Synchronization Patterns

  • Peter A. Tass
Part of the Springer Series in Synergetics book series (SSSYN)


Chapter 2 was devoted to an ensemble of oscillators interacting via random forces. We studied the oscillators’ spontaneous behavior and their reaction to stimulation. In this way we were able to investigate type 0 and type 1 resetting of an ensemble of oscillators. Moreover by encountering burst splitting we learned how difficult it may be to assign the ensemble’s collective activity to a single phase value. However, in the ensemble scenario one important dynamical feature was completely missing: No self-organized patterns of synchronization occured. Actually, self-synchronized collective activity abounds in physiological systems (see, for instance, Steriade, Jones, Llinás 1988). For this reason it is not sufficient to model the oscillators’ mutual interactions by means of random forces. Rather we have to take into account synchronizing couplings among the oscillators.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abeles, M. (1982): Local cortical circuits. An elektrophysiological study, Springer, BerlinCrossRefGoogle Scholar
  2. Aertsen, A. (ed.) (1993): Brain theory, Elsevier, AmsterdamGoogle Scholar
  3. Aizawa, Y. (1976): Synergetic approach to the phenomena of mode-locking in nonlinear systems, Prog. Theor. Phys. 56, 703–716ADSCrossRefGoogle Scholar
  4. Arbib, A. (ed.) (1995): The handbook of brain theory and neural networks, MIT Press, CambridgeGoogle Scholar
  5. Arnold, V.I. (1983): Geometrical methods in the theory of ordinary differential equations, Springer, HeidelbergzbMATHCrossRefGoogle Scholar
  6. Aulbach, B. (1984): Continuous and Discrete Dynamics near Manifolds of Equilibria, LNM 1058, Springer, HeidelbergGoogle Scholar
  7. Beurle, R.L. (1956): Properties of a mass of cells capable of regenerating pulses, Philos. Trans. Soc. London, Ser. A 240, 55–94ADSCrossRefGoogle Scholar
  8. Braitenberg, V., Schüz, A. (1991): Anatomy of the Cortex, Springer, BerlinGoogle Scholar
  9. Carr, J. (1981): Applications of Centre Manifold Theory,Appl. Math. Sciences 35, SpringerGoogle Scholar
  10. Chawanya, T., Aoyagi, T., Nishikawa, I., Okuda, K., Kuramoto, Y. (1993): A model for feature linking via collective oscillations in the primary visual cortex, Biol. Cybern. 68, 483–490Google Scholar
  11. Cowan, J.D. (1987): Brain mechansims underlying visual hallucinations. In: Paines, D. (ed.), Emerging syntheses in science, Addison-Wesley, New York, 123–131 Creutzfeldt, O.D. (1983): Cortex Cerebri, Springer, BerlinGoogle Scholar
  12. Crick, F. (1984): Function of the thalamic reticular complex: the searchlight hypothesis, Proc. Natl. Acad. Sci. USA 81, 4586–4590ADSCrossRefGoogle Scholar
  13. Daido, H. (1992a): Quasientrainment and slow relaxation in a population of oscillators with random and frustrated interactions, Phys. Rev. Lett. 68, 1073–1076.ADSCrossRefGoogle Scholar
  14. Daido, H. (1992b): Order function and macroscopic mutual entrainment in uniformly coupled limit-cycle oscillators, Prog. Theor. Phys. 88, 1213–1218.ADSCrossRefGoogle Scholar
  15. Daido, H. (1993): A solvable model of coupled limit-cycle oscillators exhibiting partial perfect synchrony and novel frequency spectra, Physica D 69, 394–403.ADSzbMATHCrossRefGoogle Scholar
  16. Daido, H. (1994): Generic scaling at the onset of macroscopic mutual entrainment in limit-cycle oscillators with uniform all-to-all coupling, Phys. Rev. Lett. 73, 760–763.ADSCrossRefGoogle Scholar
