Oscillators, globally coupled

Globally coupled oscillators) Consider the following system of N identical oscillators ... 

The model of globally coupled oscillators is commonly used as a simplest model of neural synchrony. We illustrate this using a computationally efficient neuronal model, proposed by Rulkov [42, 43]. In this model a neuron is described by a 2D map. In spite of its simplicity, this model reproduces most regimes exhibited by the full Hodgkin-Huxley model, but at essentially lower computational costs, thus allowing detailed analysis of the dynamics of large ensembles. The model reads... 

J. D. Crawford. Amplitude expansions for instabilities in populations of globally-coupled oscillators. J. Stat. Phys., 74(5/6) 1047-1084, 1994. 

Let us now consider the behavior of the system when the Kerr coupling constant is switched on (e12 / 0). For brevity and clarity, we restrict our discussion to the question of how the attractors in Fig. 20 change when both oscillators interact with each other. To answer this question, let us have a look at the joint auto-nomized spectrum of Lyapunov exponents for the two oscillators A,j, A,2, L3, A-4, L5 versus the interaction parameter 0 < ( 2 < 0.7. The spectrum is seen in Fig. 32 and describes the dynamical properties of our oscillators in a global sense. The dynamics of individual oscillators can be glimpsed at the appropriate phase portraits. Let us now fix our attention on a detailed analysis of Fig. 32. For the limit value ei2 = 0, the dynamics of the uncoupled oscillators has already been presented in Fig. 20. In the case of very weak interaction 0 < C 2 < 0.0005, the system of coupled oscillators manifests chaotic behavior. For C 2 = 0.0005 we obtain the spectrum 0.06,0.00, —0.21, 0.54, 0.89. It is interesting to... 

Fig. 67. Current as a function of space and time for an array consisting of 64 Ni electrodes exhibiting relaxation oscillations and different strength of the global coupling. Dark corresponds to high current. Top left i — 0 top right e — 0.28 bottom left e — 0.67 and bottom right e — 0.95. (Reproduced with permission from I. Z. Kiss, W. Wang, J. L. Hudson, J. Phys. Chem. B 103 (1999) 11433-11444, 1999, American Chemical Society.)...

Behavior similar to the oscillator death described above has been observed as the suppression of oscillations due to stirring in some experiments with oscillatory chemical reactions, and was also found in a few other systems without mixing, like globally coupled phase-amplitude oscillators or oscillators with delayed coupling (Mirollo and Strogatz, 1990 Ramana Reddy et al., 1998). 

Thus, the oscillators in a globally coupled ensemble are driven by a common force. Clearly, this force can entrain many oscillators if their frequencies are close. The problem is that this force (the mean field) is not predetermined, but arises from interaction witbiii the ensemlile, This force determines wliether the systems synchronize, but it itself depends on their oscillation - it is a tyjjical example of self-organization [20], To explain qualitati ely the appearance of tliis force (or to compute it, as is done ill [28, 38j) one should consider this problem self-consistently. 

The physical mechanism we described is known as the Kuramoto selfsynchronization transition [27]. The scenario of this transition does not depend on the origin of the oscillators (biological, electronic, etc.) or on the origin of interaction. In the above presented examples the coupling occurred via an optical or acoustic field. Global coupling of electronic systems... 

Fig. 15.10. Standard deviation of frequencies

When macroscopic oscillations occur (as monitored, e.g., with a Kelvin probe or by a mass spectrometer tuned to CO2), the partial pressures change periodically with time. Since the mean free path of the gas molecules is at the applied pressures considerably larger than the vessel diameter, this gives rise to a global coupling mechanism [71]. The coupling is essentially mediated by CO because O2 is present in excess (and CO2 is just an inert product). As... 

One can see that the coupling between the two oscillators is proportional to qiq2, and that the global minimum of Eq. (7.55) occurs when qt = l. The potential function (7.55) can be rewritten in a variety of ways by making appro-... 

---