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A Synthetic Approach to Exotic Kinetics (With Examples)

  • Conference paper
Physics and Mathematics of the Nervous System

Part of the book series: Lecture Notes in Biomathematics ((LNBM,volume 4))

Abstract

The approach to be outlined in the following is “classical” in the sense that only concentrations of individual substances are considered, rather than potentials of chemical reactions; furthermore, use is made of the “nonexplicit” convention for catalyzed reactions, so that the mass action law is violated in the equations; finally, the existence of “pool” substances, i.e. non-exhaustible sources, is presupposed. Hence neither the elegant tools of the recent potential-theoretic “network” approach of Oster et al. (1973), nor the recent methods of algebraic redundance-reduction in closed mass-action systems and their generalizations (cf. Feinberg, 1972; Horn & Jackson, 1972; and Horn, 1973) can be exploited. However, the reader should be offered sufficient methods in order to be able to invent new, non-trivial exotic chemical circuits for himself.

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References

  • Abakumov, A.S., A.A. Gulyayev, and G.A. Gulyayev: Propagation of excitation in an active medium, Biophysics 15, 1113–1120 (1970).

    Google Scholar 

  • Abraham, N. and F. Bloch: Mesures en valeur absolue des périodes des oscillations électriques de haute frequence, Ann. der Phys. (Ser. 9) 12, 237 (1919).

    Google Scholar 

  • Andronov, A.A., A.A. Vitt, and S.E. Khaikin: Theory of Oscillators, Pergamon Press, Oxford and New York, 1966, pp. 309–312.

    Google Scholar 

  • Anninos, P.A.: Mathematical model of memory trace and forgetfulness, Kybernetik 10, 165–170 (1972).

    Article  Google Scholar 

  • Balakhovsky, I.S.: Several modes of excitation movement in an ideal excitable tissue, Biophysics 10, 1175–1179 (1965).

    Google Scholar 

  • Belousov, B.P.: Sborn. Referat. Radiats. Meditsin. za 1958 (1958 Collection of Abstracts on Radiation Medicine), Medgiz. Publ. House, Moscow, 1959, p. 145.

    Google Scholar 

  • Bonhoeffer, K.F.: Zur Theorie des elektrischen Reizes, Naturwissenschaften 31, 270–275 (1943).

    Article  Google Scholar 

  • Bray, W.C. : J. Amer. Chem. Soc. 43 ,1262 (1921).

    Article  Google Scholar 

  • Chirlian, P.M.: Electronic Circuits: Physical Principles, Analysis and Design, McGraw-Hill, New York, 1971.

    Google Scholar 

  • Cowan, J.: Stochastic models of neuroelectric activity. In: Towards a Theoretical Biology, 4, C.H. Waddington, ed., University Press, Edinburgh, 1972, pp. 169–187.

    Google Scholar 

  • Decker, P.: Evolution in open systems: bistability and the origin of molecular asymmetry, Nature New Biology, 241, 72–74 (1973).

    Google Scholar 

  • Degn, H.: Oscillating chemical reactions in homogeneous phase, J. Chem. Educ, 49, 302–307 (1972).

    Article  Google Scholar 

  • Denzel, B.: Berechnungen zum Hystereseverhalten von Enzymsystemen mit Substrathemmung. Diploma Work, University of Tübingen (1973).

    Google Scholar 

  • Eccles, W.H. and F.W. Jordan: A trigger relay utilizing three electrode thermionic vacuum tubes, Radio Review, 1 ,143 (1919).

    Google Scholar 

  • Edelstein, B.B.: Biochemical model with multiple steady states and hysteresis, J. theor. Biol., 29, 57–62 (1970).

    Article  Google Scholar 

  • Feinberg, M.: On chemical kinetics of a certain class, Arch. Rational Mech. Anal., 46, 1–41 (1972).

