Skip to main content
SpringerLink
Log in

Time Series from a Nonlinear Dynamical Systems Perspective

  • Chapter
  • First Online:
Book cover Advanced Data Analysis in Neuroscience

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

Abstract

Nonlinear dynamics is a huge field in mathematics and physics, and we will hardly be able to scratch the surface here. Nevertheless, this field is so tremendously important for our theoretical understanding of brain function and time series phenomena that I felt a book on statistical methods in neuroscience should not go without discussing at least some of its core concepts. Having some grasp of nonlinear dynamical systems may give important insights into how the observed time series were generated. In fact, nonlinear dynamics provides a kind of universal language for mathematically describing the deterministic part of the dynamical systems generating the observed time series—we will see later (Sect. 9.3) how to connect these ideas to stochastic processes and statistical inference. ARMA and state space models as discussed in Sects. 7.2 and 7.5 are examples of discrete-time, linear dynamical systems driven by noise. However, linear dynamical systems can only exhibit a limited repertoire of dynamical behaviors and typically do not capture a number of prominent and computationally important phenomena observed in physiological recordings. In the following, we will distinguish between models that are defined in discrete time (Sect. 9.1), as all the time series models discussed so far, and continuous-time models (Sect. 9.2).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 69.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Aarts, E., Korst, J.: Simulated Annealing and Boltzmann Machines: A Stochastic Approach to Combinatorial Optimization and Neural Computing. Wiley, Chichester (1988)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  • Abarbanel, H.: Predicting the Future. Completing Models of Observed Complex Systems. Springer, New York (2013)

    MATH  Google Scholar 

  • Aksay, E., Gamkrelidze, G., Seung, H.S., Baker, R., Tank, D.W.: In vivo intracellular recording and perturbation of persistent activity in a neural integrator. Nat. Neurosci. 4, 84–193 (2001)

    Article  Google Scholar 

  • Auger-Méthé, M., Field, C., Albertsen, C.M., Derocher, A.E., Lewis, M.A., Jonsen, I.D., Mills Flemming, J.: State-space models’ dirty little secrets: even simple linear Gaussian models can have estimation problems. Scientific Rep. 6, 26677 (2016)

    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 

  • Beer, R.D.: Parameter space structure of continuous-time recurrent neural networks. Neural Comput. 18, 3009–3051 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Bertschinger, N., Natschläger, T.: Real-time computation at the edge of chaos in recurrent neural networks. Neural Comput. 16, 1413–1436 (2004)

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Brette, R., Gerstner, W.: Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. J. Neurophysiol. 94, 3637–3642 (2005)

    Article  Google Scholar 

  • Brody, C.D., Hopfield, J.J.: Simple networks for spike-timing-based computation, with application to olfactory processing. Neuron. 37, 843–852 (2003)

    Article  Google Scholar 

  • Brunton, S.L., Proctor, J.L., Kutz, J.N.: Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl. Acad. Sci. U. S. A. 113, 3932–3937 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Brody, C.D., Hernández, A., Zainos, A., Romo, R.: Timing and neural encoding of somatosensory parametric working memory in macaque prefrontal cortex. Cereb. Cortex. 13, 1196–1207 (2003)

    Article  Google Scholar 

  • Brunel, N.: Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J. Comput. Neurosci. 8, 183–208 (2000)

    Article  MATH  Google Scholar 

  • Brunel, N., Hakim, V.: Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Comput. 11, 1621–1671 (1999)

    Article  Google Scholar 

  • Brunel, N., van Rossum, M.C.W.: Lapicque’s 1907 paper: from frogs to integrate-and-fire. Biol. Cybern. 97, 337–339 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Brunel, N., Wang, X.J.: Effects of neuromodulation in a cortical network model of object working memory dominated by recurrent inhibition. J. Comput. Neurosci. 11, 63–85 (2001)

    Article  Google Scholar 

  • Brunton, B.W., Botvinick, M.M., Brody, C.D.: Rats and humans can optimally accumulate evidence for decision-making. Science. 340, 95–98 (2013)

    Google Scholar 

  • Buesing, L., Macke, J.H., Sahani, M.: Learning stable, regularised latent models of neural population dynamics. Network. 23, 24–47 (2012)

    Google Scholar 

  • Buonomano, D.V.: Decoding temporal information: a model based on short-term synaptic plasticity. J. Neurosci. 20, 1129–1141 (2000)

