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  • Springer  (436,192)
  • Wiley-Blackwell  (79,141)
  • American Association for the Advancement of Science  (28,805)
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  • 101
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    Bulletin of mathematical biology 45 (1983), S. 739-748 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The objective of this preliminary study was to develop a new quantitative method of setting the initial insulin infusion patterns in treatment of diabetic patients. The method is based upon the mathematical estimation of the insulin profile required to maintain the glucose level within the normal range after glucose loading in diabetic patients. Using our previously developed equivalent circuit model of glucose kinetics and the reported data of an intravenous glucose tolerance test (IVGTT) in two groups of normal and diabetic patients, two important physiological parameters of the model (the peripheral tissue's insulin resistivity and the hepatic sensitivity to glucose level) were computed for two clinical groups. Then the insulin profile was obtained by computing the plasma insulin concentrations required to keep the total glucose utilization rate of the tissue and the liver in the diabetic group equal to that of the normal group. The simulation result indicated that the computed insulin profile produced a plasma glucose profile which was more closely matched to the normal group's glucose profile than with the case of emulating the normal group's insulin profile in the diabetic group.
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  • 102
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    Bulletin of mathematical biology 45 (1983), S. 759-780 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper shows that the Na conductance changes can be explained quantitatively, based on the following assumptions: (1) there exist in nerve membranes the electron transfer (ET) complexes and traps, (2) there is energy migration among them. The gating mechanism is explained in physical terms. Its mathematical expression differs from the Hodgkin-Huxley equations, but resembles the Hoyt formulation. In the present model, the physical parameters for the squid axon can be estimated from currently available experimental data. The density of the ET complexes is on the order of 105/μm2, and the density of the traps is 103/μm2. The magnitude of the energy transfer rate between ET complexes is about 106/sec at large depolarization and decreases with decreasing depolarizations, as does the Na inactivation rate. The energy gap between the two stable states of the transfer electron in the ET complex is estimated to be around 0.1 eV, which is approximately the same as that for the photosynthetic systems.
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  • 103
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    Bulletin of mathematical biology 45 (1983), S. 781-792 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The role of symmetry in simplifying the theory of complex neural systems is argued. When the structural symmetries of a network are expressed as an ismorphism group, implications emerge for the dynamics. Various qualitative possibilities concerning stability of uniform motion in homogeneous nets are discussed and an approach to neural hierarchies is outlined.
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  • 104
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    Bulletin of mathematical biology 45 (1983), S. 793-805 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract By constructing appropriate Liapunov functionals, asymptotic behaviour of the solutions of various delay differential systems describing prey-predator, competition and symbiosis models has been studied. It has been shown that equilibrium states of these models are globally stable, provided certain conditions in terms of instantaneous and delay interaction coefficients are satisfied.
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  • 105
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    Bulletin of mathematical biology 45 (1983), S. 807-826 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract Sensitivity analyses have been used to examine the flow structure of two hypothetical ecosystem models. These analyses have results which relate to important aspects of ecosystem theory. Cycles are shown to increase the sensitivity of the network, while increased throughflow is shown to decrease the sensitivity. Such results indicate that several factors can be modified to decrease the sensitivity of ecosystems to environmental stress.
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  • 106
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    Bulletin of mathematical biology 45 (1983), S. 827-836 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract A continous, deterministic mathematical model is used to predict population distributions by age at any time, given the initial distribution and the variation of birth and death rates with age and time. Solutions are obtained on a computer using a semi-discretization algorithm in which time derivatives in the partial differential equations are replaced by finite-difference expressions. The resulting sets of ordinary differential equations are solved by a predictor-corrector method. Graphical results are shown for some examples.
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  • 107
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    Bulletin of mathematical biology 45 (1983), S. 849-855 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract A new formula for the complexity of graphs is proposed and applied to the points lines and ‘connections’ of some chemically relevant graphs.
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  • 108
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    Bulletin of mathematical biology 45 (1983), S. 837-847 
    ISSN: 1522-9602
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    Notes: Abstract This paper reports general and specialized results on analytical solutions to the governing phenomenological equations for chemotactic redistribution and population growth of motile bacteria. It is shown that the number of bacteria cells per unit volume,b, is proportional to a certain prescribed function ofs, the concentration of the critical substrate chemotactic agent, for steady-state solutions through an arbitrary spatial region with a boundary that is impermeable to bacteria cell transport. Moreover, it is demonstrated that the steady-state solution forb ands is unique for a prescribed total number of bacteria cells in the spatial region and a generic Robin boundary condition ons. The latter solution can be approximated to desired accuracy in terms of the Poisson-Green's function associated with the spatial region. Also, as shown by example, closed-form exact steady-state solutions are obtainable for certain consumption rate functions and geometrically symmetric spatial regions. A solutional procedure is formulated for the initialvalue problem in cases for which significant population growth is present and bacteria cell redistribution due to motility and chemotactic flow proceeds slowly relative to the diffusion of the chemoattractant substrate. Finally, a remarkably simple exact analytical solution is reported for a stradily propagating plane-wave which features motility, chemotactic motion and bacteria population growth regulated by substrate diffusion.
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  • 109
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    Bulletin of mathematical biology 45 (1983), S. 857-867 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract This paper discusses the flow of blood in large artries under the influence of linear periodic acceleration. The governing equations and boundary conditions are established and analytical solutions for the velocity, fluid acceleration, bulk flow and shear stress are obtained. The results for these physical quantitites are computed for the case of an artery the size of a normal human aorta. It is found that the flow field variables are directly proportional to the external accelerating force. The behaviour of the velocity profile along the radial distance at different stages of times at fixed applied acceleration is also shown.
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  • 110
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    Bulletin of mathematical biology 45 (1983), S. 931-968 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract The evolutionary selection circuits model of learning has been specified algorithmically. The basic structural components of the selection circuits model are enzymatic neurons, that is, neurons whose firing behavior is controlled by membrane-bound macromolecules called excitases. Learning involves changes in the excitase contents of neurons through a process of variation and selection. In this paper we report on the behavior of a basic version of the learning algorithm which has been developed through extensive interactive experiments with the model. This algorithm is effective in that it enables single neurons or networks of neurons to learn simple pattern classification tasks in a number of time steps which appears experimentally to be a linear function of problem size, as measured by the number of patterns of presynaptic input. The experimental behavior of the algorithm establishes that evolutionary mechanisms of learning are competent to serve as major mechanisms of neuronal adaptation. As an example, we show how the evolutionary learning algorithm can contribute to adaptive motor control processes in which the learning system develops the ability to reach a target in the presence of randomly imposed disturbances.
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  • 111
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    Bulletin of mathematical biology 45 (1983), S. 981-990 
    ISSN: 1522-9602
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    Notes: Abstract In the present paper we discuss the behaviour of solutions of a dynamical system describing the growth of cells in a well-mixed continuous culture where the supply of the growth-limiting nutrient depends on the activity of an enzyme outside the cell membrane. It turns out that for positive dilution rates there exists an exponentially attractive two-dimensional simplex. Furthermore, the reversed system restricted to this simplex is quasimonotone. In every case all trajectories tend to an equilibrium state.
