Detecting generalized synchrony: an improved approach

Brain connectivity analysis: a short survey

This short survey the reviews recent literature on brain connectivity studies. It encompasses all forms of static and dynamic connectivity whether anatomical, functional, or effective. The last decade has seen an ever increasing number of studies devoted to deduce functional or effective connectivity, mostly from functional neuroimaging experiments. Resting state conditions have become a dominant experimental paradigm, and a number of resting state networks, among them the prominent default mode network, have been identified. Graphical models represent a convenient vehicle to formalize experimental findings and to closely and quantitatively characterize the various networks identified. Underlying these abstract concepts are anatomical networks, the so-called connectome, which can be investigated by functional imaging techniques as well. Future studies have to bridge the gap between anatomical neuronal connections and related functional or effective connectivities.

An analytic criterion for generalized synchronization in unidirectionally coupled systems based on the auxiliary system approach

An analytic criterion is developed to investigate generalized synchronization (GS) in unidirectionally coupled systems based on the auxiliary system approach. The criterion is derived by transforming the existence problem of generalized synchronization into an eigenvalue problem. Numerical simulations show that the analytic criterion is almost as accurate as the response Lyapunov exponents method, and may provide an estimation of the threshold of strong generalized synchronization. A significant result can be deduced from our analysis that the more the number of equilibria of the unidirectionally coupled systems, the greater the chance of generalized synchronization, but the harder it may be for strong generalized synchronization to occur.

Order parameter for the transition from strong to weak generalized synchrony from empirical mode decomposition analysis

We examine driven nonlinear dynamical systems that are known to be in a state of generalized synchronization with an external drive. The chaotic time series of the response system are subject to empirical mode decomposition analysis. The instantaneous intrinsic mode frequencies (and their variance) present in these signals provide suitable order parameters for detecting the transition between the regimes of strong and weak generalized synchrony. Application is made to a variety of chaotically driven flows as well as maps.

Filters display inverse limit spaces

A rigorous proof that linear filters display the inverse limit spaces of chaotic maps is given.

The structure of synchronization sets for noninvertible systems

Unidirectionally coupled systems (x,y) --> (f(x),g(x,y)) occur naturally, and are used as tractable models of networks with complex interactions. We analyze the structure and bifurcations of attractors in the case the driving system is not invertible, and the response system is dissipative. We discuss both cases in which the driving system is a map, and a strongly dissipative flow. Although this problem was originally motivated by examples of nonlinear synchrony, we show that the ideas presented can be used more generally to study the structure of attractors, and examine interactions between coupled systems.

Robust measure for characterizing generalized synchronization

Generalized synchronization between two coupled systems can be characterized by recently proposed interdependency measures calculated from two simultaneously observed time series from them. However, numerical tests have shown that these measures cannot consistently indicate the direction of the coupling for strongly coupled systems or in situations with a large phase space neighbor size. An interdependency measure is proposed here quantifying how close a conditional neighbor is to a true neighbor in terms of the degree of alignment of their principal axes. Numerical tests are carried out on time series generated from a coupled Hénon map and a Lorenz model driven by a Rossler model. Given that a driving system is more dependent on a response system, the results show that the direction of the coupling is consistently detected by using the proposed measure even in those unfavorable cases for the measures mentioned above.

Detectability of nondifferentiable generalized synchrony

Generalized synchronization of chaos is a type of cooperative behavior in directionally coupled oscillators that is characterized by existence of stable and persistent functional dependence of response trajectories from the chaotic trajectory of driving oscillator. In many practical cases this function is nondifferentiable and has a very complex shape. The generalized synchrony in such cases seems to be undetectable, and only the cases in which a differentiable synchronization function exists are considered to make sense in practice. We show that this viewpoint is not always correct and the nondifferentiable generalized synchrony can be revealed in many practical cases. Conditions for detection of generalized synchrony are derived analytically, and illustrated numerically with a simple example of nondifferentiable generalized synchronization.