Journal of Computational Neuroscience

, Volume 32, Issue 3, pp 521–538 | Cite as

Detecting effective connectivity in networks of coupled neuronal oscillators

  • Erin R. Boykin
  • Pramod P. Khargonekar
  • Paul R. Carney
  • William O. Ogle
  • Sachin S. Talathi


The application of data-driven time series analysis techniques such as Granger causality, partial directed coherence and phase dynamics modeling to estimate effective connectivity in brain networks has recently gained significant prominence in the neuroscience community. While these techniques have been useful in determining causal interactions among different regions of brain networks, a thorough analysis of the comparative accuracy and robustness of these methods in identifying patterns of effective connectivity among brain networks is still lacking. In this paper, we systematically address this issue within the context of simple networks of coupled spiking neurons. Specifically, we develop a method to assess the ability of various effective connectivity measures to accurately determine the true effective connectivity of a given neuronal network. Our method is based on decision tree classifiers which are trained using several time series features that can be observed solely from experimentally recorded data. We show that the classifiers constructed in this work provide a general framework for determining whether a particular effective connectivity measure is likely to produce incorrect results when applied to a dataset.


Granger causality Partial directed coherence Phase dynamics modeling Effective connectivity Neuronal oscillators Decision tree classifiers 



This work was partially funded from the start-up funding to SST through the Dept of Pediatrics at UF and the startup funding to WOO and alumina fellowship to ERB through the Dept of Biomedical Eng at UF. SST and PRC were partially funded through the Children’s Miracle Network Funds and Eve and B. Wilder Center of Excellence for Epilepsy Research. PPK was partially funded through the Eckis Professor Endowment at the University of Florida. We appreciate constructive feedback from Dr. Alex Cadotte. We also acknowledge the generosity of Dr CJ Frazier for allowing access to his electrophysiology rig set up and Ms. Aishwarya Parthasarthy for assistance in setting up the dynamic clamp and collection of experimental datasets. We finally acknowledge assistance from Mr. Kyungpyo Hong in generating Fig. 2 of the manuscript.


