is examined.
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    URL: http://dx.doi.org/10.1007/BF02461609
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  • 126
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    A comment on the Richardson-Miekisz field theory of thermodynamics (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 425-426 
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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/BF02461612
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  • 127
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    Multiple speciation, competitive exclusion, evolutionary pattern, and the grand ensemble in volterra eco-dynamics (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 387-410 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The general means are set forth by which, in a highly speciated Volterra eco-system, the bifurcation of a species into two in a competitive-exclusion configuration, provides a description of odd as well as even systems. The the Gibbs ensemble theory earlier developed is extended to grand ensembles in which the number of ecotypical species in genera-like clusters is allowed to be variable. The probability law for the sizes of clusters is deduced, and shown very closely to imply independent Poisson statistics for the cluster sizes, in answer to a fundamental issue of bioevolution. The role of the eco-analogs of Gibbs’ chemical potentials is brought forth, with a suggestion of how to describe cluster-to-cluster eco-genetic flows, and therewith a definite relationship between the degrees of speciation of genetically linked clusters on the one hand, and amplitudes of population fluctuations of member species within the clusters on the other hand.
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    URL: http://dx.doi.org/10.1007/BF02461610
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  • 128
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    The stochastic theory of compartments—a mammillary system (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 427-427 
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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/BF02461613
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  • 129
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    Ionic concentration profiles and charge distribution for the domain of a polarized spherical cell (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 411-424 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Solutions of the Poisson-Boltzmann equation yield potential profiles and equilibrium distributions of ions on either side of a spherical shell membrane across which there exists a separation of ionic charges. For the special case in which the membrane is permeable to only one ion the total charge separation is analyzed in terms of the potential difference given by the Nernst equation. Potential profiles and ionic charge distributions are also given for situations involving a uniform distribution of fixed charges within the membrane.
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    URL: http://dx.doi.org/10.1007/BF02461611
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    An elementary approach to catastrophe theory (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 429-450 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The recently proved classification theorem of René Thom has led to the development of the new science of Catastrophe Theory. This theory has been widely held to be particularly relevant in biological applications. The present paper presents a simplified proof of Thom’s Theorem and assesses its importance in biology.
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    URL: http://dx.doi.org/10.1007/BF02460770
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  • 131
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    Mechanical behaviour of ventricular aneurysms (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 451-464 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A mathematical model describes the mechanical behaviour of ventricular aneurysms assuming a spherical geometry for the left ventricle. Employing pressure-volume data obtained from normal dog hearts 1–2 hours after infarction, conditions are obtained on infarct thickness and angle of damage for ventricular rupture to occur. The results indicate that rupture is more likely to occur in the early period following infarction and that the dominant factor is the per cent thickness of the infarct.
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    URL: http://dx.doi.org/10.1007/BF02460771
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    A stochastic approach to predator-prey models (1978)
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    Bulletin of mathematical biology 40 (1978), S. 483-498 
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    Notes: Abstract A deterministic investigation of a linear differential equation system which describes predator vs prey behavior as a function of equilibrium densities and reproductive rates is given. A more realistic structure of this model in a stochastic framework is presented. The reproductive rates and initial population sizes are considered to be random variables and their probabilistic behavior characterized by various joint probability distributions. The deterministic behaviors of the prey and predator species as functions of time are compared with the mean behaviors in the stochastic model.
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    Effects of external pressures on the pressure-volume relation of the left ventricle (1978)
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    Bulletin of mathematical biology 40 (1978), S. 465-482 
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    Notes: Abstract The effects of ventricular geometry, muscle mass, muscle elasticity and external pressures on the pressure-volume and muscle stiffness-stress relations have been quantitated on the basis of a theoretical model. Data taken from patients before and after interventions with nitroprusside and angiotensin were applied to the model in order to explain the possible causes for the marked shifts in the pressure-volume relations. The results indicate that (a) ventricular geometry does not markedly alter the pressure-volume and stiffness-stress relations unless there is a drastic change from a spherical shape to an ellipsoidal shape orvice versa, (b) increases in muscle mass and muscle elasticity of the order of 30% result in significant alterations in the P-V relations but are not the cause for the parallel shifts unless accompanied by substantial external pressures, (c) the parallel shifts in the pressure-volume relations can be accounted for entirely by the presence of external pressures without changes in muscle mass or muscle elasticity. Thus manipulation of right ventricular pressures or pericardial pressures by drug interventions may be useful in the treatment of left heart disease and the presence of such pressures must be considered in the analysis of ventricular function curves.
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    Qualitative similarities between the behavior of coupled oscillators and circadian rhythms (1978)
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    Bulletin of mathematical biology 40 (1978), S. 675-692 
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    Notes: Abstract This paper summarizes in a nonmathematical way the major properties of coupled oscillators which relate to circadian rhythms. For certain values of the coupling strength it is far easier to maintain synchrony than to achieve it among the various interacting units. This property not only simulates the free run period lability but also the effects of critical pulses.
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    The effects of asymptomatic attacks on the spread of infectious disease: A deterministic model (1978)
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    Bulletin of mathematical biology 40 (1978), S. 707-718 
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    Notes: Abstract A deterministic model for a multi-agent disease epidemic with asymptomatic attacks is proposed and investigated. The limitations inherent in the assumptions of the model are discussed in connection with specific agents of disease. The mathematical treatment of the model is separated into analyses of the equilibrium situation and the transient behavior of the disease outbreak. Explicit formulas are derived for the number of susceptibles in the population as well as for the numbers of each type of infective—those with and without symptoms. These theoretical results are followed by a discussion of the practical considerations which must be taken into account to obtain useful information from the model.
