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Synchronization patterns in coupled optoelectronic oscillators
Caitlin R. S. WilliamsUniversity of Maryland
Dissertation DefenseTuesday 13 August 2013
My Research• Random Number Generation:
– C. R. S. Williams, J. C. Salevan, X. Li, R. Roy, and T. E. Murphy. “Fast physical random number generator using amplified spontaneous emission.” Optics Express, 18(23):23584-23597 (2010).
• Optoelectronic Oscillators and Synchronization:– T. E. Murphy, A. B. Cohen, B. Ravoori, K. R. B. Schmitt, A. V. Setty, F. Sorrentino, C. R. S.
Williams, E. Ott, and R. Roy. “Complex dynamics and synchronization of delayed-feedback nonlinear oscillators.” Phil. Trans. R. Soc. A, 368(1911):343-366 (2010).
– C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, “Group Synchrony in an Experimental System of Delay-coupled Optoelectronic Oscillators,” Conference Proceedings of NOLTA2012, 70-73 (2012).
– C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll. “Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators.” Phys. Rev. Lett., 110:064104 (2013).
– C. R. S. Williams, F. Sorrentino, T. E. Murphy, and R. Roy. “Synchronization States and Multistability in a Ring of Periodic Oscillators: Experimentally Variable Coupling Delays.” Manuscript submitted.
Outline
• Introduction: Dynamical Systems and Synchronization
• Synchrony of periodic oscillators in a unidirectional ring
• Group synchrony of chaotic oscillators
3
Pendulum: The Simplest Dynamical System
4
• For an ideal, small amplitude oscillation:
€
θ(t) = θ0 cos(2πt
T)
T = 2πL
g
• Not so simple for large amplitudes or real pendulum!
Image: Wikipedia.org
Weather: Example of Chaos
5
Lorenz System:
• Deterministic• Sensitive to initial
conditions€
˙ x = σ (y − x)
˙ y = x(ρ − z) − y
˙ z = xy − βz
R. C. Hilborn, Chaos and Nonlinear Dynamics.
Image: Wikipedia.org
Synchronization of Periodic Oscillators
6
Metronome Synchronization (IkeguchiLab on YouTube)
Synchronization Example: Millennium Bridge
Bridge-pedestrian coupling created pedestrian synchrony and bridge swaying!
7S. H. Strogatz, D. M. Abrams, A. McRobie, B. Eckhardt and E. Ott, Nature 438, 43-44 (2005).
Synchronization of Brain Signals
8
Image: Wikipedia.org
Experiment
9
Experiment
• Insert photo of experiment hereLaser
10
Mach-Zehnder Modulator
Digital Signal Processing (DSP) Board
Photoreceivers andVoltage Amplifier
Experimental Diagram
11
Nonlinearity
12
-4 -2 0 2 40
0.5
1
VRF
(V)
tran
sm
issio
n
V
V
oo V
VPP
2
cos2
Transmission:
P
V
Image: B. Ravoori
Single Node Block Diagram
13
Dynamics of a Single Node
β
14
B. Ravoori, Ph.D. Dissertation, 2011.A. B. Cohen, Ph.D. Dissertation, 2011.T. E. Murphy, et al., PTRSA (2010).
Dynamics of a Single Node
15
B. Ravoori, Ph.D. Dissertation, 2011.A. B. Cohen, Ph.D. Dissertation, 2011.T. E. Murphy, A. B. Cohen, B. Ravoori, K. R. B. Schmitt, A. V. Setty, F. Sorrentino, C. R. S. Williams, E. Ott, PTRSA 368 (2010).
Four Node Network: Flexible Experiment
Synchronization TypesIdentical,
isochronalPhase Lag
(amplitude)
17
Phase Synchrony States
• Control of phase synchronization states in coupled oscillators is interesting because of neurological disorders and other phenomena observed in coupled neurons
• Interested in controlling synchronization in coupled oscillators from complete synchrony, cluster synchrony, and different types of lag synchrony, specifically ‘splay phase’ synchrony
18C. R. S. Williams, F. Sorrentino, T. E. Murphy, and R. Roy, Manuscript submitted.
Coupled Periodic Oscillators
• Coupled Neurons: Transitions from lag to isochronal synchrony
Unidirectional Ring of Neurons
20B. Adhikari, et al. Chaos 21, 023116 (2011).
v is membrane potentialh, m are membrane channel gating variables
Background
• In numerical and analytical studies, changing the coupling delay has produced different synchronization states
21C. Choe, et al., PRE 81, 025205 (2010).