  1. Abarbanel, H., Gibb, L., Huerta, R., & Rabinovich, M. (2003). Biophysical model of synaptic plasticity dynamics. Biological Cybernetics, 89, 214.PubMedCrossRefGoogle Scholar
  2. Akaike, H. (1969). Fitting autoregressive models for prediction. Annals of the institute of Statistical Mathematics, 21, 243.CrossRefGoogle Scholar
  3. Astolfi, L., Cincotti, F., Mattia, D., Marciani, M., Baccala, L., Fallani, F., et al. (2007). Comparison of different cortical connectivity estimators for high-resolution eeg recordings. Human Brain Mapping, 28, 143.PubMedCrossRefGoogle Scholar
  4. Baccala, L., & Sameshima, K. (2001). Partial directed coherence: a new concept in neural structure determination. Biological Cybernetics, 84(6), 463–474.PubMedCrossRefGoogle Scholar
  5. Baccala, L., Sameshima, K., Ballester, G., Valle, A. D., & Timo-Iaria, C. (1998). Studying the interaction between brain structures via directed coherence and granger causality. Applied Signal Processing, 5(1), 40.CrossRefGoogle Scholar
  6. Baccala, L., Sameshima, K., & Takahashi, D. (2007). Generalized partial directed coherence. In Proceedings of the 15th international conference on digital signal processing, Cardiff, Wales, UK (pp. 163–166).Google Scholar
  7. Balenzuela, P., & García-Ojalvo, J. (2005). Role of chemical synapses in coupled neurons with noise. Physical Review. E, Statistical, Nonlinear and Soft Matter Physics, 72(2 Pt 1), 021901.CrossRefGoogle Scholar
  8. Bennett, M. (1997). Gap junctions as electrical synapses. Journal of Neurocytology, 26, 249.CrossRefGoogle Scholar
  9. Bezruchko, B., Ponomarenko, V., Rosenblum, M., & Pikovsky, A. (2003). Characterizing direction of coupling from experimental observations. Chaos: An Interdisciplinary Journal of Nonlinear Science, 13, 179.CrossRefGoogle Scholar
  10. Blair, R., & Karniski, W. (1993). An alternative method for significance testing of waveform difference potentials. Psychophysiology, 30, 518.PubMedCrossRefGoogle Scholar
  11. Box, G., Jenkins, G., & Reinsel, G. (2008). Time series analysis: Forecasting and control (4th ed.). Hoboken: Wiley.Google Scholar
  12. Breiman, L., Friedman, J., Stone, C., & Olshen, R. (1984). Classification and regression trees. New York: Chapman and Hall.Google Scholar
  13. Brockwell, P., & Davis, R. (1991). Time series: Theory and methods. New York: Springer.CrossRefGoogle Scholar
  14. Brovelli, A., Ding, M., Ledberg, A., Chen, Y., Nakamura, R., & Bressler, S. (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(26), 9849–9854.PubMedCrossRefGoogle Scholar
  15. Buzsaki, G. (2006). Rhythms of the brain. Oxford: Oxford University Press.CrossRefGoogle Scholar
  16. Buzsaki, G., & Draguhn, A. (2004). Neuronal oscillations in cortical networks. Science, 304(5679), 1926–1929.PubMedCrossRefGoogle Scholar
  17. Cadotte, A., DeMarse, T., Mareci, T., Parekh, M., Talathi, S., Hwang, D. U., et al. (2010). Granger causality relationships between local field potentials in an animal model of temporal lobe epilepsy. Journal of Neuroscience Methods, 189, 121–129.PubMedCrossRefGoogle Scholar
  18. Chen, Y., Bressler, S., & Ding, M. (2006). Frequency decomposition of conditional granger causality and application to multivariate neural field potential data. Journal of Neuroscience Methods, 150(2), 228.PubMedCrossRefGoogle Scholar
  19. Chow, C., & Kopell, N. (2000). Dynamics of spiking neurons with electrical coupling. Neural Computation, 12(7), 1643.PubMedCrossRefGoogle Scholar
  20. Davidson, R., & MacKinnon, J. (2003). Econometric theory and methods. Oxford: Oxford University Press.Google Scholar
  21. Deister, C., Teagarden, M., Wilson, C., & Paladini, C. (2009). An intrinsic neuronal oscillator underlies dopaminergic neuron bursting. Journal of Neuroscience, 29, 15888–15897.PubMedCrossRefGoogle Scholar
  22. Ding, M., Chen, Y., & Bressler, S. (2006). Granger causality: basic theory and application to neuroscience. In B. Schelter, M. Winterhalder, & J. Timmer (Eds.), Handbook of time series analysis (p. 451). Weinheim: Wiley-VCH.Google Scholar
  23. Dorval, A. D., Christini, D. J., & White, J. A. (2001). Real-time linux dynamic clamp: A fast and flexible way to construct virtual ion channels in living cells. Annals of Biomedical Engineering, 29(10), 897–907.PubMedCrossRefGoogle Scholar
