Oscillators abound in physics (Haken 1970, 1983a) and many other branches of the natural sciences such as chemistry (Kuramoto 1984), biology and medicine (Freeman 1975, Freund 1983, Steriade, Jones, Llinás 1990). These days the theory of oscillators is a well-established branch of nonlinear physics and mathematics (cf. Haken 1977, 1983a, Winfree 1980, Arnold 1983, Kuramoto 1984, Guckenheimer and Holmes 1990). It revealed many important insights into the interactions of oscillators and the reactions of an oscillator to external perturbations.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adam, D.R., Smith, J.M., Akselrod, S., Nyberg, S., Powell, A.O., Cohen, R.J. (1984): Fluctuations in T-wave morphology and susceptibility to ventricular fibrillation, J. Electrocardiol. 17, 209–218CrossRefGoogle Scholar
  2. Allessie, M.A., Lammers, W.J.E.P., Bonke, F.I.M., Hollen, J. (1985): Experimental evaluation of Moe’s multiple wavelet hypothesis of atrial fibrillation, In: Cardiac Electrophysiology and Arrhythmias, Zipes, D.P., Jalife, J. (eds.), Grune and Stratton, OrlandoGoogle Scholar
  3. Andy, O.J. (1983): Thalamic stimulation for control of movement disorders, Appl. Neurophysiol. 46, 107–113Google Scholar
  4. Arnold, V.I. (1983): Geometrical methods in the theory of ordinary differential equations, Springer, HeidelbergzbMATHCrossRefGoogle Scholar
  5. Basar, E. (1998a): Brain Oscillations, Springer, Berlin; (1998b): Integrative Brain Function, Springer, BerlinGoogle Scholar
  6. Benabid, A.L., Pollak, P., Gervason, C., Hoffmann, D., Gao, D.M., Hommel, M., Perret, J.E., De Rougemont, J. (1991): Long-term suppression of tremor by chronic stimulation of the ventral intermediate thalamic nucleus, The Lancet 337, 403–406CrossRefGoogle Scholar
  7. Benabid, A.L., Pollak, P., Seigneuret, E., Hoffmann, D., Gay, E., Perret, J. (1993): Chronic VIM thalamic stimulation in Parkinson’s disease, essential tremor and extra-pyramidal dyskinesia, Acta Neurochir. Suppl. (Wien) 58, 39–44Google Scholar
  8. Benabid, A.L., Pollak, P., Gao, D., Hoffmann, D., Limousin, P., Gay, E., Payen, I., Benazzouz, A. (1996): Chronic electrical stimulation of the ventralis intermedius nucleus of the thalamus as a treatment of movement disorders, J. Neurosurg. 84, 203–214CrossRefGoogle Scholar
  9. Blond, S., Caparros-Lefebvre, D., Parker, F., Assaker, R., Petit, H., Guieu, J.-D., Christiaens, J.-L. (1992): Control of tremor and involuntary movement disorders by chronic stereotactic stimulation of the ventral intermediate thalamic nucleus, J. Neurosurg. 77, 62–68CrossRefGoogle Scholar
  10. Caparros-Lefebvre, D., Ruchoux, M.M., Blond, S., Petit, H., Percheron, G. (1994): Long term thalamic stimulation in Parkinson’s disease, Neurology 44, 1856–1860CrossRefGoogle Scholar
  11. Cohen, A.H., Rossignol, S., Grillner, S. (eds.) (1988): Neural Control of Rhythmic Movements in Vertebrates, John Wiley and Sons, New YorkGoogle Scholar
  12. Eckhorn, R., Bauer, R., Jordan, W., Brosch, M., Kruse, W., Munk, M., Reitboeck, H.J. (1988): Coherent oscillations: a mechanism of feature linking in the visual cortex?, Biol. Cybern. 60, 121–130CrossRefGoogle Scholar
  13. Elble, R.J., Koller, W.C. (1990): Tremor, The Johns Hopkins University Press, BaltimoreGoogle Scholar
  14. Engel, J., Pedley, T. A. (eds.) (1997): Epilepsy: A comprehensive textbook, Lippincott-Raven, PhiladelphiaGoogle Scholar
  15. Freeman, W.J. (1975): Mass action in the nervous system, Academic Press, New YorkGoogle Scholar