  17. Daido, H. (1996): Onset of cooperative entrainment in limit-cycle oscillators with uniform all-to-all interactions: bifurcation of the order function, Physica D 91, 24–66.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 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
  19. Eckhorn, R., Frien, A., Bauer, R., Woelbern, T., Kehr, H. (1993): High Frequency 60–90 Hz oscillations in primary visual cortex of awake monkey, Neuro Rep. 4, 243–246Google Scholar
  20. Edelman, G.M. (1992): Bright air, brilliant fire, Penguin Books, London Eggermont, J.J.Google Scholar
  21. Edelman, G.M. (1992): (1990): The correlative brain. Theory and experiment in neural interaction, Springer, BerlinGoogle Scholar
  22. Elphik, C., Tirapegui, E., Brachet, M., Coullet, P. Iooss, G. (1987): A simple global characterization for normal forms of singular vector fields, Physica D 29, 95–127MathSciNetADSCrossRefGoogle Scholar
  23. Engel, A.K., König, P., Gray, C.M., Singer, W. (1990): Synchronization of oscillatory responses: a mechanism for stimulus-dependent assembly formationin cat visual cortex. In: Parallel Processing in Neural Systems and Computers, Eck-miller, R., Hartmann, G., Hauske, G. (eds.), Elsevier, North HollandGoogle Scholar
  24. Ermentrout, G.B., Cowan, J. (1979): A mathematical theory of visual hallucination patterns, Biol. Cybern. 34, 137–150MathSciNetzbMATHCrossRefGoogle Scholar
  25. Ermentrout, G.B., Rinzel, J. (1981): Waves in a simple, excitable or oscillatory reaction-diffusion model, J. Math. Biol. 11, 269–294Google Scholar
  26. FitzHugh, R. (1961): Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1, 445–466Google Scholar
  27. Freeman, W.J. (1975): Mass action in the nervous system, Academic Press, New YorkGoogle Scholar
  28. Gerstner, W., Ritz, R., Hemmen, J.L. van (1993): A biologically motivated and analytically soluble model of collective oscillations in the cortex, Biol. Cybern. 68, 363–374zbMATHCrossRefGoogle Scholar
  29. Glass, L., Mackey, M.C. (1988): From Clocks to Chaos, The Rhythms of Life,Princeton University PressGoogle Scholar
  30. Golomb, D., Hansel, D., Shraiman, B., Sompolinsky, H. (1992): Clustering in globally coupled phase oscillatorsGoogle Scholar
  31. Golomb, D., Wang, X.J., Rinzel, J. (1996): Propagation of spindle waves in a tha- lamic slice model, J. Neurophysiol. 75, 750–769 Phys. Rev. A 45, 3516–3530CrossRefGoogle Scholar
  32. Gray, C.M., Singer, W. (1987): Stimulus specific neuronal oscillations in the cat visual cortex: a cortical function unit, Soc. Neurosci. 404, 3Google Scholar
  33. Gray, C.M., Singer, W. (1987): (1989): Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex, Proc. Natl. Acad. Sci. USA 86, 1698–1702Google Scholar
  34. Gray, C.M., König, P., Engel, A.K., Singer, W. (1989): Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties, Nature 338, 334–337ADSCrossRefGoogle Scholar
  35. Griffith, J.S. (1963): A field theory of neural nets: I: Derivation of field equations, Bull. Math. Biophys. 25, 111–120MathSciNetzbMATHCrossRefGoogle Scholar
  36. Griffith, J.S. (1965): A field theory of neural nets: II: Properties of field equations, Bull. Math. Biophys. 27, 187 195MathSciNetGoogle Scholar
  37. Grossberg, S., Somers, D. (1991): Synchronized oscillations during cooperative feature linking in a cortical model of visual perception, Neural Networks 4, 453–466CrossRefGoogle Scholar
  38. Guckenheimer, J., Holmes, P. (1990): Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Berlin, HeidelbergGoogle Scholar