    Google Scholar 

  • Buzsaki, G.: Rhythms of the Brain. Oxford University Press, Oxford (2011)

    MATH  Google Scholar 

  • Buzsaki, G., Draguhn, A.: Neuronal oscillations in cortical networks. Science. 304, 1926–1929 (2004)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Chow, T.W.S., Li, X.-D.: Modeling of continuous time dynamical systems with input by recurrent neural networks. IEEE Transactions on Circuits and Systems—I: Fundamental Theory and Applications. 47, 575–578 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Cohen, A.H., Holmes, P.J., Rand, R.H.: The nature of the coupling between segmental oscillators of the lamprey spinal generator for locomotion: a mathematical model. J. Math. Biol. 13, 345–369 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  • Daunizeau, J., Stephan, K.E., Friston, K.J.: Stochastic dynamic causal modelling of fMRI data: should we care about neural noise? NeuroImage. 62, 464–481 (2012)

    Article  Google Scholar 

  • Dayan, P., Abott, L.F.: Theoretical Neuroscience. Computational and Mathematical Modeling of Neural Systems. MIT Press, Cambridge, MA (2001)

    Google Scholar 

  • Dong, Y., Mihalas, S., Russell, A., Etienne-Cummings, R., Niebur, E.: Estimating parameters of generalized integrate-and-fire neurons from the maximum likelihood of spike trains. Neural Comput. 23, 2833–2867 (2011)

    Article  MATH  Google Scholar 

  • Durbin, J., Koopman, S.J.: Time Series Analysis by State Space Methods (Oxford Statistical Science). Oxford University Press, Oxford (2012)

    MATH  Google Scholar 

  • Durstewitz, D.: Self-organizing neural integrator predicts interval times through climbing activity. J. Neurosci. 23, 5342–5353 (2003)

    Google Scholar 

  • Durstewitz, D.: A state space approach for piecewise-linear recurrent neural networks for identifying computational dynamics from neural measurements. PLoS Comput. Biol. 13, e1005542 (2017)

    Article  Google Scholar 

  • Durstewitz, D.: Implications of synaptic biophysics for recurrent network dynamics and active memory. Neural Netw. 22, 1189–1200 (2009)

    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.: The computational role of dopamine D1 receptors in working memory. Neural Netw. 15, 561–572 (2002)

    Article  Google Scholar 

  • Durstewitz, D., Seamans, J.K.: The dual-state theory of prefrontal cortex dopamine function with relevance to catechol-o-methyltransferase genotypes and schizophrenia. Biol. Psychiatry. 64, 739–749 (2008)

    Article  Google Scholar 

  • Durstewitz, D., Koppe, G., Toutounji, H.: Computational models as statistical tools. Curr. Opin. Behav. Sci. 11, 93–99 (2016)

    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 

  • Elman, J.L.: Finding structure in time. Cognitive Sci. 14, 179–211 (1990)

    Article  Google Scholar 

  • Engel, A.K., Fries, P., Singer, W.: Dynamic predictions: oscillations and synchrony in top-down processing (Review). Nat. Rev. Neurosci. 2, 704–716 (2001)

    Article  Google Scholar 

  • Fahrmeir, L., Tutz, G.: Multivariate Statistical Modelling Based on Generalized Linear Models. Springer, New York (2010)

    MATH  Google Scholar 

  • Fries, P., Reynolds, J.H., Rorie, A.E., Desimone, R.: Modulation of oscillatory neuronal synchronization by selective visual attention. Science. 291, 1560–1563 (2001)

    Article  Google Scholar 

  • Friston, K.J., Harrison, L., Penny, W.: Dynamic causal modelling. Neuroimage. 19, 1273–1302 (2003)

    Article  Google Scholar 

  • Funahashi, K.-I., Nakamura, Y.: Approximation of dynamical systems by continuous time recurrent neural networks. Neural Netw. 6, 801–806 (1993)

    Article  Google Scholar 

  • Funahashi, S., Bruce, C.J., Goldman-Rakic, P.S.: Mnemonic coding of visual space in the monkey’s dorsolateral prefrontal cortex. J. Neurophysiol. 61, 331–349 (1989)

