Skip to main content Accessibility help
×
×
Home

Chaos in biological systems

  • Lars Folke Olsen (a1) and Hans Degn (a1)
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.

Copyright
References
Hide All
Aihara, K., Matsumoto, G. & Ikegaya, Y. (1984). Periodic and non-periodic responses of a periodically forced Hodgkin-Huxley oscillator. J. theor. Biol. 109, 249269.
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. 137. London: Chapman & Hall.
Anderson, R. M. & May, R. M. (1979). Population biology of infectious diseases: I. Nature. 280, 361367.
Aron, J. L. & Schwartz, I. B. (1984). Seasonality and period-doubling bifurcations in an epidemic model. J. theor. Biol. 110, 665679.
Bartlett, M. S. (1960). Stochastic Population Models in Ecology and Epidemiology. London: Methuen.
Beeler, G. W. & Reuter, H. (1977). Reconstruction of the action potential of ventricular myocardial fibres. J. Physiol. 268, 177210.
Benettin, G., Galgani, L. & Strelcyn, J.-M. (1976). Kolmogorov entropy and numerical experiments. Phys. Rev. A 14, 23382345.
Bier, M. & Bountis, T. C. (1984). Remerging Feigenbaum Trees in Dynamical Systems. Physics. Lett. 104A, 239244.
Chance, B., Hess, B. & Betz, A. (1964). DPNH oscillations in a cell-free extract of S. carlsbergensis. Biochem. biophys. Res. Commun. 16, 182187.
Chay, T. R. (1984). Abnormal discharges and chaos in a neuronal model system. Biol. Cybernet. 50, 301311.
Chay, T. R. & Rinzel, J. (1985). Bursting, beating and chaos in an excitable membrane model. Biophys. J. 47, 357366.
Chirikov, B. V. (1979). A universal instability of many-dimensional oscillator systems. Physics. Rep. 52, 263379.
Collet, P. & Eckmann, J.-P. (1980). Iterated maps on the interval as dynamical systems. Basel: Birkhauser.
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, 69176921.
Decroly, O. & Goldbeter, A. (1984). Multiple periodic regimes and final state sensitivity in a biochemical system. Phys. Lett. 105A, 259262.
Degn, H. (1968). Bistability caused by substrate inhibition of peroxidase in an open reaction system. Nature. 217, 10471050.
Degn, H. (1982). Discrete chaos is reversed random walk. Phys. Rev. A 26, 711712.
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. 364370. Amsterdam: Elsevier.
Degn, H. & Mayer, D. (1969). Theory of oscillations in peroxidase catalyzed oxidation reactions in open system. Biochim. biophys. Acta. 180, 291301.
Degn, H., Olsen, L. F. & Perram, J. W. (1979). Bistability, Oscillations and chaos in an enzyme reaction. Ann. N. Y. Acad. Sci. 316, 623637.
Durston, A. J. (1974). Pacemaker mutants of Dictyostelium discoideum. Devl. Biol. 38, 308319.
Eckmann, J.-P. (1981). Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys. 53, 643654.
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, 657660.
Farmer, J. D., Ott, E. & Yorke, J. A. (1983). The dimension of chaotic attractors. Physica 7D, 153180.
Feigenbaum, M. J.(1978). Quantitative universality for a class of non-linear transformations. J. Statist. Phys. 19, 2552.
Ford, J. (1983). How random is a coin toss? In Physics Today, April 4047.
Fraser, S. & Kapral, R. (1982). Analysis of flow hysteresis by a one-dimensional map. Phys. Rev. A. 25, 32233233.
Ghosh, A. & Chance, B. (1964). Oscillations of glycolytic intermediates in yeast cells. Biochem. biophys. Res. Commun. 16, 174181.
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, 455475.
Glass, L., Guevara, M. R., Belair, J. & Shrier, A. (1984). Global bifurcations of a periodically forced biological oscillator. Phys. Rev. A. 29, 13481357.
Glass, L., Guevara, M. R., Shrier, A. & Perez, R. (1983). Bifurcation and chaos in a periodically stimulated cardiac oscillator. Physica 7D, 89101.
Glass, L. & Mackey, M. C. (1979). Pathological conditions resulting from instabilities in physiological control systems. Ann. N. Y. Acad. Sci. 316, 214235.