  39. Haken, H. (1964): A nonlinear theory of laser noise and coherence I, Z. Phys. 181, 96–124CrossRefGoogle Scholar
  40. Haken, H. (1970): Laser Theory, Springer, Berlin; (ed. )Google Scholar
  41. Haken, H. (1973): Synergetics (Proceedings of a Symposium on Synergetics, Elmau 1972 ), B.G. Teubner, StuttgartGoogle Scholar
  42. Haken, H. (1975): Generalized Ginzburg-Landau equations for phase transition-like phenomena in lasers, nonlinear optics, hydrodynamics and chemical reactions, Z. Phys. B 21, 105–114Google Scholar
  43. Haken, H. (1977): Synergetics, An Introduction, Springer, BerlinzbMATHCrossRefGoogle Scholar
  44. Haken, H. (1979): Pattern formation and pattern recognition — an attempt at a synthesis. In: Pattern formation by dynamic systems and pattern recognition, H. Haken (ed.), Springer, Berlin, 2–13CrossRefGoogle Scholar
  45. Haken, H. (1983) Advanced Synergetics, Springer, BerlinzbMATHGoogle Scholar
  46. Haken, H. (1988): Information and Self-Organization, Springer, BerlinzbMATHGoogle Scholar
  47. Haken, H. (1991): Synergetic computers and cognition, Springer, BerlinzbMATHCrossRefGoogle Scholar
  48. Haken, H. (1996a): Principles of Brain Functioning, A Synergetic Approach to Brain Activity, Behavior and Cognition, Springer, BerlinzbMATHCrossRefGoogle Scholar
  49. Haken, H. (1996b): Slaving principle revisited, Physica D 97, 95–103MathSciNetzbMATHCrossRefGoogle Scholar
  50. Haken, H., Graham, R. (1971): Synergetik — Die Lehre vom Zusammenwirken, Umschau 6, 191Google Scholar
  51. Haken, H., Wunderlin, A. (1982): Slaving principle for stachastic differential equations with additive and multiplicative noise and for discrete noisy maps, Z. Phys. B 47, 179–187MathSciNetCrossRefGoogle Scholar
  52. Haken, H., Kelso, J.A.S., Bunz, H. (1985): A theoretical model of phase transitions in human hand movements, Biol. Cybern. 51, 347–356MathSciNetzbMATHCrossRefGoogle Scholar
  53. Hakim, V., Rappel, W. (1992): Dynamics of the globally coupled complex Ginzburg-Landau equation, Phys. Rev. A 46, R7347 - R7350CrossRefGoogle Scholar
  54. Han, S. K., Kurrer, C., Kuramoto, Y. (1995): Dephasing and bursting in coupled neural oscillators, Phys. Rev. Lett. 75, 3190–3193ADSCrossRefGoogle Scholar
  55. Hebb, D.O. (1949): Organization of Behavior, Wiley, New YorkGoogle Scholar
  56. Hansel, D., Mato, G., Meunier, C. (1993a): Clustering and slow switching in globally coupled phase oscillators, Phys. Rev. E 48, 3470–3477Google Scholar
  57. Hansel, D., Mato, G., Meunier, C. (1993b): Phase dynamics of weakly coupled Hodgkin-Huxley neurons, Europhys. Lett., 23, 367–372ADSCrossRefGoogle Scholar
  58. Hirsch, M.W., Smale, S. (1974): Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, San DiegozbMATHGoogle Scholar
  59. Hirsch, M., Pugh, C., Shub, M. (1976): Invariant Manifolds, Lecture Notes Math. 583, Springer, BerlinGoogle Scholar
  60. Hodgkin, A.L., Huxley, A. F. (1952): A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (London) 117, 500–544Google Scholar
  61. Hopfield, J.J. (1982): Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. 79, 2554–2558MathSciNetADSCrossRefGoogle Scholar
  62. Hoppensteadt, F.C., Izhikevich, E.M. (1997): Weakly Connected Neural Networks, Springer, BerlinCrossRefGoogle Scholar
  63. Hubel, D.H., Wiesel T.N. (1959): Receptive fields of single neurones in the cat’s striate cortex, J. Physiol. 148, 574–591Google Scholar
