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Analyzing multiple spike trains with nonparametric granger causality

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Abstract

Simultaneous recordings of spike trains from multiple single neurons are becoming commonplace. Understanding the interaction patterns among these spike trains remains a key research area. A question of interest is the evaluation of information flow between neurons through the analysis of whether one spike train exerts causal influence on another. For continuous-valued time series data, Granger causality has proven an effective method for this purpose. However, the basis for Granger causality estimation is autoregressive data modeling, which is not directly applicable to spike trains. Various filtering options distort the properties of spike trains as point processes. Here we propose a new nonparametric approach to estimate Granger causality directly from the Fourier transforms of spike train data. We validate the method on synthetic spike trains generated by model networks of neurons with known connectivity patterns and then apply it to neurons simultaneously recorded from the thalamus and the primary somatosensory cortex of a squirrel monkey undergoing tactile stimulation.

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References

  • Albo, Z., Di Prisco, G. V., Chen, Y., Rangarajan, G., Truccolo, W., Feng, J., et al. (2004). Is partial coherence a viable technique for identifying generators of neural oscillations? Biological Cybernetics, 90, 318–326. doi:10.1007/s00422-004-0475-5.

    Article  PubMed  Google Scholar 

  • Baccala, L. A., & Sameshima, K. (2001). Partial directed coherence: A new concept in neural structure determination. Biological Cybernetics, 84, 463–474. doi:10.1007/PL00007990.

    Article  PubMed  CAS  Google Scholar 

  • Bartlett, M. S. (1963). The spectral analysis of point processes. Journal of the Royal Statistical Society B, 25, 264–280.

    Google Scholar 

  • Bernasconi, C., & Konig, P. (1999). On the directionality of cortical interactions studied by structural analysis of electrophysiological recordings. Biological Cybernetics, 81, 199–210. doi:10.1007/s004220050556.

    Article  PubMed  CAS  Google Scholar 

  • Bernasconi, C., von Stein, A., Chiang, C., & Konig, P. (2000). Bi-directional interactions between visual areas in the awake behaving cat. Neuroreport, 11, 689–692.

    Article  PubMed  CAS  Google Scholar 

  • Bollimunta, A., Chen, Y., Schroeder, C. E., & Ding, M. (2008). Neuronal mechanisms of cortical alpha oscillations in awake-behaving macaques. The Journal of Neuroscience, 28, 9976–9988.

    Article  PubMed  CAS  Google Scholar 

  • Boudjellaba, H., Dufour, J., & Roy, R. (1992). Testing causality between two vectors in multivariate autoregressive moving average models. Journal of the American Statistical Association, 87, 1082–1090. doi:10.2307/2290645.

    Article  Google Scholar 

  • Brillinger, D. R. (1992). Nerve-cell spike train data-analysis—A progression of technique. Journal of the American Statistical Association, 87, 260–271. doi:10.2307/2290256.

    Article  Google Scholar 

  • Brovelli, A., Ding, M., Ledberg, A., Chen, Y. H., Nakamura, R., & Bressler, S. L. (2004). Beta oscillations in a large-scale sensorimotor cortical network: Directional influences revealed by Granger causality. Proceedings of the National Academy of Sciences of the United States of America, 101, 9849–9854. doi:10.1073/pnas.0308538101.

    Article  PubMed  CAS  Google Scholar 

  • Brown, E. N., Kass, R. E., & Mitra, P. P. (2004). Multiple neural spike train data analysis: state-of-the-art and future challenges. Nature Neuroscience, 7, 456–461. doi:10.1038/nn1228.

    Article  PubMed  CAS  Google Scholar 

  • Chen, Y. H., Rangarajan, G., Feng, J. F., & Ding, M. Z. (2004). Analyzing multiple nonlinear time series with extended Granger causality. Physics Letters Part A, 324, 26–35. doi:10.1016/j.physleta.2004.02.032.

    Article  CAS  Google Scholar 

  • Chen, Y., Bressler, S. L., & Ding, M. (2006a). Stochastic modeling of neurobiological time series: Power, coherence, Granger causality, and separation of evoked responses from ongoing activity. Chaos (Woodbury, N.Y.), 16, 026113. doi:10.1063/1.2208455.

