## Phase transition in crowd synchrony of delay-coupled multilayer laser networks |

Optics Express, Vol. 20, Issue 18, pp. 19683-19689 (2012)

http://dx.doi.org/10.1364/OE.20.019683

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### Abstract

An analogy between crowd synchrony and multi-layer neural network architectures is proposed. It indicates that many non-identical dynamical elements (oscillators) communicating indirectly via a few mediators (hubs) can synchronize when the number of delayed couplings to the hubs or the strength of the couplings is large enough. This phenomenon is modeled using a system of semiconductor lasers optically delay-coupled in either a fully connected or a diluted manner to a fixed number of non-identical central hub lasers. A universal phase transition to crowd synchrony with hysteresis is observed, where the time to achieve synchronization diverges near the critical coupling independent of the number of hubs.

© 2012 OSA

1. T. Danino, O. Mondragón-Palomino, L. Tsimring, and J. Hasty, “A synchronized quorum of genetic clocks,” Nature **463**(7279), 326–330 (2010). [CrossRef] [PubMed]

2. J. Garcia-Ojalvo, M. B. Elowitz, and S. H. Strogatz, “Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing,” Proc. Natl. Acad. Sci. U.S.A. **101**(30), 10955–10960 (2004). [CrossRef] [PubMed]

3. K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy, “Observation of antiphase states in a multimode laser,” Phys. Rev. Lett. **65**(14), 1749–1752 (1990). [CrossRef] [PubMed]

6. J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett. **105**(26), 264101 (2010). [CrossRef] [PubMed]

7. S. De Monte, F. d’Ovidio, S. Danø, and P. G. Sørensen, “Dynamical quorum sensing: Population density encoded in cellular dynamics,” Proc. Natl. Acad. Sci. U.S.A. **104**(47), 18377–18381 (2007). [CrossRef] [PubMed]

8. A. F. Taylor, M. R. Tinsley, F. Wang, Z. Huang, and K. Showalter, “Dynamical quorum sensing and synchronization in large populations of chemical oscillators,” Science **323**(5914), 614–617 (2009). [CrossRef] [PubMed]

9. J. Gómez-Gardeñes, S. Gómez, A. Arenas, and Y. Moreno, “Explosive synchronization transitions in scale-free networks,” Phys. Rev. Lett. **106**(12), 128701 (2011). [CrossRef] [PubMed]

10. I. Leyva, R. Sevilla-Escoboza, J. M. Buldú, I. Sendiña-Nadal, J. Gómez-Gardeñes, A. Arenas, Y. Moreno, S. Gómez, R. Jaimes-Reátegui, and S. Boccaletti, “Explosive first-order transition to synchrony in networked chaotic oscillators,” Phys. Rev. Lett. **108**(16), 168702 (2012). [CrossRef] [PubMed]

6. J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett. **105**(26), 264101 (2010). [CrossRef] [PubMed]

*Q*, equal to the normalized time-averaged coherent intensity summed over all lasers,

*Q*=

*<I>/M*, where

*<>*denotes the average over a time window T.

*Q*is a constant in the absence of synchronization and grows with

*M*as crowd synchrony emerges. The critical crowd,

*M*, was defined where the order parameter changes abruptly. Though this provides a convenient measure for synchronization for simple architectures, for more general architectures, such as those containing multiple HUs, this criterion is less appropriate, since cluster synchronization of a subset of the lasers might emerge. In addition, since this criterion looks at the change of

_{c}*Q*(d

*Q*/d

*M*) one cannot determine if a single simulation with a given

*M*is synchronized or not. Finally, since

*M*is an integer one cannot investigate the continuous mathematical nature of the transition to synchronization.

*ρ*, of the output intensity amongst all pairs of lower layer lasers over a time window of Τ:

*I*

_{i}(t) = |

*E*

_{i}(t)|

^{2}is the

*ith*laser's intensity at time

*t*and

*I*(t) over

_{i}*Τ*.

*ρ(i,j)*using one of the following threshold criteria: the mean value of all the correlations in the histogram or the minimal correlation value in the histogram (which is dominated by the correlation between laser pairs on opposite extremes of the frequency distribution). Using any of these criteria in our simulations yields similar qualitative results. To minimize computational complexity, our selected calculation criteria for synchronization was that the minimal value of correlations in the histogram be above a threshold value,

*ρ*= 0.7. Once synchronized, all lower level lasers receive roughly the same coupling amplitude, however, lasers on the extremes of the frequency distribution are less effectively coupled and as a result, their lasing intensities are lower. The time dependent intensity fluctuations of all lasers, however, are almost identical reflected by

*ρ*approaching unity.

