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  • Articles  (22,542)
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  • Articles  (22,542)
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  • 2010-2014
  • 2005-2009  (9)
  • 1985-1989  (13,130)
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  • 1
    Electronic Resource
    Electronic Resource
    Memory but no suppression in low-dimensional symmetric idiotypic networks (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 223-246 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We present a new symmetric model of the idiotypic immune network. The model specifies clones of B-lymphocytes and incorporates: (1) influx and decay of cells; (2) symmetric stimulatory and inhibitory idiotypic interactions; (3) an explicit affinity parameter (matrix); (4) external (i.e. non-idiotypic) antigens. Suppression is the dominant interaction, i.e. strong idiotypic interactions are always suppressive. This precludes reciprocal stimulation of large clones and thus infinite proliferation. Idiotypic interactions first evoke proliferation, this enlarges the clones, and may in turn evoke suppression. We investigate the effect of idiotypic interactions on normal proliferative immune responses to antigens (e.g. viruses). A 2-D, i.e. two clone, network has a maximum of three stable equilibria: the virgin state and two asymmetric immune states. The immune states only exist if the affinity of the idiotypic interaction is high enough. Stimulation with antigen leads to a switch from the virgin state to the corresponding immune state. The network therefore remembers antigens, i.e. it accounts for immunity/memory by switching beteen multiple stable states. 3-D systems have, depending on the affinities, 9 qualitatively different states. Most of these also account for memory by state switching. Our idiotypic network however fails to account for the control of proliferation, e.g. suppression of excessive proliferation. In symmetric networks, the proliferating clones suppress their anti-idiotypic suppressors long before the latter can suppress the former. The absence of proliferation control violates the general assumption that idiotypic interactions play an important role in immune regulation. We therefore test the robustness of these results by abandoning our assumption that proliferation occurs before suppression. We thus define an “escape from suppression” model, i.e. in the “virgin” state idiotypic interactions are now suppressive. This system erratically accounts for memory and never for suppression. We conclude that our “absence of suppression from idiotypic interactions” does not hinge upon our “proliferation before suppression” assumption.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02458444
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  • 2
    Electronic Resource
    Electronic Resource
    Book reviews (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 287-291 
    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/BF02458448
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  • 3
    Electronic Resource
    Electronic Resource
    Editorial: Software survey section (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. I 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02458449
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  • 4
    Electronic Resource
    Electronic Resource
    Upper and lower bounds from the maximum principle. Intracellular diffusion with Michaelis-Menten kinetics (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 325-335 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Analytical bounding functions for diffusion problems with Michaelis-Menten kinetics were recently presented by Anderson and Arthurs, 1985 (Bull. math. Biol. 47, 145–153). Their methods, successful to some extent for a small range of parameters, has the disadvantage of providing a weak upper bound. The optimal approach for the use of one-line bounding kinetics is presented. The use of two-line bounding kinetics is also shown, in order to give, sufficient accuracy in those cases where the one-line approach does not provide satisfactory results. The bounding functions provide excellent upper and lower bounds on the true solution for the entire range of kinetic and transport parameters.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02460111
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  • 5
    Electronic Resource
    Electronic Resource
    The threshold of survival for systems in a fluctuating environment (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 311-323 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Thresholds for survival and extinction are important for assessing the risk of mortality in systems exposed to exogeneous stress. For generic, rudimentary population models and the classical resource-consumer models of Leslie and Gallopin, we demonstrate the existence of a survival threshold for situations where demographic parameters are fluctuating, generally, in a nonperiodic manner. The fluctuations are assumed, to be generated by exogenous, anthropogenic stresses such as toxic chemical exposures. In general, the survival threshold is determined by a relationship between mean stress measure in organisms to the ratio of the population intrinsic growth rate and stress response rate.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02460110
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  • 6
    Electronic Resource
    Electronic Resource
    Plant strategies and the dynamics and structure of plant communities (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 409-411 
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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/BF02460116
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  • 7
    Electronic Resource
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    Editorial: Viewpoint (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 415-415 
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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/BF02460080
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  • 8
    Electronic Resource
    Electronic Resource