  64. Hubel, D.H., Wiesel T.N. (1962): Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex, J. Physiol. 160, 106–154Google Scholar
  65. Hubel, D.H., Wiesel T.N. (1963): Shape and arrangement of columns in cat’s striate cortex, J. Physiol. 165, 559–568Google Scholar
  66. Iooss, G. (1987): Global characterization of the normal form for a vector field near a closed orbit, J. Diff. Equ. 76, 47–76MathSciNetCrossRefGoogle Scholar
  67. boss, G., Adelmeyer, M. (1992): Topics in Bifurcation Theory and Applications, Advanced Series in Nonlinear Dynamics, Vol. 3, World Scientific, SingaporeGoogle Scholar
  68. Jirsa, V.K., Haken, H. (1996): Field theory of electromagnetic brain activity, Phys. Rev. Lett. 77, 960–963ADSCrossRefGoogle Scholar
  69. Jirsa, V.K., Haken, H. (1997): A derivation of a macroscopic field theory of the brain from the quasi-microscopic neural dynamics, Physica D 99, 503–526zbMATHCrossRefGoogle Scholar
  70. Julesz, B. (1991): Early vision and focal attention, Rev. Mod. Phys. 63, 735–772ADSCrossRefGoogle Scholar
  71. Kelley, A. (1967): The stable, center-stable, center, center-unstable and unstable manifolds, J. Diff. Equ. 3, 546–570MathSciNetADSzbMATHCrossRefGoogle Scholar
  72. Kelso, J.A.S. (1981): On the oscillatory basis of movements, Bulletin of Psychonomic Society 18, 63Google Scholar
  73. Kelso, J.A.S. (1984): Phase transitions and critical behavior in human bimanual coordination, American Journal of Physiology: Regulatory, Integrative and Comparative Physiology 15, R1000 - R1004Google Scholar
  74. Kirchgässner, K. (1982): Wave-solutions of reversible systems and applications, J. Diff. Equations 45, 113–127zbMATHCrossRefGoogle Scholar
  75. Koch, C., Segev, I. (1989): Methods in Neuronal Modeling, From Synapses to Networks, MIT Press, CambridgeGoogle Scholar
  76. König, P., Engel, A.K., Singer, W. (1996): Integrator or coincidence detector? The role of the cortical neuron revisited, TINS 19, 130–137Google Scholar
  77. Kreiter, A.K., Singer, W. (1992): Oscillatory neuronal responses in the visual cortex of awake macaque monkey, Eur. J. Neurosci. 4, 369–375CrossRefGoogle Scholar
  78. Kuramoto, Y. (1991): Collective synchronization of pulse-coupled oscillators and excitable units, Physica D 50, 15–30Google Scholar
  79. Kuramoto, Y. (1984): Chemical Oscillations, Waves, and Turbulence, Springer, BerlinzbMATHCrossRefGoogle Scholar
  80. Langenberg, U., Kessler, K., Hefter, H., Cooke, J.D., Brown, S.H., Freund, H.-J. (1992): Effects of delayed visual feedback during sinusoidal visuomotor tracking, Soc. Neurosci. Abstr. Suppl. 5, 209Google Scholar
  81. Livingstone, M.S. (1991): Visually evoked oscillations in monkey striate cortex, Soc. Neurosci. Abstr. 17, 73Google Scholar
  82. Malsburg, C. von der, Schneider, W. (1986): A neural cocktail-party processor, Biol. Cybern. 54, 29–40CrossRefGoogle Scholar
  83. Marr, D. (1976): Early processing of visual information, Philos. Trans. R. Soc. Lond. [Biol] 275, 483–524ADSCrossRefGoogle Scholar
  84. Matthews, P.C., Strogatz, S.H. (1990): Phase diagram for the collective behavior of limit-cycle oscillators, Phys. Rev. Lett. 65, 1701–1704MathSciNetADSzbMATHCrossRefGoogle Scholar
  85. McCulloch, W., Pitts, W. (1943): A logical calculus of the ideas immanent in nervous activity, Bull. Math. Biophys. 5, 115–133MathSciNetzbMATHCrossRefGoogle Scholar
  86. Meinhardt, H. (1982): Models of Biological Pattern Formation, Academic Press, LondonGoogle Scholar
