1. http://ifisc.uib.es - Mallorca - Spain Workshop on Network Synchronization: from dynamical systems to neuroscience Lorentz Center, Leiden, 19-30 May 2008 Excitability mediated by dissipative solitons Pere Colet Adrian Jacobo, Damià Gomila, Manuel Matías Claudio J. Tessone, Alessandro Sciré, Raúl Toral

2. http://ifisc.uib.es Introduction Dissipative solitons in a Kerr cavity Soliton instabilities Soliton excitability Effect of a localized pump Interaction of oscillating & excitable solitons Collective firing induced by noise or diversity. Outline

3. http://ifisc.uib.es Dissipative solitons Localized excitations in a vertically vibrated granular layer. P.B. Umbanhowar, F. Melo & H.L. Swinney Nature 382, 793 (1996). Soliton in a Vertical Cavity Surface Emitting Laser S. Barland et al., Nature, 419, 699 (2002). Dissipative solitons are localized spatial structures that appear in certain dissipative media: Chemical reactions: J.E. Pearson, Science 261, 189 (1993); K.J. Lee & H.L. Swinney, Science 261, 192 (93). Gas discharges: I. Müller, E. Ammelt & H.G. Purwins, Phys. Rev. Lett. 73, 640, (1994). Fluids: O. Thual & S. Fauve, J. Phys. 49, 1829 (1988). N. Akhmediev & A. Ankiewicz (eds), “Dissipative solitons”, Lecture Notes in Physics 661 (Springer, Berlin, 2005); “Dissipative Solitons: From Optics to Biology and Medicine”, (Springer 2008)

4. http://ifisc.uib.es Pattern formation in nonlinear optical cavities 1.Driving 2.Dissipation 3.Nonlinearity 4.Spatial coupling Spontaneous pattern formation Pump field Nonlinear medium Sodium vapor cell with single mirror feedback Liquid crystal light valve T. Ackemann and W. Lange, Appl. Phys. B 72, 21 (2001) P.L. Ramazza et al., J. Nonlin. Opt. Phys. Mat. 8, 235 (1999) P.L. Ramazza, S. Ducci, S. Boccaletti & F.T. Arecchi, J. Opt. B 2, 399 (2000) F.T. Arecchi, S. Boccaletti & P.L. Ramazza, Phys. Rep. 318, 1 (1999). L.A. Lugiato, M. Brambilla & A. Gatti, Adv. Atom. Mol. Opt. Phys. 40, 229 (1999) N.N. Rosanov, “Spatial Hysteresis and Optical Patterns”, Springer 2002.

5. http://ifisc.uib.es Dissipative solitons versus propagation solitons “Dissipative solitons” Dissipative. Unique once the parameters of the system are fixed. Potentially useful for optical storage & information processing. Propagation solitons Conservative Continuous family of solutions depending on energy. Useful for optical communication systems N.N. Rosanov in Progress in Optics, 35 (1996). M. Segev (ed.) Special Issue on Solitons, Opt. Photonics News 13(27), 2002 L.A. Lugiato (ed), Feature section on Cavity Solitons, IEEE J. Quantum Electron. 39(2) (2003); N. Akhmediev & A. Ankiewicz (eds), “Dissipative solitons”, Lecture Notes in Physics 661 (Springer, Berlin, 2005). Ackemann-Lange

6. http://ifisc.uib.es Scenarios for dissipative solitons Stable droplets: Localized structures stabilised by nonlinear domain wall dynamics due curvature. Exist in 2d systems. D. Gomila et al, PRL 87, 194101 (2001) Homogeneous Solution Control Parameter Amplitude Bistability Homogeneous Solution Homogeneous Solution Localized structures as single spot of a cellular pattern. Exist in 1d & 2d systems. W.J. Firth & A. Lord, J. Mod. Opt. 43, 1071 (1996) Homogeneous Solution Hexagonal Pattern Subcritical Cellular Pattern Control Parameter Amplitude Localized structures stabilised by interaction of oscillatory tails. Exist in 1d & 2d systems. P. Coullet, et al PRL 58, 431(1987) G.-L.Oppo et al. J. Opt. B 1, 133 (1999) G.-L.Oppo et al. J. Mod Opt. 47, 2005 (2000) P. Coullet, Int. J. Bif. Chaos 12, 2445 (2002) Excitability mediated by localized structures