    Article  Google Scholar 

  • Chen, Y. H., Bressler, S. L., & Ding, M. (2006b). Frequency decomposition of conditional Granger causality and application to multivariate neural field potential data. Journal of Neuroscience Methods, 150, 228–237. doi:10.1016/j.jneumeth.2005.06.011.

    Article  PubMed  Google Scholar 

  • Conway, B. A., Halliday, D. M., & Rosenberg, J. R. (1993). Detection of weak synaptic interactions between single Ia-afferent and motor-unit spike trains in decerebrate cats. The Journal of Physiology, 471, 379–409.

    PubMed  CAS  Google Scholar 

  • Dahlhaus, R., Eichler, M., & Sandkuhler, K. (1997). Identification of synaptic connections in neural ensembles by graphical methods. Journal of Neuroscience Methods, 77, 93–107. doi:10.1016/S0165-0270(97)00100-3.

    Article  PubMed  CAS  Google Scholar 

  • Daley, D., & Vere-Jones, D. (2003). An introduction to the theory of point processes, vol. I: Elementary theory and methods. New York: Springer.

    Google Scholar 

  • Daley, D., & Vere-Jones, D. (2007). An introduction to the theory of point processes, vol. II: General theory and structure. New York: Springer.

    Google Scholar 

  • Dhamala, M., Rangarajan, G., & Ding, M. (2008a). Estimating Granger causality from Fourier and wavelet transforms of time series data. Physical Review Letters, 100, 018701.1–018701.4.

    Article  CAS  Google Scholar 

  • Dhamala, M., Rangarajan, G., & Ding, M. (2008b). Analyzing information flow in brain networks with nonparametric Granger causality. NeuroImage, 41, 354–362. doi:10.1016/j.neuroimage.2008.02.020.

    Article  PubMed  Google Scholar 

  • Ding, M., Bressler, S. L., Yang, W., & Liang, H. (2000). Short-window spectral analysis of cortical event-related potentials by adaptive multivariate autoregressive modeling: data preprocessing, model validation, and variability assessment. Biological Cybernetics, 83, 35–45. doi:10.1007/s004229900137.

    Article  PubMed  CAS  Google Scholar 

  • Ding, M., Chen, Y., & Bressler, S. L. (2006). Granger causality: Basic theory and applications to neuroscience. In B. Schelter, M. Winterhalder, & J. Timmer (Eds.), Handbook of Time Series Analysis (pp. 437–460). Weinheim: Wiley-VCH.

    Chapter  Google Scholar 

  • Fanselow, E. E., Sameshima, K., Baccala, L. A., & Nicolelis, M. A. L. (2001). Thalamic bursting in rats during different awake behavioral states. Proceedings of the National Academy of Sciences of the United States of America, 98, 15330–15335. doi:10.1073/pnas.261273898.

    Article  PubMed  CAS  Google Scholar 

  • Geweke, J. (1982). Measurement of linear dependence and feedback between multiple time series. Journal of the American Statistical Association, 77, 304–313. doi:10.2307/2287238.

    Article  Google Scholar 

  • Geweke, J. (1984). Measures of conditional linear dependence and feedback between time series. Journal of the American Statistical Association, 79, 907–915. doi:10.2307/2288723.

    Article  Google Scholar 

  • Goebel, R., Roebroeck, A., Kim, D. S., & Formisano, E. (2003). Investigating directed cortical interactions in time-resolved fMRI data using vector autoregressive modeling and Granger causality mapping. Magnetic Resonance Imaging, 21, 1251–1261. doi:10.1016/j.mri.2003.08.026.

    Article  PubMed  Google Scholar 

  • Granger, C. (1969). Investigating causal relations by econometric models and cross-spectral methods. Econornetrica, 37, 424–438. doi:10.2307/1912791.

    Article  Google Scholar 

  • Halliday, D. M., Rosenberg, J. R., Amjad, A. M., Breeze, P., Conway, B. A., & Farmer, S. F. (1995). A framework for the analysis of mixed time series/point process data-theory and application to the study of physiological tremor, single motor unit discharges and electromylograms. Progress in Biophysics and Molecular Biology, 64, 237–238. doi:10.1016/S0079-6107(96)00009-0.

    Article  PubMed  CAS  Google Scholar 

  • Hesse, W., Moller, E., Arnold, M., & Schack, B. (2003). The use of time-variant EEG Granger causality for inspecting directed interdependencies of neural assemblies. Journal of Neuroscience Methods, 124, 27–44. doi:10.1016/S0165-0270(02)00366-7.