  16. Freund, H.-J. (1983): Motor unit and muscle activity in voluntary motor control, Physiological Reviews 63, 387–436; (1987): Abnormalities of motor behavior after cortical lesions in man, In: The Nervous System: Higher Functions of the Brain, Mountcastle, V.B. (Section ed.), Plum, F. (Vol. ed.), Sect. 1, vol. 5 of Handbook of Physiology, Williams and Wilkins, Baltimore 763–810Google Scholar
  17. Glass, L., Mackey, M.C. (1988): From Clocks to Chaos. The Rhythms of Life,Princeton University PressGoogle Scholar
  18. Golomb, D., Hansel, D., Shraiman, B., Sompolinsky, H. (1992): Clustering in globally coupled phase oscillators, Phys. Rev. A 45, 3516–3530CrossRefGoogle Scholar
  19. Gray, C.M., Singer, W. (1987): Stimulus specific neuronal oscillations in the cat visual cortex: a cortical function unit, Soc. Neurosci. 404, 3Google Scholar
  20. Gray, C.M., Singer, W. (1989): Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex, Proc. Natl. Acad. Sci. USA 86, 1698–1702ADSCrossRefGoogle Scholar
  21. Guckenheimer, J., Holmes, P. (1990): Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Berlin, HeidelbergGoogle Scholar
  22. Hämäläinen, M., Hari, R., Ilmoniemi, R.J.,Knuutila, J., Lounasmaa, O.V. (1993): Magnetoencephalography — theory, instrumentation, and applications to noninvasive studies of the working human brain, Rev. Mod. Phys. 65, 413–497Google Scholar
  23. Haken, H. (1970): Laser Theory, Springer, Berlin; (1977): Synergetics, An Introduction, Springer, BerlinGoogle Scholar
  24. Haken, H. (1983a): Advanced Synergetics, Springer, Berlin; (1983b): Synopsis and Introduction, In: Synergetics of the brain, Basar, E., Flohr, H., Haken, H. (eds.), Springer, BerlinGoogle Scholar
  25. Haken, H. (1996): Principles of Brain Functioning, A Synergetic Approach to Brain Activity, Behavior and Cognition, Springer, BerlinzbMATHCrossRefGoogle Scholar
  26. Haken, H., Koepchen, H.P. (eds.) (1991): Rhythms in Physiological Systems, Springer, BerlinGoogle Scholar
  27. Hakim, V., Rappel, W. (1992): Dynamics of the globally coupled complex Ginzburg—Landau equation, Phys. Rev. A 46, R7347 — R7350CrossRefGoogle Scholar
  28. Hansel, D., Mato, G., Meunier, C. (1993): Clustering and slow switching in globally coupled phase oscillators, Phys. Rev. E 48, 3470–3477Google Scholar
  29. Hari, R., Salmelin, R. (1997): Human cortical oscillations: a neuromagnetic view through the skull, TINS 20, 44–49Google Scholar
  30. Hildebrandt, G. (1982): The time structure of autonomous processes, In: Biological Adaptation, Hildebrandt, G., Hensel, H. (eds.), Georg Thieme, Stuttgart (1987): The autonomous time structure and its reactive modifications in the human organism, In: Temporal Disorder in Human Oscillatory Systems, Rensing, L., an der Heiden, U., Mackey, M.C. (eds.), Springer, BerlinGoogle Scholar
  31. Holden, A.V. (1997): The restless heart of a spiral, Nature 387, 655–657ADSCrossRefGoogle Scholar
  32. Kelley, A. (1967): The stable, center-stable, center, center-unstable and unstable manifolds, J. Diff. Equ. 3, 546–570MathSciNetADSzbMATHCrossRefGoogle Scholar
  33. Koepchen, H.P. (1991): Physiology of rhythms and control systems: an integrative approach. In: Haken and Koepchen ( 1991 ), pp. 3–20Google Scholar
  34. Krinskii, V.I. (1966): Spread of excitation in an inhomogeneous medium, Biofizika 11, 676–683; (1968): Fibrillation in excitable media, Systems Theory Research (Prob. Kyb.) 20, 46–65 (1978): Mathematical models of cardiac arrhythmias (spiral waves), Pharmac. Ther. B. 3, 539–555Google Scholar