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  • 112
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    Bulletin of mathematical biology 45 (1983), S. 991-1004 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract We present a Gause predator-prey model incorporating mutual interference among predators, a density-dependent predator death rate and a time lag due to gestation. It is well known that mutual interference is stabilizing, whereas time delays are destabilizing. We show that in combining the two, a long time-lag usually, but not always, destabilizes the system. We also show that increasing delays can cause a bifurcation into periodic solutions.
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  • 113
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    Bulletin of mathematical biology 45 (1983), S. 969-980 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract The cycle structure of enzymatic neural networks may be characterized in terms of number of cycles exhibited, size of cycle state sets and cycle lengths. Simulation experiments show that the stability properties of these networks have some unusual features which are not exhibited by networks of two-state switching elements or by randomly constructed ecosystem models. The behavioral and structural stability of these systems decreases with their structural complexity, as measured by the number of components. The behavioral and structural stability of enzymatic neural networks also decreases with structural complexity, as measured by the number of excitase types, but only up to the middle level of excitases per neuron. This is the point of highest potential responsiveness of the system to environmental stimuli. Beyond this point the behavioral and structural stability increase. This is due to the fact that the number of possible states increases up to this point and decreases beyond it. The number of possible states, not the number of components, serves as the useful measure of complexity in these types of systems. The selection circuits learning algorithm has been used to evolve networks whose cycle structures have desired features.
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  • 114
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    Bulletin of mathematical biology 45 (1983), S. 1005-1011 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract Similarity criteria of the functional design of the mammalian cardiovascular system are scant. For the analysis of mammalian cardiac energetics physiological parameters such as mean arterial blood pressure, stroke volume, heart rate, metabolic rate and heart and body weights are considered pertinent. Based on these parameters, a new similarity principle is established via allometric equations, dimensional analysis and Buckingham's pi-theorem. The principle states that the ratio of left ventricular external work to metabolic rate is inversely proportional to resting heart rates of mammals. The proportionality constant is dimensionless and is invariant of mammalian body weights.
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  • 115
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    Bulletin of mathematical biology 45 (1983), S. 1029-1045 
    ISSN: 1522-9602
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    Notes: Abstract The mathematical theory of categories is used as a tool in the description of the structure and function of natural systems. The connections between the category of natural systems, with observables and dynamics, and the phenomenological calculus of response tensors, duality- and adjoint-invariance diagrams are established. The unified theory is applied to the analysis of hierarchies, pattern generation and the structure and dynamics of proteins.
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  • 116
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    Bulletin of mathematical biology 45 (1983), S. 1047-1072 
    ISSN: 1522-9602
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    Notes: Abstract This is an investigation of natural systems from the standpoint of the mathematical theory of categories. It examines the relationships which exist between different descriptions through measurement of observables and dynamical interactions. We begin with a category theory of formal systems with observables, and then proceed to a category theory of dynamical systems. The two categories are then combined to represent natural systems. Topological considerations enter in the study of stability and bifurcation phenomena. Special emphasis is placed on natural systems which model biological processes. The categorical system theory developed is applied to the analysis of several biological problems and biological system theories.
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  • 117
    ISSN: 1522-9602
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    Notes: Abstract Tetanic hyperpolarization for theXenopus node is simulated by means of iterative solutions of the Frankenhaeuser-Huxley excitation equations together with an active transport current density term which is dependent on sodium and potassium levels as well as the ADP/ATP ratio. All time-dependent variables at the end of one interspike interval are introduced as initial conditions for the next response, whereupon all time-dependent changes in voltage and permeability factors appear identical for the third and fourth responses of a sequence. Net change in internal sodium concentration is zero throughout the third and fourth intervals if sodium loading of the system is initially adjusted to a critical level. Extent of tetanic hyperpolarization is a function of the pump conductance.
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  • 118
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    Bulletin of mathematical biology 45 (1983), S. 1097-1097 
    ISSN: 1522-9602
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  • 119
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    Bulletin of mathematical biology 45 (1983), S. 1073-1096 
    ISSN: 1522-9602
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    Notes: Abstract The properties of nonlinear equations describing the solute and solvent transport across a simplified Patlak-Goldstein-Hoffman model (two membranes in series without unstirred layers) are investigated both analytically and numerically. The analysis shows that the principal coefficients measured in transport experiments in the presence of active transport are dependent on the experimental conditions. These ‘apparent’ system parameters are extensions of the corresponding parameters determined both in passive systems and in the linear Kedem-Katchalsky theory. Moreover, they are related to the local phenomenological coefficients of the single membranes of the array. Several relationships between measurable quantities and the local system parameters are indicated, allowing the planning of experiments aimed at the measurement of the latter. Data in the literature have been used to check the proposed volume flow equation.
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  • 120
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    Bulletin of mathematical biology 46 (1984), S. 11-17 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract Linear birth and death processes are used to derive simple expressions for sequential extinction times and gene fixation probabilities in asexual populations.
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  • 121
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    Bulletin of mathematical biology 46 (1984), S. 1-10 
    ISSN: 1522-9602
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    Notes: Abstract We are here concerned with the functionf which assigns to each pointP of an object the numberf(P) which is the shortest distance fromP to the border. This function appears in various guises in diverse biological studies. The functionf(P) is itself a measure of shape—or more precisely, an infinite set of measures, one for each point (and hence, in view of its geometric definition, usually in a form inconvenient for use). Thus in this paper we sought a reasonable representative of this infinite set of measures, namely themean of the numbersf(P) asP ranges over all points of the entity. Computability studies are developed for various classes of shapes. For example, (1) the mean for a lamina bounded by a polygon circumscribable about a circle of radiusr isr/3; (2) the mean for a domain bounded by a polyhedron circumscribable about a sphere of radiusr isr/4. The transition from pointwise to piecewisef(P), especially in the non-convex case, requires working with inequalities.
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  • 122
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    Bulletin of mathematical biology 46 (1984), S. 19-40 
    ISSN: 1522-9602
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    Notes: Abstract A mathematical model for traveling bands of motile and chemotactic bacteria in the presence of cell growth and death is examined. It is found that asymptotic traveling wave solutions exist in the absence of chemotaxis, due to the balance of growth, death and random motility. Thus random motility confers the ecological advantage of population propagation through migration into nutrient-rich regions. The presence of chemotaxis amplifies this advantage by moving more cells into higher nutrient concentration regions, resulting in larger and faster bands. Therefore there seem to be two types of traveling bands that can be attained by chemotactic bacteria in the presence of growth and death: (1) these growth/death/motility bands; and (2) pure chemotactic ‘Keller-Segel'-type bands. Comparison to experimental observations by Chapman in 1973 indicate that the latter seem to be formed. The relationship between these two types of solution is at present uncertain. The growth/death/motility bands may have relevance on longer time or distance scales characteristic of microbial ecological systems.
