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Nonlinear Concepts in Time Series Analysis

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Advanced Data Analysis in Neuroscience

Part of the book series: Bernstein Series in Computational Neuroscience ((BSCN))

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Abstract

In biology, neuroscience in particular, the dynamical processes generating the observed time series will commonly be (highly) nonlinear. A prominent example is the very essence of neural communication itself, the action potential, which is generated by the strongly nonlinear feedbacks between sodium and potassium channel gating and membrane potential (or interactions among channels themselves; Naundorf et al. 2006). Stable oscillations as frequently encountered in neural systems, detected, e.g., in EEG or local field potentials, are nonlinear phenomena as well. This does not imply that linear time series analysis is not useful. Linear models, especially in very noisy or chaotic (see below) situations, may still provide a good approximation; they may still be able to capture the bulk of the deterministic dynamics in a sea of noise and explain most of the deterministic variance of the process (Perretti et al. 2013). Even if they do not capture a too large proportion of the deterministic fluctuations in the data, they could still be harvested as hypothesis testing tools in some situations. But linear systems are very limited from a computational point of view (arguably the most important biological purpose of brains) and won’t be able to capture a number of prominent biophysical phenomena.

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References

  • Abarbanel, H.: Analysis of Observed Chaotic Data. Springer, New York (1996)

    Book  MATH  Google Scholar 

  • Abeles, M., Bergman, H., Gat, I., Meilijson, I., Seidemann, E., Tishby, N., Vaadia, E.: Cortical activity flips among quasi-stationary states. Proc. Natl. Acad. Sci. U S A. 92, 8616–8620 (1995)

    Article  Google Scholar 

  • Amit, D.J., Brunel, N.: Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex. Cereb. Cortex. 7, 237–252 (1997)

    Article  Google Scholar 

  • Balaguer-Ballester, E., Lapish, C.C., Seamans, J.K., Daniel Durstewitz, D.: Attractor dynamics of cortical populations during memory-guided decision-making. PLoS Comput. Biol. 7, e1002057 (2011)

    Article  Google Scholar 

  • Basseville, M., Nikiforov, I.V.: Detection of Abrupt Changes: Theory and Application. Prentice Hall Information and System Sciences Series. Prentice Hall, Englewood Cliffs, NJ (1993)

    Google Scholar 

  • Baum, L.E., Petrie, T., Soules, G., Weiss, N.: A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Ann. Math. Stat. 41, 164–171 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  • Bauwens, L., Rombouts, J.V.: On marginal likelihood computation in change-point models. Comput. Stat. Data Anal. 56, 3415–3429 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Bhattacharya, P.K.: Some aspects of change-point analysis. In: Change-Point Problems, IMS Lecture Notes – Monograph Series, vol. 23 (1994)

    Google Scholar 

  • Bishop, C.M.: Pattern Recognition and Machine Learning. Springer, New York (2006)

    MATH  Google Scholar 

  • Bouezmarni, T., Rombouts, J.V.: Nonparametric density estimation for positive time series. Comput. Stat. Data Anal. 54, 245–261 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Cao, L., Mees, A., Judd, K.: Dynamics from multivariate time series. Physica D. 121, 75–88 (1998)

    Article  MATH  Google Scholar 

  • De Gooijer, J.G.: Detecting change-points in multidimensional stochastic processes. Comp. Stat. Data Anal. 51, 1892–1903 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Demanuele, C., Bähner, F., Plichta, M.M., Kirsch, P., Tost, H., Meyer-Lindenberg, A., Durstewitz, D.: A statistical approach for segregating cognitive task stages from multivariate fMRI BOLD time series. Front. Human Neurosci. 9, 537 (2015a)

    Article  Google Scholar 

  • Durstewitz, D., Gabriel, T.: Dynamical basis of irregular spiking in NMDA-driven prefrontal cortex neurons. Cereb. Cortex. 17, 894–908 (2007)

    Article  Google Scholar 

  • Durstewitz, D., Seamans, J.K., Sejnowski, T.J.: Dopamine-mediated stabilization of delay-period activity in a network model of prefrontal cortex. J. Neurophysiol. 83, 1733–1750 (2000a)

