Correlation dimension for two experimental time series

A method for detecting the dimension of a dynamical system encompassing simultaneously two distinct discrete time series is presented. The time series are provided by the same observable taking distinct and independent initial conditions or they can be formed by realizations of different observables measured simultaneously in a symmetric attractor. The method is derived from an extension of the technique introduced in [18, 19] for single time series and allows to evaluate the common correlation dimension of the chaotic attractor. The correlation dimension associated to two time series is computed for some mathematical models. In particular the two-dimensional standard mapping is analysed; a dissipative four-dimensional Hénon-like mapping is introduced and analyses with single and multiple time series are performed. The double series method provides a more accurate and efficient evaluation of the embedding and correlation dimensions in all experimental cases. The method is also applied to discrete time series derived from multiple single unit electrophysiological recordings. Several examples of significant dynamics have been revealed. The results are confirmed by the computation of the (double series) entropy and compared to usual time domain analyses performed in Neuroscience. The double series method is proposed as a complementary method for investigation of dynamical properties of cell assemblies and its potential usefulness for detecting higher order cognitive processes is discussed.