    Stochastic models for subpopulation emergence in heterogeneous tumors (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 731-747 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A stochastic analog to a deterministic model describing subpopulation emergence in heterogeneous tumors is developed. The resulting system is described by the Fokker-Planck or forward Kolmogorov equation. A finite element approach for the numerical solution to this equation is described. Four biological and clinical scenarios are simulated (emergence of heterogeneity, exclusion of a subpopulation, and induction of drug resistance in both pure and heterogeneous tumors). The results of the simulations show that the stochastic model describes the same basic dynamics as its deterministic counterpart via a convective component, but that for each simulation a distribution of tumor sizes and mixes can also be derived from a diffusive component in the model. These distributions yield estimates for subpopulation extinction probabilities. The biological and clinical relevance of these results are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02459658
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  • 9
    Electronic Resource
    Electronic Resource
    Deductive biology—Some cautious steps (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 45-58 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract For certain environments, the Darwinian model allows unique prediction of a function that any surviving system adapted to such an environment has to perform. This is the case for those environments that determine a “survival functional” of position in space-time of known shape. Purely temporal survival functionals can be distinguished from spatial and mixed ones. In each case, there exists an optimum path in combined physical and (reduced) metabolic space. Dependent on the admissible error, approximate solutions of different complexity are sufficient. All solutions possess an afferent, a central, and an efferent part. Within this general frame, specific, “probably simplest”, solutions are proposed for adaptive chemotaxis, insect locomotion, lower vertebrates locomotion, higher vertebrates locomotion, chronobiological systems, and immune systems, respectively—or rather, for the underlying functionals.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02463129
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  • 10
    Electronic Resource
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    The role of glucagon in the regulation of blood glucose: Model studies (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 59-77 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Mathematical models afford a procedure of unifying concepts and hypotheses by expressing quantitative relationships between observables. The model presented indicates the roles of both insulin and glucagon as regulators of blood glucose, albeit in different ranges of the blood glucose concentrations. Insulin secretion is induced during hyperglycemia, while glucagon secretion results during hypoglycemia. These are demonstrated by simulations of a mathematical model conformed to data from the oral glucose tolerance test and the insulin infusion test in normal control subjects and stable and unstable diabetic patients. The model studies suggest the parameters could prove of value in quantifying the diabetic condition by indicating the degree of instability.
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    URL: http://dx.doi.org/10.1007/BF02463130
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  • 11
    Electronic Resource
    Electronic Resource
    A note on dynamics of population with history-dependent birth rate (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 123-131 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A model for the dynamics of a single-species population whose birth rate depends on densities of previous generations is introduced. A difference equation formulation is proposed and the solutions classified for the various parameter values. Data from an experimental population of mice growing in limited space is cited and compared with the model predictions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02463134
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  • 12
    Electronic Resource
    Electronic Resource
    Tension field theories for soft tissues (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 161-182 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract All soft tissues are modeled as either one-dimensionalstrings, two-dimensionalmembranes, or three-dimensionalsolids. Attention is restricted to tissues in which one of the principal stress components is large and positive in comparison with the other negligible components. Results indicate the following: (1) If a deformed string isconstrained to lie on a surface and is free of tangential pressure, the tension is carried by rays which are geodesics of the surface. If a string or membrane isfree to deform in space without normal pressure, the tension rays are straight lines. If a membrane deforms without tangential surface loads, the tension rays are always geodesics on the deformed surface. If a solid deforms without body forces, the tension rays are straight lines. (2) The stress in a string is a constant if the string is free of tangential pressure and has constant cross-sectional area. The stress in flat tension fields free of tangential surface loads decays inversely with distance along a tension ray from the edge of regression. The stress in a spherically symmetric tension field free of body forces decays inversely with the square of the distance from the center of the sphere. (3) Stress singularities can occur in soft tissues, such as at the corners of a closed rectangular hole in a flat membrane strip. (4) The tension rays in the torsion of soft annular membranes are more steeply inclined from the radial direction than the tension rays for hard metals equally displaced.
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    URL: http://dx.doi.org/10.1007/BF02461433
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  • 13
    Electronic Resource
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    Algorithms for the optimal identification of segment neighborhoods (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 39-54 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Two algorithms for the efficient identification of segment neighborhoods are presented. A segment neighborhood is a set of contiguous residues that share common features. Two procedures are developed to efficiently find estimates for the parameters of the model that describe these features and for the residues that define the boundaries of each segment neighborhood. The algorithms can accept nearly any model of segment neighborhood, and can be applied with a broad class of best fit functions including least squares and maximum likelihood. The algorithms successively identify the most important features of the sequence. The application of one of these methods to the haemagglutinin protein of influenza virus reveals a possible mechanism for conformational change through the finding of a break in a strong heptad repeat structure.