  87. Milner, P.M. (1974): A model for visual shape recognition, Psychol. Rev. 81, 52 1535Google Scholar
  88. Mirollo, R. E., Strogatz, S. H. (1990): Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math. 50, 1645–1662MathSciNetzbMATHGoogle Scholar
  89. Müller, B., Reinhardt, J. (1990): Neural Networks, An Introduction, Springer, BerlinzbMATHGoogle Scholar
  90. Murray, J. D. (1989): Mathematical Biology, Springer, BerlinzbMATHCrossRefGoogle Scholar
  91. Nagumo, J.S., Arimoto, S., Yoshizawa, S. (1962): An active pulse transmission line simulating nerve axon, Proc. IRE 50, 2061–2071CrossRefGoogle Scholar
  92. Murthy, V.N., Fetz, E.E. (1992): Coherent 25- to 35-Hz oscillations in the sensorimotor cortex of awake behaving monkeys, Proc. Natl. Acad. Sci. USA 89, 5670–5674ADSCrossRefGoogle Scholar
  93. Nakagawa, N., Kuramoto, Y. (1993): Collective chaos in a population of globally coupled oscillators, Prog. Theor. Phys. 89, 313–323ADSCrossRefGoogle Scholar
  94. Neuenschwander, S., Varela, F.J. (1993): Visually triggered neuronal oscillations in birds: an autocorrelation study of tectal activity, Eur. J. Neurosci. 5, 870–881CrossRefGoogle Scholar
  95. Nicholis, S., Wiesenfeld, K. (1992): Ubiquitous neutral stability of splay-phase states, Phys. Rev. A 45, 8430–8435CrossRefGoogle Scholar
  96. Nicolis, G., Prigogine, I. (1977): Self-Organization in Nonequilibrium Systems, Wiley, New YorkzbMATHGoogle Scholar
  97. Niebur, E., Schuster, H.G., Kammen, D.M. (1991): Collective frequencies and metastability in networks of limit-cycle oscillators with time delay, Phys. Rev. Lett. 67, 2753–2756ADSCrossRefGoogle Scholar
  98. Nunez, P.L. (1974): The brain wave equation: a model for the EEG, Math. Biosci. 21, 279–297zbMATHCrossRefGoogle Scholar
  99. Nunez, P.L. (1981): Electric fields of the brain, Oxford University Press; ( 1995 ): Neocortical dynamics and human EEG rhythms, Oxford University PressGoogle Scholar
  100. Okuda, K (1993): Variety and generality of clustering in globally coupled oscillators, Physica D 63, 424–436ADSzbMATHCrossRefGoogle Scholar
  101. Omidvar, O.M. (ed.) (1995): Progress in neural networks, Vol. 3, Ablex Publishing Corporation, Norwood, New JerseyGoogle Scholar
  102. Orban, G.A. (1984): Neuronal Operations in the Visual Cortex, Springer, BerlinCrossRefGoogle Scholar
  103. Perkel, D.H., Bullock, T.H. (1968): Neural coding, Neurosci. Res. Prog. Sum. 3, 405–527Google Scholar
  104. Plant, R.E. (1978): The effects of calcium on bursting neurons, Biophys. J. 21, 217–237CrossRefGoogle Scholar
  105. Plant, R.E. (1981): Bifurcation and resonance in a model for bursting nerve cells, J. Math. Biol. 11, 15–32MathSciNetzbMATHCrossRefGoogle Scholar
  106. 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
  107. Ramachandran, V.S. (1988): Perception of shape from shading Nature 331, 163–166Google Scholar
  108. Reichardt, W.E., Poggio, T (eds.) (1981): Theoretical approaches in neurobiology, MIT Press, CambridgeGoogle Scholar
  109. Rieke, F., Warland, D., de Ruyter van Stevenick, R., Bialek, W. (1997): Spikes: Exploring the Neural Code, MIT Press, CambridgeGoogle Scholar
  110. Rinzel, J. (1986): On different mechanisms for membrane potential bursting, Proc. Sympos. on Nonlinear Oscillations in Biology and Chemistry, Salt Lake City 1985, Lect. Notes in Biomath. Springer, Berlin 66, 19–33MathSciNetGoogle Scholar
  111. Roelfsema, P.R., Engel, A.K., König, P., Singer, W. (1997): Visuomotor integration is associated with zero time-lag synchronization among cortical areas, Nature 385, 157–161ADSCrossRefGoogle Scholar