  24. Ermentrout, B. (1996). Type 1 membranes, phase resetting curves, and synchrony. Neural Computation, 8(5), 979–1001.PubMedCrossRefGoogle Scholar
  25. Fanselow, E. E., Sameshima, K., Baccala, L. A., & Nicolelis, M. A. (2001). Thalamic bursting in rats during different awake behavioral states. Proceedings of the National Academy of Sciences of the United States of America, 98(26), 15330–15335. doi: 10.1073/pnas.261273898.PubMedCrossRefGoogle Scholar
  26. Fries, P. (2009). Neuronal gamma-band synchronization as a fundamental process in cortical computation. Annual Review of Neuroscience, 32, 209–224.PubMedCrossRefGoogle Scholar
  27. Friston, K. (2002). Functional integration and inference in the brain. Progress in Neurobiology, 68(2), 113–143.PubMedCrossRefGoogle Scholar
  28. Geweke, J. (1982). Measurement of linear dependence and feedback between multiple time series. Journal of the American Statistical Association, 77(378), 304.Google Scholar
  29. Geweke, J. (1984). Measures of conditional linear dependence and feedback between time series. Journal of the American Statistical Association, 79(388), 907–915.Google Scholar
  30. Granger, C. (1969). Investigating causal relations by econometric models and cross-spectral methods. Econometrica: Journal of the Econometric Society, 37(3), 424.CrossRefGoogle Scholar
  31. Granger, C. (1980). Testing for causality: A personal viewpoint. Journal of Economic Dynamics & Control, 2, 329–352.CrossRefGoogle Scholar
  32. Hammond, C., Bergman, H., & Brown, P. (2007). Pathological synchronization in Parkinson’s disease: networks, models and treatments. Trends in Neuroscience, 30(7), 357–364.CrossRefGoogle Scholar
  33. Havlicek, M., Jan, J., Brazdil, M., & Calhoun, V. (2010). Dynamic granger causality based on kalman filter for evaluation of functional network connectivity in FMRI data. NeuroImage, 53, 65–77.PubMedCrossRefGoogle Scholar
  34. Izhikevich, E. (2007). Dynamical systems in neuroscience: The geometry of excitability and bursting. Cambridge: MIT Press.Google Scholar
  35. Kaminski, M., & Blinowska, K. (1991). A new method of the description of the information flow in the brain structures. Biological Cybernetics, 65, 203–210.PubMedCrossRefGoogle Scholar
  36. Kamiński, M., Ding, M., Truccolo, W., & Bressler, S. (2001). Evaluating causal relations in neural systems: Granger causality, directed transfer function and statistical assessment of significance. Biological Cybernetics, 85(2), 145.PubMedCrossRefGoogle Scholar
  37. Kayser, A., Sun, F., & D’Esposito, M. (2009). A comparison of granger causality and coherency in FMRI-based analysis of the motor system. Human Brain Mapping, 30(11), 3475.PubMedCrossRefGoogle Scholar
  38. Lecar, H. (2007). Morris–lecar model. Scholarpedia, 2, 1333.
  39. Liao, W., Mantini, D., Zhang, Z., Pan, Z., Ding, J., Gong, Q., et al. (2010). Evaluating the effective connectivity of resting state networks using conditional granger causality. Biological Cybernetics, 102(1), 57–69.PubMedCrossRefGoogle Scholar
  40. Lindsly, C., & Frazier, C. J. (2010). Two distinct and activity-dependent mechanisms contribute to autoreceptor-mediated inhibition of gabaergic afferents to hilar mossy cells. Journal of Physiology, 588(Pt 15), 2801–22. doi: 10.1113/jphysiol.2009.184648.PubMedCrossRefGoogle Scholar
  41. Lungarella, M., Ishiguro, K., Kuniyoshi, Y., & Otsu, N. (2007). Methods for quantifying the causal structure of bivariate time series. International Journal of Bifurcation and Chaos, 17, 903–921.CrossRefGoogle Scholar
  42. Lütkepohl, H. (2010). New introduction to multiple time series analysis. New York: Springer.Google Scholar
  43. MacKinnon, J. (2006), Bootstrap methods in econometrics. Economic Record, 82, S2.CrossRefGoogle Scholar
  44. Mormann, F., Lehnertz, K., David, P., & Elger, C. (2000). Mean phase coherence as a measure for phase synchronization and its application to the eeg of epilepsy patients. Physica D: Nonlinear Phenomena, 144, 358.CrossRefGoogle Scholar
  45. Morris, C., & Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophysical Journal, 35(1), 193–213.PubMedCrossRefGoogle Scholar
  46. Nedungadi, A. G., Rangarajan, G., Jain, N., & Ding, M. (2009). Analyzing multiple spike trains with nonparametric granger causality. Journal of Computational Neuroscience, 27(1), 55–64. doi: 10.1007/s10827-008-0126-2.PubMedCrossRefGoogle Scholar