Experiment on Unidirectional Ring
22
€
dui (t)
dt=Eui (t) −Fβ cos2(x i (t − τ f ) + φ0)
x i (t) =G(ui (t) + ε K ij[u j (t − τ c + τ f ) −ui (t)])j
∑
Mathematical Model
€
i =1,2,3,4 (node)
€
K =
0 0 0 1
1 0 0 0
0 1 0 0
0 0 1 0
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
(coupling matrix)
23
Mathematical Model
€
β = 1.2 ( )feedback strength
ε =0.8 (coupling strength)
τ f =1.4 ms (feedback delay)
τ c ≥1.4 ms (coupling delay)
φ0 = π4 (Modulator bias)
ωL = 2π • 2.5kHz
ωH = 2π • 0.1kHz
€
E =−(ωH + ωL ) −ωL
ωH 0
⎛
⎝ ⎜
⎞
⎠ ⎟
F =ωL
0
⎛
⎝ ⎜
⎞
⎠ ⎟
G = (1 0)
€
dui (t)
dt=Eui (t) −Fβ cos2(x i (t − τ f ) + φ0)
x i (t) =G(ui (t) + ε K ij[u j (t − τ c + τ f ) −ui (t)])j
∑
24
Isochronal Synchrony(Phase = 0)
Tuning Coupling Delay
Experiment Simulation
fc
25
Splay-phase (Lag) Synchrony
(Phase = π/2)
€
c =1.3τ f
Tuning Coupling Delay
Experiment Simulation
26
€
c =1.5τ f
Cluster (Lag) Synchrony(Phase = π)
Tuning Coupling Delay
Experiment Simulation
27
€
c =1.8τ f
Splay-phase (Lag) Synchrony
(Phase = 3 π/2)
Tuning Coupling Delay
Experiment Simulation
28
Varying Coupling Delay
c (ms)
Ph
ase
Re
latio
nsh
ip
1.4 1.8 2.2 2.6
3pi/2
pi
pi/2
00
20
40
60
80
100
2
3
2
0
Experiment10 Measurements per delay
Simulation2000 Random initial conditions per delay
c (ms)
Ph
ase
Re
latio
nsh
ip
1.4 1.8 2.2 2.6
3pi/2
pi
pi/2
0
0
20
40
60
80
100
Frequency of Occurrence (%
)
c (ms)
Ph
ase
Re
latio
nsh
ip
1.4 1.8 2.2 2.6
3pi/2
pi
pi/2
0
0
20
40
60
80
100
29
Predicted Stability
30
€
ΔX(t) = ΔX(t0)λmax ( t−t0 )
e
€
λmax
Coupled Chaotic Oscillators
• Groups of different oscillators• Intra-group identical synchrony, but not inter-
group• This has been studied numerically and
analytically, but previously not in an experiment
Group Synchrony
32Dahms, Lehnert, and Schöll, PRE 86, 016202 (2012)C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, PRL 110, 064104 (2013)
• Special case of group synchrony with identical nodes
Cluster Synchrony
33
Motivation
• Neurons can display a variety of dynamical behaviors, and they are coupled to each other
34J. Lapierre, et al., Journal of Neuroscience 27 (44), 2007.
Experimental Network Structure
35
Synchrony of Coupled Groups
36
Mathematical Model
j
mi
mj
mij
mi
mi
mi
mmi
mi
ttKttx
txtdt
td
))]()(()([)(
))((cos)()(
)()'()()()(
0)(2)()(
)(
uuuG
FEuu
ε
β
i 1,2 (node)m,m' A,B (group) (m' m)
€
K =0 K(A )
K(B ) 0
⎛
⎝ ⎜
⎞
⎠ ⎟=
1
2
0 0 1 1
0 0 1 1
1 1 0 0
1 1 0 0
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
(coupling matrix)
37
Mathematical Model
€
β (A ),β (B ) from 0 to 10 (feedback strength)
ε = 0.8 (coupling strength)
τ =1.4 ms (feedback and coupling delay)
φ0 = π4 (Modulator bias)
ωL = 2π • 2.5kHz
ωH = 2π • 0.1kHz
€
E =−(ωH + ωL ) −ωL
ωH 0
⎛
⎝ ⎜
⎞
⎠ ⎟
F =ωL
0
⎛
⎝ ⎜
⎞
⎠ ⎟
G = (1 0)
j
mi
mj
mij
mi
mi
mi
mmi
mi
ttKttx
txtdt
td
))]()(()([)(
))((cos)()(
)()'()()()(
0)(2)()(
)(
uuuG
FEuu
ε
β
38
Stability of Group Synchrony
39C. R. S. Williams, et al., PRL 110 (2013).
Global Synchronyβ(A)=β(B) = 3.3 Simulation
Experiment
40
Cluster Synchronyβ(A)=β(B) = 7.6 Simulation
Experiment
41
Group Synchronyβ(A)=7.6β(B) = 3.3
Simulation
Experiment
42
Dissimilar Nodesβ(A) = 7.6 β(B) = 3.3
43
Autocorrelation Function Autocorrelation Function
Coupled Nodes
44
Cross-correlation Function
Group Synchrony and Time-lagged Phase Synchrony
Group B Traces Delayed
45
Group Sync for Different Structures
46
Group Sync for Different Structures
47
Group Sync for Different Structures
48
Larger Networks
49
Conclusions I
• Shown transitions between isochronal, cluster, and splay-phase synchrony by varying coupling delays between periodic oscillators
• Have an experiment with tunable coupling delay
• Tested stability calculations and predictions with experiments and simulations
50
Conclusions II
• Experimental demonstration of global, cluster and group synchrony
• Stability calculations extended to group synchrony with time-delayed systems, used to correctly predict experimental results of this optoelectronic system, with coupled non-identical nodes
• Results can be generalized to groups of different sizes, and to different coupling configurations
51
Acknowledgements
• Thomas E. Murphy, Rajarshi Roy (University of Maryland)
• Francesco Sorrentino (Mechanical Engineering, University of New Mexico)
• Thomas Dahms, Eckehard Schöll (Tecnische Universität Berlin)
• MURI grant ONR N000140710734 (CRSW, TEM, RR)• DFG in the framework of SFB 910 (TD, ES)• Adam Cohen and Bhargava Ravoori• Hien Dao and Aaron Hagerstrom
52
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