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    URL: http://dx.doi.org/10.1007/BF02458835
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  • 14
    Electronic Resource
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    Approximate matching of regular expressions (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 5-37 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Given a sequenceA and regular expressionR, theapproximate regular expression matching problem is to find a sequence matchingR whose optimal alignment withA is the highest scoring of all such sequences. This paper develops an algorithm to solve the problem in timeO(MN), whereM andN are the lengths ofA andR. Thus, the time requirement is asymptotically no worse than for the simpler problem of aligning two fixed sequences. Our method is superior to an earlier algorithm by Wagner and Seiferas in several ways. First, it treats real-valued costs, in addition to integer costs, with no loss of asymptotic efficiency. Second, it requires onlyO(N) space to deliver just the score of the best alignment. Finally, its structure permits implementation techniques that make it extremely fast in practice. We extend the method to accommodate gap penalties, as required for typical applications in molecular biology, and further refine it to search for substrings ofA that strongly align with a sequence inR, as required for typical data base searches. We also show how to deliver an optimal alignment betweenA andR in onlyO(N+logM) space usingO(MN logM) time. Finally, anO(MN(M+N)+N 2logN) time algorithm is presented for alignment scoring schemes where the cost of a gap is an arbitrary increasing function of its length.
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    URL: http://dx.doi.org/10.1007/BF02458834
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  • 15
    Electronic Resource
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    Codon preference and primary sequence structure in protein-coding regions (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 95-115 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The stochastic complexity of a data base of 365 protein-coding regions is analysed. When the primary sequence is modeled as a spatially homogeneous Markov source, the fit to observed codon preference is very poor. The situation improves substantially when a non-homogeneous model is used. Some implications for the estimation of species phylogeny and substitution rates are discussed.
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    URL: http://dx.doi.org/10.1007/BF02458838
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  • 16
    Electronic Resource
    Electronic Resource
    Tutorial on large deviations for the binomial distribution (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 125-131 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We present, in an easy to use form, the large deviation theory of the binomial distribution: how to approximate the probability ofk or more successes inn independent trials, each with success probabilityp, when the specified fraction of successes,a≡k/n, satisfies 0〈p〈a〈1.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02458840
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  • 17
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    Editorial: Software survey section (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. I 
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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/BF02458844
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  • 18
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    A continuous analog for RNA folding (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 167-171 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A linear segment in which a number of pairs of intervals of equal length are identified as potential stems is the subject of a folding problem analogous to inference of RNA secondary structure. A quantity of free energy (or equivalently, energy per unit length) is associated with each stem, and the various types of loops are assigned energy costs as a function of their lengths. Inference of stable structures can then be carried out in the same way as in RNA folding. More important, perturbation of stem lengths and energy densities (modelling various mutational processes affecting nucleotide sequences) allows the delineation of domains of stability of various foldings, through the explicit calculation of their boundaries, in a low-dimensional parameter space.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02458842
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  • 19
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    Prediction of linear and non-linear responses of MGB neurons by system identification methods (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 337-346 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In sensory physiology, various System Identification methods are implemented to formalized stimulus-response relationships. We applied the Volterra approach for characterizing input-output relationships of cells in the medial geniculate body (MGB) of an awake squirrel monkey. Intraspecific communication calls comprised the inputs and the corresponding cellular evoked responses—the outputs. A set of vocalization was used to calculate the kernels of the transformation, and these kernels subserved to predict the responses of the cell to a different set of vocalizations. It was found that it is possible to predict the response (PSTH) of MGB cells to natural vocalizations, based on envelopes of the spectral components of the vocalization. Some of the responses could be predicted by assuming a linear transformation function, whereas other responses could be predicted by non-linear (second order) kernels. These two modes of transformation, which are also reflected by a distinct spatial distribution of the linearvis-à-vis non-linear responding cells, apparently representa new revelation of parallel processing of auditory information.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02460112
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  • 20
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    Diffusion limited immunochemical sensing (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 359-379 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The time-dependent surface coverage of antigen-antibody complexes for a sensor in which antigens are bound to surface immobilized antibodies is determined analytically. Assuming a reversible first order reaction between the antigens and antibodies, a model is derived describing the dynamical response of the sensor. The surface coverage is related explicitly to the antigen concentration which is of special interest in experimental situations. The stationary state and short time behaviour are determined explicitly. Several illustrations of the full solution are provided.