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  • 123
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    Bulletin of mathematical biology 46 (1984), S. 115-125 
    ISSN: 1522-9602
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    Notes: Abstract Based on the principle of minimum power, a mathematical model of the pathological functional state of the oxygen transport system is presented. The model is used to determine the optimal functional parameters of the oxygen transport system in hyperthyroidism, anemia and hypertension. Theoretical results are compared with clinical data.
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  • 124
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    Bulletin of mathematical biology 46 (1984), S. 139-153 
    ISSN: 1522-9602
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    Notes: Abstract A modified SIRS model is developed as a framework for the study of epizootiological dynamics in an insect-pathogen system. Linearized stability analysis reveals that the system with one immune and one susceptible host class can exhibit stable, periodic or unstable behavior depending on model parameters. In general, high pathogenicity, short pathogen propagule lifespan and high host reproductive rate are stabilizing influences. Pathogen transmissibility and propagule production/host do not influence local stability. The effect of seasonal host reproduction is studied because most insect hosts are seasonal in temperate climates. The basic stability dependence on model parameters holds except as modified by the length of the reproduction interval. The results of this study are compared with the recent work of Anderson and May.
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  • 125
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    Bulletin of mathematical biology 46 (1984), S. 175-184 
    ISSN: 1522-9602
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    Notes: Abstract The random predator-prey type interactions of the population species in a random varying environment have been investigated. The Fokker-Planck equation for the transition probability, corresponding to the stochastic differential equations established from Lotka-Volterra equations by the introduction of randomness and variability, has been integrated in the form of a path integral. The transition probabilities for extinction or survival of one or several species have been approximately evaluated and investigated.
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    Bulletin of mathematical biology 46 (1984), S. 155-174 
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    Notes: Abstract If two microbial populations compete for a single resource in a homogeneous environment with time invariant inputs they cannot coexist indefinitely if the resource competed for is not renewed by biological activity within the system. Mathematical studies have shown that in a predator-prey system, where the resource (prey) is self-renewing, the two competitors (predators) can coexist in a limit cycle. This suggests that if the resource competed for is renewed by biological activity within the system coexistence can occur in any microbial system provided that it exhibits the same features as, but without being, a predator-prey one. A food chain involving commensalism, competition and amensalism is presented here. Two subcases are considered. It is only when maintenance effects are taken into account that coexistence, in limit cycles, can occur for this system. Limit cycle solutions for the system are demonstrated with the help of computer simulations. Some necessary conditions for coexistence are presented, as are some speculations regarding the possible physical explanations of the results.
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    Bulletin of mathematical biology 46 (1984), S. 127-137 
    ISSN: 1522-9602
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    Notes: Abstract The branching structure of the mammalian arterial tree has been known to be close to that of an optimal conduit system of the minimum work model characterized as the branch system of constant wall shear rate. The physiological mechanism producing such construction was considered to be based on the local response of arterial caliber induced by the wall shear stress (shear rate × blood viscosity) and thereby maintaining this stress constant, which was previously observed at the canine common carotid artery shunted to the external jugular vein. The stress levels at various parts of the arterial system estimated from available data fell within ±50% of the mean (15 dyn/cm2), which was consistent with the value predicted from the model. Theoretical analyses on the cost function of the model indicated that the suspected variation of shear rate levels in the arterial tree due to the anomalous changes in blood viscosity which might bring about 3- to 4-fold differences between the minimum and maximum shear rates would cause less than 10% increase in the total energy cost. It was concluded that a local adaptive response to wall shear stress is the mechanism which effectively optimizes the design of the arterial tree.
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    Bulletin of mathematical biology 46 (1984), S. 185-185 
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    Bulletin of mathematical biology 46 (1984), S. 187-203 
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    Notes: Abstract The structure of the genetic code is related to a Gray code, which is a plausible theoretical model for an amino acid code. The proposed model implies that the most important factor in shaping the code was the effects of mistakes in translation, not effects of mutations. Another possible implication is that the preservation of stiffness and flexibility at appropriate places in a protein chain is as important in protein structure as the appropriate placement of hydrophilic (external) and hydrophobic (internal) residues. Other results are a simple conceptualization of the relationships among the 20 amino acids and their relations to their codons. The detailed relationships are summarized in the following ‘similarity alphabet’: ala, thr, gly, pro, ser; asp, asn, glu, gln, lys; his, arg, trp, tyr, phe; leu, met, ile, val, cys; (ATGPS DNEQK HRWYF LMIVC in the one-letter code). This alphabet falls into four groups of amino acids: small, external, large, internal. The approximate relation of the groups to their codons is expressed as: the first base of a codon controls size—a purine means a small amino acid, a pyrimidine means large; the middle base controls cloisterednes—purine means external, pyrimidine means internal. These relationships express the minimum change principle upon which the code appears to be founded.
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    Bulletin of mathematical biology 46 (1984), S. 269-282 
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    Notes: Abstract A theoretical study of the Brusselator model with non-uniform distribution of component A and a concentration-dependent diffusion coefficient has been performed. Numerical simulation reveals that a variable diffusion coefficient alters the bifurcation pattern and the stability properties of the steady-state as well as periodic solutions. A simple approximate method, based on one-point collocation, has been proposed to analyze the bifurcation phenomena for the case of fixed boundary conditions and low system size.
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    Bulletin of mathematical biology 46 (1984), S. 283-294 
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    Notes: Abstract In this paper the effects of the occurrence of cut trees in the topological analysis of branching patterns have been studied. It is assumed that branches are removed at random from the trees. We prove that, for both the segmental and terminal growth models, the probability distributions of the cut trees are identical to those of complete trees.
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    Bulletin of mathematical biology 46 (1984), S. 247-268 
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    Notes: Abstract The goal of this work is an examination of capillary exchange models as mathematical operators. The concentration function relations for the Krogh cylinder of a single capillary, basic to many organ models, are studied via the theory of operators on the Lebesgue normed spacesL p[0,∞], (1〈-p〈-∞). A discussion is included of theL p -normsvis-à-vis the coefficient of variation currently used in finding capillary parameters and evaluating parameter searches. The capillary model determines two operators on the space of locally integrable functions: O K (relating extravascular concentration to intravascular) and K a, k (relating intravascular concentration to input), wherek is the ratio of permeabilitysurface area (PS) to extravascular volume, and α is the ratio of PS to flow. These operators are shown to induce contractive (‖O K ‖ p 〈-1, ‖K a, k ‖ p 〈-1), isotone, linear operators onL p . The uniform convergence relation $$K_{a,k} = \mathop {\lim _{(p)} }\limits_{N \to \infty } \left( {\sum\limits_{n = 0}^N {P_n (a)O_k^n } } \right)$$ (as operators onL p) is derived, whereP n (a) is the Poisson probabilitye −a a n /n!. For the important special cases ofp=∞, 1, 2 the norms are found (‖Ok‖=‖Ka,k‖p=1). Consideration is also given to the norms and operators when the functions involved are limited to a finite interval of time.
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    Bulletin of mathematical biology 46 (1984), S. 295-326 
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    Notes: Abstract One particular kind of structure offers possible explanations, for long-term memory, efficient consolidation of stored information from the environment, clustering of data strings and multimodal functioning. It is a possible model for pieces of neural structure and its use offers a uniform method for both studying and constructing an extensive class of mechanisms.