  47. Perkel, D., Gerstein, G., & Moore, G. (1967). Neuronal spike trains and stochastic point processes: I. the single spike train. Biophysical Journal, 7(4), 391–418.PubMedCrossRefGoogle Scholar
  48. Pikovsky, A., Rosenblum, M., & Kurths, J. (2002). Synchronization: A universal concept in nonlinear sciences. Cambridge: Cambridge University Press.Google Scholar
  49. Rosenblum, M., & Pikovsky, A. (2001). Detecting direction of coupling in interacting oscillators. Physical Review E, 64(4), 45202.CrossRefGoogle Scholar
  50. Sato, J., Takahashi, D., Arcuri, S., Sameshima, K., Morettin, P., & Baccalá, L. (2009). Frequency domain connectivity identification: An application of partial directed coherence in FMRI. Human Brain Mapping, 30(2), 452.PubMedCrossRefGoogle Scholar
  51. Schaffer, C. (1993). Selecting a classification method by cross-validation. Machine Learning, 13, 135–143.Google Scholar
  52. Schelter, B., Winterhalder, M., Eichler, M., Peifer, M., Hellwig, B., Guschlbauer, B., et al. (2006). Testing for directed influences among neural signals using partial directed coherence. Journal of Neuroscience Methods, 152(1–2), 210–219.PubMedCrossRefGoogle Scholar
  53. Schneider, T., & Neumaier, A. (2001). Algorithm 808: Arfit—a matlab package for estimation and spectral decomposition of multivariate autoregressive models. ACM Transactions on Mathematical Software, 27, 58–65.CrossRefGoogle Scholar
  54. Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6, 461–464.CrossRefGoogle Scholar
  55. Smirnov, D., & Andrzejak, R. (2005). Detection of weak directional coupling: Phase-dynamics approach versus state-space approach. Physical Review E, 71(3), 36207.CrossRefGoogle Scholar
  56. Smirnov, D., & Bezruchko, B. (2009). Detection of couplings in ensembles of stochastic oscillators. Physical Review E, 79(4), 046204.CrossRefGoogle Scholar
  57. Smirnov, D., Schelter, B., Winterhalder, M., & Timmer, J. (2007). Revealing direction of coupling between neuronal oscillators from time series: Phase dynamics modeling versus partial directed coherence. Chaos: An Interdisciplinary Journal of Nonlinear Science, 17, 013111.CrossRefGoogle Scholar
  58. Somers, D., & Kopell, N. (1993). Rapid synchronization through fast threshold modulation. Biological Cybernetics, 68, 393.PubMedCrossRefGoogle Scholar
  59. Sporns, O. (2010). Networks of the brain. Cambridge: MIT Press.Google Scholar
  60. Talathi, S., Hwang, D. U., Carney, P., & Ditto, W. (2010). Synchrony with shunting inhibition in a feedforward inhibitory network. Journal of Computational Neuroscience, 28, 305.PubMedCrossRefGoogle Scholar
  61. Uhlhaas, P., & Singer, W. (2010). Abnormal neural oscillations and synchrony in schizophrenia. Nature Reviews. Neuroscience, 11, 100–113.PubMedCrossRefGoogle Scholar
  62. Wang, X. J., & Buzsáki, G. (1996). Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. Journal of Neuroscience, 16(20), 6402–6413.PubMedGoogle Scholar
  63. Ward, L. (2003). Synchronous neural oscillations and cognitive processes. Trends in Cognitive Science, 7(12), 553–559.CrossRefGoogle Scholar
  64. Winterhalder, M., Schelter, B., Hesse, W., Schwab, K., Leistritz, L., Klan, D., et al. (2005). Comparison of linear signal process techniques to infer directed interactions in multivariate neural systems. Signal Processing, 85(11), 2137–2160.CrossRefGoogle Scholar
  65. Winterhalder, M., Schelter, B., & Timmer, J. (2007). Detecting coupling directions in multivariate oscillatory systems. International Journal of Bifurcation and Chaos, 17, 3725–3739.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Erin R. Boykin
    • 1
  • Pramod P. Khargonekar
    • 1
  • Paul R. Carney
    • 3
  • William O. Ogle
    • 2
  • Sachin S. Talathi
    • 4
  1. 1.Department of Electrical and Computer EngineeringUniversity of FloridaGainesvilleUSA
  2. 2.J. Crayton Pruitt Family Department of Biomedical EngineeringUniversity of FloridaGainesvilleUSA
  3. 3.Department of Pediatrics, Neurology, Neuroscience, and Biomedical EngineeringUniversity of FloridaGainesvilleUSA
  4. 4.Department of Pediatrics, Neuroscience, and Biomedical EngineeringUniversity of FloridaGainesvilleUSA

Personalised recommendations