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
http://ifisc.uib.es - Mallorca - Spain
Outline
•
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.
http://ifisc.uib.es
Dissipative solitons
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).
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).
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)
http://ifisc.uib.es
Pattern formation in nonlinear optical cavities
Pump
field
Nonlinear
medium
Sodium vapor cell with
single mirror feedback
1. Driving
2. Dissipation
T. Ackemann and W. Lange, Appl. Phys. B 72, 21 (2001)
Liquid crystal
light valve
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)
3. Nonlinearity
4. Spatial coupling
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.
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
http://ifisc.uib.es
Scenarios for dissipative solitons
Amplitude
Bistability
Homogeneous
Solution
Homogeneous
Solution
Homogeneous
Solution
Control Parameter
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)
Stable droplets: Localized structures
stabilised by nonlinear domain wall
dynamics due curvature.
Exist in 2d systems.
D. Gomila et al, PRL 87, 194101 (2001)
Subcritical Cellular Pattern
Amplitude
Hexagonal Pattern
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)
Excitability mediated by
localized structures
Control Parameter
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).
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.
http://ifisc.uib.es
Self-focusing Kerr cavity
y
input field E0

3 
x
output field
z
 
 


E x , z, t   E x , t  ei k z z  t

x  x, y  field envelope
Lugiato-Lefever model
E
2
2
 1  i E  i E  E0  i E E
t
Homogeneous solution
E0  Es [1  i ( I s   )], I s  Es
Control parameters
E0: pump
 : detuning
2
It becomes unstable at Is=1 leading to a subcritical hexagonal pattern
L.A. Lugiato & R. Lefever, PRL 58, 2209 (1988).
http://ifisc.uib.es
Self-focusing Kerr cavity solitons
Cavity soliton
Can be seen as a solution connecting a cell of
the pattern with the homogeneous solution
W.J. Firth & A. Lord, J. Mod. Opt. 43, 1071 (1996);
W.J. Firth, A. Lord & A.J. Scroggie, Phys. Scripta, T67, 12 (96)
E  Es (1  A)

 2 1  
A
2
2
*
2






1

i

A

i

A

iI
2
A

A

A

2
A

A
A
Radial equation:
s
  2r r r 
t


A
A

0
Soliton profile can be found solving the l.h.s. equated to zero with
r r 0 r r 

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)
http://ifisc.uib.es
Stability of Kerr cavity solitons
Soliton
amplitude
Azimuth inst.
  1.3
m=6
Hopf
m=5
Stable
Unstable
Hopf
SaddleNode
Is
No solitons
Hom. solution
SaddleNode
Is
Hopf instability observed in
W.J. Firth, A. Lord & A.J. Scroggie, Phys. Scripta,T67,12 (96)
W.J. Firth, G.K. Harkness, A. Lord, J. McSloy,
D. Gomila & P. Colet, JOSA B 19, 747 (2002)
http://ifisc.uib.es
Azimuth instabilities
m=6
t
m=5
Unstable Eigenmode
http://ifisc.uib.es
Hopf instability
Azimuth inst.
  1.3, I  0.9
Hopf
No solitons
Saddle-node
middle branch soliton
Cross-section
Oscillating soliton still useful for
applications since its amplitude
is bounded below by middle
branch soliton.
http://ifisc.uib.es
Saddle-loop bifurcation
Is =0.9
oscillating cavity soliton
L
C
=1.3047
middle-branch
cavity soliton
max(|E|)
=1.3
Hopf
SN
Saddle-loop
Homogeneous solution