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    Bulletin of mathematical biology 46 (1984), S. 327-332 
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    Notes: Abstract Levenshtein dissimilarity measures are used to compare sequences in application areas including coding theory, computer science and macromolecular biology. In general, they measure sequence dissimilarity by the length of a shortest weighted sequence of insertions, deletions and substitutions required, to transform one sequence into another. Those Levenshtein dissimilarity measures based on insertions and deletions are analyzed by a model involving valuations on a partially ordered set. The model reveals structural relationships among poset, valuation and dissimilarity measure. As a consequence, certain Levenshtein dissimilarity measures are shown to be metrics characterized by betweenness properties and computable in terms of well-known measures of sequence similarity.
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    Bulletin of mathematical biology 46 (1984), S. 337-337 
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    Bulletin of mathematical biology 46 (1984), S. 333-336 
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    Notes: Abstract It is pointed out that the methane flux measured experimentally for certain ponds and swamps is quantitatively consistent with a commensal dependence of Methanobacteria on O2-chemotactic motile aerobic bacteria. The Methano species is thereby shielded from oxygen and provided with carbon dioxide for the anaerobic production of methane.
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    Bulletin of mathematical biology 46 (1984), S. 357-370 
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    Notes: Abstract A sufficient condition is given for stochastic boundedness persistence of a top predator in generalized Lotka-Volterra-type stochastic food web models in arbitrary bounded regions of state space. The main result indicates that persistence in the corresponding deterministic system is preserved in the stochastic system if the intensities of the random fluctuations are not too large.
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    Bulletin of mathematical biology 46 (1984), S. 371-377 
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    Notes: Abstract One of Bobisud's models for the evolution of cannibalism is reanalyzed by applying the method of finding evolutionarily stable strategies (or ESS's). It is demonstrated that ‘no cannibalism’ never will be an ESS if the initial rate of cannibalism is too large. It is further demonstrated that individual selection may even result in the evolution of cannibalism during food abundance. Some empirical case studies are briefly discussed in relation to this model.
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    Bulletin of mathematical biology 46 (1984), S. 379-387 
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    Notes: Abstract A computer algorithm is presented which equiprobably generates any member of the set of all directed trees withk labeled terminal nodes and unlabeled interior nodes. The algorithm requires roughlyk 2 /2 storage locations. The one-time initialization requiresO(k 2 ) time, while generating each tree requiresO(k) time.
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    Bulletin of mathematical biology 46 (1984), S. 515-527 
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    Notes: Abstract The comparison of several sequences is central to many problems of molecular biology. Finding consensus patterns that define genetic control regions or that determine structural or functional themes are examples of these problems. Previously proposed methods, such as dynamic programming, are not adequate for solving problems of realistic size. This paper gives a new and practical solution for finding unknown patterns that occur imperfectly above a preset frequency. Algorithms for finding the patterns are given as well as estimates of statistical significance.
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    Bulletin of mathematical biology 46 (1984), S. 501-514 
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    Notes: Abstract A new development is introduced here in the use of dynamic programming in finding pattern similarities in genetic sequences, as was first done by Needleman and Wunsch (1969). A condition of pattern similarity is defined and an algorithm is given which scans any set of similarities and screens out those which fail to meet the condition. When the set to be scanned contains every pair of segments, one from each of two given sequences of lengthsm andn (i.e. every possible location for a pattern similarity), then it completes the scan in a number of computational steps proportional tom·n, leaving those pairs of segments which satisfy the similarity condition. The algorithm is based on the concept of match density, as suggested by Goad and Kanehisa (1982).
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    Bulletin of mathematical biology 46 (1984), S. 529-543 
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    Notes: Abstract This paper concerns sequences of letters in which certain “distinguished” words are of interest. Such sequences arise as data in numerous fields including genetics and neuroscience. A probability distribution is given for the number of occurrences of a chosen word in a randomized sequence of letters. Such words are considered “favored” if they occur more than expected at random. Favored words have been discovered in nerve impulse trains and may reflect a neural coding scheme.
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    Bulletin of mathematical biology 46 (1984), S. 545-552 
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    Notes: Abstract As the volume of protein sequence data grows, rapid methods for searching the protein sequence database become of primary importance. Rigorous comparison of sequences is obtained with the well-known dynamic programming algorithms. However, these algorithms are not rapid enough to use for routinely searching the entire database. In this paper we discuss some methods that can be used for rapid searches.
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    Bulletin of mathematical biology 46 (1984), S. 553-566 
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    Notes: Abstract We give algorithms for computing the extent of similarity between two or three sequences of letters. The similarity measures we consider include a penalty for inserting gaps within the sequence in order to enhance similarity. The magnitude of the penalty for gaps is assumed to be independent of their size in order to accommodate certain biological applications. Our algorithm for three sequence comparisons, which is based on solving a system of recursive equations, improves upon the efficiency of existing methods. Although the system of recursive equations utilized by the algorithm is quite complicated as it stands, it has none the less been simplified by appeal to combinatorial considerations.
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    Bulletin of mathematical biology 46 (1984), S. 567-577 
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    Notes: Abstract Well-known dynamic programming algorithms exist for comparing two finite sequences inO(N 2) time and storage, whereN is the common sequence length. Extensions to the comparison ofM finite sequences requireO((2N) M) time and storage, making such algorithms difficult even forM=3. A simple generalization of the sequences makes it possible to obtain some results about the geometry of sequence alignments. These ideas suggest heuristic approaches to problems of comparing several sequences. IfM sequences are known to be related by a binary tree, they can be aligned inO(MN 2) time andO(N 2+NM) storage.
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    Bulletin of mathematical biology 46 (1984), S. 661-672 
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    Notes: Abstract Protein sequences of the Dayhoff databank of 1984 have been analyzed to evaluate the occurrences of the 400 dipeptides and 8000 tripeptides. Expected values and standard deviations for the di- and tripeptides were determined by Monte Carlo and binomial approximation. A condensed format containing this information, labeled a uniqueness diagram, is presented and made available in the form of a microfiche.
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    Bulletin of mathematical biology 46 (1984), S. 827-844 
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    Notes: Abstract In two-state sliding filament models of muscle contraction a partial differential equation must be solved to find the cross-bridge distribution functionn(x, t). In this paper the analytical form of this function is obtained by integration along the characteristic line and special cases are presented in which the explicit expression forn(x, t) can be completely determined. These analytical solutions provide a direct mathematical connection between the microscopic contraction parameters contained in the kinetic theories and macroscopic muscle dynamics and are thus used to investigate what parameters influence the transient contractile tension in typical experimental conditions. The results of this investigation are consistent with relevant aspects of muscle physiology.