    Article  MathSciNet  Google Scholar 

  • Field, R.J., E. Körös, and R.M. Noyes: Oscillations in chemical systems. II. Thorough analysis of temporal oscillation in the bromate-cerium-malonic-acid system, J. Amer. Chem. Soc., 94, 8649–8664 (1972).

    Article  Google Scholar 

  • Fitzhugh, R.: Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 ,445–466 (1961).

    Article  Google Scholar 

  • Franck, U.F.: Modelle zur biologischen Erregung, Studium Generale, 18, 313–329 (1965).

    Google Scholar 

  • Franck, U.F.: Oscillatory behavior, excitability and propagation phenomena on membranes and membrane-like interfaces. In: Biological and Biochemical Oscillators, ed. by B. Chance, A.K. Ghosh, E.K. Pye, and B. Hess, pp. 7–30. Academic Press, New York, 197 3.

    Google Scholar 

  • Frank, F.C.: On spontaneous asymmetric synthesis, Biochim. et Biophysica Acta 11, 459–463 (1953).

    Article  Google Scholar 

  • Gause, G.I.: The Struggle for Existence, Williams and Wilkins, Baltimore, 1934.

    Book  Google Scholar 

  • Gerisch, G.: Periodische Signale steuern die Musterbildung in Zellverbänden, Naturwissenschaften 58, 430–438 (1971).

    Article  Google Scholar 

  • Gierer, A. and H. Meinhardt: A theory of biological pattern formation, Kybernetik 12, 30–39 (1972).

    Article  Google Scholar 

  • Gul’ko, F.B. and A.A. Petrov: Mechanism of formation of closed pathways of conduction in excitable media, Biophysics 17, 271–281 (1972).

    Google Scholar 

  • Güttinger, W.: Catastrophe geometry in physics and biology, These Proceedings (1974).

    Google Scholar 

  • Horn, F.: Stability and complex balancing in mass-action systems with three short complexes, Proc. R. Soc. Lond. A, 334, 331–342 (1973).

    Article  MathSciNet  Google Scholar 

  • Horn, F. and R. Jackson: General mass action kinetics, Arch. Rational Mech. Anal. 47, 81–116 (1972).

    Article  MathSciNet  Google Scholar 

  • Hodgkin, A.L. and A.F. Huxley: A qualitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117, 500–544 (1952).

    Google Scholar 

  • Hurley, R.B.: Transistor Logic Circuits, Wiley, New York, 1961.

    MATH  Google Scholar 

  • Kacser, H.: Kinetic models in development and heredity, Symp. Soc. Exptl. Biol. 14, 13–27 (1960).

    Google Scholar 

  • Karfunkel, H.R.: A computer program for the simulation of the temporal and spatial behavior of multi-component reaction-diffusion systems, Comput. Biol. Med. (in press) (1974).

    Google Scholar 

  • Karfunkel, H.R. and F.F. Seelig: Reversal of inhibition of enzymes and the model of a spike oscillator, J. theor. Biol. 36, 237–253 (1972).

    Article  Google Scholar 

  • Kettner, F.: Zur Theorie und Phänomenologie elektrischer, chemischer, hydraulischer und mathematischer Analogmodelle biologischer Oszillatoren, Doctoral Thesis, Technical University of Aachen (1969).

    Google Scholar 

  • Krinsky, V.I.: Spread of excitation in an inhomogeneous medium (state similar to cardiac fibrillation), Biophysics 11, 776–784 (1966).

    Google Scholar 

  • Licko, V.: Some biochemical threshold mechanisms, Bull. math. Biophysics 34, 103–112 (1972).

    Article  Google Scholar 

  • Liénard, A.: Etude des oscillations entretenues (2 parts), Revue Generale de l’Electricité 23, 901–912; 946–954 (1929).

    Google Scholar 

  • Novak, B. and F.F. Seelig: Phase-shift model for the aggregation of cellular slime mold, a computer study, J. theor. Biol. (submitted) (1974).

    Google Scholar 

  • Noyes, R.M., R.J. Field, and E. Körös: Oscillations in chemical systems. I. Detailed mechanism in a system showing temporal oscillations, J. Amer. Chem. Soc. 94, 1394–1395 (1972).