    Google Scholar 

  • Durstewitz, D., Seamans, J.K., Sejnowski, T.J.: Neurocomputational models of working memory. Nat. Neurosci. 3(Suppl), 1184–1191 (2000b)

    Article  Google Scholar 

  • Durstewitz, D., Vittoz, N.M., Floresco, S.B., Seamans, J.K.: Abrupt transitions between prefrontal neural ensemble states accompany behavioral transitions during rule learning. Neuron. 66, 438–448 (2010)

    Article  Google Scholar 

  • Fan, J., Yao, Q.: Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York (2003)

    Book  MATH  Google Scholar 

  • Frick, K.F., Munk, A., Sieling, A.: Multiscale change point inference. J. Roy. Stat. Soc. B. 76, 495–580 (2014)

    Article  MathSciNet  Google Scholar 

  • Ghahramani, Z.: An introduction to Hidden Markov Models and Bayesian networks. Int. J. Pattern Recog. Artif. Intell. 15, 9–42 (2001)

    Article  Google Scholar 

  • Ghahramani, Z., Hinton, G.E.: Variational learning for switching state-space models. Neural Comput. 12, 831–864 (2000)

    Article  Google Scholar 

  • Hamilton, J.D.: Time Series Analysis. Princeton UP, Princeton, NJ (1994)

    MATH  Google Scholar 

  • Hart, J.D.: Automated kernel smoothing of dependent data by using time series cross-validation. J. Roy. Stat. Soc. Ser. B (Methodological). 56, 529–542 (1994)

    MathSciNet  MATH  Google Scholar 

  • Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning (Vol. 2, No. 1) Springer, New York (2009)

    Google Scholar 

  • Hertz, J., Krogh, A.S., Palmer, R.G.: Introduction to the theory of neural computation. Addison-Wesley, Reading, MA (1991)

    Google Scholar 

  • Hušková, M., Kirch, C.: Bootstrapping confidence intervals for the change-point of time series. J. Time Series Anal. 29, 947–972 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Jirak, M.: Change-point analysis in increasing dimension. J. Multivar. Anal. 111, 136–159 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Jones, L.M., Fontanini, A., Sadacca, B.F., Miller, P., Katz, D.B.: Natural stimuli evoke dynamic sequences of states in sensory cortical ensembles. Proc. Natl. Acad. Sci. U S A. 104, 18772–18777 (2007)

    Article  Google Scholar 

  • Kantz, H., Schreiber, T.: Nonlinear Time Series Analysis. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  • Kirch, C.: Block permutation principles for the change analysis of dependent data. J. Stat. Plan. Inference. 137, 2453–2474 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Kirch, C.: Resampling in the frequency domain of time series to determine critical values for change-point tests. Stat. Decis. 25, 237–261 (2008)

    MathSciNet  MATH  Google Scholar 

  • Kirch, C., Kamgaing, J.T.: Detection of change points in discrete-valued time series. In: Davis, R.A., Holan, S.H., Lund, R., Ravishanker, N. (eds.) Handbook of Discrete-Valued Time Series. Chapman and Hall/CRC Press, New York (2015)

    Google Scholar 

  • Mazzucato, L.: Dynamics of multistable states during ongoing and evoked cortical activity. J. Neurosci. 35, 8214–8231 (2015)

    Article  Google Scholar 

  • McFarland, J.M., Hahn, T.T., Mehta, M.R.: Explicit-duration hidden Markov model inference of UP-DOWN states from continuous signals. PLoS One. 6, e21606 (2011)

    Article  Google Scholar 

  • Messer, M., Kirchner, M., Schiemann, J., Roeper, J., Neininger, R., Schneider, G.: A multiple filter test for the detection of rate changes in renewal processes with varying variance. Ann. Appl. Stat. 8, 2027–2067 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Miller, P., Katz, D.B.: Stochastic transitions between neural states in taste processing and decision-making. J. Neurosci. 30, 2559–2570 (2010)

    Article  Google Scholar 

  • Naundorf, B., Wolf, F., Volgushev, M.: Unique features of action potential initiation in cortical neurons. Nature. 20, 1060–1063 (2006)

