Abstract
Abstract. 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.
Sommario: Si presenta un metodo per determinare la dimensione di un sistema dinamico comprendente simultaneamente due diverse serie temporali discrete. Le serie temporali sono costituite dallo stesso osservabile, prendendo condizioni iniziali distinte e indipendenti, oppure sono formate da realizzazioni di differenti osservabili misurati simultaneamente in un attrattore simmetrico. Il metodo deriva da un'estensione della tecnica introdotta in [18,19] per singole serie temporali e consente di valutare la comune dimensione di correlazione dell'attrattore caotico. La dimensione di correlazione associata a due serie temporali è calcolata per alcuni modelli matematici. In particolare si analizza la mappa standard bidimensionale; si introduce inoltre una mappa a 4 dimensioni analoga alla mappa di Hénon si analizzano serie temporali singole e multiple. Il metodo delle serie multiple consente una valutazione più accurata ed efficiente delle dimensioni di immersione e di correlazione nei casi sperimentali. Si applica inoltre il metodo a serie discrete associate a registrazioni elettrofisiologiche. Sono stati determinati svariati esempi di dinamica significativa. I risultati sono confermati dal calcolo dell'entropia con il metodo delle doppie serie e sono stati confrontati con le analisi standard che si utilizzano nel campo delle Neuroscienze. Si propone il metodo della doppia serie come uno strumento complementare per lo studio delle proprietà dinamiche di gruppi di cellule e per l'analisi di processi cognitivi.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abeles, M., ‘Quantification, smoothing, and confidence limits for single unit hystogram’. J. Neurosc. Methods 5(1982) 317–325.
Abeles, M., Gerstein, G.L., ‘Detecting spatiotemporal firing patterns among simultaneously recorded single neurones’. J. Neurophysiol. 60(1988) 909–924.
Abeles, M., Bergman, H., Margalit, E., Vaadia, E., ‘Spatiotemporal firing patterns in the frontal cortex of behaving monkeys’. J. Neurophysiol. 70(1993) 1629–1638.
Aertsen, A.M., Gerstein, G.L., Habib, M.K., Palm, G., ‘Dynamics of neuronal firing correlation: modulation of ‘effective connectivity’. J. Neurophysiol. 61(1989) 900–917.
Babloyantz, A., Salazar, J.M., ‘Evidence of chaotic dynamics of brain activity during the sleep cycle’. Phys. Letters A 111(1985) 152–156.
Brillinger, D.R., Villa, A.E.P., ‘Examples of the investigation of neural information processing by point process analysis’ In: Advanced Methods of Physiological System Modeling, vol. 3, Plenum Press/ New York 1994, pp. 111–127.
Brillinger, D.R., Bryant, H.L. Jr., Segundo, J.P., ‘Identification of synaptic interactions’. Biol. Cybernetics 22(1976) 213–228.
Celletti, A., Froeschlè, C., ‘On the determination of the stochasticity threshold of invariant curves’. Int. J. of Bifurcation and Chaos 5(6) (1995) 1713.
Celletti, A., Villa, A.E.P., ‘Low dimensional chaotic attractors in the rat brain’. Biol. Cybernetics 74(5) (1996) 387.
Chirikov, B.V., ‘A universal instability of many-dimensional oscillator systems’. Phys. Rep. 52(1979) 263.
Dayhoff, J., Gerstein, G.L., ‘Favored patterns in spike trains. I. Detection’. J. Neurophysiol. 49(1983) 1334–1348.
Eckmann, J.P., Kamphorst Oliffson, S., Ruelle, D., Ciliberto, S., ‘Lyapunov exponents from time series’. Phys. Rev. A34(1986) 4971–4979.
Eckmann, J.P., Ruelle, D., ‘Ergodic theory of chaos and strange attractors’. Rev. Mod. Phys. 57(1985) 617–656.
Eckmann, J.P., Ruelle, D., ‘Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems’. Physica D 56(1992) 185–187.
Friston, K.J., ‘Neuronal transients’. Proc. Royal Soc. London – Ser. B261(1995) 401–405.
Gerstein, G.L., Perkel, D.H., ‘Mutual temporal relationships among neuronal spike trains. Statistical techniques for display and analysis’. Biophys. J. 12(1972) 453–473.
Grassberger, P., Hegger, R., Kantz, H., Schaffrath, C., ‘On noise reduction methods for chaotic data’. Chaos 3(2) (1993) 127–141.
Grassberger, P., Procaccia, I., ‘Estimation of the Kolmogorov entropy from a chaotic signal’. Phys. Rev. A28(1983) 2591–2593.
Grassberger, P., Procaccia, I., ‘Characterization of strange attractors’. Phys. Rev. Letters 50(5) (1983) 346–349.