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    URL: http://dx.doi.org/10.1007/BF02460114
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  • 21
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    An informational approach to reaction times (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 347-358 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Simple reaction time is the minimum time required to respond to a signal such as a steady light or tone. Such a reaction time is taken to be the time required for transmission of a fixed quantity of information, ΔH, from stimulus to subject. That is, information summation replaces energy summation. This information is calculated from consideration of the quantum nature of the stimulus. The theoretically derived equation for reaction time is fitted to experimental data. Piéron's empirical law for reaction time is obtained as an approximation from a proposed informational equation. The exponent in Piéron's law is found to be the same as the exponent in the power law of sensation. Threshold appears to be the smallest stimulus capable of transmitting the quantity of information ΔH.
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    URL: http://dx.doi.org/10.1007/BF02460113
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  • 22
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    The Third World Conference on Mathematics at the Service of Man (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 413-413 
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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/BF02460117
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  • 23
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    Editorial: Software survey section (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. I 
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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/BF02460118
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  • 24
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    Design of FIR filters with reduced computations (1989)
    Springer
    Circuits, systems and signal processing 8 (1989), S. 17-23 
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    ISSN: 1531-5878
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology
    Notes: Abstract The performance of a symmetric nonrecursive filter can be improved by multiple use of the same filter. The method is based on an Amplitude Change Function (ACF). An approach to the design of nonrecursive filters using an ACF is discussed in this paper. The prototype filter chosen is a Recursive Running Sum (RRS) filter which does not require any multipliers for its implementation. The required filter specifications are met by multiple use of the RRS filters. The overall filter requires a much smaller number of multiplications and adders than the one designed using the conventional method. It is shown that this method provides reduced noise due to coefficient quantization and product quantization compared with the conventional design technique.
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    URL: http://dx.doi.org/10.1007/BF01598743
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  • 25
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    Statistical analysis of two nonlinear least-squares estimators of sine-wave parameters in the colored-noise case (1989)
    Springer
    Circuits, systems and signal processing 8 (1989), S. 3-15 
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    ISSN: 1531-5878
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology
    Notes: Abstract This paper establishes the large-sample accuracy properties of two nonlinear least-squares estimators (NLSE) of sine-wave parameters: the basic NLSE, which ignores the possible correlation of the noise, and the optimal NLSE, which, besides the sine-wave parameters, also estimates the noise correlation (appropriately parametrized). It is shown that these two NLS estimators have thesame accuracy in large samples. This result provides complete justification for preferring the computationally less expensive basic NLSE over the “optimal” NLSE. Both estimators are shown to achieve the Cramér-Rao Bound (CRB) as the sample size increases. A simple explicit expression for the CRB matrix is provided, which should be useful in studying the performance of sine-wave parameter estimators designed to work in the colored-noise case.
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    URL: http://dx.doi.org/10.1007/BF01598742
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  • 26
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    Synthesis of two-dimensional losslessm-Ports with prescribed scattering matrix (1989)
    Springer
    Circuits, systems and signal processing 8 (1989), S. 97-119 
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    ISSN: 1531-5878
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology
    Notes: Abstract Multidimensional lossless networks are of special interest for use as reference structures for multidimensional wave digital filters [l]–[3]. The starting point of the presented synthesis procedure for two-dimensional representatives of the networks mentioned is a scattering matrix description of the desired multiport. This given matrix is assumed to have those properties which have turned out to be necessary [9], [10] for any scattering matrix of a multidimensional lossless network. The method presented for the synthesis of 2-D reactancem-ports is based mainly on known properties of block-companion matrices and the factorization of a univariable rational matrix which is discrete para-Hermitian and nonnegative definite on the unit circle. The resulting network always contains only a minimal number of frequency-dependent building elements. No restrictions are made concerning the coefficients of the rational entries of the scattering matrix; they may be either real or complex, so as to include even complex networks which are of special interest for multi-dimensional wave digital filters [3].
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    URL: http://dx.doi.org/10.1007/BF01598747
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  • 27
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    Systems which possessg- andh- Representations (1989)
    Springer
    Circuits, systems and signal processing 8 (1989), S. 145-162 
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    ISSN: 1531-5878
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology
    Notes: Abstract Examples are given concerning the range of applicability of recent representation results that provide a means of studying the input-output properties of nonlinear systems in terms of the familiar impulse-response concept, and which extend the concept of integral transformation to nonlinear maps. We show that such representations, which we call “g-” and “h-representations,” exist for important classes of systems governed by nonlinear integral equations. In particular, it is proved that a large class of maps that have Volterra series representations also have these representations.