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    On a model and the kinetics of photo-enhanced enzyme reactions (1978)
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    Bulletin of mathematical biology 40 (1978), S. 719-726 
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    Notes: Abstract The recently observed enhancement, by laser irradiation, of the specific activity of the enzyme chymotrypsin (which hydrolyses Benzoyl-L-tyrosine-ethyl ester) at low enzyme concentration is considered. The enhancement of the reaction rate is attributed to a coherently excited state of the enzyme molecule (activated through Raman scattering of the laser light) following a prediction due to Fröhlich. The model is described, the kinetics of the process is framed and the observed enzyme-concentration dependence of the specific activity is reproduced. Predictions of the model are delineated to urge verification of the main contentions through further experimentation.
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    A two-component theory of aerosol deposition in lung airways (1978)
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    Bulletin of mathematical biology 40 (1978), S. 693-706 
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    Notes: Abstract The deposition of aerosol particles in the human lung airways is due to two distinct mechanisms. One is by direct deposition resulting from diffusion, sedimentation and impaction as the aerosol moves in and out of the lung. The other is an indirect mechanism by which particles are transported mechanically from the tidal air to the residential air and eventually captured by the airways due to intrinsic particle motion. This last mechanism is not well understood at present. Using a trumpet airway model constructed from Weibel's data, a two-component theory is developed. In this theory, the particle concentrations in the airways and the alveoli at a given airway depth are considered to be quantitatively different. This difference in concentrations will cause a net mixing between the tidal and residential aerosol as the aerosol is breathed in and out. A distribution parameter is then introduced to account for the distribution of ventilation. The effect of intrinsic particle motion on the aerosol mixing is also included. From this theory, total and regional deposition in the lung at the steady mouth breathing without pause is calculated for several different respiratory cycles. The results agree reasonably well with the experimental data.
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    On the kinetics of enzyme reactions (1978)
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    Bulletin of mathematical biology 40 (1978), S. 727-734 
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    Notes: Abstract The Abel type differential equation governing the kinetics of the enzyme reactions is derived. Approximate solutions of this equation corresponding to the transient phase of the reaction, before a steady state is reached, are considered. It is shown that in several cases it is possible to obtain explicit, approximate solutions to the transient phase.
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    Complementary variational principles for the steady-state finite cable model of nerve membranes (1978)
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    Bulletin of mathematical biology 40 (1978), S. 735-742 
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    Notes: Abstract Maximum and minimum principles for the steady-state finite cable model of nerve membranes are derived from the canonical theory of complementary variational principles. An accurate variational solution is obtained in an illustrative calculation.
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    Dynamical systems under constant organization I. Topological analysis of a family of non-linear differential equations —A model for catalytic hypercycles (1978)
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    Bulletin of mathematical biology 40 (1978), S. 743-769 
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    Notes: Abstract The paper presents a qualitative analysis of the following systems ofn differential equations: $$\dot x_i = x_i x_j - x_i \sum\nolimits_r^n { = 1} x_r x_s {\mathbf{ }}(j = i - 1 + n\delta _{i1} {\mathbf{ }}and{\mathbf{ }}s = r - 1 + n\delta _{r1} )$$ , which show cyclic symmetry. These dynamical systems are of particular interest in the theory of selforganization and biological evolution as well as for application to other fields.
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    Turing's theory in morphogenesis (1978)
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    Bulletin of mathematical biology 40 (1978), S. 771-789 
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    Notes: Abstract Bifurcation theoretical and numerical analyses of one of Turing's models are performed. It is shown that at the first instability point of the homogeneous state the bifurcating branches aresubcritical, and thus emerge as unstable solutions. This, together with the presence of concentration-independent sink terms is responsible for the solutions becoming negative ast→∞. It is pointed out that this deficiency is an accident related to the choice of the model, and that the general idea of symmetry-breaking is perfectly compatible with the generation of regular morphogenetic patterns.
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    On phase locking of pulse encoders (1978)
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    Bulletin of mathematical biology 40 (1978), S. 807-821 
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    Notes: Abstract Several models have been proposed to underststand the patterns of nerve impulses produced by periodic stimuli. This paper shows that for a very large class of such models there exists a pattern of phases that repeats periodically after a finite number of pulses; the actual pulses produced by the model depend on its initial condition, but in all cases they either follow such a pattern or approach it asymptotically.
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    Global flow equations for membrane transport from local equations of motion: I. The general case for (n−1) nonelectrolyte solutes plus water (1978)
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    Bulletin of mathematical biology 40 (1978), S. 791-805 
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    Notes: Abstract The global force-flow equations of nonequilibrium thermodynamics are obtained by solving the local equations of motion for a system ofn-components (n−1 solutes plus water) passing through a membrane. When viscous forces and position dependent membranepermeating species frictional interactions are considered, it becomes more difficult to obtain the result because even the stationary state problem becomes one of solving a second order linear ordinary differential equation for the barycentric velocity in a space divided inton+1 regions. Using the continuity of the boundary velocities and their gradients as well as the usual boundary conditions for the hydrodynamic problem, a set of 2n+1 linear equations in the intergration constants can be obtained and a closed form solution is possible. The resultant global description of the system does not obey Onsager reciprocity. What is more, the interpretation of global phenomenological coefficients in terms of local interactions in any simple way is next to impossible. This makes the hope of a molecular level interpretation of phenomenological membrane transport coefficients very slim. The relevance of this finding to the validity of reductionist approaches to biological transport is discussed.
