Abstract. We describe a new algorithm for the detection of dynamical interdependence in bivariate time-series data sets. By using geometrical and dynamical argu- ments, we produce a method that can detect dynamical interdependence in weakly coupled systems where pre- vious techniques have failed. We illustrate this by comparison of our algorithm with another commonly used technique when applied to a system of coupled He´non maps. In addition, an improvement of 20% in the detection rate is observed when the technique is applied to human scalp EEG data, as compared with existing techniques. Such an improvement may assist an understanding of the role of large-scale nonlinear processes in normal brain function. 1 Introduction When presented with time-series data, the detection of underlying nonlinear structure is an important step toward understanding complex system behavior. Usual- ly this task is nontrivial since both nonlinear and high- dimensional stochastic behavior often coexist in the same system, and the measurement procedure used to obtain the data is typically corrupted by a degree of external noise. One therefore seeks techniques that maximally exploit the properties of nonlinear systems to measure the relative contributions of these processes. Nonlinear forecasting techniques have been widely employed for this purpose (Farmer and Sidorowich 1987; Cadagli 1989; Sugihara and May 1990; Sauer 1993). The basic aim of these techniques is to predict the evolution of a state space vector through observations of nearby orbits. A question that has generated much recent interest is, given time-series data from two arbitrary systems, to what extent (if any) does one system influence the other? To address this, nonlinear forecast- ing techniques were generalized by Schiff et al. (1995, 1997) to test for the existence of dynamical interdepen- dence (Pecora et al. 1996) in bivariate time-series data. These techniques have found many applications in the physical sciences, including mapping the spread of epileptic discharges in human EEG (Le Van Quyen et al. 1999). In this paper it is shown how geometrical and dynamical considerations motivate a modification to the technique widely employed and that the resulting algorithm offers a significant improvement in the ability to detect dynamical interdependence. In conjunction with amplitude and phase-randomized surrogate data sets, it is demonstrated that this modification permits the detection of weak interdependencies that would have been otherwise missed. Given that nonlinear structure (Rombouts et al. 1995; Pritchard et al. 1995; Stam et al. 1999; Quian Quiroga et al. 2000a,b, 2002) and interde- pendence (Muller-Gerking et al. 1996) may be only weakly or occasionally present in many physical and biological data sets, such an improvement is clearly desirable and has great potential across many subdisci- plines. Nonlinear prediction algorithms fall into two classes: global and local, depending on whether one seeks single approximate functions for the whole data set or separate local graphs for different neighborhoods. Local tech- niques are widely employed and usually fit polynomials of small degree d piecewise to the data (Farmer and Sidorowich 1987; Cadagli 1989) to approximate the future behavior of orbits. These local graphs are con- structed from interpolating the evolutions of each state space vector’s k-nearest neighbors. In most algorithms, d ¼ 1, in which case the local predictors are linear (Sauer 1993). Subsequently, a prediction error is obtained by measuring the difference between the predicted and the observed local orbits. For low-dimensional chaotic sys- tems, the growth of this error has a sequential rate approximately equal to the system’s largest Lyapunov exponent. [The actual growth of errors is greater than expected because local expansion rates fluctuate and the largest tend to be overemphasized (Cadagli 1989)]. The growth of the error reaches a plateau once it approaches Correspondence to: J.R. Terry (e-mail: J.R.Terry@lboro.ac.uk) Biol. Cybern. 88, 129–136 (2003) DOI 10.1007/s00422-002-0368-4 Ó Springer-Verlag 2003 An improved algorithm for the detection of dynamical interdependence in bivariate time-series John R. Terry 1 , Michael Breakspear 2;3 1 Department of Mathematical Sciences, Loughborough University, Leices, LEII 3TU, UK 2 School of Physics, University of Sydney, Sydney, NSW, Department of Psychological Medicine, Faculty of Medicine, University of Sydney, Sydney, NSW, Australia 3 Brain Dynamics Centre, Westmead Hospital, Westmead, NSW, Australia Received: 28 December 2001 ⁄ Accepted in revised form: 2 October 2002