7. http://ifisc.uib.es Excitability. General ideas Excitability: has origin in Biology (action potential of nerve cells; also heart), also found in reaction-diffusion systems. Simplest minimum ingredients in phase space for excitability: Stable fixed point Threshold Reinjection mechanism in phase space (that leads to refractory period). Different responses to sub/supra-threshold perturbations. Three simplest excitability routes (2-D phase space), occur close to bifurcations leading to oscillatory behavior: a) saddle-node in invariant circle (Andronov-Leontovich) (Adler equation) b) saddle-loop (homoclinic) bifurcation c) fast-slow systems with S nullcline (slow manifold): canard (Fitzhugh-Nagumo ) Excitable media: spatially extended systems in which the local dynamics is excitable. J.D. Murray, Mathematical Biology, Springer 2002, 3rd ed. E. Meron; Pattern formation in excitable media; Phys. Rep. 218, 1 (1992). B. Lindner, J. García-Ojalvo, A. Neiman & L. Schimansky-Geier; Effects of noise in excitable systems; Phys. Rep. 392, 321 (2004).

8. http://ifisc.uib.es Excitability in optical systems Some examples of excitability in optical systems (mostly active systems): Systems with thermal effects (slow variable) that interplay with a hysteresis cycle of a fast variable. Leads to (c), FHN-like excitability. Cavity with T-dependent absorption (Lu et al, PRA 58, 809 (1998)). Semiconductor optical amplifier (Barland et al, PRE 68, 036209 (2003)). Lasers with saturable absorber (Dubbeldam et al, PRE 60, 6580 (1999)); lasers with optical feedback (Giudicci et al, PRE 55, 6414 (1997); Yacomotti et al, PRL 83, 292 (1999)); lasers with injected signal (Coullet et al, PRE 58, 5347 (1998); Goulding et al, PRL 98, 153903 (2007)). These lead to (a): saddle-node in an invariant circle. Lasers with intracavity saturable absorber (Plaza et al, Europhys. Lett. 38, 85 (1997)). Excitability mediated by a saddle-loop bifurcation. Semiconductor DFB laser (interaction of 2 modes) (Wuensche et al, PRL 88, 023901 (2002)). Homoclinic bifurcation slightly different than (b). Possible applications: optical switch (responding to sufficiently high optical input signals); optical communications: pulse reshaping.

9. http://ifisc.uib.es Self-focusing Kerr cavity x output field input field E 0 z y  : detuning Homogeneous solution E 0 : pump Control parameters Lugiato-Lefever model L.A. Lugiato & R. Lefever, PRL 58, 2209 (1988). It becomes unstable at I s =1 leading to a subcritical hexagonal pattern field envelope

10. http://ifisc.uib.es Self-focusing Kerr cavity solitons W.J. Firth & A. Lord, J. Mod. Opt. 43, 1071 (1996); W.J. Firth, A. Lord & A.J. Scroggie, Phys. Scripta, T67, 12 (96) Cavity soliton Can be seen as a solution connecting a cell of the pattern with the homogeneous solution Radial equation: Soliton profile can be found solving the l.h.s. equated to zero with Numerical solutions with arbitrary precision: Discretize r set of nonlinear ordinary eqs. Spatial derivatives computed in Fourier space Solve using Newton-Raphson Continuation methods can be used Linear stability analysis can be performed W.J. Firth & G.K. Harkness, Asian J. Phys 7, 665 (1998); G.-L. Oppo, A.J. Scroggie & W.J. Firth, PRE 63, 066209 (2001); J.M. McSloy, W.J. Firth, G.K. Harkness & G.-L. Oppo, PRE 66, 046606 (2002)