    Article  PubMed  Google Scholar 

  • Izhikevich, E. M. (2003). Simple model of spiking neurons. IEEE Transactions on Neural Networks, 14, 1569–1572. doi:10.1109/TNN.2003.820440.

    Article  PubMed  CAS  Google Scholar 

  • Jain, N., Qi, H.-X., & Kaas, J. H. (2001). Long-term chronic multichannel recordings from sensorimotor cortex and thalamus of primates. Prog Brain Res, 130, 63–72. doi:10.1016/S0079-6123(01)30006-7.

    Article  Google Scholar 

  • Jarvis, M. R., & Mitra, P. P. (2001). Sampling properties of the spectrum and coherency of sequences of action potentials. Neural Computation, 13, 717–749. doi:10.1162/089976601300014312.

    Article  PubMed  CAS  Google Scholar 

  • Kaas, J. H., Jain, N., & Qi, H.-X. (2002). The organization of the somatosensory system in primates. In R. J. Nelson (Ed.), The somatosensory system-deciphering the brain’s own body image (pp. 1–25). New York: CRC.

    Google Scholar 

  • Kaminski, M., Ding, M., Truccolo, W. A., & Bressler, S. L. (2001). Evaluating causal relations in neural systems: Granger causality, directed transfer function and statistical assessment of significance. Biological Cybernetics, 85, 145–157. doi:10.1007/s004220000235.

    Article  PubMed  CAS  Google Scholar 

  • Kus, R., Kaminski, M., & Blinowska, K. J. (2004). Determination of EEG activity propagation: Pair-wise versus multichannel estimate. IEEE Transactions on Bio-Medical Engineering, 51, 1501–1510. doi:10.1109/TBME.2004.827929.

    Article  PubMed  Google Scholar 

  • Lungarella, M., & Sporns, O. (2006). Mapping information flow in sensorimotor networks. PLoS Computational Biology, 2, 1301–1312.

    Article  CAS  Google Scholar 

  • Mitra, P. P., & Pesaran, B. (1999). Analysis of dynamic brain imaging data. Biophysical Journal, 76, 691–708.

    Article  PubMed  CAS  Google Scholar 

  • Nowak, L. G., & Bullier, J. (2000). Crosscorrelograms for neuronal spike trains. Different types of temporal correlation in neocortex, their origin and significance. In R. Miller (Ed.), Time and the Brain: Conceptual Advances in Brain Research (pp. 53–96). Amsterdam: Harwood Academic.

    Google Scholar 

  • Okatan, M., Wilson, M. A., & Brown, E. N. (2005). Analyzing functional connectivity using a network likelihood model of ensemble neural spiking activity. Neural Computation, 17, 1927–1961. doi:10.1162/0899766054322973.

    Article  PubMed  Google Scholar 

  • Pierce, D. A. (1979). R2 measures for time series. Journal of the American Statistical Association, 74, 901–910. doi:10.2307/2286421.

    Article  Google Scholar 

  • Roebroeck, A., Formisano, E., & Goebel, R. (2005). Mapping directed influence over the brain using Granger causality and fMRI. NeuroImage, 25, 230–242. doi:10.1016/j.neuroimage.2004.11.017.

    Article  PubMed  Google Scholar 

  • Rosenberg, J. R., Amjad, A. M., Breeze, P., Brillinger, D. R., & Halliday, D. M. (1989). The Fourier approach to the identification of functional coupling between neuronal spike trains. Progress in Biophysics and Molecular Biology, 53, 1–31. doi:10.1016/0079-6107(89)90004-7.

    Article  PubMed  CAS  Google Scholar 

  • Rosenberg, J. R., Halliday, D. M., Breeze, P., & Conway, B. A. (1998). Identification of patterns of neuronal connectivity-partial spectra, partial coherence, and neuronal interactions. Journal of Neuroscience Methods, 83, 57–72. doi:10.1016/S0165-0270(98)00061-2.

    Article  PubMed  CAS  Google Scholar 

  • Salazar, R. F., Konig, P., & Kayser, C. (2004). Directed interactions between visual areas and their role in processing image structure and expectancy. The European Journal of Neuroscience, 20, 1391–1401. doi:10.1111/j.1460-9568.2004.03579.x.