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    A mathematical model of transmural transport of oxygen to the retina (1978)
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    Bulletin of mathematical biology 40 (1978), S. 823-837 
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    Notes: Abstract A mathematical model of transmural transport of oxygen to a metabolizing retina is presented based on the equations of fluid dynamics. The equations of oxygen transfer are derived and then solved subject to the condition that the capillaries begin to transport oxygen at an initial time. The resulting transient analysis gives us insight into how diffusive and filtrative processes lead to the oxygen distributions both inside and outside capillaries. On the other hand, the steady state solution allows us to predict the cutoff intraocular pressure above which no oxygen is transferred to retinal tissue. It also gives quantitative relationships which allow us to postulate how intracapillary hypertension counterbalances elevated intraocular pressures and how low pressure glaucoma may arise from ineffective diffusive and filtrative processes of oxygen transport.
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    Interactions among biological systems: An analysis of asymptotic stability (1978)
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    Bulletin of mathematical biology 40 (1978), S. 839-851 
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    Notes: Abstract An analysis of the interactions among asymptotically stable dynamical systems is formulated by making use of the dynamical system theory. Some results coming from previous mathematical analyses have been slightly modified to take into account some typical biological constraints as the boundedness properties of the solutions. In particular it has been shown that when the “coupling” among the subsystems is “loose” enough (in a sense that has to be made mathematically precise) the asymptotic behaviour of a complex system is the same of that of its individual components. The mathematical theory has been used to analyze two systems of biological significance: the coupling among chemical reactions and the stability properties of a 4-dimensional system describing the kinetics of a chemical transmitter.
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    Steady state bifurcation analysis of reaction-diffusion equations-A critique (1978)
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    Bulletin of mathematical biology 40 (1978), S. 865-872 
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    Notes: Abstract This note supplies missing details and corrects some mathematical errors in the recent significant paper by Auchmuty and Nicolis (1975).
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    On a two urn model of Polya-type (1981)
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    Bulletin of mathematical biology 43 (1981), S. 33-45 
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    Notes: Abstract A two urn Polya-type scheme is considered in whichr black balls (corresponding to the stable form of an element) are added to urn one at every stage and the same number of balls are removed at random at every stage from the same urn. In between these two operations, which form a stage or iteration, a fixed number of balls is exchanged at random between urns one and two. Urn one has a given initial number of white balls (corresponding to a radioactive form of the same element). The problem of interest is to study the stochastic aspect of the number of white balls remaining in urn one (and/or urn two) aftern iterations.
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    Nonlinear biological oscillators: Action-angle variable approach with an application to Cowan's model of neuroelectric activity (1981)
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    Bulletin of mathematical biology 43 (1981), S. 21-32 
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    Notes: Abstract We obtain within the action-angle variable approach new expressions, involving the Dirac delta function, for time periods and time averages of dynamical variables which are useful for nonlinear biological oscillator problems. We combine these with Laplace transformation techniques for evaluating the required perturbation expansions. The radii of convergence of these series are determined through a complex variable approach. The method is powerful enough to yield explicit results for such systems as the two species Volterra model, Goodwin's model of protein synthesis etc. and as an illustration, is applied here to Cowan's model of neuroelectric activity. We also point out the usefulness of the action integral in the case where parameters occurring in dynamics have slow time variations.
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    A generalized Fokker-Planck equation in the case of the Volterra model (1981)
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    Bulletin of mathematical biology 43 (1981), S. 69-79 
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    Notes: Abstract Zwanzig and Mori's projection-operator method is used in order to derive a generalized nonlinear Fokker-Planck equation for one “relevant” species in the many species conservative Volterra model. The deterministic, autonomous, Markovian equations of motion, when averaged over a suitable ensemble of initial conditions in general give rise to a non-autonomous, non-Markovian stochastic process for the evolution of this relevant species. Moreover, this relevant species may show irreversible damping, although self-interaction terms are absent in the many species model.
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    A hydrodynamical study of the flow in renal tubules (1981)
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    Bulletin of mathematical biology 43 (1981), S. 151-163 
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    Notes: Abstract The hydrodynamical problem of flow in proximal renal tubule is investigated by considering axisymmetric flow of a viscous, incompressible fluid through a long narrow tube of varying cross-section with reabsorption at the wall. Two cases for reabsorption have been studied (i) when the bulk flow,Q, decays exponentially with the axial distancex, and (ii) whenQ is an arbitrary function ofx such thatQ-Q 0 can be expressed as a Fourier integral (whereQ 0 is the flux atx=0). The analytic expressions for flow variables have been obtained by applying perturbation method in terms of wall parameter ε. The effects of ε on pressure drop across the tube, radial velocity and wall shear have been studied in the case of exponentially decaying bulk flow and it has been found that the results are in agreement with the existing ones for the renal tubules.
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    Some comments on the conductance of slow inward current in cardiac muscle (1981)
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    Bulletin of mathematical biology 43 (1981), S. 183-199 
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    Notes: Abstract Voltage clamp experiments, which determine the kinetic parameters of calcium conductance of cardiac muscle, (d ∞,f ∞, τ d and τ f ) are analyzed with a generally accepted expression for slow inward currentI s=g sdf (E-E R). Activation (d) and inactivation (f) reach the final valuesd ∞ andf ∞ with time constants τ d and τ f respectively. The analysis indicates that the measuredf ∞ agrees with the theoreticalf ∞, but the measuredd ∞ differs from the theoreticald ∞ by a factor which depends on τ d . The peak tension can be made to correlate closely with the theoreticald ∞ after a correction factor is applied to the raw measurements of activation. It can be shown that experiments designed to measure τ f can also be used to determine τ d with greater accuracy.
