Coherence and Synchronization of Noisy-Driven Oscillators Denis S. - - PowerPoint PPT Presentation

Coherence and Synchronization

• f Noisy-Driven Oscillators

Denis S. Goldobin

Institut für Physik, Universität Potsdam

Scientific advisor:

• Prof. Dr. Arkady Pikovsky

Potsdam - Aug 27, 2007

2

Outline

• 1. Coherence of Oscillators with Delayed Feedback
• 2. Synchronization of Oscillators by Common Noise
• 3. Effects of Delayed Feedback on Kuramoto Transition
• 4. Conclusion

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• 1. Coherence of Oscillators with Delayed Feedback
• oscillation coherence & phase diffusion
• the effect of delayed feedback
• analytical theory of the effect
• 2. Synchronization of Oscillators by Common Noise
• 3. Effects of Delayed Feedback on Kuramoto Transition
• 4. Conclusion
• D. Goldobin, M. Rosenblum, & A. Pikovsky, Controlling oscillator coherence

by delayed feedback, Phys. Rev. E 67(6), 061119 (2003);

• D. Goldobin, M. Rosenblum, & A. Pikovsky, Coherence of noisy oscillators

with delayed feedback, Physica A 327(1–2), 124–128 (2003).

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1.1. Oscillation coherence & phase diffusion

For chaotic systems with oscillatory-like behav- ior the oscillation phase can be introduced as well as for periodic self-sustained systems. E.g., for the Lorenz system

1

( ) ( ) arctan , [ , ). ( )

n n

z t z t n t t t u t u φ π

+

− = + ∈ −

For chaotic and noisy limit-cycle systems the phase does not grow uniformly, but diffuses. Coherence, or "constancy" of oscillation fre- quency, may be quantified by the phase diffu- sion coefficient D

2

( ( ) ( ) / ) t T t d dt T T φ φ φ + − − ≈ D

. (the greater D the less coherent oscillations)

5

Coherence determines the quality of clocks, electronic generators, lasers, etc.; the predisposition of an oscillatory system to synchronization; the susceptibility to control (driving) (the improvement of the coherence is of interest in all these cases). Utilizing a delayed feedback appears to be an effective way to control coherence. The key feature: We intend to control phase diffusion but not suppress chaos, and, therefore, use a quite weak feedback.

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1.2. The effect of delayed feedback

The chaotic system studied is the Lorenz system:

( ), , . x y x y rx y xz z bz xy σ = − = − − =− + ( ( ) ( ))

• k z t

z t + − − τ

Here

10 σ = , 8/3 b = , 32 r = , k : the feedback strength, τ : the delay time.

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• Fig. 1: The diffusion constant D(τ/T0, k) is plotted for the Lorenz system.

T0 ≈ 0.69 is the average oscillation period without delay.

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k =

0.2, 0.3 k τ = = 0.2, 0.65 k τ = =

spectra of z

• Fig. 2:

Feedback makes the spec- tral peak essentially more broad (enhanced diffusion)

• r more narrow (suppressed

diffusion), whereas practi- cally no changes can be seen in the phase portraits.

9

Entrainment of the Lorenz system by a harmonic force with

2 E =

:

( ( ) ( ) sin ) z bz xy k z t z t τ =− + + − −

• E

t + ν

• Fig. 3: Right graph: without feedback the mean oscillator frequency

φ < >

• is

not locked to the driving frequency ν . Left graph: the feedback makes the os- cillator coherent, what results in the appearance of the synchronization region

φ ν < >≈

• .

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The noisy limit cycle system studied is the Van der Pol oscillator:

2

(1 ) ( ). x x x x t μ ζ − − + = + ( ( ) ( ))

• k x t

x t − − τ

Here ( )

t ζ

is a δ-correlated Gaussian noise:

ζ =

,

2

( ) ( ') 2 ( ') t t d t t ζ ζ δ = −

.

• Fig. 4:

D(τ/T0, k) for μ = 0.7, d = 0.1

(T0 ≈ 6.61)

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1.3. Analytical theory of the effect

The phase approximation provides opportunities for an analytical treatment of the effect for the noisy limit cycle systems (and qualitative understanding of the effect for chaotic ones). Utilizing the Gaussian approximation makes it possible to find the diffusion constant and the mean frequency shift.