Greene, J.M., ‘A method for determining a stochastic transition’. J. of Math. Phys. 20(1979) 1183.
Hénon, M., ‘A two-dimensional mapping with a strange attractor’. Comm. Math. Phys. 50(1976) 69–77.
Kantz, H., ‘Quantifying the closeness of fractal measures’. Phys. Rev. E49(6) (1994) 5091–5097.
Kantz, H., Schreiber, T., Hoffmann, I., Buzug, T., Pfister, G., Flepp, L.G., Simonet, J., Badii, R., Brun, E., ‘Nonlinear noise reduction: a case study on experimental data’. Phys. Rev. E48(2) (1993) 1529–1538.
Laskar, J., Froeschlé, C., Celletti, A., ‘The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping’. Physica D 56(1992) 253–269.
Martignon, L., Von Hasseln, H., Grun, S., Aertsen, A., Palm, G., ‘Detecting higher-order interactions among the spiking events in a group of neurons’. Biol. Cybernetics 73(1995) 69–81.
Mpitsos, G.J., Burton, R.M. Jr., Creech, H.C., Soinila, S.O., ‘Evidence for chaos in spike trains of neurons that generate rhythmic motor patterns’. Brain Res. Bull. 21(1988) 529–538.
Mpitsos, G.J., Creech, H.C., Cohan, C.S., Mendelson, M., ‘Variability and chaos: neurointegrative principles in self-organization of motor patterns’ In: Kelso JAS, Mandell AJ, Shlesinger MF (eds), Dynamic Patterns in Complex Systems, World Scientific, Singapore, 1988, pp. 162–190.
Packard, N.H., Crutchfield, J.P., Farmer, J.D., Shaw, R.S., ‘Geometry from a time series’. Phys. Rev. Lett. 45(1980) 712–716.
Rapp, P.E., Albano, A.M., Schmah, T.I., Farwell, L.A., ‘Filtered noise can mimic low-dimensional chaotic attractors’. Phys. Rev. E47(1993) 2289–2297.
Rapp, P.E., Zimmerman, I.D., Albano, A.M., Deguzman, G.C., Greenbaun, N.N., ‘Dynamics of spontaneous neural activity in the simian motor cortex: the dimension of chaotic neurons’. Phys. Letters A 110(1985) 335–338.
Tetko, I.V., Villa, A.E.P., ‘Fast combinatorial methods for estimation of complex temporal patterns of spikes’. Biol. Cybern 76(1997) 397–407.
Theiler, J., Rapp, P.E., ‘Re-examination of the evidence for low-dimensional, nonlinear structure in the human electroencephalogram’. Preprint(1994).
Vaadia, E., Haalman, I., Abeles, M., Bergman, H., Prut, Y., Slovin, H., Aertsen, A., ‘Dynamics of neuronal interactions in monkey cortex in relation to behavioural events’. Nature 373(1995) 515–518.
Villa, A.E.P., ‘Physiological differentiation within the auditory part of the thalamic reticular nucleus of the cat’. Brain Res. Reviews 15(1990) 25–40.
Villa, A.E.P., Abeles, M., ‘Evidence for spatio-temporal firing patterns within the auditory thalamus of the cat’. Brain Res. 509(1990) 325–327.
Villa, A.E.P., Fuster, J.M., ‘Temporal correlates of information processing during visual short-term memory’. NeuroReport 3(1992) 113–116.
Villa, A.E.P., Tetko, I.V., Riehle, A., Requin, J., ‘Selective spatio-temporal firing patterns in the motor cortex during a reaction-time task’. Soc. Neurosci. Abst. 21(1995) 516.
Ville, A.E.P., Bajo Lorenzone, V.M., Vantini, G., ‘Nerve growth factor modulates information processing in the auditory thalamus’. Brain Res. Bull. 39(3) (1996) 139–147.
Villa, A.E.P., Bajo Lorenzana, V.M., ‘Ketamine modulation of the temporal pattern of discharges and spike train interactions in the rat substantia nigra pars reticulata’. Brain Res. Bull. 43(1997) 525–535.
Wilson, C.J., Young, S.J., Groves, P.M., ‘Statistical properties of neuronal spike trains in the substantia nigra: cell types and their interactions’. Brain Res. 136(1977) 243–260.
Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A., ‘Determining Lyapunov exponents from a time series’. Physica D 16(1985) 285–317.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Celletti, A., Bajo Lorenzana, V.M. & Villa, A.E.P. Correlation Dimension for Two Experimental Time Series. Meccanica 33, 381–396 (1998). https://doi.org/10.1023/A:1004355114814
Issue Date:
DOI: https://doi.org/10.1023/A:1004355114814