    Google Scholar 

  • Fusi, S., Asaad, W.F., Miller, E.K., Wang, X.J.: A neural circuit model of flexible sensorimotor mapping: learning and forgetting on multiple timescales. Neuron. 54, 319–333 (2007)

    Article  Google Scholar 

  • Fuster, J.M.: Unit activity in prefrontal cortex during delayed-response performance: neuronal correlates of transient memory. J. Neurophysiol. 36, 61–78 (1973)

    Google Scholar 

  • Gerstner, W., Kempter, R., van Hemmen, J.L., Wagner, H.: A neuronal learning rule for sub-millisecond temporal coding. Nature. 383, 76–81 (1996)

    Article  Google Scholar 

  • Ghahramani, Z., Roweis, S.: Learning nonlinear dynamical systems using an EM algorithm. In: Kearns, M.S., Solla, S.A., Cohn, D.A. (eds.) Advances in Neural Information Processing Systems, vol. 11, pp. 599–605. MIT Press, Cambridge, MA (1999)

    Google Scholar 

  • Graves, A., Wayne, G., Reynolds, M., Harley, T., Danihelka, I., et al.: Hybrid computing using a neural network with dynamic external memory. Nature. 538, 471–476 (2016)

    Article  Google Scholar 

  • Grün, S., Diesmann, M., Aertsen, A.: Unitary events in multiple single-neuron spiking activity: I. Detection and significance. Neural Comput. 14, 43–80 (2002a)

    Article  MATH  Google Scholar 

  • Grün, S., Diesmann, M., Aertsen, A.: Unitary events in multiple single-neuron spiking activity: II. Nonstationary data. Neural Comput. 14, 81–119 (2002b)

    Article  MATH  Google Scholar 

  • Gütig, R., Sompolinsky, H.: The tempotron: a neuron that learns spike timing-based decisions. Nat. Neurosci. 9, 420–428 (2006)

    Article  Google Scholar 

  • Hartig, F., Dormann, C.F.: Does model-free forecasting really outperform the true model? Proc. Natl. Acad. Sci. U S A. 110, E3975 (2013)

    Article  Google Scholar 

  • Hartig, F., Calabrese, J.M., Reineking, B., Wiegand, T., Huth, A.: Statistical inference for stochastic simulation models—Theory and application. Ecol. Lett. 14, 816–827 (2011)

    Article  Google Scholar 

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

    Google Scholar 

  • Hegger, R., Kantz, H., Schreiber, T.: Practical implementation of nonlinear time series methods: the TISEAN package. Chaos. 9, 413–435 (1999)

    Article  MATH  Google Scholar 

  • Hertäg, L., Hass, J., Golovko, T., Durstewitz, D.: An approximation to the adaptive exponential integrate-and-fire neuron model allows fast and predictive fitting to physiological data. Front. Comput. Neurosci. 6, 62 (2012)

    Article  Google Scholar 

  • Hertäg, L., Durstewitz, D., Brunel, N.: Analytical approximations of the firing rate of an adaptive exponential integrate-and-fire neuron in the presence of synaptic noise. Front. Comput. Neurosci. 8, 116 (2014)

    Article  Google Scholar 

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

    Google Scholar 

  • Hinton, G.E., Sejnowski, T.J.: Learning and relearning in Boltzmann machines. In: Rumelhart, D.E., McClelland, J.L. (eds.) Parallel Distributed Processing: Explorations in the Microstructure of Cognition Foundations, vol. 1. MIT Press, Cambridge, MA (1986)

    Google Scholar 

  • Holden, A.V., Ramadan, S.M.: Repetitive activity of a molluscan neurone driven by maintained currents: a supercritical bifurcation. Biol. Cybern. 42, 79–85 (1981)

    Article  MathSciNet  Google Scholar 

  • Hochreiter, S., Schmidhuber, J.: Long short-term memory. Neural Comput. 9, 1735–1780 (1997)

    Article  Google Scholar 

  • Hopfield, J.J.: Pattern recognition computation using action potential timing for stimulus representation. Nature. 376, 33–36 (1995)

    Article  Google Scholar 

  • Hopfield, J.J., Brody, C.D.: What is a moment? “Cortical” sensory integration over a brief interval. Proc. Natl. Acad. Sci. 97, 13919–13924 (2000)

