Globally Coupled Oscillator Networks

  • Eric Brown
  • Philip Holmes
  • Jeff Moehlis


We study a class of permutation-symmetric globally-coupled, phase oscillator networks on N-dimensional tori. We focus on the effects of symmetry and of the forms of the coupling functions, derived from underlying Hodgkin-Huxley type neuron models, on the existence, stability, and degeneracy of phase-locked solutions in which subgroups of oscillators share common phases. We also estimate domains of attraction for the completely synchronized state. Implications for stochastically forced networks and ones with random natural frequencies are discussed and illustrated numerically. We indicate an application to modeling the brain structure locus coeruleus: an organ involved in cognitive control.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Arnold, V. [ 1973 ], Ordinary Differential Equations. MIT Press, Boston.MATHGoogle Scholar
  2. Ashwin, P. and J. Swift [ 1992 ], The dynamics of N weakly coupled identical oscillators, J. Nonlin. Sci. 2, 69–108.MathSciNetMATHCrossRefGoogle Scholar
  3. Aston-Jones, G., J. Rajkowski, and T. Alexinsky [ 1994 ], Locus coeruleus neurons in the monkey are selectively activated by attended stimuli in a vigilance task., J. Neurosci. 14, 4467–4480.Google Scholar
  4. Bressloff, P. and S. Coombes [1998], Spike train dynamics underlying pattern formation in integrate-and-fire oscillator networks, Phys. Rev. Lett. 81, 2384 2387.Google Scholar
  5. Brown, E., J. Moehlis, P. Holmes, G. Aston-Jones, and E. Clayton [ 2002 ], The influence of spike rate on response in locus coeruleus, Unpublished manuscript, Program in Applied and Computational Mathematics, Princeton University, 2002.Google Scholar
  6. Chow, C. and N. Kopell [ 2000 ], Dynamics of spiking neurons with electrotonic coupling, Neural Comp. 12, 1643–1678.CrossRefGoogle Scholar
  7. Crawford, J. [ 1995 ], Scaling and singularities in the entrainment of globally coupled oscillators, Phys. Rev. Lett. 74, 4341–4344.CrossRefGoogle Scholar
  8. Crawford, J. and K. Davies [ 1999 ], Synchronization of globally coupled phase oscillators: singularities and scaling for general couplings, Physica D 125 (12), 1-46.Google Scholar
  9. Daido, H. [ 1994 ], Generic scaling at the onset of macroscopic mutual entrainment in limit cycles with uniform all-to-all coupling, Phys. Rev. Lett. 73(5), 760763.Google Scholar
  10. E, W. [2001], personal communication, 2001.Google Scholar
  11. Ermentrout, B. [ 1996 ], Type I membranes, phase resetting curves, and synchrony, Neural Comp. 8, 979–1001.CrossRefGoogle Scholar
  12. Ermentrout, G. and N. Kopell [ 1990 ], Oscillator death in systems of coupled neural oscillators, SIAM J. on Appl. Math. 50, 125–146.MathSciNetMATHCrossRefGoogle Scholar
  13. Fenichel, N. [ 1971 ], Persistence and smoothness of invariant manifolds for flows, Ind. Univ. Math. J. 21, 193–225.MathSciNetMATHCrossRefGoogle Scholar
  14. Freidlin, M. and A. Wentzell [ 1998 ], Random perturbations of dynamical systems. Springer, New York.MATHCrossRefGoogle Scholar
  15. Gerstner, W., L. van Hemmen, and J. Cowan [ 1996 ], What matters in neuronal locking?, Neural Comp. 8, 1653–1676.CrossRefGoogle Scholar
  16. Colomb, D., D. Hansel, B. Shraiman, and H. Sompolinsky [ 1992 ], Clustering in globally coupled phase oscillators, Phys. Rev. A 45 (6), 3516–3530.CrossRefGoogle Scholar
  17. Golubitsky, M., I. Stewart, and D. Schaeffer [ 1988 ], Singularities and Groups in Bifurcation Theory, Vol. 2. Springer, New York.MATHCrossRefGoogle Scholar
  18. Grant, S., G. Aston-Jones, and D. Redmond [ 1988 ], Responses of primate locus coeruleus neurons to simple and complex sensory stimuli., Brain Res. Bull. 21 (3), 401–410.Google Scholar
  19. Guckenheimer, J. and P. Holmes [ 1983 ], Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, New York.MATHGoogle Scholar
  20. Hansel, D., G. Mato, and C. Meunier [ 1995 ], Synchrony in excitatory neural networks, Neural Comp. 7, 307–337.CrossRefGoogle Scholar
  21. Hodgkin, A. and A. Huxley [ 1952 ], A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117, 500–544.Google Scholar
  22. Honeycutt, R. [ 1992 ], Stochastic Runge-Kutta Algorithms I. White Noise, Phys. Rev. A 45, 600–603.CrossRefGoogle Scholar
  23. Hoppensteadt, F. and E. Izhikevich [ 1997 ], Weakly Connected Neural Networks. Springer-Verlag, New York.CrossRefGoogle Scholar
  24. Izhikevich, E. [ 2000 ], Phase equations for relaxation oscillators, SIAM J. on Appl. Math. 60, 1789–1804.MathSciNetMATHCrossRefGoogle Scholar
