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

Coherence and Synchronization of 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

3 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).

4 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 − z t ( ) z φ = + π ∈ t 0 n t t t ( ) arctan , [ , ). + n n 1 − u t ( ) u 0 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 φ + − φ − φ 2 ≈ D ( ( t T ) ( ) t d / dt T ) T . 0 0 (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 .

6 1.2. The effect of delayed feedback The chaotic system studied is the Lorenz system : = σ − � x ( y x ), = − − � y rx y xz , = − + + − τ − � z bz xy k z t z t . ( ( ) ( )) σ = b = r = 10 , 8/3 , 32 , Here k : the feedback strength, τ : the delay time.

7 Fig. 1: The diffusion constant D ( τ / T 0 , k ) is plotted for the Lorenz system . T 0 ≈ 0.69 is the average oscillation period without delay.

8 k = = τ = = τ = 0 k 0.2, 0.3 k 0.2, 0.65 spectra of z Fig. 2: Feedback makes the spec- tral peak essentially more broad (enhanced diffusion) or more narrow (suppressed diffusion), whereas practi- cally no changes can be seen in the phase portraits.

9 E = 2 Entrainment of the Lorenz system by a harmonic force with : � = − + + − τ − + ν z bz xy k z t ( ( ) z t ( ) ) E sin t � < φ > Fig. 3: Right graph: without feedback the mean oscillator frequency 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 .

10 The noisy limit cycle system studied is the Van der Pol oscillator : − μ − 2 + = � − τ − � + ζ �� � x (1 x ) x x k x t x t ( ). t ( ( ) ( )) ζ is a δ -correlated Gaussian noise: ζ = ζ ζ = 2 δ − Here ( ) t 0 ( ) ( ') t t 2 d ( t t ') , . Fig. 4: D ( τ / T 0 , k ) for μ = 0.7 , d = 0.1 ( T 0 ≈ 6.61 )

11 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.

12 The mechanism of the effect is as follows: τ T 0 τ T /2 0 φ −τ − ( t ) φ −τ φ ( ) + ( t ) t φ ( ) t φ −τ t + ( ) φ −τ − ( t ) The faster phase speed during the last pe- The slower phase speed during the last pe- riod leads to slowing down the phase speed riod leads to slowing down the phase speed in comparison to the mean phase speed , in comparison to the mean phase speed , while the slower one does to speeding up. while the faster one does to speeding up. This results in homogenization of phase This results in destabilization of phase growth and the tendency to anticorrela- growth and the tendency to a monoto- � � � < φ + φ > − < > < φ 2 nously decaying correlation function . ( t T ) ( ) t 0 tions : . 0

13 ρ Fig. 6: Autocorrelation functions ( ) u for the sequences of the Poincaré return k = 0.2 times in the Lorenz system ( ).

14 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).

15 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.

16 • "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. δ ln x ( ) t ( ) ( ) � = ξ δ � = ξ δ λ = x F x , , x J x ( ), ( ) t t i x , lim . For noisy systems: t →∞ t λ < : identical oscillators driven by common noise are synchronized ; 0 λ > 0 : 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

17 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: π 2 ϕ M d + ∑ ∫ ( ) 2 = ω σ ϕ ξ ϕ ϕ = ξ ξ − = δ δ f ( ) t f ( )d 1 ( ) t ( t t ') 2 ( ') t , , . 0 k k k k j k jk d t k = 1 0 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: π 2 2 δϕ � N σ d ∑ 2 [ ] ∫ ( ) λ = = − ϕ ϕ + k f ' d ... . k δϕ π d 2 k = 1 0 J. Teramae & D. Tanaka, Phys. Rev. Lett. 93 , 204103 (2004) D.S. Goldobin & A.S. Pikovsky, Physica A 351 (1), 126–132 (2005)

18 • Numerical simulation – Lyapunov exponent Van der Pol–Duffing oscillator with additive noise �� − μ − 2 � + + 3 = σξ t x (1 x x ) x bx ( ) • for weak noise, λ σ ( ) resembles the ana- lytical law • for moderate noise, desynchronization is possible λ σ for the Van der Pol–Duffing oscillator at μ = 0.2 The dependence ( )

19 • Van der Pol–Duffing oscillator: non-perfect cases (i) small frequency mismatch: �� − μ − 2 � + ± Ω + 3 = σξ x (1 x ) x (1 ) x bx ( ), t 1,2 1,2 1,2 1,2 1,2 μ = Ω = 0.2 0.002 Pair of nonidentical VdP-D oscillators with and

20 (ii) weak different intrinsic noise: �� − μ − 2 � + + 3 = σξ ± T η ( ) t x (1 x ) x x bx ( ) t , 1,2 1,2 1,2 1,2 1,2 μ = T σ = 0.2 / 0.01 Pair of VdP-D oscillators ( ) subject to intrinsic noises, 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).

21 2.3. Neural oscillators We present here results for a FitzHugh-Nagumo model ε − ⎡ ⎤ ( ) � = 1 − 2 − + σξ t ( ) v 3 v v w , ⎢ ⎥ ⎣ ⎦ � = − w v v . 0 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)