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    Bulletin of mathematical biology 46 (1984), S. 845-857 
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    Notes: Abstract Experiments on random binary, ternary, etc. (P=2, 3,…, 10) switching nets are reported. Behavioral cycle lengths are examined as functions of output variety,P, input connectance,K, and net size,N. Overall, output variety appears an influential, well-behaved net property. Strong, but well-behaved interactions appear among net variables. In high connectance nets, median cycle length grows approx. asP N/2. Other factors constant, one-connected nets show the shortest cycles, and connectance effects appear to converge asymptotically aroundN. Data for cycle length as a function of net size suggest a concavity not compatible with the Kauffman “square root law” (Kauffman, 1969). Evidence of a positive relationship between cycle length and run-in length is found in two-input nets; weaker evidence is obtained that in higher connectance nets this relationship becomes negative in sign. The “modular complexity” ofP〉2 nets is examined briefly.
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    Bulletin of mathematical biology 46 (1984), S. 869-877 
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    Notes: Abstract The influence of topographical situation on the spread of infection is studied. The investigation is based on a multigroup model. The population under consideration is thought to be divided into subpopulations living in regions that are separated from each other by natural barriers (mountains). Infection is carried from one region to another by migrating infectives. Migration is possible only along the river system so that the structure of the epidemiological network is that of a symmetric tree. The results allow comparison of the velocity of propagation of the epidemic for different geographical situations and allow quantification of the “channel-effect”, according to which mountainous regions are channels rather than barriers to the spread of an epidemic.
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    Circuits, systems and signal processing 1 (1982), S. 93-104 
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    Notes: Abstract A stochastic approximation problem for polynomic operators is formulated. The performance of polynomic operators of various degrees is compared. The effect of causality constraints is also examined.
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    Circuits, systems and signal processing 1 (1982), S. 137-169 
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    Notes: Abstract A recently developed algebraic approach to the feedback system design problem is reviewed via the derivation of the theory in the single-variate case. This allows the simple algebraic nature of the theory to be brought to the fore while simultaneously minimizing the complexities of the presentation. Rather than simply giving a single solution to the prescribed design problem we endeavor to give a complete parameterization of the set of compensators which meet specifications. Although this might at first seem to complicate our theory it, in fact, opens the way for a sequential approach to the design problem in which one parameterizes the subset of those compensators which meet the second specification...etc. Specific problems investigated include feedback system stabilization, the tracking and disturbance rejection problem, robust design, transfer function design, pole placement, simultaneous stabilization, and stable stabilization.
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    Circuits, systems and signal processing 1 (1982), S. 267-268 
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    Circuits, systems and signal processing 1 (1982), S. 433-445 
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    Notes: Abstract We describe a recently developed framework for exploring the structure of linear time-invariant models of large systems, and for constructing interpretable or physically-based, reduced-order models that reproduce selected modes of the original systems to a desired accuracy. Application of this framework to constructing lumped approximations for interconnections of lumped and distributed systems is briefly explored.
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    The Geneva risk and insurance review 10 (1978), S. 50-66 
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    The Geneva risk and insurance review 10 (1978), S. 44-49 
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    Notes: Conclusion For all these reasons, the rediscovery of the equivalence theorem, first stated by David Ricardo in 1817, must be rejected as an adequate basis for policy, just as Ricardo had denied its applicability to the real world. Correspondingly, the concern with the adverse consequences of unfunded social insurance wealth for the supply of national saving, capital intensity, and living standards remains well founded. p ]If, as a practical matter, public pension and social security programs will never be funded actuarially in the United States and most other postindustrial countries, then government-supervised substitution of private for public retirement plans is the only way to achieve at least partial funding. If such substitution follows the British model of allowing employers to contract out of the earnings-related part of the state scheme if equivalent pensions are provided by the company plan, payroll taxes and social security wealth decline so as to reduce their adverse impact on capital formation.
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    The Geneva risk and insurance review 10 (1978), S. 73-84 
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    The Geneva risk and insurance review 10 (1978), S. 85-95 
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    The Geneva risk and insurance review 10 (1978), S. 96-98 
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    The Geneva risk and insurance review 11 (1979), S. 5-13 
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    The Geneva risk and insurance review 10 (1978), S. 67-72 
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    Notes: Summary: Social Policy in the Italian Economy Favourable social and economic conditions constitute the essential framework for a stable development of savings. Saving in the form of insurance becomes advantageous for the individual, and private insurance can thus extend its activity, when social attitudes and the economic situation favour the propensity to save. If conditions change, the State can take over the coverage of risks through social insurance. By means of this institutions, an anti-cyclical policy can be pursued: the amount of social security contributions, for instance, can be increased during the expansion of the cycle and the amounts thus accumulated can be used to grant benefits during the recession period, when contributions can be fixed at a lower percentage of wages. Another type of policy can be pursued by government authorities: that of adjusting social security contributions to industrial profits, thereby directing the subsequent effects on economic growth. Inflation cab cause instability in decisional policies taken by private insurance companies. A solution to the unbalanced increase of costs can be found in index-linking. Life policies of this kind, for instance, cab be closely related to investments in houses, to be bought by the insured themselves in price-linked instalments. After a reference to present developments regarding risk instability and to possibilities of new forms of insurance, this paper considers the competition resulting from the opening of the EEC insurance markets as an opportunity for the Italian market to strengthen its structures.
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    Bulletin of mathematical biology 39 (1977), S. 663-678 
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    Notes: Abstract A number of apparently different lines of inquiry into fundamental biological processes point to the central role played by the notion of observation in the theory of biological systems. Not only do we use the results of our own observations to obtain the system descriptions which are the starting-points for an understanding of biological processes, but it is a basic postulate of physics that the interactions between biological systems themselves can be regarded as observations. On this basis, it is clear that we cannot properly understand biological interactions unless the observables we employ for system description are the same as those involved in the interactions we are describing. To do this requires a general theory of observables and system description, establishing the relationship between different modes of description. A sketch of such a theory is developed in the present paper, using only two postulates: (a) that all interactions are determined by the values of observables of a system evaluated on specific states, and (b) that real-valued observables suffice. As an application, a specific test is proposed whereby it can be determined whether the observables employed to describe interacting systems are sufficient to specify the observables involved in the interaction itself.
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    Bulletin of mathematical biology 39 (1977), S. 679-691 
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    Notes: Abstract Differential equations are derived whose solution gives the cross-sectional shape of a flexible tube as a function of the transmural pressure. These equations are solved digitally to produce a series of closed curves, each curve representing the shape of a cross section for a particular set of conditions. These are then applied to the case of systemic arteries, pulmonary arteries, and large veins. The results predict that systemic arteries must always be circular, even when the internal and external pressures are equal. In veins, a small positive internal pressure causes them to become circular, regardless of their initial state, with negligible stretching. Further increases in internal pressure cause the area of the cross section to increase due only to stretching, the shape remaining essentially circular. With pulmonary arteries, known to be noncircular, changes in the cross-sectional area result from a combination of stretching and changes of shape.