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    On the relevance of stochastic compartmental models to pharmacokinetic systems (1981)
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    Bulletin of mathematical biology 43 (1981), S. 245-247 
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    Identification of silent infections in SIR epidemics (1981)
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    Bulletin of mathematical biology 43 (1981), S. 249-257 
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    Notes: Abstract A deterministic model for an SIR epidemic with silent infections is investigated. It is shown for the model studied that the extent to which silent infections are present may be determined from data concerning only those individuals with symptomatic infection.
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    A reinforcement-depletion urn problem—I. Basic theory (1981)
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    Bulletin of mathematical biology 43 (1981), S. 327-340 
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    Notes: Abstract An urn contains balls of different colors. Specified numbers of each color are added and form a reinforcement. The total reinforcement is randomly removed, forming a depletion. The process, not necessarily with the same reinforcements, is performed a number of times. The factorial moment generating function of the urn configurations at any stage is given in terms of multivariate difference operators. Cases when the reinforcement vector is defined as a stochastic variable are considered. The problem is a generalization of an urn model associated with radioactive atoms and stable atoms proposed by S. R. Bernard. The solutions given here have a definite application to the problem of modelling tracers in compartmental systems.
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    Stochastic compartmental model with branching particles (1981)
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    Bulletin of mathematical biology 43 (1981), S. 347-360 
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    Notes: Abstract A multi-compartmental model with particles producing offspring according to the Markov branching process has been studied. Explicit results are given for the two-compartmental system and for irreversible general multicompartmental systems. The known models in stochastic compartmental analysis are shown to be particular cases of this model and applications are cited.
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    On Wei's fundamental equation for nerve excitation (1981)
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    Bulletin of mathematical biology 43 (1981), S. 371-372 
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    Phase-plane analyses of the Michaelis-Menten reaction equations (1981)
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    Bulletin of mathematical biology 43 (1981), S. 361-370 
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    Notes: Abstract The irreversible Michaelis-Menten scheme may be reduced to a pair of autonomous first-order differential equations. The phase-plane behaviour of these is investigated.
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    Reply to Chyuan-Yih Lee (1981)
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    Bulletin of mathematical biology 43 (1981), S. 372-373 
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    Further considerations in a theoretical description of gas transport in lung airways (1981)
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    Bulletin of mathematical biology 43 (1981), S. 517-548 
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    Notes: Abstract A discrete one-dimensional model of convection-diffusion in branching alveolar ducts is described and it is shown that, for a suitable choice of effective axial dispersion, the solution closely approximates that for an axially symmetric representation, at least for Peclet numbers Pe〈1. Following earlier work a composite model of a uniform lung is formed by matching such a respiratory pathway (now having the more convenient one-dimensional form) onto a trumpet representation of the conducting airways. Enhanced mixing due to heart action, and isotropic volume changes of trumpet (in addition to the pathway) during breathing are additional factors included. Calculations are made of O2 concentrations during steady-state breathing and of the concentration of inert gas during single breath wash-out of a gas mixture containing it. Predicted alveolar levels in each case agree extremely well with published data, although no alveolar slope is obtained for the inert gas.
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    Effects of convective and dispersive interactions on the stability of two species (1981)
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    Bulletin of mathematical biology 43 (1981), S. 593-610 
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    Notes: Abstract The evolution and local stability of a system of two interacting species in a finite two-dimensional habitat is investigated by taking into account the effects of self- and cross-dispersion and convection of the species. In absence of cross-dispersion, an equilibrium state which is stable without dispersion is always stable with dispersion provided that the dispersion coefficients of the two species are equal. However, when the dispersion coefficients of the two species are different, the possibility of self-dispersive instability arises. It is also pointed out that the cross-dispersion of species may lead to stability or instability depending upon the nature and the magnitude of the cross-dispersive interactions in comparison to the self-dispersive interactions. The self-convective movement of species increases the stability of the equilibrium state and can stabilize an otherwise unstable equilibrium state. The effect of cross-convection (in absence of self-dispersion and self-convection) is to stabilize the equilibrium state in a prey-predator model with positive cross-dispersion coefficients for the prey species. Finally, it is shown that if the system is stable under homogeneous boundary conditions it remains so under non-homogeneous boundary conditions.
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    Role of the hypergeometric distribution in the transmission of excitation through sympathetic ganglia (1981)
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    Bulletin of mathematical biology 43 (1981), S. 611-618 
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    Notes: Abstract Various observations suggest that sympathetic ganglia act as local integrative centers redistributing preganglionic excitation (i.e. the information issued by the central nervous system) to the postganglionic fibers (and effector organs). In order to support this concept a simple mathematical model of the elementary integration process, treating the case of a single preganglionic compound action potential, has been developed. This quantitative description, based on a few elementary assumptions, shows a possible way of processing preganglionic excitation in the ganglion. It is shown that on a particular ganglion cell the probability distribution of the number of activated synapses obeys hypergeometric distribution and hence, the postganglionic compound action potential is built up of several compound action potentials occurring at different times. The former correspond to different groups of firing cells. The model discloses modes of structural and temporal pattern generation performed by the sympathetic ganglion.