*P*= 3 is slowly increased, by Δσ = 3·10

^{−3}σ

_{c}every 10τ, from 0.975σ

_{c}, through the critical point, to σ = 1.0275σ

_{c}. Once the system is fully synchronized, the coupling is slowly decreased back to its initial value (Fig. 3(b)). A hysteresis loop is formed, as expected for a first order phase transition [16], whose width is ~1-2% of the normalized sigma (the width depends on the rate of modification of the coupling strength).

*M·*. To check this hypothesis we measure the life-time of the hysteresis loop as a function of

*M*. The network is first synchronized at

*σ*~1.01

*σ*

_{c}and then the coupling strength is abruptly decreased to be within the hysteresis loop,

*σ*= 0.995

*σ*

_{c}. We next measure the decay of synchronization as a function of time, Fig. 3(c). Results indicate a good collapse of the data for all

*M*in the range [20,100] and the estimated decay exponent is almost a constant independent of

*M*as shown in Fig. 3(d). This lack of dependence of the decorrelation time on the system size indicates an unusual

*discontinuous dynamical transition*which might be in the spirit of dynamical first order phase transition measured for spin-glass systems with the lack of latent heat [17

17. T. R. Kirkpatrick and D. Thirumalai, “Dynamics of the structural glass transition and the p-spin-interaction spin-glass model,” Phys. Rev. Lett. **58**(20), 2091–2094 (1987). [CrossRef] [PubMed]

18. D. J. Gross, I. Kanter, and H. Sompolinsky, “Mean-field theory of the Potts glass,” Phys. Rev. Lett. **55**(3), 304–307 (1985). [CrossRef] [PubMed]

*P*/

*M*lasers. Substituting Eq. (6) in Eq. (5) one finds a self-consistent equation for

^{0.5}in accordance with Eq. (4).

## Acknowledgments

## References and links

1. | T. Danino, O. Mondragón-Palomino, L. Tsimring, and J. Hasty, “A synchronized quorum of genetic clocks,” Nature |

2. | J. Garcia-Ojalvo, M. B. Elowitz, and S. H. Strogatz, “Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing,” Proc. Natl. Acad. Sci. U.S.A. |

3. | K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy, “Observation of antiphase states in a multimode laser,” Phys. Rev. Lett. |

4. | S. H. Strogatz, |

5. | P. Dallard, A. Fitzpatrick, A. Flint, S. Le Bourva, A. Low, R. Ridsdill Smith, and M. Willford, “The London millennium footbridge,” Structural Engineer |

6. | J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett. |

7. | S. De Monte, F. d’Ovidio, S. Danø, and P. G. Sørensen, “Dynamical quorum sensing: Population density encoded in cellular dynamics,” Proc. Natl. Acad. Sci. U.S.A. |

8. | A. F. Taylor, M. R. Tinsley, F. Wang, Z. Huang, and K. Showalter, “Dynamical quorum sensing and synchronization in large populations of chemical oscillators,” Science |

9. | J. Gómez-Gardeñes, S. Gómez, A. Arenas, and Y. Moreno, “Explosive synchronization transitions in scale-free networks,” Phys. Rev. Lett. |

10. | I. Leyva, R. Sevilla-Escoboza, J. M. Buldú, I. Sendiña-Nadal, J. Gómez-Gardeñes, A. Arenas, Y. Moreno, S. Gómez, R. Jaimes-Reátegui, and S. Boccaletti, “Explosive first-order transition to synchrony in networked chaotic oscillators,” Phys. Rev. Lett. |

11. | A. Engel and C. Broeck, |

12. | M. Opper and W. Kinzel, “Statistical mechanics of generalization,” in |

13. | F. Rosenblatt, “The perceptron: A probabilistic model for information storage and organization in the brain,” Psychol. Rev. |

14. | M. Opper, “Learning in neural networks: Solvable dynamics,” Europhys. Lett. |

15. | A. Priel, M. Blatt, T. Grossmann, E. Domany, and I. Kanter, “Computational capabilities of restricted two-layered perceptrons,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

16. | G. Bertotti, |

17. | T. R. Kirkpatrick and D. Thirumalai, “Dynamics of the structural glass transition and the p-spin-interaction spin-glass model,” Phys. Rev. Lett. |

18. | D. J. Gross, I. Kanter, and H. Sompolinsky, “Mean-field theory of the Potts glass,” Phys. Rev. Lett. |