    Article  Google Scholar 

  • Page, E.S.: Continuous inspection scheme. Biometrika. 41, 100–115 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  • Perretti, C.T., Munch, S.B., Sugihara, G.: Model-free forecasting outperforms the correct mechanistic model for simulated and experimental data. PNAS. 110, 5253–5257 (2013)

    Article  Google Scholar 

  • Pillow, J.W., Ahmadian, Y., Paninski, L.: Model-based decoding, information estimation, and change-point detection techniques for multineuron spike trains. Neural Comput. 23, 1–45 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Rabiner, L.: A tutorial on hidden Markov models and selected applications in speech recognition. Proc. IEEE. 77, 257–286 (1989)

    Article  Google Scholar 

  • Rabinovich, M.I., Rabinovich, M.I., Huerta, R., Varona, P., Afraimovich, V.S.: Transient cognitive dynamics, metastability, and decision making. PLoS Comput. Biol. 2, e1000072 (2008)

    Article  MathSciNet  Google Scholar 

  • Radons, G., Becker, J.D., DĂĽlfer, B., KrĂĽger, J.: Analysis, classification, and coding of multielectrode spike trains with hidden Markov models. Biol. Cybern. 71, 359–373 (1994)

    Article  MATH  Google Scholar 

  • Rainer, G., Miller, E.K.: Neural ensemble states in prefrontal cortex identified using a hidden Markov model with a modified EM algorithm. Neurocomputing. 32–33, 961–966 (2000)

    Article  MATH  Google Scholar 

  • Richter, S.H., Zeuch, B., Lankisch, K., Gass, P., Durstewitz, D., Vollmayr, B.: Where have I been? Where should I go? Spatial working memory on a radial arm maze in a rat model of depression. PLoS One. 8, e62458 (2013)

    Article  Google Scholar 

  • Ruelle, D.: Strange Attractors. Mathematical Intelligencer 1980, vol. 2, pp. 126–137. Springer, New York (1980)

    MATH  Google Scholar 

  • Russo, E., Treves, A.: Cortical free-association dynamics: distinct phases of a latching network. Phys. Rev. E. Stat. Nonlin. Soft Matter Phys. 85(5 Pt 1), 051920 (2012)

    Article  Google Scholar 

  • Schreiber, T., Schmitz, A.: Improved surrogate data for nonlinearity tests. Phys. Rev. Lett. 77, 635 (1996)

    Article  Google Scholar 

  • Schreiber, T., Schmitz, A.: Surrogate time series. Physica D: Nonlinear Phenomena. 142, 346–382 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Seidemann, E., Meilijson, I., Abeles, M., Bergman, H., Vaadia, E.: Simultaneously recorded single units in the frontal cortex go through sequences of discrete and stable states in monkeys performing a delayed localization task. J. Neurosci. 16, 752–768 (1996)

    Google Scholar 

  • Takiyama, K., Okada, M.: Detection of hidden structures in nonstationary spike trains. Neural Comput. 23, 1205–1233 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Tran, L.T.: On multivariate variable-kernel density estimates for time series. Can. J. Stat./La Revue Canadienne de Statistique. 19, 371–387 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Verduzco-Flores, S.O., Bodner, M., Ermentrout, B.: A model for complex sequence learning and reproduction in neural populations. J. Comput. Neurosci. 32, 403–423 (2012)

    Article  MathSciNet  Google Scholar 

  • Vlachos, I., Kugiumtzis, D.: State space reconstruction for multivariate time series prediction. Nonlinear Phenomena Complex Syst. 11, 241–249 (2008)

    MathSciNet  Google Scholar 

  • Wang, X.J.: Synaptic basis of cortical persistent activity: the importance of NMDA receptors to working memory. J. Neurosci. 19, 9587–9603 (1999)

    MathSciNet  Google Scholar 

Download references

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  1. Department of Theoretical Neuroscience, Central Institute of Mental Health, Medical Faculty Mannheim of Heidelberg University, Mannheim, Germany

    Daniel Durstewitz

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Durstewitz, D. (2017). Nonlinear Concepts in Time Series Analysis. In: Advanced Data Analysis in Neuroscience. Bernstein Series in Computational Neuroscience. Springer, Cham. https://doi.org/10.1007/978-3-319-59976-2_8

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