Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-14T23:08:18.679Z Has data issue: false hasContentIssue false
  • English
  • Français

Chaos in biological systems

Published online by Cambridge University Press:  17 March 2009

Lars Folke Olsen  and
Hans Degn
Lars Folke Olsen
Affiliation:
Institute of Biochemistry, Odense University, Denmark
Hans Degn
Affiliation:
Institute of Biochemistry, Odense University, Denmark
Get access Rights & Permissions [Opens in a new window]

Extract

Chaos is a widespread and easily recognizable phenomenon that hardly anybody took notice of until recently. The reason may be that chaos has something profoundly counterintuitive about it. It will not fit easily into any familiar cause–effect frame. The best introduction to chaos is by the way of an example. Consider a leaking faucet (Shaw, 1984). When the weight of the accumulating drop exceeds the surface tension the drop falls and a new drop begins to form. If the leak is small and the pressure in the faucet is constant, the time taken for the drop to reach the critical weight is constant. The dripping is perfectly periodic, the period depending on the leak rate. If the leak is slightly increased, the period of dripping will decrease slightly and vice versa. However, somewhere beyond this point the leaking faucet becomes a nuisance. When the leak is increased beyond a certain point the dripping looses its regularity. The time interval between the drops will first alternate periodically between a short and a long time interval. After a further increase of the leak this double periodic pattern will become unstable and change into a new pattern where four different time intervals between the drops alternate periodically. As the leak is further increased the period will double again and again and finally the dripping becomes completely irregular without any repeating pattern. When this occurs we are observing chaos. At the same time we are posed with the problem of understanding how such a ridiculously simple system can show random behaviour.

Type
Research Article
Information
Quarterly Reviews of Biophysics , Volume 18 , Issue 2 , May 1985 , pp. 165 - 225
Copyright
Copyright © Cambridge University Press 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aihara, K., Matsumoto, G. & Ikegaya, Y. (1984). Periodic and non-periodic responses of a periodically forced Hodgkin-Huxley oscillator. J. theor. Biol. 109, 249–269.CrossRefGoogle ScholarPubMed
Anderson, R. M. (1982). Directly transmitted viral and bacterial infections of man. In The Population Dynamics of Infectious Diseases: Theory and Applications (ed. Anderson, R. M.), pp. 1–37. London: Chapman & Hall.Google Scholar
Anderson, R. M. & May, R. M. (1979). Population biology of infectious diseases: I. Nature. 280, 361–367.CrossRefGoogle Scholar
Aron, J. L. & Schwartz, I. B. (1984). Seasonality and period-doubling bifurcations in an epidemic model. J. theor. Biol. 110, 665–679.CrossRefGoogle Scholar
Bartlett, M. S. (1960). Stochastic Population Models in Ecology and Epidemiology. London: Methuen.Google Scholar
Beeler, G. W. & Reuter, H. (1977). Reconstruction of the action potential of ventricular myocardial fibres. J. Physiol. 268, 177–210.CrossRefGoogle ScholarPubMed
Benettin, G., Galgani, L. & Strelcyn, J.-M. (1976). Kolmogorov entropy and numerical experiments. Phys. Rev. A 14, 2338–2345.CrossRefGoogle Scholar
Bier, M. & Bountis, T. C. (1984). Remerging Feigenbaum Trees in Dynamical Systems. Physics. Lett. 104A, 239–244.CrossRefGoogle Scholar
Chance, B., Hess, B. & Betz, A. (1964). DPNH oscillations in a cell-free extract of S. carlsbergensis. Biochem. biophys. Res. Commun. 16, 182–187.CrossRefGoogle Scholar
Chay, T. R. (1984). Abnormal discharges and chaos in a neuronal model system. Biol. Cybernet. 50, 301–311.CrossRefGoogle Scholar