Grebogi, C., Ott, E. & Yorke, J. A. (1982). Chaotic attractors in crisis. Phys. Rev. Lett. 48, 15071510.
Grebogi, C., Ott, E. & Yorke, J. A. (1983). Crises, sudden changes in chaotic attractors, and transient chaos. Physica 7D, 181200.
Grossmann, S. & Thomae, S. (1977). Invariant distributions and stationary correlation functions of one-dimensional discrete processes. Z. Naturf. 32a, 13531363.
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, 123.
Guevara, M. R., Glass, L., Mackey, M. C. & Shrier, A. (1983). Chaos in neurobiology. IEEE Trans. Sys. Man Cyber. 13, 790798.
Guevara, M. R., Glass, L. & Shrier, A. (1981). Phase locking period-doubling bifurcations, and irregular dynamics in periodically stimulated cardiac cells. Science. 214, 13501353.
Guttman, R., Feldman, L. & Jakobsson, E. (1980). Frequency entrainment of squid axon membrane. J. Membrane Biol. 56, 918.
Hayashi, H., Ishizuka, S. & Hirakawa, K. (1983). Transition to chaos via intermittency in the onchidium pacemaker neuron. Phys. Lett. 98A, 474476.
Hayashi, H., Ishizuka, S., Ohta, M. & Hirakawa, K. (1982). Chaotic behavior in the onchidium giant neuron under sinusoidal stimulation. Phys. Lett. 88A, 435438.
Hayashi, H., Nakao, M. & Hirakawa, K. (1982). Chaos in the self-sustained oscillation of an excitable membrane under sinusoidal stimulation. Phys. Lett. 88A, 265266.
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, 344351.
Hénon, M. (1976). A two-dimensional mapping with strange attractor. Commun. Math. Phys. 50, 6977.
Hess, B. & Markus, M. (1985 a). The diversity of biochemical time patterns. Ber. Bunsenges. Phys. Chem. 89, 642651.
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.)
Holden, A. V. & Muhamad, M. A. (1984). Chaotic activity in neural systems. In Cybernetics and Systems Research 2 (ed. Trappl, R.), pp. 245250. Amsterdam: Elsevier.
Holden, A. V. & Winlow, W. (1982). Bifurcation of periodic activity from periodic activity in a molluscan neurone. Biol. Cybern. 42, 189194.
Holden, A. V., Winlow, W. & Haydon, P. G. (1982). The induction of periodic and chaotic activity in a molluscan neurone. Biol. Cybern. 43, 169173.
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. 301310. New York: Raven Press.
Hudson, J. L. & Mankin, J. C. (1981). Chaos in the Belousov-Zhabotinskii reaction. J. Chem. Phys. 74, 61716177.
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, 269277
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.
Kaczmarek, L. K. & Babloyantz, A. (1977). Spatiotemporal patterns in epileptic seizures. Biol. Cybern. 26, 199208.
Kaplan, H. (1983). New method for calculating stable and unstable periodic orbits of one-dimensional maps. Phys. Lett. 97A, 365367.
King, R., Barchas, J. D. & Huberman, B. A. (1984). Chaotic behaviour in dopamine neurodynamics. Proc. natn Acad. Sci. U.S.A. 81, 12441247.
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. 287300. New York: Raven Press.
Kloeden, P., Deakin, M. A. B. & Tirkel, A. Z. (1976). A precise definition of chaos. Nature 264, 295.
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. 177188. New York: Raven Press.
Li, T.-Y. & Yorke, J. A. (1975). Period three implies chaos. Am. Math. Mon. 82, 985992.
London, W. P. & Yorke, J. A. (1973). Recurrent outbreaks of measles, chickenpox and mumps I: seasonal variations in contact rates. Am. J. Epidemiol. 98, 453468.
Lorenz, E. N. (1963). Deterministic nonperiodic flow. J. atmos Sci. 20, 130141.
Mackey, M. C. & Glass, L. (1977). Oscillation and chaos in physiological control systems. Science. 197, 287289.
Mandelbrot, B. (1977). Fractals: Form, Chance and Dimension. San Francisco: Freeman.
Manneville, P. & Pomeau, Y. (1979). Intermittency and the Lorenz Model. Phys. Lett. 75A, 12.
Manneville, P. & Pomeau, Y. (1980). Different ways to turbulence in dissipative dynamical systems. Physica 1D, 219226.
