counterpart is known as phase-locking value or phase coherence. The size of the neighborhood in the responder is compared with the size of the mapping in the driver. Cross-spectral matrices are estimated via a Welch procedure, that is, they are averaged. State space reconstruction of two nonlinear systems (Rössler and Lorenz, see Quian Quiroga et al., 2000) for the A. Recently, in addition to this linear approach a variety of nonlinear extensions have been proposed (e.g., Marinazzo et al., 2008).įigure 1: Nonlinear interdependence. The performance for the two models is typically evaluated by comparing the variances of their prediction errors. \frac\) are the prediction errors associated with the respective model. In this paper, we report that when a partially coherent beam carrying a cross phase propagates in free space, in a paraxial optical system or in a turbulent medium, the modulus of the far-field (focal plane) DOC acquires the same value as it has in the source plane. It is defined in the time domain as a function of the time lag \(\tau = -(N-1).,0.,N-1\) and is derived from normalized signals \(x_n\) and \(y_n\) of length \(N\) and with zero mean and unit variance as The simplest and most widely used measure of synchronization is the linear cross correlation. Furthermore, different indices of phase synchronization such as the mean phase coherence (Kuramoto, 1984 Mormann et al., 2000) have been introduced. These approaches comprise linear ones like the cross correlation or the spectral coherence function as well as nonlinear measures like mutual information (Gray, 1990), transfer entropy (Schreiber, 2000), Granger causality (Granger, 1969), or the nonlinear interdependence (Arnhold et al., 1999 Quian Quiroga et al., 2002 Andrzejak et al., 2003). Finally, phase synchronization, first described for chaotic oscillators (Rosenblum et al., 1996), is defined as the global entrainment of the phases while the amplitudes may remain uncorrelated.įollowing this variety of concepts many different approaches to quantify the degree of synchronization between two continuous signals have been proposed. The concept of generalized synchronization (Afraimovich et al., 1986) introduced for uni-directionally coupled systems, describes the presence of some functional relation between the states of the two systems. The simplest case of complete synchronization (Fujisaka and Yamada, 1983) can be attained if identical systems are coupled sufficiently strongly so that their states coincide after transients have died out. Synchronization of continuous time series can manifest itself in many different ways. A complementary class of approaches comprises measures of spike train synchrony which quantify the degree of synchrony between discrete signals. Measures of neuronal signal synchrony are estimators of the synchrony between two or sometimes more continuous time series of brain activity which yield low values for independent time series and high values for correlated time series. Thomas Kreuz, Institute for complex systems (ISC), National research council (CNR), Sesto Fiorentino, Italy
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