    Article  PubMed  Google Scholar 

  • Sameshima, K., & Baccala, L. A. (1999). Using partial directed coherence to describe neuronal ensemble interactions. Journal of Neuroscience Methods, 94, 93–103. doi:10.1016/S0165-0270(99)00128-4.

    Article  PubMed  CAS  Google Scholar 

  • Seth, A. K. (2005). Causal connectivity of evolved neural networks during behavior. Network-Computation in Neural Systems, 16, 35–54. doi:10.1080/09548980500238756.

    Article  Google Scholar 

  • Tang, A., Jackson, D., Hobbs, J., Chen, W., Smith, J. L., Patel, H., et al. (2008). A maximum entropy model applied to spatial and temporal correlations from cortical networks in Vitro. The Journal of Neuroscience, 28(2), 505–518. doi:10.1523/JNEUROSCI.3359-07.2008.

    Article  PubMed  CAS  Google Scholar 

  • Thomson, D. J. (1982). Spectrum estimation and harmonic analysis. Proceedings of the IEEE, 70, 1055–1096. doi:10.1109/PROC.1982.12433.

    Article  Google Scholar 

  • Truccolo, W., Eden, U. T., Fellows, M. R., Donoghue, J. P., & Brown, E. N. (2005). A point process framework for relating neural spiking activity to spiking history, neural ensemble, and extrinsic covariate effects. Journal of Neurophysiology, 93, 1074–1089. doi:10.1152/jn.00697.2004.

    Article  PubMed  Google Scholar 

  • Walden, A. T. (2000). A unified view of multitaper multivariate spectral estimation. Biometrika, 87, 767–788. doi:10.1093/biomet/87.4.767.

    Article  Google Scholar 

  • Wilson, G. T. (1972). The factorization of matricial spectral densities. SIAM Journal on Applied Mathematics, 23, 420–426. doi:10.1137/0123044.

    Article  Google Scholar 

  • Wu, J. H., Liu, X. G., & Feng, J. F. (2008). Detecting causality between different frequencies. Journal of Neuroscience Methods, 167, 367–375. doi:10.1016/j.jneumeth.2007.08.022.

    Article  PubMed  Google Scholar 

  • Zhang, Y., Chen, Y., Bressler, S. L., & Ding, M. (2008). Response preparation and inhibition: The role of the cortical sensorimotor beta rhythm. Neuroscience, 156, 238–246. doi:10.1016/j.neuroscience.2008.06.061.

    Article  PubMed  CAS  Google Scholar 

  • Zhu, L., Lai, Y. C., Hoppensteadt, F. C., & He, J. (2003). Probing changes in neural interaction during adaptation. Neural Computation, 15, 2359–2377. doi:10.1162/089976603322362392.

    Article  PubMed  Google Scholar 

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Acknowledgements

AN was supported by Council of Scientific and Industrial Research. GR was supported by research grants from Defence Research and Development Organization (DRDO), Department of Science and Technology (MS:419/07) and University Grants Commission (under SAP-Phase IV). GR is also associated with the Jawaharlal Nehru Centre for Advanced Scientific Research as an Honorary Faculty Member. NJ is a Wellcome Trust International Senior Research Fellow (Grant No. 063259/Z/00/Z). NJ is also supported by a grant from DRDO. MD was supported by the US National Institute of Mental Health under Grants MH071620, MH070498, and MH079388. The Matlab code used for this study is available to interested readers upon request.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Science, Bangalore, 560 012, India

    Aatira G. Nedungadi & Govindan Rangarajan

  2. National Brain Research Centre, Manesar, Haryana, 122050, India

    Neeraj Jain

  3. J. Crayton Pruitt Family Department of Biomedical Engineering, University of Florida, Gainesville, FL, 32611, USA

    Mingzhou Ding

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  1. Aatira G. Nedungadi
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  2. Govindan Rangarajan
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  3. Neeraj Jain
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  4. Mingzhou Ding
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Correspondence to Govindan Rangarajan.

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Nedungadi, A.G., Rangarajan, G., Jain, N. et al. Analyzing multiple spike trains with nonparametric granger causality. J Comput Neurosci 27, 55–64 (2009). https://doi.org/10.1007/s10827-008-0126-2

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