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    URL: http://dx.doi.org/10.1007/BF01599935
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  • 28
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    Analysis of the transient behavior of kidney models (1978)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 211-221 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02461436
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    Ecosystem-like behavior of a random-interaction model I (1978)
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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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    Mathematical analysis of cancer chemotherapy: The effects of chemotherapeutic agents on the cell cycle traverse (1978)
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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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    Global stability in a class of prey-predator models (1978)
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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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    A theoretical study of the temperature distribution in skeletal muscle undergoing rhythmic contractions (1978)
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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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    A comment on Bobisud's paper on evolution of cannibalism (1978)
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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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    Prediction of light-induced transient thermal response of the retina (1978)
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    Bulletin of mathematical biology 40 (1978), S. 547-547 
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    Dynamical similarity and the theory of biological transformations (1978)
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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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    Distributed sensory coding applied to simulations of iconic storage and metacontrast (1978)
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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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    Theoretical study of nasal deposition (1978)
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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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    What do mathematical models tell us about circadian clocks? (1978)
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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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    Electric field distribution, ionic selectivity and permeability in nerve (1978)
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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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    Self-similar solution of a distributed model for transcapillary exchange and solute removal by the liver (1978)
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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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    Steady-state distribution of bacteria chemotactic toward oxygen (1978)
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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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    On the identification by filtering techniques of a biologicaln-compartment model in which the transport rate parameters are assumed to be stochastic processes (1978)
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    Bulletin of mathematical biology 40 (1978), S. 651-660 
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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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    Galerkin-Ritz procedures for approximate solutions to systems of reaction-diffusion equations (1978)
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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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    Mathematical models in biological discovery. Lecture notes in biomathematics (1978)
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    Bulletin of mathematical biology 40 (1978), S. 873-875 
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    The alignment of protein structures in three dimensions (1989)
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    Bulletin of mathematical biology 51 (1989), S. 55-78 
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    Notes: Abstract This article extends the use of dynamic programming algorithms in molecular sequence comparison to the alignment of the α-carbon (Cα-) coordinates of two protein structures in three dimensions. The algorithm is described in detail and is applied to the comparison of α-lactalbumin with both hen egg white lysozyme and T4 lysozyme. In the first case, the structures are similar, while the second comparison is between two distantly related molecules. References are made to the usual sequence alignments. A variety of complementary methods are introduced to display the results.
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    Stochastic models for heterogeneous DNA sequences (1989)
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    Bulletin of mathematical biology 51 (1989), S. 79-94 
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    Notes: Abstract The composition of naturally occurring DNA sequences is often strikingly heterogeneous. In this paper, the DNA sequence is viewed as a stochastic process with local compositional properties determined by the states of a hidden Markov chain. The model used is a discrete-state, discreteoutcome version of a general model for non-stationary time series proposed by Kitagawa (1987). A smoothing algorithm is described which can be used to reconstruct the hidden process and produce graphic displays of the compositional structure of a sequence. The problem of parameter estimation is approached using likelihood methods and an EM algorithm for approximating the maximum likelihood estimate is derived. The methods are applied to sequences from yeast mitochondrial DNA, human and mouse mitochondrial DNAs, a human X chromosomal fragment and the complete genome of bacteriophage lambda.
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    Weak hierarchies associated with similarity measures—An additive clustering technique (1989)
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    Bulletin of mathematical biology 51 (1989), S. 133-166 
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    Notes: Abstract A new and apparently rather useful and natural concept in cluster analysis is studied: given a similarity measure on a set of objects, a sub-set is regarded as a cluster if any two objectsa, b inside this sub-set have greater similarity than any third object outside has to at least one ofa, b. These clusters then form a closure system which can be described as a hypergraph without triangles. Conversely, given such a system, one may attach some weight to each cluster and then compose a similarity measure additively, by letting the similarity of a pair be the sum of weights of the clusters containing that particular pair. The original clusters can be reconstructed from the obtained similarity measure. This clustering model is thus located between the general additive clustering model of Shepard and Arabie (1979) and the standard hierarchical model. Potential applications include fitting dendrograms with few additional nonnested clusters and simultaneous representation of some families of multiple dendrograms (in particular, two-dendrogram solutions), as well as assisting the search for phylogenetic relationships by proposing a somewhat larger system of possibly relevant “family groups”, from which an appropriate choice (based on additional insight or individual preferences) remains to be made.