Chay, T. R. & Rinzel, J. (1985). Bursting, beating and chaos in an excitable membrane model. Biophys. J. 47, 357–366.CrossRefGoogle Scholar
Chirikov, B. V. (1979). A universal instability of many-dimensional oscillator systems. Physics. Rep. 52, 263–379.CrossRefGoogle Scholar
Collet, P. & Eckmann, J.-P. (1980). Iterated maps on the interval as dynamical systems. Basel: Birkhauser.Google Scholar
Decroly, O. & Goldbeter, A. (1982). Birhythmicity, chaos and other patterns of temporal self-organization in a multiply regulated biochemical system, Proc. natn Acad. Sci. U.S.A. 79, 6917–6921.CrossRefGoogle Scholar
Decroly, O. & Goldbeter, A. (1984). Multiple periodic regimes and final state sensitivity in a biochemical system. Phys. Lett. 105A, 259–262.CrossRefGoogle Scholar
Degn, H. (1968). Bistability caused by substrate inhibition of peroxidase in an open reaction system. Nature. 217, 1047–1050.CrossRefGoogle Scholar
Degn, H. (1982). Discrete chaos is reversed random walk. Phys. Rev. A 26, 711–712.CrossRefGoogle Scholar
Degn, H. (1983). Strange attractors in linear period transfer functions with periodic perturbations. In Chemical Applications of Topology and Graph Theory (ed. King, R. B.), pp. 364–370. Amsterdam: Elsevier.Google Scholar
Degn, H. & Mayer, D. (1969). Theory of oscillations in peroxidase catalyzed oxidation reactions in open system. Biochim. biophys. Acta. 180, 291–301.CrossRefGoogle ScholarPubMed
Degn, H., Olsen, L. F. & Perram, J. W. (1979). Bistability, Oscillations and chaos in an enzyme reaction. Ann. N. Y. Acad. Sci. 316, 623–637.CrossRefGoogle Scholar
Durston, A. J. (1974). Pacemaker mutants of Dictyostelium discoideum. Devl. Biol. 38, 308–319.CrossRefGoogle ScholarPubMed
Eckmann, J.-P. (1981). Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys. 53, 643–654.CrossRefGoogle Scholar
Eschrich, K., Schellenberger, W. & Hofmann, E. (1983). Sustained oscillations in a reconstituted enzyme system containing phospho-fructokinase and fructose 1,6-bisphosphate. Archs. Biochem. Biophys. 222, 657–660.CrossRefGoogle Scholar
Farmer, J. D., Ott, E. & Yorke, J. A. (1983). The dimension of chaotic attractors. Physica 7D, 153–180.Google Scholar
Feigenbaum, M. J.(1978). Quantitative universality for a class of non-linear transformations. J. Statist. Phys. 19, 25–52.CrossRefGoogle Scholar
Ford, J. (1983). How random is a coin toss? In Physics Today, April 40–47.CrossRefGoogle Scholar
Fraser, S. & Kapral, R. (1982). Analysis of flow hysteresis by a one-dimensional map. Phys. Rev. A. 25, 3223–3233.CrossRefGoogle Scholar
Ghosh, A. & Chance, B. (1964). Oscillations of glycolytic intermediates in yeast cells. Biochem. biophys. Res. Commun. 16, 174–181.CrossRefGoogle ScholarPubMed
Glass, L., Graves, C., Petrillo, G. A. & Mackey, M. C. (1980). Unstable dynamics of a periodically driven oscillator in the presence of noise. J. theor. Biol. 86, 455–475.CrossRefGoogle ScholarPubMed
Glass, L., Guevara, M. R., Belair, J. & Shrier, A. (1984). Global bifurcations of a periodically forced biological oscillator. Phys. Rev. A. 29, 1348–1357.CrossRefGoogle Scholar
Glass, L., Guevara, M. R., Shrier, A. & Perez, R. (1983). Bifurcation and chaos in a periodically stimulated cardiac oscillator. Physica 7D, 89–101.Google Scholar
Glass, L. & Mackey, M. C. (1979). Pathological conditions resulting from instabilities in physiological control systems. Ann. N. Y. Acad. Sci. 316, 214–235.CrossRefGoogle ScholarPubMed
Grebogi, C., Ott, E. & Yorke, J. A. (1982). Chaotic attractors in crisis. Phys. Rev. Lett. 48, 1507–1510.CrossRefGoogle Scholar
Grebogi, C., Ott, E. & Yorke, J. A. (1983). Crises, sudden changes in chaotic attractors, and transient chaos. Physica 7D, 181–200.Google Scholar