    Article  Google Scholar 

  • O’Neill, S.P., M.D. Lilly, and P.N. Rowe: Multiple steady states in continuous flow stirred tank enzyme reactors, Chem. Enging. Sci. 26, 173–175 (1971).

    Article  Google Scholar 

  • Oster, G., A.S. Perelson, and A. Katchalsky: Network thermodynamics: dynamic modelling of biophysical systems, Quarterly Rev. Biophys. 6, 1–134 (1973).

    Article  Google Scholar 

  • Rashevsky, N.: Über Hysterese-Erscheinungen in physikalischen und chemischen Systemen, Z. Physik 3 ,102–106 (1929).

    Article  Google Scholar 

  • Rashevsky, N.: An approach to the mathematical biophysics of biological self-organization and of cell polarity; Further comments to the theory of cell polarity and self-regulation; and Physicomathe-matical aspects of some problems of organic form, Bull. math. Biophysics, 2, 15–25; 65–67; and 109–121 (1940).

    Article  MathSciNet  Google Scholar 

  • Rashevsky, N.: Mathematical Biophysics, 3rd ed., Vol. II, Dover Publications, New York, 1960, pp. 69–73.

    Google Scholar 

  • Reissig, G., G. Sansone, and R. Conti: Qualitative Theorie nichtlinearer Differentialgleichungen, Edizione Cremonese, Rome, 1963.

    Google Scholar 

  • Rosen, R.: Dynamical System Theory in Biology I, Wiley-Interscience, New York, 1970.

    MATH  Google Scholar 

  • Rosenschöld, P.S. Munck af: Poggendorf’s Annalen der Physikalischen Chemie, 32 ,216 (1834).

    Article  Google Scholar 

  • Rössler, O.E.: A system-theoretic model of biogenesis (in German), Z. Naturforsch. 26b, 741–746 (1971).

    Google Scholar 

  • Rössler, O.E.: Grundschaltungen von flüssigen Automaten und Reaktionssystemen (Basic circuits of fluid automata and relaxation systems), Z. Naturforsch. 27b, 333–343 (1972a).

    Google Scholar 

  • Rössler, O.E.: A principle for chemical multivibration, J. theor. Biol. 36, 413–417 (1972b).

    Article  Google Scholar 

  • Rössler, O.E.: Existenz eines Bausteinprinzips beim Entwurf von komplizierten Reaktionssystemen, Habilitationsschrift, University of Tübingen (1973).

    Google Scholar 

  • Rössler, O.E.: Chemical automata in homogeneous and reaction diffusion kinetics, These Proceedings (1974a).

    Google Scholar 

  • Rössler, O.E.: A multivibrating switching network in homogeneous kinetics, Bull. math. Biol. (in press) (1974b).

    Google Scholar 

  • Rössler, O.E. and D. Hoffmann: Repetitive hard bifurcation in a homogeneous reaction system. In: Analysis and Simulation of Biochemical Systems, H.C. Hemker & B. Hess, eds. North Holland and Elsevier, Amsterdam and New York, 1972, pp. 91–101.

    Google Scholar 

  • Rössler, O.E. and F.F. Seelig: A Rashevsky-Turing system as a two-cellular flip-flop, Z. Naturforsch. 27b, 1445–1448 (1972).

    Google Scholar 

  • Seelig, F.F.: Undamped sinusoidal oscillations in linear chemical reaction systems, J. theor. Biol. 27, 197–206 (1970).

    Article  Google Scholar 

  • Seelig, F.F.: Activated enzyme catalysis as a possible realization of the stable linear chemical oscillator model, J. theor. Biol.30, 497–514 (1971a).

    Article  Google Scholar 

  • Seelig, F.F.: Systems-theoretic model for the spontaneous formation of optical antipodes in strongly asymmetric yield, J. theor. Biol. 31, 355–361 (1971b).