  112. Sakaguchi, H., Shinomoto, S., Kuramoto, Y. (1987): Local and global self-entrainments in oscillator lattices, Prog. Theor. Phys. 77, 1005–1010ADSCrossRefGoogle Scholar
  113. Sakaguchi, H., Shinomoto, S., Kuramoto, Y. (1988): Mutual entrainment in oscillator lattices with nonvariational type interaction, Prog. Theor. Phys. 79, 1069–1079MathSciNetADSCrossRefGoogle Scholar
  114. Sandstede, B., Scheel, A., Wulff, C. (1997): Center-manifold reduction for spiral waves, C. R. Acad. Sci. Paris (Série I, Équations aux dérivées partielles/Partial Differential Equations) 324, 153–158MathSciNetADSzbMATHGoogle Scholar
  115. Schillen, T.B., König, P. (1994): Binding by temporal structure in multiple feature domains of an oscillatory neuronal network, Biol. Cybern. 70, 397–405CrossRefGoogle Scholar
  116. Schöner, G., Haken, H., Kelso, J.A.S. (1986): A stochastic theory of phase transitions in human hand movement, Biol. Cybern. 53, 247–257zbMATHCrossRefGoogle Scholar
  117. Schuster, H.G., Wagner, P. (1990a): A model for neuronal oscillations in the visual cortex. 1. Mean-field theory and derivation of the phase equations. Biol. Cybern. 64, 77–82zbMATHCrossRefGoogle Scholar
  118. Schuster, H.G., Wagner, P. (1990b): A model for neuronal oscillations in the visual cortex. 2. Phase description of the feature dependent synchronization. Biol. Cybern. 64, 83–85zbMATHCrossRefGoogle Scholar
  119. Shiino, M., Frankowicz, M. (1989): Synchronization of infinitely many coupled limit-cycle type oscillators, Physics Letters A 136, 103–108MathSciNetADSCrossRefGoogle Scholar
  120. Shimizu, H., Yamaguchi, Y., Tsuda, I., Yano, M. (1985): Pattern recognition based on holonic information dynamics: towards synergetic computers. In: Complex systems - operational approaches, Haken, H. (ed.), Springer, BerlinGoogle Scholar
  121. Singer, W. (1989): Search for coherence: a basic principle of cortical self-organization, Concepts Neurosci. 1, 1–26Google Scholar
  122. Singer, W., Gray, C.M. (1995): Visual feature integration and the temporal correlation hypothesis, Annu. Rev. Neurosci. 18, 555–586CrossRefGoogle Scholar
  123. Sompolinsky, H., Golomb, D., Kleinfeld, D. (1991): Cooperative dynamics in visual processing, Phys. Rev. A 43, 6990–7011CrossRefGoogle Scholar
  124. Stephan, K.M., Binkofski, F., Halsband, U., Dohle, C., Wunderlich, G., Schnitzler, A., Tass, P., Posse, S., Herzog, H., Sturm, V., Zilles, K., Seitz, R.J., Freund, H.-J.: The role of ventral medial wall motor areas in bimanual coordination, Brain, 122, 351–368Google Scholar
  125. Steriade, H., Jones, E.G., Llinâs, R. (1988): Thalamic Oscillations and Signaling, Wiley, New YorkGoogle Scholar
  126. Strogatz, S.H. (1994): Nonlinear Dynamics and Chaos, Addison-Wesley, Reading, MAGoogle Scholar
  127. 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, 143–168MathSciNetADSzbMATHCrossRefGoogle Scholar
  128. Strogatz, S.H., Mirollo, R.E. (1988b): Collective Synchronisation in lattices of non-linear oscillators with randomness, J. Phys. A 21, L699 - L705MathSciNetADSzbMATHCrossRefGoogle Scholar
  129. Strogatz, S.H., Mirollo, R.E. (1993): Splay states in globally coupled Josephson arrays: analytical prediction of Floquet multipliers, Phys. Rev. E 47, 220–227Google Scholar
  130. Swift, J.W., Strogatz, S.H., Wiesenfeld, K (1992): Averaging of globally coupled oscillators, Physica D 55, 239–250MathSciNetADSzbMATHCrossRefGoogle Scholar