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    Computational experience with an algorithm for tetrangle inequality bound smoothing (1989)
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    Bulletin of mathematical biology 51 (1989), S. 173-194 
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    Notes: Abstract An important component of computer programs for determining the solution conformation of proteins and other flexible molecules from nuclear magnetic resonance data are the so-called “bound smoothing algorithms”, which compute lower and upper limits on the values of all the interatomic distances from the relatively sparse set which can usually be measured experimentally. To date, the only methods efficient enough for use in large problems take account of only the triangle inequality, but an appreciable improvement in the precision of the limits is possible if the algebraic relations between the distances among each quadruple of atoms are also considered. The goal of this paper is to use a recently improved algorithm for computing these “tetrangle inequality limits” to determine just how much improvement really is possible, given the types of experimental data that are usually available.
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    Wavelength selection mechanism in the Gierer-Meinhardt model (1989)
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    Bulletin of mathematical biology 51 (1989), S. 207-216 
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    Notes: Abstract We apply the concept of marginal stability hypothesis, which has been proposed for solving the problem of dendritic crystal growth, to the pattern selection problem in the Gierer-Meinhardt models. In the case of a large system, the system selects a definite wavelength of the ultimate spatial pattern when the unstable homogeneous steady state is locally disturbed. The numerical results are analyzed theoretically by means of the marginal stability hypothesis, and they are in good agreement with it. Biologically, these results imply why for large systems the Gierer-Meinhardt model (and presumably other reaction-diffusion schemes) have the ability to explain the observation that pattern-generating mechanisms are remarkably insensitive to a wide range of environmental and experimental conditions.
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    Automata network theories in immunology: Their utility and their underdetermination (1989)
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    Bulletin of mathematical biology 51 (1989), S. 247-253 
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    Notes: Abstract Small networks of threshold automata are used to model complex interactions between populations of regulatory cells (helpers and suppressors, antigen specific and anti-idiotypic) which participate in the immune response. The models, being discrete and semiquantitative, are well adapted to the situation of incomplete information often encounteredin vivo. However, the dynamics of many different network structures usually end up in the same attractor set. Thus, many different theories are equivalent in their explicative power for the same facts. This property, known as underdetermination of the theories by the facts, is given a quantitative estimate. It appears that such an underdetermination, as a kind of irreductible complexity, can be expected in manyin vivo biological processes, even when the number of interacting and functionally coupled elements is relatively small.
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    Resilience and local stability in a nutrient-limited resource-consumer system (1989)
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    Bulletin of mathematical biology 51 (1989), S. 501-510 
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    Notes: Abstract The “paradox of enrichment” predicts that increasing the growth rate of the resource in a resource-consumer dynamic system, by nutrient enrichment, for example, can lead to local instability of the system—that is, to a Hopf bifurcation. The approach to the Hopf bifurcation is accompanied by a decrease in resilience (rate of return to equilibrium). On the other hand, studies of nutrient cycling in food webs indicate that an increase in the nutrient input rate usually results in increased resilience. Here these two apparently conflicting theoretical results are reconciled with a model of a nutrient-limited resource-consumer system in which the tightly recycled limiting nutrient is explicitly modelled. It is shown that increasing nutrient input may at first lead to increased resilience and that resilience decreases sharply only immediately before the Hopf bifurcation is reached.
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    Book reviews (1989)
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    Bulletin of mathematical biology 51 (1989), S. 537-544 
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    Centrifugal-order distributions in binary topological trees (1989)
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    Bulletin of mathematical biology 51 (1989), S. 511-536 
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    Notes: Abstract Statistical properties of topological binary trees are studied on the basis of the distribution of segments in relation to centrifugal order. Special attention is paid to the mean of this distribution in a tree as it will be used as a measure of tree topology. It will be shown how the expectation of the mean centrifugal order depends both on the size of the tree and on the mode of growth in the context of modelling the growth of tree structures. Observed trees can be characterized by their mean orders and procedures are described to find the growth mode that optimally corresponds to these data. The variance structure of the mean-order measure appears to be a crucial factor in these fitting procedures. Examples indicate that mean-order analysis is an accurate alternative to partition analysis that is based on the partitioning of segments over sub-tree pairs at branching points.
