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  • 1
    Electronic Resource
    Electronic Resource
    Existence of periodic solutions of a nerve equation (1976)
    Springer
    Biological cybernetics 21 (1976), S. 37-39 
    Details
    ISSN: 1432-0770
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Computer Science , Physics
    Notes: Abstract A recently developed continuous neuronal model is transformed into a system which is closely related to the theory of cellular control processes. In a broad region of parameter values periodic solutions of the equations describing the neuron do exist.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00326671
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  • 2
    Electronic Resource
    Electronic Resource
    Generation of the nervous impulse and periodic oscillations (1976)
    Springer
    Biological cybernetics 23 (1976), S. 211-218 
    Details
    ISSN: 1432-0770
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Computer Science , Physics
    Notes: Abstract Usually the models for the excitation and propagation of the nervous impulse are studied either in the space-clamp situation or on a model axon extended on both sides to infinity. Following Fitzhugh in the present paper the release of an impulse train at the axon hillock is studied within the scope of Fitzhugh's BVP model. The existence and stability of periodic oscillations are studied by direct methods, also the relation to Liénard's equation. The exact correspondence between the BVP model and the socalled Nagumo-equation is established. For typical examples the solutions are computed by numerical methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00340337
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  • 3
    Electronic Resource
    Electronic Resource
    A basic mathematical model of the immune response (1995)
    Woodbury, NY : American Institute of Physics (AIP)
    Chaos 5 (1995), S. 155-161 
    Details
    ISSN: 1089-7682
    Source: AIP Digital Archive
    Topics: Physics
    Notes: Interaction of the immune system with a target population of, e.g., bacteria, viruses, antigens, or tumor cells must be considered as a dynamic process. We describe this process by a system of two ordinary differential equations. Although the model is strongly idealized it demonstrates how the combination of a few proposed nonlinear interaction rules between the immune system and its targets are able to generate a considerable variety of different kinds of immune responses, many of which are observed both experimentally and clinically. In particular, solutions of the model equations correspond to states described by immunologists as "virgin state,'' "immune state'' and "state of tolerance.'' The model successfully replicates the so-called primary and secondary response. Moreover, it predicts the existence of a threshold level for the amount of pathogen germs or of transplanted tumor cells below which the host is able to eliminate the infectious organism or to reject the tumor graft. We also find a long time coexistence of targets and immune competent cells including damped and undamped oscillations of both. Plausibly the model explains that if the number of transformed cells or pathogens exeeds definable values (poor antigenicity, high reproduction rate) the immune system fails to keep the disease under control. On the other hand, the model predicts apparently paradoxical situations including an increased chance of target survival despite enhanced immune activity or therapeutically achieved target reduction. A further obviously paradoxical behavior consists of a positive effect for the patient up to a complete cure by adding an additional target challenge where the benefit of the additional targets depends strongly on the time point and on their amount. Under periodically pulsed stimulation the model may show a chaotic time behavior of both target growth and immune response. © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.166098
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  • 4
    Electronic Resource
    Electronic Resource
    On manifolds of equilibria in the selection model for multiple alleles (1975)
    Springer
    Journal of mathematical biology 1 (1975), S. 321-330 
    Details
    ISSN: 1432-1416
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary A population subjected to the Fisher-Wright-Haldane selection model for three alleles always converges to a fixed state even if the matrix of viabilities is singular. In this case there can occur lines of equilibrium points being partly attracting, partly repulsive. If there are n alleles then the manifolds of equilibrium points can be of every dimension between zero and n−1. Some criteria for such manifolds to be asymptotically stable are derived in terms of determinants and eigenvalues of the viability matrix.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00279851
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  • 5
    Electronic Resource
    Electronic Resource
    Delays in physiological systems (1979)
    Springer
    Journal of mathematical biology 8 (1979), S. 345-364 
    Details
    ISSN: 1432-1416
    Keywords: Differential-Difference equations ; Delays ; Biochemical reactions ; Lateral inhibition ; Entire functions ; Periodic and chaoticoscillations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary In comparison to most physical or chemical systems, biological systems are of extreme complexity. In addition the time needed for transport or processing of chemical components or signals may be of considerable length. Thus temporal delays have to be incorporated into models leading to differential-difference and functional differential equations rather than ordinary differential equations. A number of examples, on different levels of biological organization, demonstrate that delays can have an influence on the qualitative behavior of biological systems: The existence or non-existence of instabilities and periodic or even chaotic oscillations can entirely depend on the presence or absence of delays with appropriate duration.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275831
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  • 6
    Electronic Resource
