Correlation dimension for paired discrete time series

We present a method for detecting the dimension of a dynamical system encompassing simultaneously two distinct discrete time series. This method is an extension of the technique introduced by Grassberger and Procaccia for single time series and allows to evaluate the common correlation dimension of the chaotic attractor. The method is applied to some mathematical models and to multiple single-neuron spike trains

[1]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[2]  Eckmann,et al.  Liapunov exponents from time series. , 1986, Physical review. A, General physics.

[3]  Alessandro E. P. Villa,et al.  Ketamine Modulation of the Temporal Pattern of Discharges and Spike Train Interactions in the Rat Substantia Nigra Pars Reticulata , 1997, Brain Research Bulletin.

[4]  James P. Crutchfield,et al.  Geometry from a Time Series , 1980 .

[5]  Alessandro E. P. Villa,et al.  Nerve growth factor modulates information processing in the auditory thalamus , 1996, Brain Research Bulletin.

[6]  A. Babloyantz,et al.  Evidence of Chaotic Dynamics of Brain Activity During the Sleep Cycle , 1985 .

[7]  D. Ruelle,et al.  Ergodic theory of chaos and strange attractors , 1985 .

[8]  D. Ruelle,et al.  Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems , 1992 .

[9]  Kantz.  Quantifying the closeness of fractal measures. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  G L Gerstein,et al.  Mutual temporal relationships among neuronal spike trains. Statistical techniques for display and analysis. , 1972, Biophysical journal.

[11]  Albano,et al.  Filtered noise can mimic low-dimensional chaotic attractors. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[12]  M. Abeles.  Quantification, smoothing, and confidence limits for single-units' histograms , 1982, Journal of Neuroscience Methods.

[13]  U. Parlitz,et al.  Lyapunov exponents from time series , 1991 .

[14]  G. Mpitsos,et al.  Evidence for chaos in spike trains of neurons that generate rhythmic motor patterns , 1988, Brain Research Bulletin.

[15]  P. Grassberger,et al.  Estimation of the Kolmogorov entropy from a chaotic signal , 1983 .

[16]  John M. Greene,et al.  A method for determining a stochastic transition , 1979, Hamiltonian Dynamical Systems.

[17]  Alessandro E. P. Villa,et al.  Physiological differentiation within the auditory part of the thalamic reticular nucleus of the cat , 1990, Brain Research Reviews.