Skip to main content

Advertisement

SpringerLink
Log in

Time series analysis of hybrid neurophysiological data and application of mutual information

  • Published:
Journal of Computational Neuroscience Aims and scope Submit manuscript

Abstract

Multivariate time series data of which some components are continuous time series and the rest are point processes are called hybrid data. Such data sets routinely arise while working with neuroscience data, EEG and spike trains would perhaps be the most obvious example. In this paper, we discuss the modeling of a hybrid time series, with the continuous component being the physiological tremors in the distal phalanx of the middle finger, and motor unit firings in the middle finger portion of the extensor digitorum communis (EDC) muscle. We employ a model for the two components based on Auto-regressive Moving Average (ARMA) type models. Another major issue to arise in the modeling of such data is to assess the goodness of fit. We suggest a visual procedure based on mutual information towards assessing the dependence pattern of hybrid data. The goodness of fit is also verified by standard model fitting diagnostic techniques for univariate data.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Andrieu, C., Davy, M., & Doucet, A. (2003). Efficient particle filtering for jump Markov systems: application to time-varying autoregressions. IEEE Transactions on Signal Processing, 51, 1762–1770.

    Article  Google Scholar 

  • Antos, A., & Kontoyiannis, Y. (2001). Convergence properties of functional estimates for discrete distributions. Random Structures & Algorithms, 19, 163–193.

    Article  Google Scholar 

  • Atkinson, A. C., & Biswas, A. (2005). Bayesian adaptive biased-coin designs for clinical trials with normal responses. Biometrics, 61, 118–125.

    Article  PubMed  Google Scholar 

  • Barbieri R, Quirk M. C., Frank L. M., Wilson M. A., & Brown E. N. (2001). Construction and analysis of non-Poisson stimulus response models of neural spike train activity. Journal of Neuroscience Methods, 105, 25–37.

    Article  CAS  PubMed  Google Scholar 

  • Basharin, G. P. (1959). On a statistical estimate for the entropy of a sequence of independent random variables. Theory of Probability and its Applications, 4, 333–336.

    Article  Google Scholar 

  • Berry, M. J., Warland, D. K., & Meister, M. (1997). The structure and precision of retinal spike trains. Proceedings of the National Academy of Science of the United States of America, 94, 5411–5416.

    Article  CAS  Google Scholar 

  • Biswas, A., & Guha, A. (2007). Time series analysis of categorical data using auto-mutual information. Indian Statistical Institute Technical Report Number ASD/2007/01.

  • Box, G. E. P., & Jenkins, G. M. (1976). Time series analysis: Forecasting and control (revised ed.). Holden-Day, San Francisco.

    Google Scholar 

  • Brillinger, D. R. (2004). Some data analysis using mutual information. Brazilian Journal of Probability and Statistics, 18, 163–183.

    Google Scholar 

  • Brillinger, D. R., & Guha, A. (2007). Mutual information in the frequency domain. Journal of Statistical Planning and Inference, 137, 1076–1084.

    Article  Google Scholar 

  • Brown, E. N., Frank, L. M., Tang, D., Quirk, M. C., & Wilson, M. A. (1998). A statistical paradigm for neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal lace cells. Journal of Neuroscience, 18, 7411–7425.

    CAS  PubMed  Google Scholar 

  • Brown, E. N., Barbieri, R., Ventura, V., Kass, R. E., & Frank, L. M. (1998). The time rescaling theorem and its application to neural spike train data analysis. Neural Computation, 14, 325–346.

    Article  Google Scholar 

  • Chen Y. H., Bressler S. L., Knuth K. H., Truccolo W. A., & Ding M. Z. (2006). Stochastic modeling of neurobiological time series: Power, coherence, Granger causality, and separation of evoked responses from ongoing activity . Chaos, 16(026113).

  • Cover, T., & Thomas, J. (1991). Elements of information theory. Wiley, New York.

    Book  Google Scholar 

  • Davis, G. M., & Ensor, K. B. (2007). Multivariate time series analysis with categorical and continuous variables in an LSTR model. Journal of Time Series Analysis, 28, 867–885.

    Article  Google Scholar 

  • Elble, R. J. (1986). Physiologic and essential tremor. Neurology, 36, 225–231.

    Article  CAS  PubMed  Google Scholar 

  • Gardner, G, Harvey, A. C., & Phillips, G. D. A. (1980). An algorithm for exact maximum likelihood estimation of autoregressive-moving average models by means of Kalman filtering. Applied Statistics, 29, 311–322.

    Article  Google Scholar 

  • Granger, C., & Lin, J. L. (1994). Using the mutual information coefficient to identify lags in nonlinear models. Journal of Time Series Analysis, 15(4), 371–384.

    Article  Google Scholar 

  • Guha, A. (2005). Analysis of dependence structures of hybrid stochastic processes using mutual information. Ph.D. thesis, University of California, Berkeley.

  • Guha, A., & Biswas, A. (2008). An overview of modeling techniques for hybrid brain data. Statistica Sinica, 18(2008), 1311–1340.

    Google Scholar 

  • Halliday, D. M., Conway, B. A., Farmer, S. F., & Rosenberg, J. R. (1999). Load-independent contributions from motor-unit synchronization to human physiological tremor. Journal of Neurophysiology, 82, 664–675.

    CAS  PubMed  Google Scholar 

  • Jacobs, P. A., & Lewis, P. A. W. (1978). Discrete time series generated by mixtures. I: conditional and runs properties. Journal of the Royal Statistical Society, Series B, 40, 94–105.

    Google Scholar 

  • Jacobs, P. A., & Lewis, P. A. W. (1983). Stationary discrete autoregressive-moving average time series generated by mixtures. Journal of Time Series Analysis, 4, 19–36.

    Article  Google Scholar 

  • Joe, H. (1996). Time series models with univariate margins in the convolution-closed infinitely divisible class. Journal of Applied Probability, 33, 664–677.

    Article  Google Scholar 

  • Johnson, D. H., & Swami, A. (1983). The transmission of signals by auditory-nerve fiber discharge patterns. Journal of the Acoustical Society of America, 74, 493–501.

    Article  CAS  PubMed  Google Scholar 

  • Jørgensen, B., Lundbye-Christensen, S., Song, P. X., & Sun, L. (1996). State-space model for multivariate longitudinal data of mixed types. The Canadian Journal of Statistics, 24, 385–402.

    Article  Google Scholar 

  • Jørgensen, B., & Song, P. X.-K. (1998). Stationary time-series models with exponential dispersion model margins. Journal of Applied Probability, 35, 78–92.

    Article  Google Scholar 

  • Jung, R. C., & Tremayne, A. R. (2006). Coherent forecasting in integer time series models. International Journal of Forecasting, 22, 223–238.

    Article  Google Scholar 

  • Kanter, M. (1975). Autoregression for discrete processes mod 2. Journal of Applied Probability, 12, 371–375.

    Article  Google Scholar 

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

    Article  PubMed  Google Scholar 

  • Kedem, B., & Fokianos, K. (2002). Regression models for time series analysis. Wiley, New York.

    Book  Google Scholar 

  • Kiang, N. Y. S., Watanabe, T., Thomas, E. C. & Clark, L. F. (1965). Discharge patterns of single fibers in the cat’s auditory nerve. MIT, Cambridge.

    Google Scholar 

  • Li, W. (1990). Mutual information function versus correlation functions. Journal of Statistical Physics, 60, 823–837.

    Article  Google Scholar 

  • Li, W. (1994). Time series models based on generalized linear models: some further results. Biometrics, 50, 506–511.

    Article  CAS  PubMed  Google Scholar 

  • Mars, N. J. I., & van Arragon, G. W. (1982). Time delay estimation in non-linear systems using average amount of mutual information analysis. Signal Processing, 4, 139–153.

    Article  Google Scholar 

  • Moon, Y. I., Rajagopalan, B., & Lall, U. (1995). Estimation of mutual information using kernel density estimators. Physical Review E, 52, 2318–2321.

    Article  CAS  Google Scholar 

  • Moddemeijer, R. (1989a). Delay estimation with application to electroencephalograms in epilepsy. Ph.D. thesis, Universiteit Twente, Enschede.

  • Moddemeijer, R. (1989b). On estimation of entropy and mutual information of continuous distributions. Signal Processing, 16, 233–248.

    Article  Google Scholar 

  • Pegram, G. G. S. (1980). An autoregressive model for multilag Markov chains. Journal of Applied Probability, 17, 350–362.

    Article  Google Scholar 

  • Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27, 379–423, 623–656.

    Article  Google Scholar 

  • Steutel, F. W. & van Harn, K. (1979). Discrete analogues of self-decomposability and stability. Annals of probability, 7, 893–899.

    Article  Google Scholar 

  • Stiles, R. N., (1980). Mechanical and neural feedback factors in postural hand tremor of normal subjects. Journal of Neurophysiology, 44, 40–59.

    CAS  PubMed  Google Scholar 

  • Strong, S. P., Koberle, R., de Ruyter van Steveninck, R. R., & Bialek, W. (1998). Entropy and information in neural spike trains. Physical Review Letters, 80, 197–200.

    Article  CAS  Google Scholar 

  • Ventura, V., Carta, R., Kass, R. E., Olson, C. R., & Gettner, S. N. (2001). Statistical analysis of temporal evolution in single-neuron firing rates. Biostatistics, 1, 1–20.

    Google Scholar 

  • Waagepetersen, R. (2006). A simulation-based goodness-of-fit test for random effects in generalized linear mixed models. Scandinavian Journal of Statistics, 33, 721–731.

    Article  Google Scholar 

  • Willie, J. (1979). Analyzing relationships between a time series and a point process. Ph.D. dissertation, Department of Statistics, University of California, Berkeley.

Download references

Acknowledgements

Professor J. R. Rosenberg kindly supplied the data. The authors also wish to thank an Associate Editor and two anonymous referees for their careful reading and some constructive suggestions which led considerable improvement over an earlier version of the manuscript.

Author information

Authors and Affiliations

  1. Applied Statistics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata, 700 108, India

    Atanu Biswas

  2. School of Mathematics, University of Birmingham, Watson Building, Birmingham, B15 2TT, UK

    Apratim Guha

Authors
  1. Atanu Biswas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Apratim Guha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Apratim Guha.

Additional information

Action Editor: Rob Kass

Appendix

Appendix

The final fitted model for the discrete part was

$$\begin{array}{rll}{\mathit{\Phi}}^{-1}(s(t+8))&=&-10.32+ 0.11S_{t}- 0.0007S_{t}^2 +5.5X^0_t\\[-6pt] &&\;\;(0.76)\;\;\;(0.007)\;\;\;(8\times 10^{-5})\;\;\;(1.44)\\ &&+\,0.282X^0_{t-4}+ 7.267X^0_{t-9}+ 7.035X^0_{t-19}\\[-6pt] &&\;\;\;\;\;(1.45)\;\;\;\;\;\;\;\;\;\;\;(1.36)\;\;\;\;\;\;\;\;\;\;\;\;(1.21)\\ &&+\,3.126X^0_{t-29}-0.933X^0_{t-49},\\[-6pt] &&\;\;\;\;(1.03)\;\;\;\;\;\;\;\;\;\;\;\;(0.84) \end{array}$$

The final model for the continuous part is given by

$$\begin{array}{rll} X^0_t&=&1.80X^0_{t-1} - 0.87X^0_{t-2} + \epsilon_t+0.03\epsilon_{t-1} \\[-6pt] &&(0.004)\;\;\;\;(0.003)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(0.006)\\ &&-\, 0.43\epsilon_{t-2}+0.11\epsilon_{t-3}+ 0.05\epsilon_{t-4}- 0.16\epsilon_{t-6}\\[-6pt] &&\;\;\;(0.006)\;\;\;\;(0.005)\;\;\;\;\;\;\;(0.005)\;\;\;\;\;(0.005) \\ &&+\,0.09\epsilon_{t-7}+0.11\epsilon_{t-8}.\\[-6pt] &&\;\;\;(0.005)\;\;\;\;\;(0.005) \end{array}$$

Finally, \(V(\epsilon_t)=\sigma^2\) was estimated as 0.02893.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Biswas, A., Guha, A. Time series analysis of hybrid neurophysiological data and application of mutual information. J Comput Neurosci 29, 35–47 (2010). https://doi.org/10.1007/s10827-009-0165-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10827-009-0165-3

Keywords

Search

Navigation