    Article  Google Scholar 

  • Seelig, F.F. and B. Denzel: Hysteresis without autocatalysis: simple enzyme systems as possible binary memory elements, FEBS Letters 24., 283–286 (1972) .

    Article  Google Scholar 

  • Seelig, F.F. and F. Göbber: Stable linear reaction oscillator, J. theor. Biol. 30, 485–496 (1971).

    Article  Google Scholar 

  • Smale, S.: On periodic attractors in cell biology, Lecture held at the University of Tübingen, November, 1972.

    Google Scholar 

  • Thom, R.: Topological models in biology. In: Towards a Theoretical Biology, Vol. 3, C.H. Waddington, ed., University Press, Edinburgh, 1970, pp. 89–116.

    Google Scholar 

  • Thom, R.: Stabilité Structurelle et Morphogenese, Essai d’une Théorie Générale des Modeies, Benjamin, Reading, Mass., 1972.

    Google Scholar 

  • Turing, A.M.: The chemical basis of morphogenesis, Phil. Trans. Roy. Soc., London, Ser. B, 237, 37–72 (1952).

    Article  Google Scholar 

  • Van der Pol, B. and J. van der Mark: The heartbeat considered as relaxation oscillation and an electrical model of the heart, Phil. Mag. 6, 763 (1928).

    Google Scholar 

  • Vavilin, V.A. and A.M. Zhabotinsky: Autocatalytic oxidation of trivalent cerium by bromate ion, I, Kinetika Kataliz 10, 83–88 (in Russian) (1969) .

    Google Scholar 

  • Walter, W.G.: The Living Brain, Duckworth, London, 1953.

    Google Scholar 

  • Wiener, N. and A. Rosenblueth: The mathematical formulation of the problem of conduction of impulses on a network of excitable elements, specifically in cardiac muscle, Arch. Inst. Cardiologia de Mexico 16, 105 (1953) (re-evaluated in Abakumov et al, 1970).

    Google Scholar 

  • Winfree, A.T.: Spiral waves of chemical activity, Science 175, 634–636 (1972).

    Article  Google Scholar 

  • Winfree, A.T.: Spatial and temporal organization in the Zhabotinsky reaction. In: Aharon Katchalsky Memorial Symposium, Berkeley, California, March 21, 1973 (Preprint).

    Google Scholar 

  • Winfree, A.T.: Wave-like activity in biological and chemical media. In: Lecture Notes in Biomathematics, P. van den Driessche, ed., Springer Verlag, Berlin-Heidelberg-New York (forthcoming) (1974).

    Google Scholar 

  • Zeeman, E.C.: Topology of the Brain (Mathematics and Computer Sciences in Biology and Medicine), Medical Research Council, 1965.

    Google Scholar 

  • Zeeman, E.C: Differential equations for the heartbeat and nerve impulse. In: Towards a Theoretical Biology, Vol. 4, C.H. Waddington, ed., Edinburgh University Press, Edinburgh, 1972, pp. 8–67.

    Google Scholar 

  • Zhabotinsky, A.M.: Periodic oxidation reactions in the liquid phase, Dokl. Akad. Nauk. SSSR 157, 392–395 (1964) (Cover-to-cover translation, pp. 701–704).

    Google Scholar 

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  1. Division of Theoretical Chemistry, University of Tübingen, Tübingen, Federal Republic of Germany

    O. E. Rössler

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  1. O. E. Rössler
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  1. Institut für Informationsverarbeitung, Universität Tübingen, 74 Tübingen, Köstlinstraße 6, Deutschland

    Michael Conrad , Werner Güttinger  & Mario Dal Cin ,  & 

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Rössler, O.E. (1974). A Synthetic Approach to Exotic Kinetics (With Examples). In: Conrad, M., Güttinger, W., Dal Cin, M. (eds) Physics and Mathematics of the Nervous System. Lecture Notes in Biomathematics, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-80885-2_34

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  • DOI: https://doi.org/10.1007/978-3-642-80885-2_34

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