Synchronization likelihood: an unbiased measure of generalized synchronization in multivariate data sets

Abstract

The study of complex systems consisting of many interacting subsystems requires the use of analytical tools which can detect statistical dependencies between time series recorded from these subsystems. Typical examples are the electroencephalogram (EEG) and magnetoencephalogram (MEG) which may involve the simultaneous recording of 150 or more time series. Coherency, which is often used to study such data, is only sensitive to linear and symmetric interdependencies and cannot deal with non-stationarity. Recently, several algorithms based upon the concept of generalized synchronization have been introduced to overcome some of the limitations of coherency estimates (e.g. [Physica D 134 (1999) 419; Brain Res. 792 (1998) 24]). However, these methods are biased by the degrees of freedom of the interacting subsystems [Physica D 134 (1999) 419; Physica D 148 (2001) 147]. We propose a novel measure for generalized synchronization in multivariate data sets which avoids this bias and can deal with non-stationary dynamics.

Introduction

A central problem in the study of normal and disturbed brain function is the question how functional interactions take place between different specialized networks. Understanding the coordination between brain regions is important in the context of information processing in the healthy brain [4], [5], [6] but also in the case of neurological disease. Loss of neurons and connecting fibre systems may lead to diminished interactions and cognitive dysfunction such as in Alzheimer’s disease [7], whereas pathologically increased synchronization is the hallmark of epileptic seizures [2], [8]. Usually, functional interactions are studied by considering time series of electrical potentials (electroencephalogram (EEG)) or magnetic field strengths (magnetoencephalogram (MEG)) recorded from different brain areas. Similarities between these time series are taken to reflect functional influences between the neuronal networks generating the time series. Similarities between time series are commonly quantified with linear techniques, in particular estimates of the coherency, which is a normalized measure of linear correlation as a function of frequency (e.g. [9], [10]).

While this approach has produced a large body of knowledge on normal and pathological brain function, it has a number of limitations. First coherency estimates are not suitable to characterize non-stationary data with rapidly changing interdependencies. Possibly, modifications such as event-related coherence can overcome this limitation [11]. A more important limitation is that methods such as coherency only capture linear relations between time series, and may fail to detect non-linear interdependencies between the underlying dynamical systems. Recently a variety of methods have been proposed to detect more general types of interactions between dynamical systems. One line of research is based on the analytical signal concept [12]. Here, the instantaneous phase of both time series is computed, and interactions are quantified in terms of time-dependent n:m phase locking (n and m being integers). This approach has been successful in the study of EEG seizure data [13] and in the study of synchronization between muscle and cortical activity. However, this approach is only valid when the time series are approximately oscillatory. A more general approach is based upon the theory of non-linear dynamical systems. It was demonstrated in the 1980s and early 1990s that, contrary to intuition, two interacting chaotic systems can also display synchronization phenomena [14], [15], [16], [17]. Initially synchronization was understood as identical synchronization, implying equality of the variables of the coupled systems. In the context of unidirectionally coupled driver response systems Rulkov et al. [18] introduced the wider concept of generalized synchronization. Generalized synchronization exists between two dynamical systems X and Y when the state of the response system Y is a function of the state of the driving system X:Y=F(X). When F is continuous, and x_i, x_j are two points on the attractor of X which are very close together, then the corresponding points y_i, y_j on Y will also be close together. An important feature of generalized synchronization is that the corresponding time series need not resemble each other.

Since the concept of generalized synchronization was introduced, several algorithms have been proposed to detect this type of interdependencies in experimental time series. Rulkov et al. [18] proposed a mutual false nearest neighbours (MFNNs) parameter and another measure based upon the predictability of the response dynamics by the driver dynamics. The idea of using mutual predictions between driver and response systems was further elaborated by Schiff et al. [19] and later applied to seizure EEG data in [2], [8]. Arnhold et al. [1] proposed a very simple measure for non-linear interdependencies, which is based upon a ratio of average distances between index points, their nearest neighbours and their mutual nearest neighbours (the mutual nearest neighbours of x_i have the time indices of the nearest neighbours of y_i). All of these methods claim to be able to detect non-linear dependencies between systems as well as asymmetry due to driver response interactions. However, it has been shown that identifying the driver and the response system in the case of asymmetric interactions is not always straightforward [1], [20], [21]. Rather, the asymmetries in interdependence measures may reflect the different degrees of freedom of the two systems [1], [20]. Pereda et al. [3] have also shown that these measures do not only reflect interdependencies between the time series but are also influenced by the properties of the individual dynamical systems, in particular their dimensionality. Pereda et al. [3] propose to overcome this problem with appropriately constructed surrogate data. While this may be valid it is also somewhat cumbersome and difficult to use with non-stationary data.

In this paper we propose a synchronization likelihood measure S which avoids the bias pointed out by Pereda et al. and gives a straightforward normalized estimate of the dynamical interdependencies between two or more simultaneously recorded time series. The measure is closely related to the concept of generalized mutual information as introduced by Pawelzik and co-workers [22], [23] and this measure can also be computed in a time-dependent way, making it suitable for the analysis of non-stationary data.

This paper is organized as follows. In Section 2 we describe the synchronization likelihood S. In Section 3 we study the properties of S by applying it to a test system of two non-linearly coupled Hénon systems. The performance of S in the case of real EEG and MEG data is studied in Section 4. Finally, we discuss the main results, in particular the strengths and weaknesses of the proposed method in relation to other available algorithms, and propose some directions for future research in Section 5.