Grossmann, S. & Thomae, S. (1977). Invariant distributions and stationary correlation functions of one-dimensional discrete processes. Z. Naturf. 32a, 1353–1363.CrossRefGoogle Scholar
Guevara, M. R. & Glass, L. (1982). Phase locking, period doubling bifurcations and chaos in a mathematical model of a periodically driven oscillator: A theory for the entrainment of biological oscillators and the generation of cardiac dysrythmias. J. Math. Biol. 14, 1–23.CrossRefGoogle Scholar
Guevara, M. R., Glass, L., Mackey, M. C. & Shrier, A. (1983). Chaos in neurobiology. IEEE Trans. Sys. Man Cyber. 13, 790–798.CrossRefGoogle Scholar
Guevara, M. R., Glass, L. & Shrier, A. (1981). Phase locking period-doubling bifurcations, and irregular dynamics in periodically stimulated cardiac cells. Science. 214, 1350–1353.CrossRefGoogle ScholarPubMed
Guttman, R., Feldman, L. & Jakobsson, E. (1980). Frequency entrainment of squid axon membrane. J. Membrane Biol. 56, 9–18.CrossRefGoogle ScholarPubMed
Hayashi, H., Ishizuka, S. & Hirakawa, K. (1983). Transition to chaos via intermittency in the onchidium pacemaker neuron. Phys. Lett. 98A, 474–476.CrossRefGoogle Scholar
Hayashi, H., Ishizuka, S., Ohta, M. & Hirakawa, K. (1982). Chaotic behavior in the onchidium giant neuron under sinusoidal stimulation. Phys. Lett. 88A, 435–438.CrossRefGoogle Scholar
Hayashi, H., Nakao, M. & Hirakawa, K. (1982). Chaos in the self-sustained oscillation of an excitable membrane under sinusoidal stimulation. Phys. Lett. 88A, 265–266.CrossRefGoogle Scholar
Hayashi, H., Nakao, M. & Hirakawa, K. (1983). Entrained, harmonic, quasiperiodic and chaotic responses of the self-sustained oscillation of nitella to sinusoidal stimulation. J. phys. Soc. Japan. 52, 344–351.CrossRefGoogle Scholar
Hénon, M. (1976). A two-dimensional mapping with strange attractor. Commun. Math. Phys. 50, 69–77.CrossRefGoogle Scholar
Hess, B. & Markus, M. (1985 a). The diversity of biochemical time patterns. Ber. Bunsenges. Phys. Chem. 89, 642–651.CrossRefGoogle Scholar
Hess, B. & Markus, M. (1985 b). Dynamic coupling and time patterns in biochemical processes. In Temporal Order (ed. Rensing, L. and Jager, N. I.). Berlin: Springer-Verlag. (In the Press.)Google Scholar
Holden, A. V. & Muhamad, M. A. (1984). Chaotic activity in neural systems. In Cybernetics and Systems Research 2 (ed. Trappl, R.), pp. 245–250. Amsterdam: Elsevier.Google Scholar
Holden, A. V. & Winlow, W. (1982). Bifurcation of periodic activity from periodic activity in a molluscan neurone. Biol. Cybern. 42, 189–194.CrossRefGoogle Scholar
Holden, A. V., Winlow, W. & Haydon, P. G. (1982). The induction of periodic and chaotic activity in a molluscan neurone. Biol. Cybern. 43, 169–173.CrossRefGoogle Scholar
Hoyer, J., Park, M. R. & Klee, M. R. (1978). Changes in ionic currents associated with flurazepam-induced abnormal discharges in Aplysia neurons. In: Abnormal Neuronal Discharges (ed. Chalazonitis, N. and Boisson, M.), pp. 301–310. New York: Raven Press.Google Scholar
Hudson, J. L. & Mankin, J. C. (1981). Chaos in the Belousov-Zhabotinskii reaction. J. Chem. Phys. 74, 6171–6177.CrossRefGoogle Scholar
Jensen, J. H., Christiansen, P. L. & Scott, A. C. (1984). Chaos in the Beeler-Reuter system for the action potential of ventricular myocardial fibres. Physica 13D, 269–277Google Scholar
Jensen, J. H., Christiansen, P. L., Scott, A. C. & Skovgaard, O. (1983). Chaos in Nerve, Proceedings of the Iasted Symposium, Copenhagen, pp. 15/6–15/9.Google Scholar
Kaczmarek, L. K. & Babloyantz, A. (1977). Spatiotemporal patterns in epileptic seizures. Biol. Cybern. 26, 199–208.CrossRefGoogle ScholarPubMed
Kaplan, H. (1983). New method for calculating stable and unstable periodic orbits of one-dimensional maps. Phys. Lett. 97A, 365–367.CrossRefGoogle Scholar
