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Noise-Dependent Stability of the Synchronized State in a Coupled System of Active Rotators

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We consider a Kuramoto model for the dynamics of an excitable system consisting of two coupled active rotators. Depending on both the coupling strength and the noise, the two rotators can be in a synchronized or desynchronized state. The synchronized state of the system is most stable for intermediate noise intensity in the sense that the coupling strength required to desynchronize the system is maximal at this noise level. We evaluate the phase boundary between synchronized and desynchronized states through numerical and analytical calculations.

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S. Brandt, A. Pelster and R. Wessel, "Noise-Dependent Stability of the Synchronized State in a Coupled System of Active Rotators,"

*World Journal of Condensed Matter Physics*, Vol. 1 No. 3, 2011, pp. 88-96. doi: 10.4236/wjcmp.2011.13014.

[1] | J. F. Heagy, T. L. Carroll and L. M. Pecora, “Synchronous Chaos in Coupled Oscillator Systems,” Physical Review E, Vol. 50, No. 3, 1994, pp. 1874-1885. doi:10.1103/PhysRevE.50.1874 |

[2] | R. Roy and K. S. Thorn-burg Jr., “Experimental synchronization of chaotic lasers,” Physical Review Letters, Vol. 72, No. 13, 1994, pp. 2009-2012. doi:10.1103/PhysRevLett.72.2009 |

[3] | A. V. Ustinov, M. Cirillo and B. A. Malomed, “Fluxon Dynamics in One-Dimensional Josephson-Junction Arrays,” Physical Review B, Vol. 47, No. 13, 1993, pp. 8357-8360. doi:10.1103/PhysRevB.47.8357 |

[4] | K. Wiesenfeld, P. Colet and S. H. Strogatz, “Synchronization Transitions in a Disor-dered Josephson Series Array,” Physical Review Letters, Vol. 76, No. 3, 1996, pp. 404-407. doi:10.1103/PhysRevLett.76.404 |

[5] | J. Hertz, A. Krogh and R. Palmer, “Introduction to the Theory of Neural Computation,” Addison-Wesley, Redwood City, 1991. |

[6] | Y. Soen, N. Cohen, D. Lipson and E. Braun, “Emergence of Spontaneous Rhythm Disorders in Self-Assembled Networks of Heart Cells,” Physical Review Letters, Vol. 82, No. 17, 1999, pp. 3556-3559. doi:10.1103/PhysRevLett.82.3556 |

[7] | A. Pikovsky, M. Rosenblum and J. Kurths, “Synchronization: A Universal Concept in Nonlinear Sciences,” Cambridge University Press, Cambridge, 2003. |

[8] | S. Strogatz, “Sync: The Emerging Science of Spontaneous Order,” Hyperion, New York, 2003. |

[9] | T. Shinbrot and F. J. Muzzio, “Review Article Noise to Order,” Nature, Vol. 410, 2001, pp. 251-258. doi:10.1038/35065689 |

[10] | S. F. Brandt, B. K. Dellen and R. Wessel, “Synchronization from Disordered Driving Forces in Arrays of Coupled Oscillators,” Physical Review Letters, Vol. 96, No. 3, 2006, pp. 034104-034107. doi:10.1103/PhysRevLett.96.034104 |

[11] | R. Chaon and P. J. Martínez, “Controlling Chaotic Solitons in Frenkel-Kontorova Chains by Disordered Driving Forces,” Physical Review Letters, Vol. 98, No. 22, 2007, pp. 224102-224105. doi:10.1103/PhysRevLett.98.224102 |

[12] | M. A. Zaks, A. B. Neiman, S. Feistel and L. Schimansky-Geier, “Noise-Controlled Oscillations and Their Bifurcations in Coupled Phase Oscillators,” Physical Review E, Vol. 68, No. 6, 2003, pp. 066206-066214. doi:10.1103/PhysRevE.68.066206 |

[13] | J. A. White, J. T. Rubinstein and A. R. Kay, “Channel Noise in Neurons,” Trends in Neurosciences, Vol. 23, No. 3, 2000, pp. 131-137. doi:10.1016/S0166-2236(99)01521-0 |

[14] | L. Gammaitoni, P. H?nggi, P. Jung and F. Marchesoni, “Stochastic Resonance,” Reviews of Modern Physics, Vol. 70, No. 1, 1998, pp. 223-287. doi:10.1103/RevModPhys.70.223 |

[15] | W. H. Calvin and C. F. Stevens, “Synaptic Noise and Other Sources of Randomness in Motoneuron Interspike Intervals,” Journal of Neurophysiology, Vol. 31, 1968, pp. 574-587. |

[16] | J. K. Douglass, L. Wilkens, E. Pantazelou and F. Moss, “Noise Enhancement of Informa-tion Transfer in Crayfish Mechanoreceptors by Stochastic Resonance,” Nature, Vol. 365, 1993, pp. 337-340. doi:10.1038/365337a0 |

[17] | K. Wiesenfeld and F. Moss, “Sto-chastic Resonance and the Benefits of Noise: From Ice Ages to Crayfish and SQUIDs,” Nature, Vol. 373, 1995, pp. 33-36. doi:10.1038/373033a0 |

[18] | H. Treutlein and K. Schulten, Ber. Bunsenges. Phys. Chem. 89, 710 (1985). |