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    Bulletin of mathematical biology 39 (1977), S. 693-704 
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    Notes: Abstract Stochastic models of human reproduction are beginning to play significant roles in the evaluation of family planning programs. A class of stochastic processes called absorbing, agedependent, semi-Markov processes frequently arises in the construction of such models. The paper begins with a discussion of some technicalities regarding absorbing, age-dependent, semi-Markov processes. Then, an algorithm due to Littman, which makes possible the computerization of this class of stochastic processes, is presented. Briefly, Littman’s algorithm provides an efficient method for numerically solving systems of renewal type integral equations, provided the system does not contain a large number of equations. After setting down a concrete model for a large clinical trial of intrauterine devices conducted in Taiwan, the paper concludes with a discussion of a method for validating the model based on the data collected in the clinical trial.
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    Bulletin of mathematical biology 39 (1977), S. 743-743 
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    Bulletin of mathematical biology 39 (1977), S. 705-719 
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    Notes: Abstract Some theoretical requirements for the valid use of hemolytic plaque inhibition as a method for studying cellular selection and recognition in an immune response are reviewed. Aside from providing a rational basis for the use of this technique, the theory resolves a growing body of apparently conflicting results on the affinity regulation of IgM secreting cells and can also be used to deduce structural (in particular, morphological) features of the IgM molecule. The diverse predictions of the theory are examined in light of experimental results and find support in a wide data base. Additional specific experiments are proposed which can be used in conjunction with the theory to help clarify the relation between cellular proliferation and maturation.
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    Bulletin of mathematical biology 39 (1977), S. 721-741 
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    Notes: Abstract A theory of the cell volume is presented with emphasis on the swelling effect of high concentrations of KCl and other chloride salts. In this theory a particular cell volume represents a state of balance between the tendency of the cell water to build deeper layers of polarized water and the restraining forces exerted by the salt linkages and H-bonds. Taking into account also the different structure-breaking effects of different salts, theoretical curves can be constructed which describe the complex multiple peak-plateau of swelling curve observed in frog muscle in response to increasing concentrations of different chloride salts.
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    Bulletin of mathematical biology 40 (1978), S. 211-221 
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    Notes: Abstract Non-steady-state equations for kidney models are stated. General conservation relations for these equations are derived. Transient equations for the central core model of the renal medulla are developed. Solution of the equations by Laplace transform methods for time invariant volume flows is discussed. The general theory of solving models with time dependent flows by finite difference methods is developed.
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    Bulletin of mathematical biology 40 (1978), S. 513-524 
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    Notes: Abstract The behavior of large systems of randomly-interacting variables is examined using an intentionally simplified model. The stable positive solutions are found to exhibit to a significant degree some well-known properties of ecological systems. This resemblance (including for example the predominance of “predator-prey” interactions) is all the more striking in view of the lack of biological “data” at the input end. The findings suggest it advisable to distinguish two kinds of properties in ecosystems. One kind would depend on specifically biological mechanisms; the other would characterize a wide class of persistent systems, and arise from the need for a dynamic balance between positive and negative feedback.
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    Bulletin of mathematical biology 40 (1978), S. 499-512 
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    Notes: Abstract For the tumor model of Skipper and Zubrod, which has been analyzed previously for the theoretical FLM function and the effect of chemotherapy against tumors of known or assumed kinetic characteristics, the theoretical continuous labeling (CL) function is derived by considering an equivalent tumor (in terms of unlabeled cell populations) in which the density function of phase duration of cells inS-phasef 2(a 2)=δ(a 2−∞) and the loss functionL 2(t)=0. This mathematical concept of blocking is applied to the analysis of synchronization in tumor growth and blocking effects in cancer chemotherapy. These effects of chemical agents on the cell cycle progression are being incorporated into a previously written computer simulation program for cancer chemotherapy. Whereas, a program is written and used to simulate the CL functions for L1210 leukemia, and primary and metastatic Lewis lung carcinoma.
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    Bulletin of mathematical biology 40 (1978), S. 525-533 
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    Notes: Abstract Global stability is established in a class of prey-predator models. This includes a prey-predator model in which the predator has Type 2 functional response and no intraspecific interactions. Two simple examples demonstrate that Kolmogoroff’s theorem does not apply to some members of this class of models.
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    Bulletin of mathematical biology 40 (1978), S. 535-540 
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    Notes: Abstract The skeletal muscle is regarded as a periodically deformed plate. An equation of energy for the biological tissue in the system is used in the Lagrange variables. With the heat exchange through the above tissues to the exterior media and also with account of some relations between the physiological factors the approximate analytical solution for this equation is obtained and compared with the physiological data.
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    Bulletin of mathematical biology 40 (1978), S. 541-545 
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    Notes: Abstract One of Bobisud's (1976) models for the evolution, of cannibalism is discussed. His analysis is criticised for not being based on the principle of individual selection. Assuming the operation of that principle, we show by simulating his model that cannibals may establish themselves in a noncannibal population. This will happen both in cases where Bobisud concluded cannibalism to be optimal and in cases where he concluded cannibalism not to be optimal.
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    Bulletin of mathematical biology 40 (1978), S. 547-547 
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    Bulletin of mathematical biology 40 (1978), S. 549-579 
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    Notes: Abstract The D'Arcy Thompson concept of biological transformations is developed in a form analogous to such physical concepts as the Law of Corresponding States in thermodynamics, and the Principles of Similitude found in engineering. We find that such concepts depend on a distinction between fundamental and derived quantities, in which the values assigned to the fundamental quantities set the natural scales for the derived ones. Among other things, we see that critical phenomena, such as phase transitions, arise as an immediate consequence of this distinction. In a biological context, we explore the implications of Thompson's hypothesis that closely related organisms are phenotypically similar, assuming that the organisms we see are the result of selection processes operating on phenotypes.
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    Bulletin of mathematical biology 40 (1978), S. 605-623 
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    Notes: Abstract If information is coded in the combination of activities of many neurons operating in parallel, then information present in a network can be defined by the correlation of present network activity with the activity which had been elicited by a stimulus in the past; a high correlation indicates the presence of the previously encoded stimulus. Information is distributed in the network because coding is dependent upon the activities of all cells. A model based on Hartline-Ratliff lateral inhibition with a time delay shows that lateral inhibition can distribute information across a parallel network, reduce output noise, and also briefly store information. With no changes in model parameters, and the use of a correlation measure for recognition, the model can stimulate psychophysical results in eleven variations of the metacontrast masking paradigm.
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    Bulletin of mathematical biology 40 (1978), S. 581-604 
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    Notes: Abstract A theory is derived to calculate the regional and total deposition of aerosol particles in the nasal passages during inhalation. The particle size studied range from 0.2 to 10.0 μm diameter. The deposition is calculated in five regions; (I) the region filled with nasal hair, (II) the nasal valve, (III) the expansion region, (IV) the turbinate region and (V) the posterior bend. Equations are derived to determine the deposition caused by direct impaction on the nasal hairs and bends of the passages. The calculations show the deposition due to direct impaction does not account for the amount or location of deposited particles measured in experiments. Secondary flows have been speculated to exist in the expansion region after the nasal valve and an equation is derived to estimate the deposition caused by the secondary flows. The calculated deposition, due to direct impaction and secondary flows, shows general agreement with the experiment as to the predicted amount and location of deposited particles.