11. http://ifisc.uib.es Stability of Kerr cavity solitons Hopf instability observed in W.J. Firth, A. Lord & A.J. Scroggie, Phys. Scripta,T67,12 (96) Stable Soliton amplitude Unstable Hom. solution IsIs Saddle- Node Hopf No solitons Hopf Azimuth inst. m=5 m=6 Saddle- Node W.J. Firth, G.K. Harkness, A. Lord, J. McSloy, D. Gomila & P. Colet, JOSA B 19, 747 (2002) IsIs

12. http://ifisc.uib.es Azimuth instabilities m=6 m=5 Unstable Eigenmode t

13. http://ifisc.uib.es Cross-section middle branch soliton Oscillating soliton still useful for applications since its amplitude is bounded below by middle branch soliton. Hopf instability No solitons Azimuth inst. Hopf Saddle-node solitons

14. http://ifisc.uib.es Saddle-loop bifurcation  =1.30478592  =1.304788  =1.3047  =1.3 I s =0.9 middle-branch cavity soliton oscillating cavity soliton  max(|E|) LCLC Hopf Saddle-loop homogeneous solution SN Homogeneous solution Minimum distance of oscillating soliton to middle-branch soliton D. Gomila, M. Matias and P. Colet, Phys. Rev. Lett. 94, 063905 (2005).

15. http://ifisc.uib.es Saddle-loop bifurcation. Scaling law Close to bifurcation point: T: period of oscillation 1 unstable eigenvalue of saddle (middle-branch soliton) S.H. Strogatz, Nonlinear dynamics and chaos 2004 1/ 1 numerical simulations middle-branch soliton spectrum 1

16. http://ifisc.uib.es Phase space close to saddle-loop bifurcation Only two localized modes. uu ss middle-branch soliton spectrum Close to saddle: dynamics takes place in the plane (  u,  s ) Saddle-node index: =- s / u =2.177/0.177>1 (stable limit cycle) D. Gomila, A. Jacobo, M. Matias and P. Colet, PRA 75, 026217 (2007).  A=(E-E saddle )/E s Beyond Saddle Loop Oscillatory regime Projection onto  s Projection onto  u

17. http://ifisc.uib.es Small perturbations of homogeneous solution decay. Localized perturbations above middle branch soliton send the system to a long excursion through phase-space. Excitability D. Gomila, M. Matias and P. Colet, Phys. Rev. Lett. 94, 063905 (2005). The system is not locally excitable. Excitability emerges from spatial coupling Beyond saddle-loop bifurcation

18. http://ifisc.uib.es Takens-Bogdanov point TB Distance between saddle-node and Hopf Hopf saddle-loop saddle-node No solitons solitons oscillating solitons d → 0 for  → ∞ and I s → 0 NLSE saddle-node Hopf  =1.5  =1.6  =1.7 The Hopf frequency when it meets the saddle- node is zero. Takens-Bogdanov point. Unfolding of TB yields a Saddle-Loop  =1.5 Saddle-loop bifurcation is not generic. Why it is present here?

19. http://ifisc.uib.es Pump: Plane wave + Localized Gaussian Beam IsIs max(|E| 2 ) 1 Excitability Pattern IsIs max(|E| 2 ) 1 Excitability Pattern Oscillations Hom. pump SNIC Saddle Node Hopf I sh =0.7,  =1.34 Excitability arising from a saddle-loop bifurcation have a large threshold. To reduce the threshold we consider for the pump:

20. http://ifisc.uib.es Saddle-node in the circle (SNIC) bifurcation From the new oscillatory regime to the excitable regime. I s =0.927 I sh =0.3,  =1.45 middle-branch cavity soliton fundamental solution Close to bifurcation point: Projection onto  u Projection onto  s unstable upper branch soliton I s =0.907 I s =0.8871 I s =0.8

21. http://ifisc.uib.es Full scenario I sh =0.3 I Only fundamental solution II Stationary DS, fundamental solution stable III Oscillating DS, fundamental solution stable IV Excitable DS, fundamental solution stable V Oscillating DS, no fundamental solution Excitability can appear as a result of: Saddle loop (oscillating and middle branch solitons collide) Saddle node on the invariant circle (fundamental solution and middle branch soliton collide). Controllable excitability threshold.