  25. Johnston, D. and S. Wu [ 1997 ], Foundations of Cellular Neurophysiology. MIT Press, Cambridge, MA.Google Scholar
  26. Keener, J. and J. Sneyd [ 1998 ], Mathematical Physiology. Springer, New York.MATHGoogle Scholar
  27. Kim, S. and S. Lee [ 2000 ], Phase dynamics in the biological neural networks, Physica D 288, 380–396.CrossRefGoogle Scholar
  28. Kopell, N. and G. Ermentrout [ 1990 ], Phase transitions and other phenomena in chains of coupled oscillators, SIAM J. on Appl. Math. 50, 1014–1052.MathSciNetMATHCrossRefGoogle Scholar
  29. Kopell, N. and G. Ermentrout [ 1994 ], Inhibition-produced patterning in chains of coupled nonlinear oscillators, SIAM J. Appl. Math. 54, 478–507.MathSciNetMATHCrossRefGoogle Scholar
  30. Kopell, N., G. Ermentrout, and T. Williams [ 1991 ], On chains of osciallators forced at one end, SIAM J. on Appl. Math. 51, 1397–1417.MathSciNetMATHCrossRefGoogle Scholar
  31. Kuramoto, Y. [ 1984 ], Chemical Oscillations, Waves, and Turbulence. Springer, Berlin.MATHCrossRefGoogle Scholar
  32. Kuramoto, Y. [ 1997 ], Phase-and center-manifold reductions for large populations of coupled oscillators with application to non-locally coupled systems, Int. J. Bif. Chaos 7, 789–805.MathSciNetMATHCrossRefGoogle Scholar
  33. Murray, J. [ 2001 ], Mathematical Biology, 3rd. Ed. Springer, New York.Google Scholar
  34. Nichols, S. and K. Wiesenfeld [ 1992 ], Ubiquitous neutral stability of splay states, Phys. Rev. A 45 (12), 8430–8435.CrossRefGoogle Scholar
  35. Okuda, K. [ 1993 ], Variety and generality of clustering in globally coupled oscillators, Physica D 63, 424–436.MATHCrossRefGoogle Scholar
  36. Omurtag, A., E. Kaplan, B. Knight, and L. Sirovich [2000a], A population approach to cortical dynamics with an application to orientation tuning, Network 11, 247–260.MATHCrossRefGoogle Scholar
  37. Omurtag, A., B. Knight, and L. Sirovich [2000b], On the simulation of large populations of neurons, J. Comp. Neurosci. 8, 51–63.MATHCrossRefGoogle Scholar
  38. Ritt, J. and N. Kopell [2002], In preparation,2002.Google Scholar
  39. Rumelhart, D. and J. McClelland [ 1986 ], Parallel Distributed Processing: Explorations in the Microstructure of Cognition. MIT Press, Cambridge, MA.Google Scholar
  40. Servan-Schreiber, D., H. Printz, and J. Cohen [1990], A network model of catecholamine effects: Gain, signal-to-noise ratio, and behavior, Science 249, 89 2895.Google Scholar
  41. Sirovich, L., B. Knight, and A. Omurtag [ 2000 ], Dynamics of neuronal popula- tions: The equilibrium solution, SIAM J. on Appl. Math. 60, 2009–2028.MathSciNetMATHCrossRefGoogle Scholar
  42. Strogatz, S. [ 2000 ], From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D 143, 1–20.MathSciNetMATHCrossRefGoogle Scholar
  43. Taylor, D. and P. Holmes [ 1998 ], Simple models for excitable and oscillatory neural networks, J. Math. Biol. 37, 419–446.MathSciNetMATHCrossRefGoogle Scholar
  44. Tsang, K., R. Mirollo, and S. Strogatz [ 1991 ], Dynamics of a globally coupled oscillator array, Physica D 48, 102–112.MathSciNetMATHCrossRefGoogle Scholar
  45. Usher, M., J. Cohen, D. Servan-Schreiber, J. Rajkowski, and G. Aston-Jones [ 1999 ], The role of locus coeruleus in the regulation of cognitive performance, Science 283, 549–554.CrossRefGoogle Scholar
  46. van Vreeswijk, C., L. Abbot, and B. Ermentrout [ 1994 ], When inhibition not excitation synchronizes neural firing, J. Comp. Neurosci. 1, 313–321.CrossRefGoogle Scholar
  47. Watanabe, S. and S. Strogatz [ 1994 ], Constants of the motion for superconducting Josephson arrays, Physica D 74, 195–253.CrossRefGoogle Scholar
  48. Watanabe, S. and J. Swift [ 1997 ], Stability of periodic solutions in series arrays of Josephson Junctions with internal capacitance, J. Nordin. Sci. 7, 503–536.MathSciNetMATHCrossRefGoogle Scholar
  49. Wiesenfeld, K., P. Colet, and S. Strogatz [ 1998 ], Frequency locking in Josephson arrays: Connection with the Kuramoto model, Phys. Rev. E 57 (2), 15631569.Google Scholar
  50. Wiggins, S. [ 1994 ], Normally Hyperbolic Invariant Manifolds in Dynamical Systems. Springer, New York.MATHGoogle Scholar
  51. Williams, J., R. North, A. Shefner, S. Nishi, and T. Egan [ 1984 ], Membrane properties of rat locus coeruleus neurons, Neuroscience 13, 137–156.CrossRefGoogle Scholar
  52. Zhu, W. [ 1988 ], Stochastic averaging methods in random vibration, Appl. Mech. Rev. 41, 189–199.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 2003

Authors and Affiliations

  • Eric Brown
  • Philip Holmes
  • Jeff Moehlis

There are no affiliations available

Personalised recommendations