• Fig. 5: the Van der Pol oscillator: Symbols present the results of the direct

numerical simulation; solid lines show the corresponding theoretical results.

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The mechanism of the effect is as follows:

φ( ) t φ −τ

+(

) t φ −τ

−(

) t φ( ) t φ −τ

+(

) t φ −τ

−(

) t τ T0 τ T /2

The faster phase speed during the last pe- riod leads to slowing down the phase speed in comparison to the mean phase speed, while the slower one does to speeding up. This results in homogenization of phase growth and the tendency to anticorrela- tions:

2

( ) ( ) t T t φ φ φ < + >−< > <

• .

The slower phase speed during the last pe- riod leads to slowing down the phase speed in comparison to the mean phase speed, while the faster one does to speeding up. This results in destabilization of phase growth and the tendency to a monoto- nously decaying correlation function.

13

• Fig. 6: Autocorrelation functions ( )

u ρ

for the sequences of the Poincaré return times in the Lorenz system (

0.2 k =

).

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• 1. Coherence of Oscillators with Delayed Feedback
• 2. Synchronization of Oscillators by Common Noise
• "reliability" of neurons & Lyapunov exponent
• limit cycle oscillators
• neural oscillators
• 3. Effects of Delayed Feedback on Kuramoto Transition
• 4. Conclusion
• D.S. Goldobin & A.S. Pikovsky, Physica A 351(1), 126–132 (2005);
• D.S. Goldobin & A. Pikovsky, Phys. Rev. E 71(4), 045201 (2005);
• D.S. Goldobin & A. Pikovsky, Phys. Rev. E 73(6), 061906 (2006);
• D.S. Goldobin, in Unsolved Problems of Noise and Fluctuations: UPoN

2005, edited by L. Reggiani et al., AIP Conf. Proc. 800(1), 394–399 (2005).

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2.1. Introduction

• "Reliability" of neurons

Z.F. Mainen & T.J. Sejnowski, Reliability of Spike Timing in Neocortical Neurons, Science 268, 1503 (1995) Experimental setup: A single neuron repeatedly transforms synaptic noisy input of a prerecorded waveform into spike sequence. If these spike sequences are identical, the neuron is called reliable. (not from real experiment) Short vertical stripes denote firing events, long stripes correspond to simultaneous firing.

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• "Consistency" of neodymium-doped yttrium aluminum garnet

(Nd:YAG) lasers

• A. Uchida, R. McAllister, & R. Roy, PRL 93, 244102 (2004)
• Theoretical framework

These problems are equivalent to the problem of synchronization in an ensemble of identical oscillators (mutually uncoupled) driven by common noise. For noisy systems:

( )

( )

ln ( ) , , ( ), ( ) , lim .

t

t t t t δ ξ δ ξ δ λ

→∞

= = = x x F x x J x x

• i

λ < : identical oscillators driven by common noise are synchronized; λ >

: identical oscillators driven by common noise are desynchronized. A.S. Pikovsky, Radiophys. Quantum Electron. 27, 576 (1984) The goal is to find the Lyapunov exponent for a noise-driven oscillator

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2.2. Limit cycle oscillators

• At the limit of weak noise, phase description is valid

A limit-cycle oscillator subject to M independent Gaussian vector noises within the framework of the phase approximation:

( )

( )

M k k k k

f t t ϕ ω σ ϕ ξ

=

= +∑

1

d d

,

2 2

( )d 1

k

f

π

ϕ ϕ =

∫

,

( ) ( ') 2 ( ')

j k jk

t t t t ξ ξ δ δ − =

. A time-continuous evolution of the phase under arbitrary forcing, on a finite time inter- val gives a monotonous transformation of the phase. An attracting set of a monotonous transformation has a negative Lyapunov exponent:

( )

[ ]

N k k k

d f d

π

δϕ σ λ ϕ ϕ δϕ π

=

= = − +

∑ ∫

2 2 2 1

' d ... 2

• .
• J. Teramae & D. Tanaka, Phys. Rev. Lett. 93, 204103 (2004)

D.S. Goldobin & A.S. Pikovsky, Physica A 351(1), 126–132 (2005)

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• Numerical simulation – Lyapunov exponent

Van der Pol–Duffing oscillator with additive noise

( )

x x x x bx μ − − + +

2 3

(1 )

• = σξ t
• for weak noise,

( )

λ σ

resembles the ana- lytical law

• for moderate noise,

desynchronization is possible The dependence

( )