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    Bulletin of mathematical biology 40 (1978), S. 625-635 
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    Notes: Abstract The paper reviews a series of models for circadian clocks and discusses their conclusions and predictions. Attention is focused on Pittendrigh's empirical model, two mathematical models by the author and Winfree's work.
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    Bulletin of mathematical biology 40 (1978), S. 637-649 
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    Notes: Abstract From a model of facilitated ionic transport across axonal membranes proposed by McIlroy (1975), the equipotentials and electric field distributions in the vicinity of a potassium conducting pore and of a sodium conducting pore are computed and presented as two-dimensional mappings. The model is then extended to include the effect of impurity ions in the conducting pores viz. of potassium ions in a sodium pore and of sodium ions in a potassium pore. The ionic selectivities and permeabilities of the transported species are discussed in relation to the extended model. Bounds are deduced for the ionic selectivity coefficients for both the sodium and potassium current-carrying systems in squid giant axons and the electric-field distributions in the vicinities of the pores are computed for the extended model and compared with the impurity-free fields first calculated. Finally the permeability coefficients defined in terms of the extended model are shown to reconcile the results of attempts to measure permeability by means of radioactive tracer techniques, with the classical description of the resting nerve.
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    Bulletin of mathematical biology 40 (1978), S. 661-669 
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    Notes: Abstract The partial differential equations describing transcapillary exchange and subsequent removal of solutis from an idealized liver sinusoid are amenable to solution by similarity analysis. The power and utility of this technique, which is not widely appreciated as a method for solving biological models, is illustrated here for a system whose Laplace transforms is difficult to invert.
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    Bulletin of mathematical biology 40 (1978), S. 671-674 
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    Notes: Abstract The phenomenological theory for the chemotaxis and consumption of oxygen by motile aerobic bacteria is shown to yield a remarkably simple one-dimensional steady-state solution for a congregation of bacteria close to the surface of an oxygen-depleted aqueous medium.
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    Notes: Abstract In this paper biological compartmental models are considered which take into account the intrinsic randomness of the transport rate parameters and their possible variability in time. An identification procedure is presented for the estimation of the stochastic processes representing the transport rate parameters of a compartmental model from a noisy input-output experiment. The problem is formulated in terms of nonlinear filtering. A simple model is discussed for the case in which the transport rate parameters are independent of each other. The possibility of testing possible important features of the behavior of the transport rate parameters is also evidenced.
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    Bulletin of mathematical biology 40 (1978), S. 853-863 
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    Notes: Abstract For any essentially nonlinear system of reaction-diffusion equations of the generic form ∂ci/∂t=Di∇2ci+Qi(c,x,t) supplemented with Robin type boundary conditions over the surface of a closed bounded three-dimensional region, it is demonstrated that all solutions for the concentration distributionn-tuple function c=(c 1(x,t),...,c n (x,t)) satisfy a differential variational condition. Approximate solutions to the reaction-diffusion intial-value boundary-value problem are obtainable by employing this variational condition in conjunction with a Galerkin-Ritz procedure. It is shown that the dynamical evolution from a prescribed initial concentrationn-tuple function to a final steady-state solution can be determined to desired accuracy by such an approximation method. The variational condition also admits a systematic Galerkin-Ritz procedure for obtaining approximate solutions to the multi-equation elliptic boundary-value problem for steady-state distributions c=−c(x). Other systems of phenomenological (non-Lagrangian) field equations can be treated by Galerkin-Ritz procedures based on analogues of the differential variational condition presented here. The method is applied to derive approximate nonconstant steady-state solutions for ann-species symbiosis model.
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    Bulletin of mathematical biology 40 (1978), S. 873-875 
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    Bulletin of mathematical biology 41 (1979), S. 129-138 
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    Notes: Abstract The results of an earlier effort to provide a geometrical analysis of Hutchinsonian niche space are extended. The concept of diversity of a species in niche space is introduced and the maximization of this diversity provides a rationale for a within-niche fitness distribution which is Gaussian. Niche expansion is seen as a consequence of diffusion in niche space, and an evolutionary version of the Volterra competition equations is proposed as a way to relate niche geometry with population dynamics. Applications to topics in community evolution, species packing and the statistical fitting of species abundance data are mentioned.
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    Bulletin of mathematical biology 41 (1979), S. 203-215 
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    Notes: Abstract In this paper we use marginal probabilities to derive expressions for the means, variances and covariances ofm-compartment systems. We also present an efficient algorithm for the estimation of the parameters of the system using time series data when measurements are available fromk of them compartments. An application of the analysis and parameter estimation procedure for a model representing the results of a cancer treatment follow-up study is given.
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    Bulletin of mathematical biology 41 (1979), S. 193-201 
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    Notes: Abstract The self-organizing properties of an ensemble of interconnected units are studied by linear stability analyses. Small perturbations of a uniform steady-state may result in bifurcations to other solutions that exhibit spatial or temporal order. We show that increasing the number of connections that a unit makes with its neighbors changes the nature of these solutions and tends to destroy spatiotemporal patterns. If an unconnected system is orginally stable, the formation of multiple interconnections can never induce temporal periodicity but may, under certain circumstances, allow the emergence of stationary spatial patterns. We have verified the predictions of the linear stability analysis on a model system and comment on the implications of these results for multicellular ensembles.
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    Bulletin of mathematical biology 41 (1979), S. 217-227 
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    Notes: Abstract A performance criterion and weighting factors for the optimal cardiac assistance are investigated by applying Tellegen's network theorem and tolerance analysis on animal experimental data for left ventricular (LV) bypass on the failing heart. Two major factors with respect to cardiac assistance (total power delivered to the peripheral circulatory system, and changes in temporal pattern of ventricular contraction) are represented by two performance criteria,J 1 andJ 2 whereJ 1 relates to the sum of LV and pump power, andJ 2 relates to the “peakedness” factor of LV power. The total performance index (J) is determined as the weighted sum ofJ 1 andJ 2;J=w 1J1+w2J2. The weighting factors,w 1 andw 2, are computed as inverses of the tolerance in the performance contours with respect to improvement of stroke work per minute from pre- to post-bypass condition.
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    Bulletin of mathematical biology 41 (1979), S. 253-255 
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    Bulletin of mathematical biology 41 (1979), S. 229-251 
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    Notes: Abstract In treating the Volterra-Verhulst prey-predator system with time dependent coefficients, we ask how far this deterministic system represents or approximates the dynamics of the population evolving in a realistic environment which is stochastic in nature. We consider a stochastic system withsmall Gaussian noise type fluctuations. It is shown that the higher moments of the deviation of the deterministic system from the stochastic approach zero as the strength δ of the perturbation decays to zero. For any δ〉0 and allT〉0, ε〉0, the sample population paths that stay within ε distance from the deterministic path during [0,T] form a collection of positive probability. In comparing the stationary distributions of the two systems, we show that the weak limits of those of the stochastic system form a subset of those of the deterministic system. This is in analogy with a result of May connected with the stability of the two systems. Plant and rodent populations possess periodic parameters andexhibit periodic behaivor. We establish theoretically this periodicity under periodicity conditions on the coefficients and perturbing random forces. We also establish a central limit property for the prey-predator system.