Minimum distance of oscillating
soliton to middle-branch soliton
=1.30478592
=1.304788
homogeneous solution
D. Gomila, M. Matias and P. Colet, Phys. Rev. Lett. 94, 063905 (2005).
http://ifisc.uib.es
Saddle-loop bifurcation. Scaling law
middle-branch soliton spectrum
Close to bifurcation point:
T 
1
1
ln  c   
T: period of oscillation
1 unstable eigenvalue of saddle (middle-branch soliton)
1
S.H. Strogatz, Nonlinear dynamics and chaos 2004
numerical
simulations
1/1
http://ifisc.uib.es
Phase space close to saddle-loop bifurcation
dA=(E-Esaddle)/Es
yu
Projection onto ys
Oscillatory regime
middle-branch
soliton spectrum
Projection onto yu
Only two localized modes.
Close to saddle: dynamics takes place in the plane (yu, ys)
Beyond Saddle Loop
Projection onto ys
ys
Saddle-node index: n=-s/u=2.177/0.177>1 (stable limit cycle)
Projection onto yu
D. Gomila, A. Jacobo, M. Matias and P. Colet, PRA 75, 026217 (2007).
http://ifisc.uib.es
Excitability
Beyond saddle-loop
bifurcation
Small perturbations of
homogeneous solution decay.
Localized perturbations above
middle branch soliton send
the system to a long excursion
through phase-space.
The system is not locally excitable.
Excitability emerges from spatial
coupling
D. Gomila, M. Matias and P. Colet, Phys. Rev. Lett. 94, 063905 (2005).
http://ifisc.uib.es
Takens-Bogdanov point
Saddle-loop bifurcation is not generic. Why it is present here?
solitons
oscillating solitons
Hopf
saddle-node
=1.7
=1.6
=1.5
saddle-loop
saddle-node
Hopf
No solitons
 =1.5
Distance between saddle-node and Hopf
TB
The Hopf frequency when it meets the saddlenode is zero.
Takens-Bogdanov point.
Unfolding of TB yields a Saddle-Loop
d → 0 for  → ∞ and Is → 0
NLSE
http://ifisc.uib.es
Pump: Plane wave + Localized Gaussian Beam
Excitability arising from a saddle-loop bifurcation have a large threshold.
To reduce the threshold we consider for the pump:
H  ( I s  I sh )[1    I s  I sh 2 ]
Hopf
Saddle
Node
Is
Pattern
Oscillations
1
Excitability
SNIC
max(|E|2)
Is
Pattern
Ish=0.7, =1.34
Hom. pump
Excitability
max(|E|2)
2
2
EI  E0  He r / r0
1
http://ifisc.uib.es
Saddle-node in the circle (SNIC) bifurcation
Close to bifurcation point:
From the new oscillatory regime to the excitable regime.
Ish=0.3, =1.45

T  I s2  I sc
unstable upper
branch soliton
1/ 2
Is=0.927
Is=0.8871
Is=0.8
middle-branch
cavity soliton
Projection onto ys
Is=0.907
Projection onto yu
fundamental
solution
http://ifisc.uib.es
Full scenario
Ish=0.3
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.
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
http://ifisc.uib.es
Noise effects, coherence resonance
In excitable systems a moderate level of noise induces a more regular firing
(coherence resonance)
R
A. S. Pikovsky and J. Kurths, Phys. Rev. Lett. 78, 775 (1997).
Var ( )

Introducing white spatiotemporal noise excitable solitons show coherence resonance.
http://ifisc.uib.es
Interaction of two oscillating solitons
 =1.27, Is=0.9, homogeneous pump
Single structure period T=8.66
Oscillating solitons move until they reach equilibrium positions given by tails interaction.
Three equilibrium distances are found:
In-phase oscillation. T=8.93
Out-phase oscillation. T=8.94
Strong interaction.
In & out-phase oscillation
depending on initial condition.
Tin=8.59 < Tout=10.45
http://ifisc.uib.es
Interaction of excitable solitons
Pulse on
Pulse on
Pulse off
Firing
Firing
1
1
bit 1
1
Firing induced
by interaction
Firing induced
by interaction
Firing
1
0
1
OR logical gate
http://ifisc.uib.es
AND
1
0
1
1
Pulse on
Pulse on
Pulse off
0
1
http://ifisc.uib.es
NOT
http://ifisc.uib.es
Collective firing induced by noise or diversity
Globally coupled active rotators
wj<1 excitable.
Diversity: natural frequencies
wj>1 rotates.
Global variables:  (t )ei (t ) 
Approximate equation
1
i ( t )
e j

N j
noise
Kuramoto order parameter    (t )
 (t )  w  sin  (t )


•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).
http://ifisc.uib.es
Numerical simulations
diversity
noise
s=0
D=0.4
s=1.6
D=1.0
No firing
Synchronized firing
s=3.0
D=5.0
Desynchronized firing
Diversity and noise play a similar role and induce coherent firing.
http://ifisc.uib.es
Self-consistent approximation
i (t )
  (t )ei (t )
Shinomoto-Kuramoto order parameter    (t )e
Self-consistent
approx.
 No firing
 Collective firing
 Desynchronized firing
N=50
x N=100
N=1000
N=10000
C.J. Tessone, A. Scirè, R. Toral and P. Colet, Phys. Rev. E 75, 016203 (2007).
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  → ∞ & Is → 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.
•
•
http://ifisc.uib.es