λ σ for the Van der Pol–Duffing oscillator at 0.2 μ =

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• Van der Pol–Duffing oscillator: non-perfect cases

(i) small frequency mismatch:

2 3 1,2 1,2 1,2 1,2 1,2

(1 ) (1 ) ( ), x x x x bx t μ σξ − − + + =

• ± Ω

Pair of nonidentical VdP-D oscillators with

0.2 μ =

and

0.002 Ω=

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(ii) weak different intrinsic noise:

2 3 1,2 1,2 1,2 1,2 1,2

(1 ) ( ) , x x x x bx t μ σξ − − + + = ( )

• ± T

t η

Pair of VdP-D oscillators (

0.2 μ =

) subject to intrinsic noises,

/ 0.01 T σ =

For weak noise, an analytical theory has been developed: D.S. Goldobin and A. Pikovsky, Synchronization and desynchronization of selfsustained oscillators by common noise, PRE 71(4), 045201 (2005).

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2.3. Neural oscillators

We present here results for a FitzHugh-Nagumo model

( )

( )

1 2

3 , . v v v w w v v ε− ⎡ ⎤ = − − ⎢ ⎥ ⎣ ⎦ = −

• + σξ t

Near the transition between excitable behavior and periodic spiking, neuron-like systems are sensitive to external forcing, and regions of positive Lyapunov exponent appear.

D.S. Goldobin and A. Pikovsky, Antireliability of noise-driven neurons,

• Phys. Rev. E 73(6), 061906 (2006)

22

( )

( )

1 2

3 , , v v v w w v v ε− ⎡ ⎤ = − − = − ⎢ ⎥ ⎣ ⎦

• + σξ t

The dependence

( )

λ σ for the FitzHugh-Nagumo system at 0.05 ε =

Transition to positive Lyapunov exponent means desynchronization. Firing patterns for the ensemble of 10 FHN systems ( 1.001 v = − ) small noise (the sample is for

0.01 σ =

) the Lyapunov exponent is negative: perfect synchrony (reliability) moderate noise (

0.013 σ =

) the Lyapunov exponent is positive: asynchronous behavior (antireliability) large noise (

0.08 σ =

) the Lyapunov exponent is negative: perfect synchrony (reliability)

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The effect is slightly distorted by intrinsic noise The ensemble of 10 FHN systems with intrinsic noise of amplitude

5 int

5 10 σ

−

= ⋅ small noise (the sample is for

0.01 σ =

) the Lyapunov exponent is negative: imperfect synchrony moderate noise (

0.013 σ =

) the Lyapunov exponent is positive: asynchronous behavior large noise (

0.08 σ =

) the Lyapunov exponent is negative: imperfect synchrony

25

• Characterization with help of the event synchronization approach
• R. Quian Quiroga, T. Kreuz, and P. Grassberger, Event synchronization:

A simple and fast method to measure synchronicity and time delay patterns, Phys. Rev. E 66, 041904 (2002). At

FHN

0.05 ε =

,

1.001 v = −

the correlation function

( )

0.1

| C x y

τ=

for two FHN sys- tems driven by common noise

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• Analytical model for neural oscillators

Assumptions: – The system is especially sensitive to noise on some short part of the limit cycle. – The relaxation rate of deviations from the limit cycle is large at least

• n some pieces of the limit cycle.

The effective noise amplitude V

∝ [noise strength], ∝ [nonisochronicity], ∝ 1/[relaxation rate of

transversal deviations],

∝ [sensitivity to noise].

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• 1. Coherence of Oscillators with Delayed Feedback
• 2. Synchronization of Oscillators by Common Noise
• 3. Effects of Delayed Feedback on Kuramoto Transition
• synchronization in arrays of globally coupled oscillators
• linear feedback
• purely nonlinear feedback
• 4. Conclusion
• D.S. Goldobin & A. Pikovsky, Effects of Delaed Feedback on Kuramoto

Transition, Prog. Theor. Phys. Suppl. 161, 43–52 (2006).

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3.1. Synchronization in arrays of globally coupled oscillators

A transition to collective synchrony in an ensemble of globally coupled oscilla- tors is known as the Kuramoto transition. Application (example):

Collective dynamics of neuronal populations:

the emergence of pathological rhythmic brain activity in

Parkinson's disease, essential tremor, epilepsies

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Suppression of the collective modes by deep brain stimulation

Noninvasive techniques utilizing delayed feedback: [1] M. Rosenblum, A. Pikovsky, et al. PRL 92, 114102 (2004); PRE 70, 041904 (2004); etc.