**OCIS Codes**

(190.3100) Nonlinear optics : Instabilities and chaos

(200.4700) Optics in computing : Optical neural systems

(270.3100) Quantum optics : Instabilities and chaos

(060.4258) Fiber optics and optical communications : Networks, network topology

(060.4263) Fiber optics and optical communications : Networks, star

**ToC Category:**

Optics in Computing

**History**

Original Manuscript: June 11, 2012

Manuscript Accepted: July 28, 2012

Published: August 13, 2012

**Citation**

Elad Cohen, Michael Rosenbluh, and Ido Kanter, "Phase transition in crowd synchrony of delay-coupled multilayer laser networks," Opt. Express **20**, 19683-19689 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-19683

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### References

- T. Danino, O. Mondragón-Palomino, L. Tsimring, and J. Hasty, “A synchronized quorum of genetic clocks,” Nature463(7279), 326–330 (2010). [CrossRef] [PubMed]
- J. Garcia-Ojalvo, M. B. Elowitz, and S. H. Strogatz, “Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing,” Proc. Natl. Acad. Sci. U.S.A.101(30), 10955–10960 (2004). [CrossRef] [PubMed]
- K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy, “Observation of antiphase states in a multimode laser,” Phys. Rev. Lett.65(14), 1749–1752 (1990). [CrossRef] [PubMed]
- S. H. Strogatz, Sync: The Emerging Science of Spontaneous Order (Hyperion, 2003).
- P. Dallard, A. Fitzpatrick, A. Flint, S. Le Bourva, A. Low, R. Ridsdill Smith, and M. Willford, “The London millennium footbridge,” Structural Engineer79, 17–21 (2001).
- J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett.105(26), 264101 (2010). [CrossRef] [PubMed]
- S. De Monte, F. d’Ovidio, S. Danø, and P. G. Sørensen, “Dynamical quorum sensing: Population density encoded in cellular dynamics,” Proc. Natl. Acad. Sci. U.S.A.104(47), 18377–18381 (2007). [CrossRef] [PubMed]
- A. F. Taylor, M. R. Tinsley, F. Wang, Z. Huang, and K. Showalter, “Dynamical quorum sensing and synchronization in large populations of chemical oscillators,” Science323(5914), 614–617 (2009). [CrossRef] [PubMed]
- J. Gómez-Gardeñes, S. Gómez, A. Arenas, and Y. Moreno, “Explosive synchronization transitions in scale-free networks,” Phys. Rev. Lett.106(12), 128701 (2011). [CrossRef] [PubMed]
- I. Leyva, R. Sevilla-Escoboza, J. M. Buldú, I. Sendiña-Nadal, J. Gómez-Gardeñes, A. Arenas, Y. Moreno, S. Gómez, R. Jaimes-Reátegui, and S. Boccaletti, “Explosive first-order transition to synchrony in networked chaotic oscillators,” Phys. Rev. Lett.108(16), 168702 (2012). [CrossRef] [PubMed]
- A. Engel and C. Broeck, Statistical Mechanics of Learning (Cambridge Univ Press, 2001).
- M. Opper and W. Kinzel, “Statistical mechanics of generalization,” in Models of Neural Networks III, E. D. J. L. van Hemmen, and K. Schulten, ed. (Springer-Verlag, 1996), 151–209.
- F. Rosenblatt, “The perceptron: A probabilistic model for information storage and organization in the brain,” Psychol. Rev.65(6), 386–408 (1958). [CrossRef] [PubMed]
- M. Opper, “Learning in neural networks: Solvable dynamics,” Europhys. Lett.8(4), 389–392 (1989). [CrossRef]
- A. Priel, M. Blatt, T. Grossmann, E. Domany, and I. Kanter, “Computational capabilities of restricted two-layered perceptrons,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics50(1), 577–595 (1994). [CrossRef] [PubMed]
- G. Bertotti, Hysteresis in Magnetism: For Physicists, Materials Scientists, and Engineers (Academic Press, 1998).
- T. R. Kirkpatrick and D. Thirumalai, “Dynamics of the structural glass transition and the p-spin-interaction spin-glass model,” Phys. Rev. Lett.58(20), 2091–2094 (1987). [CrossRef] [PubMed]
- D. J. Gross, I. Kanter, and H. Sompolinsky, “Mean-field theory of the Potts glass,” Phys. Rev. Lett.55(3), 304–307 (1985). [CrossRef] [PubMed]

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