  35. Krinskii, V.I., Kholopov, A.V. (1967a): Conduction of impulses in excitable tissue with continuously distributed refractoriness, Biofizika 12, 669–675Google Scholar
  36. Krinskii, V.I., Kholopov, A.V. (1967b): Echo in excitable tissue, Biofizika 12, 524–528Google Scholar
  37. Kuramoto, Y. (1984): Chemical Oscillations, Waves, and Turbulence, Springer, BerlinzbMATHCrossRefGoogle Scholar
  38. Llinâs, R., Jahnsen, H. (1982): Electrophysiology of mammalian thalamic neurons in vitro, Nature 297, 406–408ADSCrossRefGoogle Scholar
  39. Moe, G.K., Abildskov, J.A. (1959): Atrial fibrillation as a self-sustaining arrhythmia independent of focal discharge, Am. Heart J. 58, 59–70Google Scholar
  40. Moe, G.K., Rheinboldt, W.C., Abildskov, J.A. (1964): A computer model of atrial fibrillation, Am. Heart J. 67, 200–220CrossRefGoogle Scholar
  41. Nakagawa, N., Kuramoto, Y. (1993): Collective chaos in a population of globally coupled oscillators, Prog. Theor. Phys. 89, 313–323ADSCrossRefGoogle Scholar
  42. Niedermeyer, E., Lopes da Silva, F. (1987): Electroencephalography - Basic Principles, Clinical Applications and related Fields,2nd ed., Urban and Schwarzenberg, BaltimoreGoogle Scholar
  43. Okuda, K. (1993): Variety and generality of clustering in globally coupled oscillators, Physica D 63, 424–436ADSzbMATHCrossRefGoogle Scholar
  44. Panfilov, A.V., Holden, A.V. (eds.) (1997): Computational Biology of the Heart, Wiley, ChichesterzbMATHGoogle Scholar
  45. Pare, D., Curro’Dossi, R., Steriade, M. (1990): Neuronal basis of the parkinsonian resting tremor: a hypothesis and its implications for treatment, Neuroscience 35, 217–226CrossRefGoogle Scholar
  46. Perlitz, V., Schmid-Schönbein, H., Schulte, A., Dolgner, J., Petzold, E.R., Kruse, W. (1995): Effektivität des autogenen Trainings, Therapiewoche 26, 1536–1544Google Scholar
  47. Pertsov, A.M., Davidenko, J.M., Salomonsz, R., Baxter, W.T., Jalife, J. (1993): Spiral waves of excitation underlie reentrant activity in isolated cardiac muscle, Circ. Res. 72, 631–650Google Scholar
  48. Pliss, V. (1964): Principal reduction in the theory of stability of motion, Izv. Akad. Nauk. SSSR Math. Ser. 28, 1297–1324 (in Russian)MathSciNetzbMATHGoogle Scholar
  49. Sakaguchi, H., Shinomoto, S., Kuramoto, Y. (1987): Local and global self-entrainments in oscillator lattices, Prog. Theor. Phys. 77, 1005–1010Google Scholar
  50. Sakaguchi, H., Shinomoto, S., Kuramoto, Y. (1988): Mutual entrainment in oscillator lattices with nonvariational type interaction, Prog. Theor. Phys. 79, 1069–1079MathSciNetADSCrossRefGoogle Scholar
  51. Schmid-Schönbein, H., Ziege, S. (1991): The high pressure system of the mammalian circulation as a dynamic self-organizing system. In: Haken and Koepchen ( 1991 ), pp. 77–96Google Scholar
  52. Schmid-Schönbein, H., Ziege, S., Rütten, W., Heidtmann, H. (1992): Active and passive modulation of cutaneous red cell flux as measured by Laser Doppler anemometry, Vasa 32, 38–47 (Suppl.)Google Scholar
  53. Sears, T.A., Stagg, D. (1976): Short-term synchronization of intercostal motoneurone activity, J. Physiol. (London) 263, 357–381Google Scholar
  54. Singer, W., Gray, C.M. (1995): Visual feature integration and the temporal correlation hypothesis, Annu. Rev. Neurosci. 18, 555–586CrossRefGoogle Scholar