King, R., Barchas, J. D. & Huberman, B. A. (1984). Chaotic behaviour in dopamine neurodynamics. Proc. natn Acad. Sci. U.S.A. 81, 1244–1247.CrossRefGoogle ScholarPubMed
Klee, M. R., Faber, D. S. & Hoyer, J. (1978). Doublet discharges and bistable states induced by strychnine in a neuronal soma membrane. In Abnormal Neuronal Discharges (ed. Chalazonitis, N. and Boisson, M.), pp. 287–300. New York: Raven Press.Google Scholar
Kloeden, P., Deakin, M. A. B. & Tirkel, A. Z. (1976). A precise definition of chaos. Nature 264, 295.CrossRefGoogle Scholar
Labos, E. & Lang, E. (1978). On the behavior of snail neurons in the presence of cocaine. In Abnormal Neuronal Discharges (ed. Chalazonitis, N. and Boisson, M.), pp. 177–188. New York: Raven Press.Google Scholar
Li, T.-Y. & Yorke, J. A. (1975). Period three implies chaos. Am. Math. Mon. 82, 985–992.CrossRefGoogle Scholar
London, W. P. & Yorke, J. A. (1973). Recurrent outbreaks of measles, chickenpox and mumps I: seasonal variations in contact rates. Am. J. Epidemiol. 98, 453–468.CrossRefGoogle ScholarPubMed
Lorenz, E. N. (1963). Deterministic nonperiodic flow. J. atmos Sci. 20, 130–141.2.0.CO;2>CrossRefGoogle Scholar
Mackey, M. C. & Glass, L. (1977). Oscillation and chaos in physiological control systems. Science. 197, 287–289.CrossRefGoogle ScholarPubMed
Mandelbrot, B. (1977). Fractals: Form, Chance and Dimension. San Francisco: Freeman.Google Scholar
Manneville, P. & Pomeau, Y. (1979). Intermittency and the Lorenz Model. Phys. Lett. 75A, 1–2.CrossRefGoogle Scholar
Manneville, P. & Pomeau, Y. (1980). Different ways to turbulence in dissipative dynamical systems. Physica 1D, 219–226.Google Scholar
Markus, M. & Hess, B. (1984). Transition between oscillatory modes in a glycolytic model system. Proc. natn Acad. Sci. U.S.A. 81, 4394–4398.Google Scholar
Markus, M., Kuschmitz, D. & Hess, B. (1984). Chaotic dynamics in yeast glycolysis under periodic substrate input flux. FEBS Lett. 172, 235–238.CrossRefGoogle ScholarPubMed
Markus, M., Kuschmitz, D. & Hess, B. (1985). Properties of strange attractors in yeast glycolysis. Biophys. Chem. (In the Press.)CrossRefGoogle Scholar
Martiel, J. L. & Goldbeter, A. (1985). Autonomous chaotic behaviour of the slime mold Dictyostelium discoideum predicted by a model for cyclic AMP signalling. Nature. 313, 590–592.CrossRefGoogle Scholar
Matsumoto, G., Aihara, K., Ichikawa, M. & Tasaki, A. (1984). Periodic and nonperiodic responses of membrane potentials in squid giant axons during sinusoidal current stimulation. J. theoret. Neurobiol. 3, 1–14.Google Scholar
May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature. 261, 459–467.CrossRefGoogle ScholarPubMed
May, R. M. (1980). Non-linear phenomena in ecology and epidemiology. Ann. N.Y. Acad. Sci. 357, 267–281.CrossRefGoogle Scholar
May, R. M. & Anderson, R. M. (1979). Population biology of infectious diseases: II. Nature. 280, 455–461.CrossRefGoogle Scholar
May, R. M. & Oster, G. F. (1976). Bifurcations and dynamic complexity in simple ecological models. Am. Nat. 110, 573–599.CrossRefGoogle Scholar
Meissner, H. P. (1976). Electrical characteristics of the β-cells in pancreati islets. J. Physiol. 72, 757–767.Google Scholar
Metropolis, N., Stein, M. L. & Stein, P. R. (1973). On finite limit sets for transformations on the unit interval. J. Combinat. Theor. Ser. A. 15, 25–44.CrossRefGoogle Scholar
Nicolis, J. S., Meyer-Kress, G. & Haubs, G. (1983). Non-uniform chaotic dynamics with implications to information processing. Z. Naturf. 38a, 1157–1169.CrossRefGoogle Scholar
Nicolis, J. S., Meyer-Kress, G. & Haubs, G. (1984). Non-uniform information processing by strange attractors of chaotic maps and flows. In Stochastic Phenomena and Chaotic Behaviour in Complex Systems (ed. Schuster, P.), pp. 124–139. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Olsen, L. F. (1978). The oscillating peroxidase-oxidase reaction in an open system: Analysis of the reaction mechanism. Biochim. biophys. Acta. 527, 212–220.CrossRefGoogle Scholar