Markus, M. & Hess, B. (1984). Transition between oscillatory modes in a glycolytic model system. Proc. natn Acad. Sci. U.S.A. 81, 43944398.
Markus, M., Kuschmitz, D. & Hess, B. (1984). Chaotic dynamics in yeast glycolysis under periodic substrate input flux. FEBS Lett. 172, 235238.
Markus, M., Kuschmitz, D. & Hess, B. (1985). Properties of strange attractors in yeast glycolysis. Biophys. Chem. (In the Press.)
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, 590592.
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, 114.
May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature. 261, 459467.
May, R. M. (1980). Non-linear phenomena in ecology and epidemiology. Ann. N.Y. Acad. Sci. 357, 267281.
May, R. M. & Anderson, R. M. (1979). Population biology of infectious diseases: II. Nature. 280, 455461.
May, R. M. & Oster, G. F. (1976). Bifurcations and dynamic complexity in simple ecological models. Am. Nat. 110, 573599.
Meissner, H. P. (1976). Electrical characteristics of the β-cells in pancreati islets. J. Physiol. 72, 757767.
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, 2544.
Nicolis, J. S., Meyer-Kress, G. & Haubs, G. (1983). Non-uniform chaotic dynamics with implications to information processing. Z. Naturf. 38a, 11571169.
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. 124139. Berlin: Springer-Verlag.
Olsen, L. F. (1978). The oscillating peroxidase-oxidase reaction in an open system: Analysis of the reaction mechanism. Biochim. biophys. Acta. 527, 212220.
Olsen, L. F. (1979). Studies of the chaotic behaviour in the peroxidase—oxidase reaction. Z. Naturf. 34a, 15441546.
Olsen, L. F. (1983). An enzyme reaction with a strange attractor. Phys. Lett. 94A, 454457
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. 116123. Berlin: Springer-Verlag.
Olsen, L. F. & Degn, H. (1977). Chaos in an enzyme reaction. Nature. 267, 177178.
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, 321334.
Ott, E. (1981). Strange attractors and chaotic motion of dynamical systems. Rev. Mod. Phys. 53, 655671.
Packard, N. H., Crutchfield, J. P., Farmer, J. D. & Shaw, R. S. (1980). Geometry from a time series. Phys. Rev. Lett. 45, 712716.
Ritzenberg, A. L., Adam, D. R. & Cohen, R. J. (1984). Period multupling - evidence for non-linear behaviour of the canine heart. Nature. 307, 159161.
Rogers, T. D. (1981). Chaos in systems in population biology. Prog. theor. Biol. 6, 91146.
Rossler, O. E. (1976 a). Chaotic behaviour in simple reaction systems. Z. Naturf. 31a, 259264.
Rossler, O. E. (1976 b). An equation for continuous chaos. Phys. Lett. 57A, 397398
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. 409422. Amsterdam: North-Holland.
Schaffer, W. M. & Kot, M. (1985). Nearly one dimensional dynamics in an epidemic. J. theor. Biol. 112, 403427.
Selkov, E. E. (1980). Instability and self-oscillation in the cell energy metabolism. Ber. Buns. Ges. phys. chem. 84, 399402.
Shaw, R. (1981). Strange attractors, chaotic behavior, and information flow. Z. Naturf. 36a, 80112.
Shaw, R. (1984). The Dripping Faucet as a Model Chaotic System. Santa Cruz, CA: Aerial Press.
Simo, C. (1979). On the Henon-Pomeau attractor. J. Statist. Phys. 21, 465494.
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, 520534.
Takens, F. (1981). Detecting strange attractors in turbulence. Lect. Notes in Math. 898, 366381.
Testa, J. & Held, G. A. (1983). Study of a one-dimensional map with multiple basins. Phys. Rev. A. 28, 30853089.
Tomita, K. (1982). Chaotic response of non-linear oscillators. Phys. Rep. 86, 113167.
Yamazaki, I., Yokota, K. & Nakajima, R. (1965). Oscillatory oxidations of reduced pyridine nucleotide. Biochem. biophys. Res. Commun. 21, 582586.
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, 469482.
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, 103123.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Quarterly Reviews of Biophysics
  • ISSN: 0033-5835
  • EISSN: 1469-8994
  • URL: /core/journals/quarterly-reviews-of-biophysics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed