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