22. http://ifisc.uib.es Noise effects, coherence resonance A. S. Pikovsky and J. Kurths, Phys. Rev. Lett. 78, 775 (1997). Introducing white spatiotemporal noise excitable solitons show coherence resonance. In excitable systems a moderate level of noise induces a more regular firing (coherence resonance)

23. http://ifisc.uib.es Interaction of two oscillating solitons  =1.27, I s =0.9, homogeneous pump Oscillating solitons move until they reach equilibrium positions given by tails interaction. Three equilibrium distances are found: Single structure period T=8.66 In-phase oscillation. T=8.93 Out-phase oscillation. T=8.94 Strong interaction. In & out-phase oscillation depending on initial condition. T in =8.59 < T out =10.45

24. http://ifisc.uib.es Interaction of excitable solitons Pulse on Pulse off Firing Firing induced by interaction Pulse on Firing Firing induced by interaction 1 0 1 Firing bit 1 1 1 1 OR logical gate

25. http://ifisc.uib.es AND 1 0 1 1 0 1 Pulse on Pulse off Pulse on

26. http://ifisc.uib.es NOT

27. http://ifisc.uib.es Collective firing induced by noise or diversity Globally coupled active rotators Diversity: natural frequencies noise  j <1 excitable.  j >1 rotates. Kuramoto order parameter Global variables: Approximate equation Global phase dynamics similar to individual units but with scaled frequency. A degradation in entrainment  lowers excitablity threshold allowing for synchronous firing. The precise origin of the degradation of  is irrelevant. C.J. Tessone, A. Scirè, R. Toral and P. Colet, Phys. Rev. E 75, 016203 (2007).

28. http://ifisc.uib.es Numerical simulations Diversity and noise play a similar role and induce coherent firing. diversity noise  =0  =1.6  =3.0 D=0.4 D=1.0 D=5.0 No firing Synchronized firing Desynchronized firing

29. http://ifisc.uib.es Self-consistent approximation Shinomoto-Kuramoto order parameter  No firing  Collective firing  Desynchronized firing C.J. Tessone, A. Scirè, R. Toral and P. Colet, Phys. Rev. E 75, 016203 (2007). Self-consistent approx. N=50 x N=100 N=1000 N=10000

30. http://ifisc.uib.es Summary Dissipative solitons in a nonlinear Kerr medium: subcritical cellular patterns Oscillating solitons: Still useful for applications envisioned for static solitons. New ones? Excitable regime associated with the existence of cavity solitons. Extended systems, in order to exhibit excitability, do not require local excitable behavior. Excitability in a whole new class of systems. For homogeneous pump excitability appears as a result of a saddle-loop bifurcation: oscillating and middle-branch soliton collide. Scenario organized by a Takens-Bogdanov codimension 2 point (at  → ∞ & I s → 0) For pump composed of a Gaussian localized beam on top of homogeneous background excitability also mediated by a SNIC: fundamental solution and middle branch soliton collide. Lower (controllable) excitability threshold. A suitable amount of white noise induces coherence resonance. Coupled oscillatory solitons lock to distances given by tail interaction. Depending on the locking distance solitons oscillate in or out-of-phase. For strong coupling in-phase and out-of phase oscillations coexists. Interaction of excitable solitons may be used for logical gates. In coupled excitable systems disorder can induce collective firing. Any source of disorder plays a similar role.