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    On limits to perception and pattern recognition (1989)
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    Bulletin of mathematical biology 51 (1989), S. 681-686 
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    Notes: Abstract We propose certain general conditions that we believe are reasonable for any pattern recognition algorithm. We find that these conditions give rise to paradoxical identification. The algorithms are incapable of distinguishing composite patterns and must be able to distinguish patterns at an atomistic level.
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    Microevolution in MioceneBrizalina (foraminifera) studied by canonical variate analysis and analysis of landmarks (1989)
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    Bulletin of mathematical biology 51 (1989), S. 657-679 
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    Notes: Abstract A stratigraphically oriented series of the Miocene foraminiferal speciesBrizalina mandoroveensis from Ikang, Cameroon, was analyzed both by conventional multivariate morphometric procedures and by the tensor biometric method of Bookstein (1986;Statist. Sci. 1, 181–142), a method which analyzes sets of landmark points rather than specific variables of shape or size. The conventional analysis used five size-measures upon 170 specimens from five stratigraphic levels; the tensor analysis encompassed six landmarks (12 coordinates) upon 50 specimens. Whereas certain features appeared in both analyses, such as the separation between levels one and five, the techniques did not always agree with respect to the interpretation of those findings or about most details in the sequence of mean phenotypes. The canonical variate analysis bases its ordination upon a general size factor (the meaning of which is obscured by the foreshortening of within-group variation which is built into the technique). The tensor analysis locates a similar ordination using mainly features of shape.
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    Analytic results for quantifying HIV infectivity (1989)
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    Bulletin of mathematical biology 51 (1989), S. 715-730 
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    Notes: Abstract In a separate paper, we developed a mathematical model describing HIV infection and used it to suggest experiments for quantifying characteristic viral parameters. In this paper we generalize the model to any well-mixed assay system. We also present complete and rigorous derivations of fundamental results needed for the design and analysis of HIV infectivity assays. The model is applicable to infectious agents with multiple receptors for their target cell (e.g. HIV, Epstein-Barr virus and Plasmodium), and to blockers (both reversible and irreversible), as long as blocker and target cells are the same diffusion compartment.
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    On Ebenman's model for the dynamics of a population with competing juveniles and adults (1989)
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    Bulletin of mathematical biology 51 (1989), S. 687-713 
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    Notes: Abstract A general version of a model of Ebenman for the dynamics of a population consisting of competing juveniles and adults is analyzed using methods of bifurcation theory. A very general existence results is obtained for non-trivial equilibria and non-negative synchronous two-cycles that bifurcate simultaneously at the critical valuer=1 of the inherent net reproductive rater. Stability is studied in this general setting near the bifurcation point and conditions are derived that determine which of these two bifurcating branches is the stable branch. These general results are supplemented by numerical studies of the asymptotic dynamics over wider parameter ranges where various other bifurcations and stable attractors are found. The implications of these results are discussed with respect to the effects on stability that age class competition within a population can have and whether such competition is stabilizing or destabilizing.
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    Phototransduction in cones: An inverse problem in enzyme kinetics (1989)
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    Bulletin of mathematical biology 51 (1989), S. 749-784 
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    Notes: Abstract Phototransduction is a process which links the absorption of photons by a rod or cone to the modulation of voltage across the cell membrane. An important feature of many vertebrate photoreceptors is a mechanism that adjusts the sensitivity and dynamics of the response to light according to the level of illumination. We construct a system of ordinary differential equations that models what are currently thought to be the important molecule mechanisms involved in phototransduction: this includes consideration of both intracellular enzyme kinetics and the properties of light-insensitive and light-sensitive conductances in the cone membrane. The system contains negative feedback whose functional form is determined by constraining the steady-state behaviour of the system. Despite the highly nonlinear nature of the system of ordinary differential equations, our methods permit us to derive an analytic expression for the first-order frequency response parametric in the steady-state value of only one dynamic variable, the light input. Various unknown kinetic parameters are found by fitting the model to experimental data on the first-order frequency response of cones measured at several mean light levels spanning a range of four log units. Good fits are obtained to the data, and the computed shape of the feedback function agrees qualitatively with recent experiment. Moreover, the model accounts for the dramatic speeding up of the response kinetics and the decrease in response gain with increasing light level.
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    On the geometry of space-and time-equivalent catastrophes (1978)
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    Bulletin of mathematical biology 40 (1978), S. 1-25 
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    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.
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    Prediction of limit cycles in mathematical models of biological oscillations (1978)
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    Bulletin of mathematical biology 40 (1978), S. 27-44 
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    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.
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    Least energy regulation of the arterial system (1978)
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    Bulletin of mathematical biology 40 (1978), S. 79-93 
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    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.
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    Pressure-flow relationships in a model for the arterial system (1978)
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    Bulletin of mathematical biology 40 (1978), S. 95-105 
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    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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    Conditions for periodic solutions of volterra differential systems (1978)
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    Bulletin of mathematical biology 40 (1978), S. 133-160 
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    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.
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    The kinetics of diffused substances in some biological systems (1978)
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    Bulletin of mathematical biology 40 (1978), S. 107-121 
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    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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    Heat transfer in perfused biological tissue—II. The “macroscopic” temperature distribution in tissue (1978)
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    Bulletin of mathematical biology 40 (1978), S. 183-199 
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    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.
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    A note on dipole sources in conducting biological tissues (1978)
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    Bulletin of mathematical biology 40 (1978), S. 201-209 
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    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.
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    A hydrodynamic model of aqueous flow in the posterior chamber of the eye (1978)
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    Bulletin of mathematical biology 40 (1978), S. 223-235 
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    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.
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    On the nerve impulse equation: The dynamic responses of nerve impulse (1978)
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    Bulletin of mathematical biology 40 (1978), S. 247-255 
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    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.
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    Metabolic model and dosimetric data for copper (1978)
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    Bulletin of mathematical biology 40 (1978), S. 265-269 
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    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.
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    Analytical conditions for the conservative form of the ecological equations (1978)
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    Bulletin of mathematical biology 40 (1978), S. 257-264 
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    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.
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    On the stochastic theory of compartments: III. General time-dependent reversible systems (1978)
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    Bulletin of mathematical biology 40 (1978), S. 237-246 
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    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.
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    Mathematical models of metabolic regulation, symposia biologica hungarica (1978)
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    Bulletin of mathematical biology 40 (1978), S. 271-272 
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    Optimal control of pests in the presence of predators (1978)
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    Bulletin of mathematical biology 40 (1978), S. 319-333 
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    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.
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    Transient behaviour of the single loop solute cycling model of the renal medulla (1978)
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    Bulletin of mathematical biology 40 (1978), S. 273-300 
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    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.
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    The Krylov-Bogoliubov-Mitropolskii method applied to models of population dynamics (1978)
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    Bulletin of mathematical biology 40 (1978), S. 335-345 
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    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.
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    A dynamical field theory for dissipative systems: The hierarchical structure of field thermodynamics (1978)
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    Bulletin of mathematical biology 40 (1978), S. 301-318 
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    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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    Reconstitution of the transition matrix from experimental data (1978)
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    Bulletin of mathematical biology 40 (1978), S. 369-376 
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    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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    Analytical solution to isometric mechanogram of hill’s model of striated muscle (1978)
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    Bulletin of mathematical biology 40 (1978), S. 359-368 
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    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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    Lag time in microbe growth as an age-dependent branching process with two phase types (1978)
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    Bulletin of mathematical biology 40 (1978), S. 347-358 
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    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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    Linear age-dependent population growth with harvesting (1978)
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    Bulletin of mathematical biology 40 (1978), S. 377-385 
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    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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    A comment on the Richardson-Miekisz field theory of thermodynamics (1978)
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    Bulletin of mathematical biology 40 (1978), S. 425-426 
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    Multiple speciation, competitive exclusion, evolutionary pattern, and the grand ensemble in volterra eco-dynamics (1978)
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    Bulletin of mathematical biology 40 (1978), S. 387-410 
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    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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    The stochastic theory of compartments—a mammillary system (1978)
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    Bulletin of mathematical biology 40 (1978), S. 427-427 
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    Ionic concentration profiles and charge distribution for the domain of a polarized spherical cell (1978)
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    Bulletin of mathematical biology 40 (1978), S. 411-424 
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    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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    An elementary approach to catastrophe theory (1978)
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    Bulletin of mathematical biology 40 (1978), S. 429-450 
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    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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    Mechanical behaviour of ventricular aneurysms (1978)
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    Bulletin of mathematical biology 40 (1978), S. 451-464 
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    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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    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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    URL: http://dx.doi.org/10.1007/BF02460773
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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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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02460599
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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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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 743-769 
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    Topics: Biology , Mathematics
    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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    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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)
    Springer
    Bulletin of mathematical biology 40 (1978), S. 791-805 
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    ISSN: 1522-9602
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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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    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF02460610
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