  131. Tass, P. (1995a): Cortical pattern formation during visual hallucinations, J. Biol. Phys. 21, 177–210CrossRefGoogle Scholar
  132. Tass, P. (1995b): Phase and frequency shifts of two nonlinearly coupled oscillators, Z. Phys B 99, 111–121CrossRefGoogle Scholar
  133. Tass, P. (1997a): Phase and frequency shifts in a population of phase oscillators, Phys. Rev. E 56, 2043–2060Google Scholar
  134. Tass, P. (1997b): Oscillatory cortical activity during visual hallucinations, J. Biol. Phys. 23, 21–66CrossRefGoogle Scholar
  135. Tass, P., Haken, H. (1996a): Synchronization in networks of limit cycle oscillators, Z. Phys. B 100, 303–320CrossRefGoogle Scholar
  136. Tass, P., Haken, H. (1996b): Synchronized oscillations in the visual cortex–a synergetic model, Biol. Cybern. 74, 31–39zbMATHCrossRefGoogle Scholar
  137. Tass, P., Wunderlin, A., Schanz, M. (1995): A theoretical model of sinusoidal forearm tracking with delayed visual feedback, J. Biol. Phys. 21, 83–112CrossRefGoogle Scholar
  138. Tass, P., Kurths, J., Rosenblum, M.G., Guasti, G., Hefter, H. (1996): Delay-induced transitions in visually guided movements, Phys. Rev. E 54, R2224 - R2227Google Scholar
  139. Thom, R. (1972): Stabilité structurelle et morphogénèse - Essai d’une théorie générale des modèles, W.A. Benjamin, Inc., Reading, MassachusettsGoogle Scholar
  140. Treisman, A. (1980): A feature-integration theory of attention, Cogn. Psychol. 12, 97–136Google Scholar
  141. Treisman, A. (1986): Properties, parts and objects. In: Handbook of perception and human performances, Boff, K., Kaufman, L., Thomas, I. (eds.), Wiley, New YorkGoogle Scholar
  142. Vaadia, E., Haalman, I., Abeles, M., Bergman, H., Prut, Y., Slovin, H., Aertsen, A. (1995): Dynamics of neuronal interactions in monkey cortex in relation to behavioural events, Nature 373, 515–518ADSCrossRefGoogle Scholar
  143. Vanderbauwhede, A. (1989): Center Manifolds, Normal Forms and Elementary Bifurcations, Dyn. Rep. 2, 89–169MathSciNetGoogle Scholar
  144. Wiesenfeld, K., Hadley, P. (1989): Attractor crowding in oscillator arrays, Phys. Rev. Lett. 62, 1335–1338MathSciNetADSCrossRefGoogle Scholar
  145. Wilson, H.R., Cowan, J.D. (1972): Excitatory and inhibitory interactions in localized populations of model neurons, Biophysical Journal 12, 1–24ADSCrossRefGoogle Scholar
  146. Wilson, H.R., Cowan, J.D. (1973): A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetik 13, 55–80zbMATHCrossRefGoogle Scholar
  147. Winfree, A. T. (1967): Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol. 16, 15–42CrossRefGoogle Scholar
  148. Winfree, A. T. (1980): The Geometry of Biological Time, Springer, BerlinzbMATHGoogle Scholar
  149. Wischert, W., Wunderlin, A., Pelster, A., Olivier, M., Groslambert, J. (1994): Delay-induced instabilities in nonlinear feedback systems, Phys. Rev. E 49, 203–219MathSciNetGoogle Scholar
  150. Wunderlin, A., Haken, H. (1975): Scaling theory for nonequilibrium systems, Z. Phys. B 21, 393–401Google Scholar
  151. Wunderlin, A., Haken, H. (1981): Generalized Ginzburg-Landau equations, slaving principle and center manifold theorem, Z. Phys. B 44, 135–141MathSciNetCrossRefGoogle Scholar
  152. Yamaguchi, Y., Shimizu, H. (1984): Theory of self-synchronization in the presence of native frequency distribution and external noises, Physica D 11, 212–226MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

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

Personalised recommendations