    Electronic Resource
    Nonhomogeneous spatial distributions of populations (1975)
    Springer
    Journal of mathematical biology 2 (1975), S. 165-176 
    Details
    ISSN: 1432-1416
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary Spatial inhomogeneities such as nonconstant population densities usually will be attributed to random effects or to an inhomogeneous substrate. Such an explanation may be incorrect since from certain chemical reactions it is known that the interaction of species together with diffusion may generate nonhomogeneous spatial structures. However, the effect of boundary conditions has been so far neglected. In this paper nonlinear and linear interaction-diffusion models are investigated under various side-conditions by analytic methods and by computer simulations. A remarkable fact, as compared with earlier results in the field, is the example of an interaction-diffusion process which in the whole space has only the constant as a stable limit distribution, whereas the introduction of a side condition, e.g. a population reservoir or a barrier, leads to standing spatial population waves.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275801
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  • 7
    Electronic Resource
    Electronic Resource
    The dynamics of recurrent inhibition (1984)
    Springer
    Journal of mathematical biology 19 (1984), S. 211-225 
    Details
    ISSN: 1432-1416
    Keywords: lateral inhibition ; recurrent inhibition ; delayed feedback ; multiple steady states ; chaos ; epilepsy ; hippocampus ; receptor ; physiology ; difference-differential equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A heuristic model for the dynamics of recurrent inhibition, emphasizing non-linearities arising from the stoichiometry of transmitter-receptor interactions and time delays due to finite feedback pathway transmission times, is developed and analyzed. It is demonstrated that variation in model parameters may lead to the existence of multiple steady states, and the local stability of these are analyzed as well as the occurrence of switching behaviour between them. As an example of the applicability of this model, parameters are estimated for the hippocampal mossy fibre-CA3 pyramidal cell-basket cell complex. Numerically simulated responses of this system to alterations in presynaptic drive and titration of inhibitory transmitter receptors by penicillin are presented. Numerical simulations indicate the existence of multiple bifurcations between periodic solutions, as well as the existence of bifurcations to chaotic solutions, as presynaptic drive and receptor density are varied. It is hypothesized that the model offers insight into the sequences of events recorded in single CA3 pyramidal cells following the application of penicillin, a specific inhibitory receptor blocking agent.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00277747
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  • 8
    Electronic Resource
    Electronic Resource
    Book reviews (1974)
    Springer
    Theoretical and applied genetics 45 (1974), S. 222-224 
    Details
    ISSN: 1432-2242
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00265003
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  • 9
    Electronic Resource
    Electronic Resource
    Principles of self-generation and self-maintenance (1985)
    Springer
    Acta biotheoretica 34 (1985), S. 125-137 
    Details
    ISSN: 1572-8358
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Abstract Living systems are characterized as self-generating and self-maintaining systems. This type of characterization allows integration of a wide variety of detailed knowledge in biology. The paper clarifies general notions such as processes, systems, and interactions. Basic properties of self-generating systems, i.e. systems which produce their own parts and hence themselves, are discussed and exemplified. This makes possible a clear distinction between living beings and ordinary machines. Stronger conditions are summarized under the concept of self-maintenance as an almost unique character of living systems. Finally, we discuss the far-reaching consequences that the principles of self-generation and self-maintenance have for the organization, structure, function, and evolution of single- and multi-cellular organisms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00046778
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  • 10
    Electronic Resource
    Electronic Resource
    Mathematical model and simulation of retina and tectum opticum of lower vertebrates (1987)
    Springer
    Acta biotheoretica 36 (1987), S. 179-212 
    Details
    ISSN: 1572-8358
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Abstract The processing of information within the retino-tectal visual system of amphibians is decomposed into five major operational stages, three of them taking place in the retina and two in the optic tectum. The stages in the retina involve (i) a spatially local high-pass filtering in connection to the perception of moving objects, (ii) separation of the receptor activity into ON- and OFF-channels regarding the distinction of objects on both light and dark backgrounds, (iii) spatial integration via near excitation and far-reaching inhibition. Variation of the spatial range of excitation and inhibition allows to account for typical activities observed in a variety of classes of retina ganglion cells. Mathematical description of the operations in the tectum opticum include (i) spatial summation of retinal output (mainly of class-2 and class-3 retina ganglion cells), and (ii) direct or indirect lateral inhibition between tectal cells. In the computer simulation, first the output of the mathematical retina model is computed which, then, is used as the input to the tectum model. The full spatio-temporal dynamics is taken into account. The simulations show that different combinations of strength of lateral inhibition on the one side and the response properties of the retina ganglion cells on the other side determine the response properties of tectal cell types involved in object recognition.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00052064
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