  55. Smith, J.M., Cohen, R.J. (1984): Simple finite-element models account for wide range of cardiac dysrhythmias, Proc. Natl. Acad. Sci. USA 81, 233–237ADSCrossRefGoogle Scholar
  56. Steriade, H., Jones, E.G., Llinâs, R. (1990): Thalamic Oscillations and Signaling, John Wiley and Sons, New YorkGoogle Scholar
  57. Strafella, A., Ashby, P., Munz, M., Dostrovsky, J.O., Lozano, A.M., Lang, A.E. (1997): Inhibition of Voluntary Activity by Thalamic Stimulation in Humans: Relevance for the Control of Tremor, Movement Disorders 12, 727–737CrossRefGoogle Scholar
  58. Strogatz, S.H. (1994): Nonlinear Dynamics and Chaos, Addison-Wesley, Reading, MAGoogle Scholar
  59. Strogatz, S.H., Mirollo, R.E. (1988a): Phase-locking and critical phenomena in lattices of coupled nonlinear oscillators with random intrinsic frequencies, Physica D 31, 143–168; (1988b): Collective Synchronisation in lattices of non-linear oscillators with randomness, J. Phys. A 21, L699 - L705MathSciNetADSzbMATHCrossRefGoogle Scholar
  60. Tass, P. (1996a): Resetting biological oscillators–a stochastic approach, J. Biol. Phys. 22, 27–64CrossRefGoogle Scholar
  61. Tass, P. (1996b): Phase resetting associated with changes of burst shape, J. Biol. Phys. 22, 125–155CrossRefGoogle Scholar
  62. Tass, P. (1997): Phase and frequency shifts in a population of phase oscillators, Phys. Rev. E 56, 2043–2060Google Scholar
  63. Tass, P., Haken, H. (1996): Synchronization in networks of limit cycle oscillators, Z. Phys. B 100, 303–320Google Scholar
  64. Volkmann, J., Joliot, M., Mogilner, A., Ioannides, A.A., Lado, F., Fazzini, E., Ribary, U., Llinäs, R. (1996): Central motor loop oscillations in parkinsonian resting tremor revealed by magnetoencephalography, Neurology 46, 1359–1370CrossRefGoogle Scholar
  65. Volkmann, J., Sturm, V. (1998): Indication and results of stereotactic surgery for advanced Parkinson’s disease, Crit. Rev. Neurosurg. 8, 209–216CrossRefGoogle Scholar
  66. Volkmann, J., Sturm, V., Freund, H.-J. (1998): Die subkortikale Hochfrequenz-stimulation zur Behandlung von Bewegungsstörungen, Akt. Neurologie 25, 1–9Google Scholar
  67. von Holst, E. (1935): Über den Prozess der zentralnervösen Koordination, Pflügers Archiv 236, 149–158CrossRefGoogle Scholar
  68. von Holst, E. (1939): Die relative Koordination als Phänomen und als Methode zentralnervöser Funktionsanalysen, Erg. Physiol. 42, 228–306Google Scholar
  69. Winfree, A.T. (1970): An integrated view of the resetting of a circadian clock, J. Theor. Biol. 28, 327–374CrossRefGoogle Scholar
  70. Winfree, A.T. (1980): The Geometry of Biological Time, Springer, BerlinzbMATHGoogle Scholar
  71. Winfree, A.T. (1987): When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias, Princeton University Press, PrincetonGoogle Scholar
  72. Winfree, A.T. (1994): Electrical turbulence in three-dimensional heart muscle, Science 266, 1003–1006ADSCrossRefGoogle Scholar
  73. Witkowski, F.X., Leon, L.J., Penkoske, P.A., Giles, W.R., Spano, M.L., Ditto, W.L., Winfree, A.T. (1998): Spatiotemporal evolution of ventricular fibrillation, Nature 392, 78–82ADSCrossRefGoogle Scholar
  74. Wunderlin, A., Haken, H. (1981): Generalized Ginzburg-Landau equations, slaving principle and center manifold theorem, Z. Phys. B 44, 135–141MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Peter A. Tass
    • 1
  1. 1.Neurologische KlinikHeinrich-Heine-UniversitätDüsseldorfGermany

Personalised recommendations