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    The uniqueness of protein sequences: A Monte Carlo analysis (1981)
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    Bulletin of mathematical biology 43 (1981), S. 619-639 
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    Notes: Abstract Amino acid sequences have already been examined in some detail in order to relate them to structural aspects, homology and gene duplication. This report introduces the concept of internal uniqueness of tripeptides within protein sequences and uses the Monte Carlo method to study this property. Some idea of internal uniqueness may be obtained from such an analysis using only a single sequence if the probability of the random occurrence is about 0.001 or less. This method of analysis is similar to that used in quantitative evaluations of homology. When the probability of the random occurrence is larger than 0.001 a homologous group of sequences is required and the random probabilities may be compared with the real occurrences within the group. From such an examination insulin and cytochrome c are identified as protein sequences with high internal uniqueness. A comparison of data from internal uniqueness and gene duplication analyses shows that these two properties need not be related. Results of the analysis point to internal uniqueness as an additional parameter for inclusion in speculations on why twenty amino acids are coded in protein structure.
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    Tolerance nets (1981)
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    Bulletin of mathematical biology 43 (1981), S. 641-650 
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    Notes: Abstract The concept of a tolerance net formalises simultaneously spatial closeness and nearness of neuronal activity. A method of constructing tolerance group nets is presented, leading to a means of construction of all very homogenous tolerance nets as group quotients. The dihedral group of order eight is taken as an illustrative example.
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    Relation between diameter and flow in branches of the bronchial tree (1981)
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    Bulletin of mathematical biology 43 (1981), S. 681-691 
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    Notes: Abstract Two methods are described for calculating the value of the exponentx in the equation flow =k×diameter x , as pertaining to a branch of the bronchial tree. In the lungs from three humans, two dogs, one hamster, and one rat mean values ofx between 2.419 and 2.903 were found. They lie within the range of 2.333 to 3.0 predicted by the analysis of Uylings (Bull. Math. Biol. 39, 501–519, 1977).
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    Stochastic analysis of a nonlinear model for selection of biological macromolecules (1981)
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    Bulletin of mathematical biology 43 (1981), S. 665-680 
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    Notes: Abstract A stochastic analysis of a nonlinear selection model is presented. The model, based on Eigen and Schuster's theory of selection and evolution of biological macromolecules, considers the effects of fluctuations on the individual concentrations of macromolecules as well as the total population numbers in constrained systems. Our analysis shows that one of the models most often treated deterministically (referred to as constant organization in the literature) becomes unstable when fluctuations in the total population number are considered. An alternative model which apparently has built in self-regulating properties is analyzed and proves to be stable except for some special cases of degeneracy.
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    Compartmental models with multiple sources of stochastic variability: The one-compartment models with clustering (1981)
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    Bulletin of mathematical biology 43 (1981), S. 651-664 
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    Notes: Abstract Previous compartmental models have introduced variability either at the particle or at the replicate level. This paper integrates both types of variability through the concept of clustering. The paper develops two different, general clustered models, each with time-dependent hazard rates for the clusters and for the particles within the clusters, and each with random initial number and sizes of clusters. The coefficient of variation of the total number of particles,CV[X(t)], for either model is shown to be bounded below, under very broad conditions, by the coefficient of variation of the initial number of clusters,CV[c(0)]. This high relative variability of the clustered models makes them potentially very useful in kinetic modeling. In many applications, binding and clustering are common phenomena, and two applications of the models to such phenomena are breifly outlined.
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    Some trees are not interval graphs (1981)
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    Bulletin of mathematical biology 43 (1981), S. 717-717 
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    Annual Meeting of the Cell Kinetics Society (1981)
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    Bulletin of mathematical biology 43 (1981), S. 719-719 
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    Solution of linear one-dimensional diffusion equations (1976)
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    Bulletin of mathematical biology 38 (1976), S. 1-13 
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    Notes: Abstract The paper introduces a basic mathematical form, which is characteristic of a number of linear one-dimensional diffusion equations with coefficients being represented as simple polynomials in the spatial coordinate. A number of particular diffusion equations are introduced and their corresponding exact mathematical solutions are obtained.
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    On the theory of selection of coupled macromolecular systems (1976)
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    Bulletin of mathematical biology 38 (1976), S. 15-28 
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    Notes: Abstract In this paper we examine a set of nonlinear rate equations (devised by M. Eigen (1971)) which describe the process of selection in a collection of self-reproducing, macromolecular information carriers. We construct exact solutions to the equations for the case of constant rate parameters and constant error distributions. The solutions allow the direct assessment of the effect of mutations on the “selective value” parameters discussed by Eigen as well as the distribution of the molecular species selected in steady state. In addition we show that the selection process may be characterized by an extremal principle.
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    Photon enzyme activation (1976)
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    Bulletin of mathematical biology 38 (1976), S. 29-38 
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    Notes: Abstract The electromagnetic molecular electronic resonance (EMER) frequencies of the molecular chains of α-chymotrypsin are calculated. The chain length relations and coupling positions suggest a possible energy transfer, at the EMER frequencies, from one chain to the other. Photon enzyme activation data indicate that the energy corresponding to the EMER frequencies of its molecular chains is used by α-chymotrypsin for its enzyme function.
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    Nerve excitations by the coupling of the dipoles and the membrane matrix (1976)
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    Bulletin of mathematical biology 38 (1976), S. 59-70 
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    Notes: Abstract Based on Wei's dipole flip-flop model and with the assumption that the dipole is coupled to the membrane matrix, the cathode-make-excitation, the anode-break-excitation and the cathode-gap-excitation can be explained in a systematic way. The istrength-duration relations for these three processes are derived.
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    A possible explanation of the Comorosan effects (1976)
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    Bulletin of mathematical biology 38 (1976), S. 93-94 
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    Hysteresis in systems with a single steady state (1976)
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    Bulletin of mathematical biology 38 (1976), S. 87-92 
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    Notes: Abstract It is usually stated that only systems with multiple steady states can exhibit hysteresis. For protein conformations, this would violate the idea that, in fixed environments, primary structure uniquely determines tertiary structure. It is shown that hysteresis-like phenomena can be exhibited by systems possessing only a single steady-state configuration. This property is placed in a more general theoretical setting of recognition and classification systems, and some implications for processes such as memory, learning and pattern generation (morphogenesis) are discussed.
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    The effects of harvesting on competitive systems (1976)
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    Bulletin of mathematical biology 38 (1976), S. 97-109 
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    Notes: Abstract The natural historical literature contains a considerable body of work which indicates that harvesting (or predation) can alleviate competitive instabilities. In order to arrive at an understanding of this, the appropriate bifurcation structure for a rather general family of two-dimensional competitive systems is here investigated. The results of this analysis suggest that, in more complicated ecosystems with many competing species, (1) there is a good chance that harvesting at moderate rates will increase species diversity if one species is dominant in the unharvested system, while an increase in diversity is not likely to result from harvesting from a system with no dominant species, (2) whenever harvesting does increase species diversity, maximal diversity will occur at moderate harvesting rates, with less diversity at both very high and very low harvesting rates.
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    Derivation of the Weber-Fechner law and the Loewenstein equation as the steady-state response of an Elovich solid state biological system (1976)
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    Bulletin of mathematical biology 38 (1976), S. 111-118 
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    Notes: Abstract The Weber-Fechner law (response of an organism is a linear function of log of stimulus) has been widely used for description of physiological and psychological data for many years. It is shown here that the Weber-Fechner law is derivable in a simple way from the Elovich equation (−dx/dt=m exp(nx), which is observed experimentally in numerous physiological and biochemical systems, and which has a simple derivation from solid state charge transport across interfaces in these systems. It therefore seems reasonable to interpret data conforming to the Webner-Fechner law to imply that the observed phenomenon is rate-limited by interfacial charge transport in the cell. By a similar analysis, the Loewenstein equation, which may be considered an exact form of the Weber-Fechner law applicable to data over a wider range of values of the variables, is derived from a more exact form of the Elovich equation.
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    Properties of the Leslie population matrix (1976)
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    Bulletin of mathematical biology 38 (1976), S. 199-203 
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    Notes: Abstract A simple similarity transformation of the Leslie matrix renders certain properties obvious. In particular the characteristic polynomial, characteristic vectors and principal vectors can be explicitly written out. Bounds for the dominant root are given.
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    Mathematical model of dynamic adhesion of lymphocytes on a glass-bead column (1976)
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    Bulletin of mathematical biology 38 (1976), S. 659-669 
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    Notes: Abstract A mathematical description of the process of adherence to glass of the rat thymus lymphocytes is presented. It is based on the result of previous work in which the process was studied in the course of the flow of the51Cr labelled cell suspension through the glass-bead column. The concentration of cell suspension, flow velocity and medium temperature were constant; the experiments were performed with different lengths of the bead bed. The amount of cells captured on the glass beads' surface was calculated as a function of time. Two approaches to the mathematical description of the process are presented. The first one is based on the linear equation of kinetics of cell retention on the elementary thin layer and on the transport equation of the flow of the suspension through the column. In the second one, the differential equation of the adhesion, derived from the experimental data, is discussed.
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    Extremum principles and the elastic and fluid mechanical behaviour of the human eye (1976)
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    Bulletin of mathematical biology 38 (1976), S. 83-86 
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    Notes: Abstract Complementary variational principles are derived for the pressure boundary value problems in Weinbaum's (1965) model of the human eye. These principles can be used to obtain accurate solutions for the pressure.
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    Analytical characterization of microvascular resistance distribution (1976)
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    Bulletin of mathematical biology 38 (1976), S. 71-82 
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    Notes: Abstract The bat wing is used as an experimental preparation and as a self-contained vascular bed. The number, dimensions, and distribution of the vessels of the real vascular bed are included into an analyzable, representative geometric configuration. Based on theoretical analysis and experimental data, equations are developed and utilized to characterize the pressure-flow relationships for each branching order of the vascular field. The geometric configuration and associated describing equations are used to determine the resistance distribution of the microvascular bed. The predicted values are compared with experimentally determined quantities in normal and hypertensive animals.
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    Distribution of end-points of a branching network with decaying branch length (1976)
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    Bulletin of mathematical biology 38 (1976), S. 219-237 
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    Notes: Abstract The geometry of the human bronchial tree has been described as a network formed by successive dichotomous branching with constant branching angles and geometrically decaying branch lengths. Models having these properties and with randomly distributed branching planes are constructed. The distribution of the end points of the model networks are described by computing the variance of the distributions in the direction of the axis of the network and in the transverse directions. It is found that, for a given decay ratio, there is a branching angle for which the volume filled by the end points is a maximum. The advantages of the network with the decay ratio and branching angle of the human bronchial tree are discussed.
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    Oxygen tension in thein vivo cornea (1976)
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    Bulletin of mathematical biology 38 (1976), S. 269-275 
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    Notes: Abstract The steady state distribution of oxygen tension in thein vivo cornea is estimated in the present study by using a nonlinear oxygen consumption rate equation of the Michaelis-Menten type. Such a rate expression is more accurate than the previous simplified versions in predicting the oxygen consumption rate. It is found that for an open eye with or without contact lens, the oxygen tensions predicted previously are in good agreement with these predicted in the present work. However, for a closed eye with or without contact lens, the previous predictions underestimate the oxygen tension.
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    Mathematical principles in afferent visual neurons: Differentiation, integration and transient proportionality related to receptive fields and shift-effect (1976)
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    Bulletin of mathematical biology 38 (1976), S. 253-267 
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    Notes: Abstract The dual reciprocal and antagonistic organization of B- and D-neurons of the afferent visual system is obtained using differentiation and integration as mathematical equivalents of visual information processing by an impulse frequency code. The spatial and temporal derivatives lead to the transient responses. A constant and a time-dependent term proportional to the luminance distribution describe the sustained response components and the shift-effect of retinal on- and off-center ganglion cells. Receptive field properties of lateral geniculate cells and their antagonistic shift-effect are obtained by passing the retinal output, i.e. the difference between B- and D-neurons' activity, once again through the same operations. However, the factor of proportionality is applied to the retina alone. The surprisingly small difference between retinal and geniculate receptive field properties on the one hand and the dramatic change from a synergistic to an antagonistic shift-effect on the other hand are thereby explained. The theory offers an understanding of a a possible functional significance of the shift-effect as a mechanism of transientrestoration of visual information, which prevents the system from total fading by means of shifts of the retinal image, normally produced by eye movements.
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    Connectivity in random networks (1976)
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    Bulletin of mathematical biology 38 (1976), S. 295-304 
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    Notes: Abstract Applications of randomly connected networks are reviewed briefly. The connectivity of a random network has been defined in a variety of ways including output connectivity, total or network connectivity, connectance, expected path length and radius. One or more of these definitions may prove more convenient in a given experimental system. Interrelations among these definitions are derived and displayed and asymptotic results provided in the form of two theorems. Computer simulations were used to explore the range of application of these asymptotic approximations. The results were used to determine the output connectivity of the neurons of the brain.
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    An addendum to “D'Arcy Thompson's theorems and the shape of the molluscan shell” (1976)
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    Bulletin of mathematical biology 38 (1976), S. 321-322 
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    Bifurcation analysis of reaction-diffusion equations—III. Chemical oscillations (1976)
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    Bulletin of mathematical biology 38 (1976), S. 325-350 
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    Other developments regarding the concept of energy in biological systems (1976)
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    Bulletin of mathematical biology 38 (1976), S. 547-563 
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    Notes: Abstract In order to recognize the realizability of inputs with different physical natures through a component, Yoneda's Lemma is applied. The major utility of this Lemma is when the components produce only energy. From this, it is assumed that a new material input must exist which was not recognized in the original developments in biological systems representation. Moreover, simple transfers of energy, between objects, components, and among both objects and components are developed under the generic name; energetical evolution. Thus, energetical evolution appears as anew element in the abstract representation of biological systems. These new concepts are incorporated into a new abstract diagram and a newM β category.
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    The chief abstractions of biology (1976)
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    Bulletin of mathematical biology 38 (1976), S. 569-571 
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    On peristaltic flow and its efficiency (1976)
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    Bulletin of mathematical biology 38 (1976), S. 573-596 
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    Notes: Abstract The question of efficiency in performing biological functions is raised in the context of peristaltic fluid transport. To deal with this problem a complete solution for peristaltic flow in a pipe and in a channel, assuming a given time mean flow, is developed, by a double expansion in terms of the Reynolds number and the square of the wave number. This solution is valid for arbitrary waveshapes. We resolve a long-standing problem and show that quite generally the pressure rise per wave length is constant on a cross section. We also show that for a sinusoidal wave (and others) the interaction of Reynolds number and wave number is a third-order effect for this pressure rise. Plow-type waves, nipple-type waves and the sinusoidal wave are compared for maximum efficiency and for minimum energy usage. It is found that large plows are best from mechanical efficiency considerations, but large nipples use the least energy. The biological implications of these results are discussed.
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    Two-regime model of the perfusion measuring heated thermocouple probe (1976)
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    Bulletin of mathematical biology 38 (1976), S. 633-642 
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    Notes: Abstract A theoretical analysis of the transient temperature responses of a heated thermocouple and its surroundingin vivo tissue is described. The model includes the effects of local blood perfusion, metabolic heat generation and blood pooling. The solutions presented are generalized for pulsed heating in the probe region. Inspection of these solutions reveals that for accurate experimental results precise knowledge of the tissue's thermal conductivity is necessary but that blood pooling around the probe may sometimes be regarded as an insignificant parameter.
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    Periodic and traveling wave solutions to Volterra-Lotka equations with diffusion (1976)
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    Bulletin of mathematical biology 38 (1976), S. 643-658 
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    Notes: Abstract Analytic and numerical solutions to two coupled nonlinear diffusion equations are studied. They are the modified equations of Volterra and Lotka for the spatially stratified predatorprey population model. In a bounded domain with the reflecting boundary, equilibrium, stability, and transition to time-periodic solutions are analyzed. For a wide class of initial states, the solutions to the initial boundary-value problem evolve into their corresponding stable, space-homogeneous, periodic oscillations. In an unbounded domain, a family of traveling wave solutions is found for certain exponential, initial distributions in the limit as the diffusion coefficientv 1 of the prey tends to zero. In the presence of both diffusions, the results of a numerical simulation to an initial-value problem showed the rapid formation of the Pursuit-Evasion Waves whose speed of propagation and amplitudes increase with the diffusion coefficientv 1.
