Phase synchronization

The local dynamics of tire systems considered tluis far has been eitlier steady or oscillatory. However, we may consider reaction-diffusion media where tire local reaction rates give rise to chaotic temporal behaviour of tire sort discussed earlier. Diffusional coupling of such local chaotic elements can lead to new types of spatio-temporal periodic and chaotic states. It is possible to find phase-synchronized states in such systems where tire amplitude varies chaotically from site to site in tire medium whilst a suitably defined phase is synclironized tliroughout tire medium 51. Such phase synclironization may play a role in layered neural networks and perceptive processes in mammals. Somewhat suriDrisingly, even when tire local dynamics is chaotic, tire system may support spiral waves... 

Single-phase Motors. Alternating-current single-phase motors are usually induction or series motors, although single-phase synchronous motors are available in the smaller ratings. 

The notion of synchronizing pulses over adjacent frames in overlap-add for time-scale modification was first introduced by Roucos and Wilgus in the speech context [Roucos and Wilgus, 1985]. This method relies on cross-correlation of adjacent frames to align pulses and not on phase synchronization of a subband decomposition. 

The implications of individual neuron dynamics on neuronal network synchronization is evident. In Fig. 7.9 (from Schneider et al., unpublished data) this is demonstrated with network simulations (10 x 10 neurons) of nearest neighbor gap-junction coupling. It is illustrated in quite a simple form which, in a similar way, can also be experimentally used with the local mean field potential (LFP). In the simulations LFP simply is the mean potential value of all neurons. In the nonsynchronized state LFP shows tiny, random fluctuations. In the completely in-phase synchronized states the spikes should peak out to their full height... 

Neighboring nephrons also communicate with one another. Experiments performed by Holstein-Rathlou show how nephrons that share a common interlobular artery tend to adjust their TGF mediated pressure oscillations so as to produce a state of in-phase synchronization [7]. Holstein-Rathlou also demonstrated how microperfusion of one nephron (with artificial tubular fluid) affects the amplitude of the pressure oscillation in a neighboring nephron. This provides a method to determine the strength of the nephron-nephron interaction. 

Let us examine the situation for large values of a where the individual nephron exhibits chaotic dynamics. Figure 12.15a shows a phase plot for one of the nephrons in our two-nephron model for a = 32, T = 16 s, e = 0.0, and y = 0.2. Here we have introduced a slight mismatch AT = 0.2 s in the delay times between the two nephrons and, as illustrated by the tubular pressure variations of Fig. 12.15b, the nephrons follow different trajectories. However, the average period is precisely the same. This is a typical example of phase synchronization of two chaotic oscillators. 

Fig. 12.15 (a) Phase plot for one of the nephrons and (b) temporal variation of the tubular pressures for both nephrons in a pair of coupled chaotically oscillating units, a = 32, T = 16 s, and e = y = 0.2. The figure illustrates the phenomenon of chaotic phase synchronization. By virtue of their mutual coupling the two chaotic oscillators adjust their (average) periods to be identical. The amplitudes, however, vary incoherently and in a chaotic manner [27],... 

After a further 16 h repeat the second stage when the cells will be released into S-phase synchronously. 

In the general case of second harmonic generation, it can be shown that the conversion efficiency Izo/lm is strongly dependent on the phase synchronization factor 1/2) where Ak = i 2o) respectively,... 

Grosjean, F., Batard, P., Jordan, M. and Wurm, F.M. (2002) S-phase synchronized CHO cells show elevated transfection efficiency and expression using CaPi. Cytotech-nology 38, 1/2 57-62. 

A. Neiman, L. Schimansky-Geier, A. Cornell-Bell, and F. Moss. Noise-enhanced phase synchronization in excitable media. Phys. Rev. Lett., 83 4896-4899, 1999. 

J. Casado-Pascual, J. Gomez-Ordonez, M. Morillo, J. Lehmann an I. Goy-chuk, and P. Hanggi. Theory of frequency and phase synchronization in a rocked bistable stochastic system. Physical Review E, 71 011101, 1 2005. 

J. Flreund, A.B. Neimann, and L.Schimansky-Geier. Analytic description of noise-induced phase synchronization. Europhysics Letters, 50(1) 8-14, 2000. 

On the other hand, semiconductor lasers exhibit rich nonlinear behavior and a variety of collective synchronization phenomena. Among these phenomena, one can mention complete [11, 18, 27, 28] and phase synchronization [13, 26], retarded and anticipated synchronization [9, 19, 23], inverse synchronization [20], etc. The corresponding evidences exist for both experimental systems and mathematical models. 

A. Pikovsky and M. Rosenblum. Phase synchronization of regular and chaotic self-sustained oscillators. In A. Pikovsky and Yu. Maistrenko, editors, Synchronization Theory and Application, pages 187-219. Kluwer, Dordrecht, 2003. 

Chaotic systems. Here the mere notion of synchrony is non-trivial, and several concepts have been developed. The effect of phase synchronization is a direct extension of the classical theory to the case of a subclass of self-sustained continuous time chaotic oscillators which admit a description in terms of phase. Synchronization of these systems can be described as a phase and frequency locking, in analogy to the theory of synchronization of noisy systems. An alternative approach considers a synchronization of arbitrary chaotic systems as a coincidence of their state variables (complete synchronization) or as an onset of a functional relationship between state variables of two unidirection-ally coupled systems (generalized synchronization). Although physical mechanisms behind the two latter phenomena essentially differ from the mechanisms of phase and frequency locking, all these effects constitute the field of application of the modern synchronization theory. 

B. Blasius, A. Huppert, and L. Stone. Complex dynamics and phase synchronization in spatially extended ecological systems. Nature, 399 354 (1999). 

As shown in Fig. 15.1(1 in a spatial lattice of patches, only small levels of local migration are required to induce broad-scale phase synchronization. The re.sult of the simulation in the phase synchronized regime is visualized ill Fig. 15.11, which demonstrates that all populations in the lattice are phaselocking to the same collective rhythm. Similar to the synchronized oscillations of the Canadian lynx, also in the lattice simulation despite the... 

B. Blasius L. Stone. Chaos and phase synchronization in ecological sys-... 

B. Blasius, E. Montbrio, J. Kurths. Anomalous phase synchronization in populations of nonidentical oscillators. Phys. Rev. E 67, 035204 (2003). 

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