    Article  Google Scholar 

  • Hopfield, J.J., Brody, C.D.: What is a moment? Transient synchrony as a collective mechanism for spatiotemporal integration. Proc. Natl. Acad. Sci. 98, 1282–1287 (2001)

    Article  Google Scholar 

  • Huk, A.C., Shadlen, M.N.: Neural activity in macaque parietal cortex reflects temporal integration of visual motion signals during perceptual decision making. J. Neurosci. 25, 10420–10436 (2005)

    Article  Google Scholar 

  • Humphries, M.D.: Spike-train communities: finding groups of similar spike trains. J. Neurosci. 31, 2321–2336 (2011)

    Article  Google Scholar 

  • Hurtado, J.M., Rubchinsky, L.L., Sigvardt, K.A.: Statistical method for detection of phase-locking episodes in neural oscillations. J. Neurophysiol. 91, 1883–1898 (2004)

    Article  Google Scholar 

  • Hutcheon, B., Yarom, Y.: Resonance, oscillation and the intrinsic frequency preferences of neurons. Trends Neurosci. 23, 216–222 (2000)

    Article  Google Scholar 

  • Huys, Q.J.M., Paninski, L.: Smoothing of, and parameter estimation from, noisy biophysical recordings. PLoS Comput. Biol. 5, e1000379 (2009)

    Article  MathSciNet  Google Scholar 

  • Hyman, J.M., Ma, L., Balaguer-Ballester, E., Durstewitz, D., Seamans, J.K.: Contextual encoding by ensembles of medial prefrontal cortex neurons. Proc. Natl. Acad. Sci. USA. 109(13), 5086–5091 (2012)

    Article  Google Scholar 

  • Izhikevich, E.M.: Dynamical Systems in Neuroscience. MIT Press, Cambridge, MA (2007)

    Google Scholar 

  • Jaeger, H., Haas, H.: Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication. Science. 304, 78–80 (2004)

    Article  Google Scholar 

  • Jahr, C.E., Stevens, C.F.: Voltage dependence of NMDA-activated macroscopic conductances predicted by single-channel kinetics. J. Neurosci. 10, 3178–3182 (1990)

    Google Scholar 

  • Jones, M.W., Wilson, M.A.: Theta rhythms coordinate hippocampal–prefrontal interactions in a spatial memory task. PLoS Biol. 3, e402 (2005)

    Article  Google Scholar 

  • Judd, K.: Failure of maximum likelihood methods for chaotic dynamical systems. Phys. Rev. E. Stat. Nonlin. Soft Matter Phys. 75, 036210 (2007)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  • Kass, R.E., Ventura, V., Brown, E.N.: Statistical issues in the analysis of neuronal data. J. Neurophysiol. 94, 8–25 (2005)

    Article  Google Scholar 

  • Kennel, M., Brown, R., Abarbanel, H.D.: Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys. Rev. A. 45, 3403–3411 (1992)

    Article  Google Scholar 

  • Kimura, M., Nakano, R.: Learning dynamical systems by recurrent neural networks from orbits. Neural Netw. 11, 1589–1599 (1998)

    Article  Google Scholar 

  • Koch, K.R.: Parameter Estimation and Hypothesis Testing in Linear Models. Springer Science & Business Media, Berlin (1999a)

    Book  MATH  Google Scholar 

  • Koch, C.: Biophysics of Computation: Information Processing in Single Neurons. Oxford University Press, New York (1999b)

    Google Scholar 

  • Komura, Y., Tamura, R., Uwano, T., Nishijo, H., Kaga, K., Ono, T.: Retrospective and prospective coding for predicted reward in the sensory thalamus. Nature. 412, 546–549 (2001)

    Article  Google Scholar 

  • Kostuk, M., Toth, B.A., Meliza, C.D., Margoliash, D., Abarbanel, H.D.: Dynamical estimation of neuron and network properties. II: Path integral Monte Carlo methods. Biol. Cybern. 106, 155–167 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Lankarany, M., Zhu, W.P., Swamy, M.N.S., Toyoizumi, T.: Inferring trial-to-trial excitatory and inhibitory synaptic inputs from membrane potential using Gaussian mixture Kalman filtering. Front. Comput. Neurosci. 7, 109 (2013)

    Article  Google Scholar 

  • Lapish, C.C., Balaguer-Ballester, E., Seamans, J.K., Phillips, A.G., Durstewitz, D.: Amphetamine exerts dose-dependent changes in prefrontal cortex attractor dynamics during working memory. J. Neurosci. 35, 10172–10187 (2015)

    Article  Google Scholar 

  • Lee, H., Simpson, G.V., Logothetis, N.K., Rainer, G.: Phase locking of single neuron activity to theta oscillations during working memory in monkey extrastriate visual cortex. Neuron. 45, 147–156 (2005)

    Article  Google Scholar 

  • Legenstein, R., Maass, W.: Edge of chaos and prediction of computational performance for neural circuit models. Neural Netw. 20, 323–334 (2007)

    Article  MATH  Google Scholar 

  • Li, K.: Approximation theory and recurrent networks. Proc. 1992 IJCNN. II, 266–271 (1992)

    Google Scholar 

  • Li, Z., Li, X.: Estimating temporal causal interaction between spike trains with permutation and transfer entropy. PLoS One. 8, e70894 (2013)

    Article  Google Scholar 

  • Lindner, M., Vicente, R., Priesemann, V., Wibral, M.: TRENTOOL: a Matlab open source toolbox to analyse information flow in time series data with transfer entropy. BMC Neurosci. 12, 119 (2011)

    Article  Google Scholar 

  • Lisman, J.E., Fellous, J.M., Wang, X.J.: A role for NMDA-receptor channels in working memory. Nat. Neurosci. 1, 273–275 (1998)

    Article  Google Scholar 

  • Liu, Z., Bai, L., Dai, R., Zhong, C., Wang, H., You, Y., Wei, W., Tian, J.: Exploring the effective connectivity of resting state networks in mild cognitive impairment: an fMRI study combining ICA and multivariate Granger causality analysis. Conf. Proc. IEEE Eng. Med. Biol. Soc. 2012, 5454–5457 (2012)

    Google Scholar 

  • London, M., Roth, A., Beeren, L., Häusser, M., Latham, P.E.: Sensitivity to perturbations in vivo implies high noise and suggests rate coding in cortex. Nature. 466, 123–127 (2010)

    Article  Google Scholar 

  • Maass, W., Natschläger, T., Markram, H.: Real-time computing without stable states: a new framework for neural computation based on perturbations. Neural Comput. 14, 2531–2560 (2002)

    Article  MATH  Google Scholar 

  • Machens, C.K., Romo, R., Brody, C.D.: Flexible control of mutual inhibition: a neural model of two-interval discrimination. Science. 307, 1121–1124 (2005)

    Article  Google Scholar 

  • Macke, J.H., Buesing, L., Sahani, M.: Estimating state and parameters in state space models of spike trains. In: Chen, Z. (ed.) Advanced State Space Methods for Neural and Clinical Data. Cambridge University Press, Cambridge (2015)

    Google Scholar 

  • Mader, W., Linke, Y., Mader, M., Sommerlade, L., Timmer, J., Schelter, B.: A numerically efficient implementation of the expectation maximization algorithm for state space models. Appl. Math. Comput. 241, 222–232 (2014)

    MathSciNet  MATH  Google Scholar 

  • Mandic, D.P., Chambers, J.A.: Recurrent Neural Networks for Prediction: Learning Algorithms, Architectures and Stability. Wiley, Chichester (2001)

    Book  Google Scholar 

  • Mante, V., Sussillo, D., Shenoy, K.V., Newsome, W.T.: Context-dependent computation by recurrent dynamics in prefrontal cortex. Nature. 503, 78–84 (2013)

    Article  Google Scholar 

  • Markram, H., Lübke, J., Frotscher, M., Sakmann, B.: Regulation of synaptic efficacy by coincidence of postsynaptic APs and EPSPs. Science. 275, 213–215 (1997)

    Article  Google Scholar 

  • May, R.M.: Simple mathematical models with very complicated dynamics. Nature. 261, 459–467 (1976)

    Article  MATH  Google Scholar 

  • Mazor, O., Laurent, G.: Transient dynamics versus fixed points in odor representations by locust antennal lobe projection neurons. Neuron. 48, 661–673 (2005)

    Article  Google Scholar 

  • Meyer-Lindenberg, A., Poline, J.B., Kohn, P.D., Holt, J.L., Egan, M.F., Weinberger, D.R., Berman, K.F.: Evidence for abnormal cortical functional connectivity during working memory in schizophrenia. Am. J. Psychiatry. 158, 1809–1817 (2001)

    Article  Google Scholar 

  • Miller, E.K., Erickson, C.A., Desimone, R.: Neural mechanisms of visual working memory in prefrontal cortex of the macaque. J. Neurosci. 16, 5154–5167 (1996)

    Google Scholar 

  • Mnih, V., Kavukcuoglu, K., Silver, D., Rusu, A.A., Veness, J., Bellemare, M.G., Graves, A., Riedmiller, M., Fidjeland, A.K., Ostrovski, G., Petersen, S., Beattie, C., Sadik, A., Antonoglou, I., King, H., Kumaran, D., Wierstra, D., Legg, S., Hassabis, D.: Human-level control through deep reinforcement learning. Nature. 518, 529–533 (2015)

    Article  Google Scholar 

  • Mongillo, G., Barak, O., Tsodyks, M.: Synaptic theory of working memory. Science. 319, 1543–1546 (2008)

    Article  Google Scholar 

  • Niessing, J., Friedrich, R.W.: Olfactory pattern classification by discrete neuronal network states. Nature. 465, 47–54 (2010)

    Article  Google Scholar 

  • Ostojic, S.: Interspike interval distributions of spiking neurons driven by fluctuating inputs. J. Neurophysiol. 106, 361–373 (2011)

    Article  Google Scholar 

  • Ostwald, D., Kirilina, E., Starke, L., Blankenburg, F.: A tutorial on variational Bayes for latent linear stochastic time-series models. J. Math. Psychol. 60, 1–19 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  • Paninski, L.: Maximum likelihood estimation of cascade point-process neural encoding models. Network. 15, 243–262 (2004)

    Article  Google Scholar 

  • Paninski, L., Vidne, M., DePasquale, B., Ferreira, D.G.: Inferring synaptic inputs given a noisy voltage trace via sequential Monte Carlo methods. J. Comput. Neurosci. 33, 1–19 (2012)

    Article  MathSciNet  Google Scholar 

  • Park, M., Bohner, G., Macke, J. (2016) Unlocking neural population non-stationarity using a hierarchical dynamics model. In: Advances in Neural Information Processing Systems 28, Twenty-Ninth Annual Conference on Neural Information Processing Systems (NIPS 2015), pp. 1–9

    Google Scholar 

  • Penny, W.D., Mattout, J., Trujillo-Barreto, N.: Chapter 35: Bayesian model selection and averaging. In: Friston, K., Ashburner, J., Kiebel, S., Nichols, T., Penny, W. (eds.) Statistical Parametric Mapping: The Analysis of Functional Brain Images. Elsevier, London (2006)

    Google Scholar 

  • Pearlmutter, B.A.: Learning state space trajectories in recurrent neural networks. Neural Comput. 1, 263–269 (1989)

    Article  Google Scholar 

  • Pearlmutter, B.A.: Dynamic recurrent neural networks. Technical Report CMU-CS-90-19, School of Computer Science, Carnegie Mellon University (1990)

    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 

  • Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  • Pozzorini, C., Mensi, S., Hagens, O., Naud, R., Koch, C., Gerstner, W.: Automated high-throughput characterization of single neurons by means of simplified spiking models. PLoS Comput. Biol. 11(6), e1004275 (2015)

    Article  Google Scholar 

  • Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes: The Art of Scientific Computing, 3rd edn. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  • Quintana, J., Fuster, J.M.: From perception to action: temporal integrative functions of prefrontal and parietal neurons. Cereb. Cortex. 9, 213–221 (1999)

    Article  Google Scholar 

  • Ratcliff, R., McKoon, G.: The diffusion decision model: theory and data for two-choice decision tasks. Neural Comput. 20, 873–922 (2008)

    Article  MATH  Google Scholar 

  • Rinzel, J., Ermentrout, B.: Analysis of neural excitability and oscillations. In: Koch, C., Segev, I. (eds.) Methods in Neuronal Modeling, pp. 251–292. MIT Press, Cambridge, MA (1998)

    Google Scholar 

  • Risken, H.: The Fokker-Planck Equation: Methods of Solution and Applications. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  • Roweis, S.T., Ghahramani, Z.: An EM algorithm for identification of nonlinear dynamical systems. In: Haykin, S. (ed.) Kalman Filtering and Neural Networks. http://citeseer.ist.psu.edu/306925.html (2001)

  • Russo, E., Durstewitz, D.: Cell assemblies at multiple time scales with arbitrary lag constellations. Elife. 6, e19428 (2017)

    Article  Google Scholar 

  • Sastry, P.S., Unnikrishnan, K.P.: Conditional probability-based significance tests for sequential patterns in multineuronal spike trains. Neural Comput. 22, 1025–1059 (2010)

    Article  MATH  Google Scholar 

  • Sauer, T.D.: Attractor reconstruction. Scholarpedia. 1(10), 1727 (2006)

    Article  Google Scholar 

  • Sauer, T.D., Sauer, K., Davies, D.G.: Embedology. J. Stat. Phys. 65, 579–616 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Schmidhuber, J.: Deep learning in neural networks: an overview. Neural Netw. 61, 85–117 (2015)

    Article  Google Scholar 

  • Schneidman, E., Berry, M.J., Segev, R., Bialek, W.: Weak pairwise correlations imply strongly correlated network states in a neural population. Nature. 440, 1007–1012 (2006)

    Article  Google Scholar 

  • Ratcliff, R.: A theory of memory retrieval. Psychol. Rev. 85, 59–108 (1978)

    Article  Google Scholar 

  • Shinomoto, S., Shima, K., Tanji, J.: Differences in spiking patterns among cortical neurons. Neural Comput. 15, 2823–2842 (2003)

    Article  MATH  Google Scholar 

  • Schreiber, T.: Measuring information transfer. Phys. Rev. Lett. 85, 461–464 (2000)

    Article  Google Scholar 

  • Seung, H.S., Lee, D.D., Reis, B.Y., Tank, D.W.: Stability of the memory of eye position in a recurrent network of conductance-based model neurons. Neuron. 26, 259–271 (2000)

    Article  Google Scholar 

  • Shadlen, M.N., Newsome, W.T.: The variable discharge of cortical neurons: implications for connectivity, computation, and information coding. J. Neurosci. 18, 3870–3896 (1998)

    Google Scholar 

  • Shimazaki, H., Amari, S.I., Brown, E.N., Grün, S.: State-space analysis of time-varying higher-order spike correlation for multiple neural spike train data. PLoS Comput. Biol. 8, e1002385 (2012)

    Article  MathSciNet  Google Scholar 

  • Singer, W., Gray, C.M.: Visual feature integration and the temporal correlation hypothesis. Annu. Rev. Neurosci. 18, 555–586 (1995)

    Article  Google Scholar 

  • Smith, A.C., Brown, E.N.: Estimating a state-space model from point process observations. Neural Comput. 15, 965–991 (2003)

    Article  MATH  Google Scholar 

  • Smith, A.C., Smith, P.: A set probability technique for detecting relative time order across multiple neurons. Neural Comput. 18, 1197–1214 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Smith, A.C., Nguyen, V.K., Karlsson, M.P., Frank, L.M., Smith, P.: Probability of repeating patterns in simultaneous neural data. Neural Comput. 22, 2522–2536 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Staude, B., Rotter, S., Grün, S.: CuBIC: cumulant based inference of higher-order correlations in massively parallel spike trains. J. Comput. Neurosci. 29, 327–350 (2009)

    Article  MathSciNet  Google Scholar 

  • Staude, B., Grün, S., Rotter, S.: Higher-order correlations in non-stationary parallel spike trains: statistical modeling and inference. Front. Comput. Neurosci. 4, 16 (2010)

    Article  Google Scholar 

  • Stein, R.: A theoretical analysis of neuronal variability. Biophys. J. 5, 173–194 (1965)

    Article  Google Scholar 

  • Stiefel, K.M., Englitz, B., Sejnowski, T.J.: Origin of intrinsic irregular firing in cortical interneurons. Proc. Natl. Acad. Sci. U S A. 110, 7886–7891 (2013)

    Article  MATH  Google Scholar 

  • Stopfer, M., Bhagavan, S., Smith, B.H., Laurent, G.: Impaired odour discrimination on desynchronization of odour-encoding neural assemblies. Nature. 390, 70–74 (1997)