Linear feedback Stabilization of the absolutely nonsynchronous state

[2] O.V. Popovich, C. Hauptmann & P.A. Tass, PRL 94, 164102 (2005); etc.

Strong nonlinear feedback Diminishing amplitudes of collective modes

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We study an ensemble of N phase oscillators:

f 1 1 2

• f

sin( ( ) ( )) linear delayed feedb "purely" nonlinear | ( ) | | ( ) | delayed feedback sin[ sin( ( ) ( )) ( ac 2 ( ) ( ) ( ) ) k ]

N i i j i i i i j N j j

R t T R t t t T t t t T t t N N t ε ε ϕ ω ϕ ϕ ξ θ θ ν ϕ ϕ ϕ ε

= =

+ − + − × − = + − + − − − +

∑ ∑

• and are interested in the order parameter

( ) ( ) 1

( ) | |

j

i t i t j

R t R e N e

ϕ θ −

≡ ≡

∑

. Bifurcation analysis of the absolutely nonsynchronous state for

f

• f

ε ε = = :

J.D. Crawford, Amplitude Expansions for Instabilities in Populations of Globally-Coupled Oscillators, J. Stat. Phys. 74(5-6), 1047-1084 (1994).

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3.2. Linear delayed feedback

Distribution of natural frequencies ( )

g ω may be

• Nonsymmetric ( )

( ) g g ω ω ≠ −

The only possible bifurcation is a Hopf one:

2

( ) 2 (1 ( ))

i t

R t Ae O A π

Ω

= +

,

2

| | A A P A A λ − + =

• .

ϕ R Ω

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• Symmetric ( )

( ) g g ω ω = −

[1] Steady-state bifurcation:

2 2 5

(2 ) | | ( ) R R P R R O R λ π − − + + =

• ; ,

; arg . P R const λ ∈ =

• [2] Hopf bifurcation (terminology after Crawford):

3 3

( ) 2 ( ) ( ) ( )

A B

R t O A O B π = + + +

• i

i t t

Be Ae Ω

Ω

,

2 2

[ | | | | ] A P A Q B A λ + − + + =

• ,

* * 2 * 2

[ | | | | ] B P B Q A B λ + − + + =

• .

Depending on P and Q , we have

R Ω Ae

i t Ω

Be

− Ω i t

−Ω

(1)

R =

; (2)

4 cos R A t π ≈ Ω (A B =

); (3) multistability between

2

i t

R Ae π

Ω

≈

(

B =

) and

2

i t

R Be π

− Ω

≈

(

A = ).

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3.3. Purely nonlinear delayed feedback

• For nonsymmetric

( ) ( ) g g ω ω ω ω < > +Δ ≠ < >−Δ

the only possible bifurcation is a Hopf one.

• Symmetric (

) ( ) g g ω ω ω ω < >+Δ = < >−Δ

[1] "Steady-state" bifurcation:

i t

R R e

ω < >

=

. [2] Hopf bifurcation (terminology after Crawford): (1)

R =

; (2) quasiperiodic:

4 cos

i t

R Ae t

ω

π

< >

≈ Ω (A B =

); (3) periodic: multistability between

( )

2

i t

R Ae

ω

π

< >+Ω

≈

(

B =

) and

( )

2

i t

R Be

ω

π

< >−Ω

≈

(

A = ).

34

• 4. Conclusions

Coherence of Oscillators with Delayed Feedback

• relatively small delayed feedback can strongly affect the phase diffusion
• f chaotic and stochastic systems and influence their predisposition to

synchronization and susceptibility to control.

Synchronization of Oscillators by Common Noise

• smooth limit cycle systems are synchronized by common noise;
• moderate noise may desynchronize some systems;
• neuronal oscillators can be desynchronized only close to the transition

from excitable behavior to periodic spiking.

Effects of Delayed Feedback on Kuramoto Transition

• the linear delayed feedback effectively influences both the transition

point and the amplitude of collective modes and additionally may change the bifurcation type (super-/subcritical);

• the purely nonlinear feedback allows controlling the amplitude of collec-

tive oscillations without any effect on the linear stability properties.