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    Numerical integration of a stochastic model for the Volterra-Lotka reaction (1976)
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    Bulletin of mathematical biology 38 (1976), S. 535-546 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Time-integration of the master equation governing the birth-and-death model of the Volterra-Lotka reaction is carried out for three different initial conditions, with the results: (i) Fluctuations destroy the deterministic steady state in a manner quantitatively predicted from a cumulant expansion; (ii) The sustained oscillatory behavior predicted by the deterministic model degenerated after 1/4 cycle in the stochastic model; (iii) It is possible to select initial distributions such that the asymptotic distribution is a spike at the origin of the plane of reactants.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02459552
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    Lecture notes in biomathematics 4. Physics and mathematics of the nervous system (1976)
    Springer
    Bulletin of mathematical biology 38 (1976), S. 565-568 
    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/BF02459554
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    Large deformation analysis of some basic problems in biophysics (1976)
    Springer
    Bulletin of mathematical biology 38 (1976), S. 701-712 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This study is concerned with the application of a possible form of a strain-energy function suitable for soft biological tissues. Two problems considered here are related to simultaneous extension and inflation of cylindrical arteries, and inflation of the left ventricle under a given internal pressure. The values of the material constants are obtained via comparison of theoretical results with experimental findings. Some details concerning the wall stresses and the elastic stiffness are also given in the paper. For each case, it is seen that experiment and theory are in good agreement.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02458644
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    On some Markov models of certain interacting populations (1976)
    Springer
    Bulletin of mathematical biology 38 (1976), S. 723-738 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We give a stochastic foundation to the Volterra prey-predator population in the following case. We take Volterra's predator equations and let a free host birth and death process support the evolution of the predator population. The purpose of this article is to present a rigorous population sample path construction of this interacted predator process and study the properties of this interacted process. The constructions yields a strong Markov process. The existence of steady-state distribution for the interacted predator process means the existence of equilibrium population level. We find a necessary and sufficient condition for the existence of a steady-state distribution. Next we see that if the host process possesses a steady-state distribution, so does the interacted predator process and this distribution satisfies a difference equation. For special choices of the auto death and interaction parametersa andb of the predator, whenever the host process visits the particular statea *=a/b the predator takes rest (saturates) from its evolution. We find the probability of asymptotic saturating of the predator.
    Type of Medium: Electronic Resource
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    A stochastic compartmental model with continuous infusion (1976)
    Springer
    Bulletin of mathematical biology 38 (1976), S. 695-700 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper deals with stochasticm-compartmental systems with continuous time-dependent infusions into all compartments and reversible time-independent flows between any two compartments. A methodology for the first two moments of the distribution of the number of units in the different compartments at any point in time is outlined without resorting to the usual techniques of generating functions and inverse Laplace transforms. A possible application to a systems analysis of the kidney transplant system is discussed.
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    Kinetics of cell deposition under the action of an external field (1976)
    Springer
    Bulletin of mathematical biology 38 (1976), S. 713-721 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The kinetics of cellular depositon from a stagnant solution to a surface are studied, taking into account the combined effect of an interaction field between the cells and the surface and of an external field. Since the forces involved in the adhesion of cells to a surface are short ranged, the cells are conveyed to the vicinity of the surface only by the external field. The equations developed are general, in the sense that they are independent of any particular form of the potential energy function, provided that it presents an appreciable potential barrier between the cells and the deposition surface. The characteristic shape of the curve representing the decay of the fraction of cells in solution with time is shown to be affected by the value ofPt *, consisting of the probability per unit time,P, for the escape over the potential barrier, and the sedimentation time,t *. A simple inspection procedure of this curve can disclose the relative significance of the external field and of the potential barrier in the overall kinetics of deposition. In addition, such an inspection can reveal the existence of alterations in the cellular adhesiveness with increasing coverage of the deposition surface. By matching the equation obtained to experimental results, the cellular adhesiveness, in term ofP, and the sedimentation rate (in the case of very slow sedimentation) can be evaluated.
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    Bounds for generalized Dreitlein-Smoes models (1976)
    Springer
    Bulletin of mathematical biology 38 (1976), S. 739-743 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The analysis of a previous paper obtaining bounds on the total population number of species (chemical or biological) described by the recently proposed Dreitlein-Smoes model of oscillatory kinetic systems, including diffusion, is extended to generalized models of the Dreitlein-Smoes type, describing a system ofS components withS〉2. The results for such generalized models are analogous to those of the previous case. It is found that the effects of diffusion serve to restrict the region in the concentration space available to limitcycle type oscillations.
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    Introduction (1976)
    Springer
    The Geneva risk and insurance review 2 (1976), S. 3-4 
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    ISSN: 1554-9658
    Source: Springer Online Journal Archives 1860-2000
    Topics: Economics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02420932
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    A macro-economic approach to risk management (1976)
    Springer
    The Geneva risk and insurance review 2 (1976), S. 5-9 
    Details
    ISSN: 1554-9658
    Source: Springer Online Journal Archives 1860-2000
    Topics: Economics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02420933
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