    Google Scholar 

  • Strogatz, S.H.: Nonlinear Dynamics and Chaos. Addison-Wesley, Reading, MA (1994)

    Google Scholar 

  • Sugihara, G., May, R., Ye, H., Hsieh, C.H., Deyle, E., Fogarty, M., Munch, S.: Detecting causality in complex ecosystems. Science. 338, 496–500 (2012)

    Article  MATH  Google Scholar 

  • Sussillo, D., Abbott, L.F.: Generating coherent patterns of activity from chaotic neural networks. Neuron. 63, 544–557 (2009)

    Article  Google Scholar 

  • Takens, F.: Detecting strange attractors in turbulence. Lecture Notes in Mathematics 898, pp. 366–381. Springer, Berlin (1981)

    Google Scholar 

  • Terman, D.: The transition from bursting to continuous spiking in excitable membrane models. J. Nonlinear Sci. 2, 135–182 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Toth, B.A., Kostuk, M., Meliza, C.D., Margoliash, D., Abarbanel, H.D.: Dynamical estimation of neuron and network properties I: Variational methods. Biol. Cybern. 105, 217–237 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Tsodyks, M.: Attractor neural networks and spatial maps in hippocampus. Neuron. 48, 168–169 (2005)

    Article  Google Scholar 

  • Turner, B.M., Van Zandt, T.: A tutorial on approximate Bayesian computation. J. Math. Psychol. 56, 69–85 (2012)

    Article  MathSciNet  MATH  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 

  • Wackerly, D., Mendenhall, W., Scheaffer, R.: Mathematical statistics with applications. Cengage Learning. (2008)

    Google Scholar 

  • Walter, E., Pronzato, L.: On the identifiability and distinguishability of nonlinear parametric models. Math. Comput. Simul. 42, 125–134 (1996)

    Article  MATH  Google Scholar 

  • Wang, X.J.: Probabilistic decision making by slow reverberation in cortical circuits. Neuron. 36, 955–968 (2002)

    Article  Google Scholar 

  • Wibral, M., Rahm, B., Rieder, M., Lindner, M., Vicente, R., Kaiser, J.: Transfer entropy in magnetoencephalographic data: quantifying information flow in cortical and cerebellar networks. Prog. Biophys. Mol. Biol. 105, 80–97 (2011)

    Article  Google Scholar 

  • Williams, R.J., Zipser, D.: A learning algorithm for continually running fully recurrent neural networks. Neural Comput. 1, 256–263 (1990)

    Google Scholar 

  • Wills, T.J., Lever, C., Cacucci, F., Burgess, N., O’Keefe, J.: Attractor dynamics in the hippocampal representation of the local environment. Science. 308, 873–876 (2005)

    Article  Google Scholar 

  • Wilson, H.: Spikes, Decisions, and Actions: The Dynamical Foundations of Neuroscience. Oxford University Press, Oxford (1999)

    MATH  Google Scholar 

  • Wilson, H.R., Cowan, J.D.: Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12, 1–24 (1972)

    Article  Google Scholar 

  • Wilson, H.R., Cowan, J.D.: A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik. 13(2), 55–80 (1973)

    Article  MATH  Google Scholar 

  • Wood, S.N.: Statistical inference for noisy nonlinear ecological dynamic systems. Nature. 466, 1102–1104 (2010)

    Article  Google Scholar 

  • Wu, C.F.J.: On the convergence properties of the EM algorithm. Ann. Stat. 11, 95–103 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  • Yu, B.M., Afshar, A., Santhanam, G., Ryu, S.I., Shenoy, K.V.: Extracting dynamical structure embedded in neural activity. Adv. Neural Inf. Process. Syst. 18, 1545–1552 (2005)

    Google Scholar 

  • Zipser, D., Kehoe, B., Littlewort, G., Fuster, J.: A spiking network model of short-term active memory. J. Neurosci. 13, 3406–3420 (1993)

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Daniel Durstewitz

Authors
  1. Daniel Durstewitz
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Durstewitz, D. (2017). Time Series from a Nonlinear Dynamical Systems Perspective. In: Advanced Data Analysis in Neuroscience. Bernstein Series in Computational Neuroscience. Springer, Cham. https://doi.org/10.1007/978-3-319-59976-2_9

Download citation

Publish with us

Policies and ethics

Search

Navigation