CORRELATION DIMENSION FOR 2 EXPERIMENTAL TIME-SERIES

A method for detecting the dimension of a dynamical system encompassin g simultaneously two distinct discrete time series is presented. The t ime series are provided by the same observable taking distinct and ind ependent initial conditions or they can be formed by realizations of d ifferent observables measured simultaneously in a symmetric attractor. The method is derived from an extension of the technique introduced i n [18, 19] for single time series and allows to evaluate the common co rrelation dimension of the chaotic attractor. The correlation dimensio n associated to two time series is computed for some mathematical mode ls. In particular the two-dimensional standard mapping is analysed; a dissipative four-dimensional Henon-like mapping is introduced and anal yses with single and multiple time series are performed. The double se ries method provides a more accurate and efficient evaluation of the e mbedding and correlation dimensions in all experimental cases. The met hod is also applied to discrete time series derived from multiple sing le unit electrophysiological recordings. Several examples of significa nt dynamics have been revealed. The results are confirmed by the compu tation of the (double series) entropy and compared to usual time domai n analyses performed in Neuroscience. The double series method is prop osed as a complementary method for investigation of dynamical properti es of cell assemblies and its potential usefulness for detecting highe r order cognitive processes is discussed.