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    Bulletin of mathematical biology 41 (1979), S. 257-282 
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    Notes: Abstract Adaptation of repetitively firing sensory neurons and nerve models is correlated with specific inhibitory feedback phenomena—an electrogenic sodium pump, and post synaptic self inhibition. The quality of the adaptive responses depends on the excitation properties of the neuron in the interspike interval, or the description of these properties by the underlying impulse encoder model. THis model dependence is demonstrated by comparisons of the behavior of two classes of models; the “leaky integrator models” which assume a passive neural membrane, and the “variable-γ models”, for which the neural state of excitation varies according to first order differential equations. The complexity inherent in the variable-γ models is effectively boiled down to mathematically simple relationships which are derived from studies of the neural- and model frequency responses to small amplitude sinusoidal stimuli. It is argued, and supported with examples, that these relationships hold for impulse frequency transients resulting from more general stimulus conditions. Expressions are then derived which permit feedback parameters to be determined from impulse frequency data. In this connection, recent studies of neural dynamics are brought to bear to resolve ambiguities in data interpretation.
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    Bulletin of mathematical biology 41 (1979), S. 283-304 
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    Notes: Abstract A setQ(n) of noncongruent connected shapes, constructed from a fixed numbern of congruent regular hexagons laid edge to edge, is defined. Various measures of shape are applied toQ(n) and the results are numerically analyzed in the special casesn≦9. The concept of spatial entropy is introduced which affords a measure of the “complexity” of shape. Possible biological applications include the analysis of ecological cover,stigmergy or the complex nest-building activities of social insects and morphogenesis. Essentially any comparative study of shape, regardless of the specific application, might be carried out along the lines suggested in the paper.
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    Bulletin of mathematical biology 41 (1979), S. 305-324 
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    Notes: Abstract This paper uses optimal control theory in conjunction with a Gompertzian type model for cellular growth to determine the optimal method of administering cycle non-specific chemotherapy or more generally the optimal durations of treatment and rest periods during chemotherapy. The performance critera employed to determine the relative merits of the therapy include not only the destruction of malignant cells, but also the sparing of a critical normal tissue. Since these criteria are at odds with one another, the solutions are found which satisfy the Pareto optimality conditions.
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    Bulletin of mathematical biology 41 (1979), S. 325-342 
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    Notes: Abstract Oscillations governed by generalized Volterra-Gause-Witt equations to include retardation effects in population dynamics are considered. Models containing either small or significant time delay are discussed. The Krylov-Bogoliubov-Mitropolskii perturbation method and its extension for differential equations with retarded argument is used.
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    Bulletin of mathematical biology 41 (1979), S. 357-364 
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    Notes: Abstract A general solution to the dynamical equation for the probability distribution associated withn interacting species is obtained by employing the author's generic canonical expression for the rate functions. Interacting species models with limit-cycle dynamics and no stable equilibrium points feature probability distributions that are asymptotic for large values oft to Dirac δ-distributions concentrated on the limit-cycles, as illustrated here for an analytically solvable two-species model. For ann-species Volterra model, a stationary or temporally-averaged probability distribution should generally be much more complicated than the specialized Poisson form studied by Kerner and others.
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    Bulletin of mathematical biology 41 (1979), S. 365-385 
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    Notes: Abstract A set of coupled nonlinear differential equations, determining the concentration profiles and electric potentials valid for isothermal transport of ions and molecules across a diffusion barrier are formulated, using a correction to the limiting expression for chemical potential gradients and the molecular expression for frictional force. These differential equations are similar to Nernst-Planck equations and reduce to these under appropriate approximations. Solutions of these equations valid under specified conditions are presented. Expressions for permeability, concentration profiles of many ion systems are included.
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    Bulletin of mathematical biology 41 (1979), S. 629-640 
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    Notes: Abstract The global flow equations of nonequilibrium thermodynamics for a single nonelectrolyte solute and water passing through a membrane are obtained by solving the local equations of motion. The method follows that developed for the general,n-solute case in the previous paper (Mikulecky, 1978). It is easily seen in this simple case that the passage from local interactions, formulated as position dependent frictional interactions in the equations of motion, to ghe global result involves a loss of any simple way of identifying particulars about local information. Two particular cases are analyzed in further detail: the case of no interaction within the pore and the case of constant interaction for both solute and solvent across the pore. In the former case, Onsager reciprocity survives in the global result if a self-consistent definition of the partial viscosity coefficients is used, while in the latter case, reciprocity is lost. Since, in many biologically interesting cases, the presence of interaction of the type considered here is likely to occur, the reciprocity condition should not automatically be assumed to hold.
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    Bulletin of mathematical biology 41 (1979), S. 665-686 
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    Notes: Abstract This study is related to a model describing the behavior of barium-treatedAplysia neurons generating regular burst-plateau patterns. The model is represented by an autonomous dynamical system, defined inR 4 and depending on a small parameter. This paper is restricted to the qualitative study of three “reduced systems” deduced from the “complete system”. Part of the study is performed with the use of the qualitative theory of singular perturbations. The predicted behaviors are compared with experimental results.
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    Bulletin of mathematical biology 41 (1979), S. 641-664 
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    Notes: Abstract Using linear stability analysis, the qualitative stability properties of open nonlinear chemical systems, in which reactions of any order may occur, will be studied. Systems will be classified in three fundamental classes: trees, cycles and loops, according to their knot graphs. The study of the Jacobian matrix for the kinetic equations of the system shows that the symmetrizability by a particular procedure (calledD-symmetrizability) is a sufficient condition for stability. It has been proved that tree-graphs always satisfy the above condition. For the cycle-graphs, theD-symmetrizability condition leads to a cyclic relation between forward and reverse steady state flows. The stability may be assured, even if the cyclic relation is not satisfied, providing that the “symmetry breaking” be lower than an upper bound; further alternative criteria for stability of cycles have been derived. All these results are independent of the number of diffusive exchanges with the environment.
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    Bulletin of mathematical biology 41 (1979), S. 687-705 
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    Notes: Abstract The basis of an analytical description of the behaviour of large random nets of binary elements of the type first investigated in detail by S. A. Kauffman is presented. It is shown that information about the network dynamics can be deduced from quite general considerations of the properties of the state transition graph and matrix. An expression for the matrix elements of the state transition matrix in terms of the Boolean function specification of the net is derived. Using these ideas the distribution of limit cycle lengthsl for a completely random net is calculated and shown to bex 1/l, a result which agrees well with experimental data.
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    Bulletin of mathematical biology 41 (1979), S. 707-724 
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    Notes: Abstract The state-transition matrix description of Kauffman binary networks described in the previous paper is further developed to obtain an analytical expression for the fraction of states involved in limit cycles as a function of the network size and connectivity. The result obtained for totally connected networks agrees with that derived from quite different considerations by other workers. For low connectivity networks the results are in qualitative agreement with the experimental data of Kauffman but there is a quantitative discrepancy which remains to be resolved.
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