Olsen, L. F. (1979). Studies of the chaotic behaviour in the peroxidase—oxidase reaction. Z. Naturf. 34a, 1544–1546.CrossRefGoogle Scholar
Olsen, L. F. (1983). An enzyme reaction with a strange attractor. Phys. Lett. 94A, 454–457Google Scholar
Olsen, L. F. (1984). The enyme and the strange attractor - comparisons of experimental and numerical data for an enzyme reaction with chaotic motion. In Stochastic Phenomena and Chaotic Behaviour in Complex Systems (ed. Schuster, P.), pp. 116–123. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Olsen, L. F. & Degn, H. (1977). Chaos in an enzyme reaction. Nature. 267, 177–178.CrossRefGoogle Scholar
Olsen, L. F. & Degn, H. (1978). Oscillatory kinetics of the peroxidase-oxidase reaction in an open system. Experimental and Theoretical Studies, Biochim. biophys. Acta. 523, 321–334.Google Scholar
Ott, E. (1981). Strange attractors and chaotic motion of dynamical systems. Rev. Mod. Phys. 53, 655–671.CrossRefGoogle Scholar
Packard, N. H., Crutchfield, J. P., Farmer, J. D. & Shaw, R. S. (1980). Geometry from a time series. Phys. Rev. Lett. 45, 712–716.CrossRefGoogle Scholar
Ritzenberg, A. L., Adam, D. R. & Cohen, R. J. (1984). Period multupling - evidence for non-linear behaviour of the canine heart. Nature. 307, 159–161.CrossRefGoogle Scholar
Rogers, T. D. (1981). Chaos in systems in population biology. Prog. theor. Biol. 6, 91–146.CrossRefGoogle Scholar
Rossler, O. E. (1976 a). Chaotic behaviour in simple reaction systems. Z. Naturf. 31a, 259–264.CrossRefGoogle Scholar
Rossler, O. E. (1976 b). An equation for continuous chaos. Phys. Lett. 57A, 397–398CrossRefGoogle Scholar
Roux, J. C., Turner, J. S., McCormick, W. D. & Swinney, H. L. (1982). Experimental observations of complex dynamics in a chemical reaction. In Non-linear Problems: Present and Future (ed. Bishop, A. R., Campbell, D. K. and Nicolaenko, B.), pp. 409–422. Amsterdam: North-Holland.CrossRefGoogle Scholar
Schaffer, W. M. & Kot, M. (1985). Nearly one dimensional dynamics in an epidemic. J. theor. Biol. 112, 403–427.CrossRefGoogle Scholar
Selkov, E. E. (1980). Instability and self-oscillation in the cell energy metabolism. Ber. Buns. Ges. phys. chem. 84, 399–402.CrossRefGoogle Scholar
Shaw, R. (1981). Strange attractors, chaotic behavior, and information flow. Z. Naturf. 36a, 80–112.Google Scholar
Shaw, R. (1984). The Dripping Faucet as a Model Chaotic System. Santa Cruz, CA: Aerial Press.Google Scholar
Simo, C. (1979). On the Henon-Pomeau attractor. J. Statist. Phys. 21, 465–494.CrossRefGoogle Scholar
Skjolding, H., Branner-Jorgensen, B., Christiansen, P. L. & Jensen, H. E. (1983). Bifurcations in discrete dynamical systems with cubic maps. SIAM Jl appl. Math. 43, 520–534.CrossRefGoogle Scholar
Takens, F. (1981). Detecting strange attractors in turbulence. Lect. Notes in Math. 898, 366–381.Google Scholar
Testa, J. & Held, G. A. (1983). Study of a one-dimensional map with multiple basins. Phys. Rev. A. 28, 3085–3089.CrossRefGoogle Scholar
Tomita, K. (1982). Chaotic response of non-linear oscillators. Phys. Rep. 86, 113–167.Google Scholar
Yamazaki, I., Yokota, K. & Nakajima, R. (1965). Oscillatory oxidations of reduced pyridine nucleotide. Biochem. biophys. Res. Commun. 21, 582–586.CrossRefGoogle ScholarPubMed
Yorke, J. A. & London, W. P. (1973). Recurrent outbreaks of measles chickenpox and mumps: systematic differences in contact rates and stochastic effects. Am. J. Epidemiol. 98, 469–482.Google Scholar
Yorke, J. A., Nathanson, N., Pianigiani, G. & Martin, J. (1979). Seasonality and the requirements for perpetuation and eradication of viruses in populations. Am. J. Epidemiol. 109, 103–123.CrossRefGoogle ScholarPubMed