[19] | C. Kurrer and K. Schulten, “Effect of Noise and Perturbations on Limit Cycle Systems,” Physica D: Nonlinear Phenomena, Vol. 50, No. 3, 1991, pp. 311-320. doi:10.1016/0167-2789(91)90001-P |

[20] | C. Kurrer and K. Schulten, “Noise-induced synchronous neuronal oscillations,” Physical Review E, Vol. 51, No. 6, 1995, pp. 6213-6218. doi:10.1103/PhysRevE.51.6213 |

[21] | C. Kurrer and K. Schulten, “Neuronal Oscillations and Stochastic Limit Cycles,” International Journal of Neural Systems, Vol. 7, No. 4, 1996, pp. 399-402. doi:10.1142/S0129065796000373 |

[22] | M. A. Zaks, X. Sailer, L. Schimansky-Geier and A. B. Neiman, “Noise Induced Com-plexity: From Subthreshold Oscillations to Spiking in Coupled Excitable Systems,” Chaos, Vol. 15, No. 2, 2005, p. 026117. doi:10.1063/1.1886386 |

[23] | B. Lindner, J. García-Ojalvo, A. Neiman and L. Schminasky-Geier, “Effects of Noise in Excit-able Systems,” Physics Reports, Vol. 392, No. 6, 2004, pp. 321-424. doi:10.1016/j.physrep.2003.10.015 |

[24] | J. Milton, “Dynamics of Small Neural Populations,” American Mathematical Society (AMS), Providence, 1996. |

[25] | S. F. Brandt, A. Pelster and R. Wessel, “Variational Calculation of the Limit Cycle and Its Frequency in a Two-Neuron Model with Delay,” Physical Re-view E, Vol. 74, No. 3, 2006, pp. 0362010-0362014. doi:10.1103/PhysRevE.74.036201 |

[26] | S. F. Brandt, A. Pelster and R. Wessel, “Synchronization in a Neuronal Feedback Loop through Asymmetric Temporal Delays,” Europhysics Letters, Vol. 79, No. 3, 2007, pp. 38001/1-5. doi:10.1209/0295-5075/79/38001 |

[27] | M. Rabinovich, A. Selverston, L. Rubchinsky and R. Huerta, Chaos 6, 288 (1996). doi:10.1063/1.166176 |

[28] | N. G. van Kampen, “Stochastic Processes in Physics and Chemistry,” 2nd Edition, North Hol-land, Amsterdam, 1992. |

[29] | T. Sakurai, E. Mihaliuk, F. Chirila and K. Showalter, Science 14, 296 (2002). |

[30] | H. J. Wünsche, O. Brox, M. Radziunas and F. Henneberger, “Excitability of a Semiconductor Laser by a Two- Mode Homoclinic Bifurcation,” Physical Review Letters, Vol. 88, No. 3, 2001, pp. 023901-023904. doi:10.1103/PhysRevLett.88.023901 |

[31] | E. S. Lobanova, E. E. Shnol and F. I. Ataullakhanov, “Complex Dynamics of the Formation of Spatially Localized Standing Structures in the Vicinity of Saddle-Node Bifurcations of Waves in the Reaction-Diffusion Model of Blood Clotting,” Physical Review E, Vol. 70, No. 3, 2004, pp. 032903-032906. doi:10.1103/PhysRevE.70.032903 |

[32] | A. V. Panfilov, S. C. Müller, V. S. Zykov and J. P. Keener, “Elimination of Spiral Waves in Cardiac Tissue by Multiple Electrical Shocks,” Physical Review E, Vol. 61, No. 4, 2000, pp. 4644-4647. doi:10.1103/PhysRevE.61.4644 |

[33] | C. Koch, “Biophysics of Computation: Information Processing in Single Neurons,” Oxford University Press, New York, 1999. |

[34] | J. D. Murray, “Mathematical Biology,” 2nd Edition, Springer, New York, 1993. doi:10.1007/b98869 |

[35] | A. S. Mikhailov, “Foundations of Synergetics I,” 2nd Edtion, Springer-Verlag, Berlin, 1994. |

[36] | T. R. Chay and J. Rinzel, “Bursting, Beating, and Chaos in an Excitable Membrane Model,” Biophysical Journal, Vol. 47, No. 3, 1985, pp. 357-366. doi:10.1016/S0006-3495(85)83926-6 |

[37] | B. Hu and C. Zhou, “Synchronization Regimes in Coupled Noisy Excitable Sys-temsphys,” Physical Review E, Vol. 63, No. 2, 2001, pp. 026201-026206. doi:10.1103/PhysRevE.63.026201 |

[38] | S. Shinomoto and Y. Kuramoto, “Phase Transitions in Active Rotator Systems,” Progress of Theoretical Physics, Vol. 75, No. 5, 1986, pp. 1105-1110. doi:10.1143/PTP.75.1105 |

[39] | H. Sakaguchi, S. Shinomoto and Y. Kuramoto, “Phase Transitions and Their Bifurcation Analysis in a Large Population of Active Rotators with Mean-Field Coupling,” Progress of Theoretical Physics, Vol. 79, No. 3, 1988, pp. 600-607. doi:10.1143/PTP.79.600 |

[40] | H. Risken, “The Fokker-Planck Equation. Methods of So- lution and Applications,” Springer-Verlag, Berlin, 1984. |

[41] | C. M. Bender and S. A. Orszag, “Advanced Mathematical Methods for Scientists and Engineers,” Springer, New York, 1999. |

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