Recent years have seen the emergence of microelectrode arrays and optical methods allowing simultaneous recording of spiking activity from populations of neurons in various parts of the nervous system. The analysis of multiple neural spike train data could benefit significantly from existing methods for multivariate time-series analysis which have proven to be very powerful in the modeling and analysis of continuous neural signals like EEG signals. However, those methods have not generally been well adapted to point processes. Here, we use our recent results on correlation distortions in multivariate Linear-Nonlinear-Poisson spiking neuron models to derive generalized Yule-Walker-type equations for fitting ‘‘hidden’’ Multivariate Autoregressive models. We use this new framework to perform Granger causality analysis in order to extract the directed information flow pattern in networks of simulated spiking neurons. We discuss the relative merits and limitations of the new method. 1. Introduction The analysis of multivariate neurophysiological signals at the cellular (spike trains) and population scales (EEG/MEG, LFP, and ECOG) has developed almost independently, largely due to the mathematical differences between continuous and point-process signals. The analysis of multiple neural spike train data [1] has gained tremendous relevance recently with the advent and widespread application of arrays of microelectrodes in both basic and applied Neurosciences. Furthermore, emerging optical methods for network activity imaging [2] and control [3] are likely to further compound this growth. Currently, the analysis of multichannel spike trains is still largely limited to single-channel analyses, to bivariate cross-correlation and metric-space analyses [4], and to spike train filtering (“decoding”). In contrast, much of EEG/MEG time series analysis has revolved around linear and nonlinear models and analyses that are essentially multivariate, most prominently the multivariate autoregressive (MVAR) model. The MVAR framework is associated with a powerful set of time- and frequency-domain statistical tools for inferring directional and causal information flow based on Granger’s framework [5], including linear and nonlinear Granger causality, directed transfer function, directed coherence, and partial directed coherence (see [6–8] for reviews). Scattered attempts at applying this general framework to neural spike trains have relied on smoothing the spike trains to obtain a continuous process that can be fit with an MVAR model [9–12]. This approach has the clear
References
[1]
E. N. Brown, R. E. Kass, and P. P. Mitra, “Multiple neural spike train data analysis: state-of-the-art and future challenges,” Nature Neuroscience, vol. 7, no. 5, pp. 456–461, 2004.
[2]
W. G？bel and F. Helmchen, “In vivo calcium imaging of neural network function,” Physiology, vol. 22, no. 6, pp. 358–365, 2007.
[3]
S. Shoham, D. H. O'Connor, D. V. Sarkisov, and S. S.-H. Wang, “Rapid neurotransmitter uncaging in spatially defined patterns,” Nature Methods, vol. 2, no. 11, pp. 837–843, 2005.
[4]
J. D. Victor, “Spike train metrics,” Current Opinion in Neurobiology, vol. 15, no. 5, pp. 585–592, 2005.
[5]
C. W. J. Granger, “Investigating causal relations by econometric models and cross-spectral methods,” Econometrica, vol. 37, no. 3, pp. 424–438, 1969.
[6]
E. Pereda, R. Q. Quiroga, and J. Bhattacharya, “Nonlinear multivariate analysis of neurophysiological signals,” Progress in Neurobiology, vol. 77, no. 1-2, pp. 1–37, 2005.
[7]
B. Gourévitch, R. L. Bouquin-Jeannès, and G. Faucon, “Linear and nonlinear causality between signals: methods, examples and neurophysiological applications,” Biological Cybernetics, vol. 95, no. 4, pp. 349–369, 2006.
[8]
M. Ding, Y. Chen, and S. L. Bressler, “Granger causality: basic theory and application to neuroscience,” in Handbook of Time Series Analysis, B. Schelter, M. Winterhalder, and J. Timmer, Eds., pp. 437–460, Wiley, Weinheim, Germany, 2007.
[9]
K. Sameshima and L. A. Baccalá, “Using partial directed coherence to describe neuronal ensemble interactions,” Journal of Neuroscience Methods, vol. 94, no. 1, pp. 93–103, 1999.
[10]
E. E. Fanselow, K. Sameshima, L. A. Baccala, and M. A. L. Nicolelis, “Thalamic bursting in rats during different awake behavioral states,” Proceedings of the National Academy of Sciences of the United States of America, vol. 98, no. 26, pp. 15330–15335, 2001.
[11]
M. Kamiński, M. Ding, W. A. Truccolo, and S. L. Bressler, “Evaluating causal relations in neural systems: granger causality, directed transfer function and statistical assessment of significance,” Biological Cybernetics, vol. 85, no. 2, pp. 145–157, 2001.
[12]
L. Zhu, Y.-C. Lai, F. C. Hoppensteadt, and J. He, “Probing changes in neural interaction during adaptation,” Neural Computation, vol. 15, no. 10, pp. 2359–2377, 2003.
[13]
A. G. Nedungadi, G. Rangarajan, N. Jain, and M. Ding, “Analyzing multiple spike trains with nonparametric granger causality,” Journal of Computational Neuroscience, vol. 27, no. 1, pp. 55–64, 2009.
[14]
M. Krumin and S. Shoham, “Generation of spike trains with controlled auto- and cross-correlation functions,” Neural Computation, vol. 21, no. 6, pp. 1642–1664, 2009.
[15]
M. Krumin, A. Shimron, and S. Shoham, “Correlation-distortion based identification of linear-nonlinear-poisson models,” Journal of Computational Neuroscience. In press.
[16]
N. Farah, I. Reutsky, and S. Shoham, “Patterned optical activation of retinal ganglion cells,” in Proceedings of the 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC '07), pp. 6368–6370, Cité Internationale, Lyon, France, August 2007.
[17]
E. M. Izhikevich, “Simple model of spiking neurons,” IEEE Transactions on Neural Networks, vol. 14, no. 6, pp. 1569–1572, 2003.
[18]
M. L. Hines, T. Morse, M. Migliore, N. T. Carnevale, and G. M. Shepherd, “ModelDB: a database to support computational neuroscience,” Journal of Computational Neuroscience, vol. 17, no. 1, pp. 7–11, 2004.
[19]
E. M. Izhikevich, “Polychronization: computation with spikes,” Neural Computation, vol. 18, no. 2, pp. 245–282, 2006.
[20]
C. K. Knox, “Signal transmission in random spike trains with applications to the statocyst neurons of the lobster,” Biological Cybernetics, vol. 7, no. 5, pp. 167–174, 1970.
[21]
P. Whittle, “On the fitting of multivariate autoregressions, and the approximate canonical factorization of a spectral density matrix,” Biometrika, vol. 50, no. 1-2, pp. 129–134, 1963.
[22]
G. N. Borisyuk, R. M. Borisyuk, A. B. Kirillov, E. I. Kovalenko, and V. I. Kryukov, “A new statistical method for identifying interconnections between neuronal network elements,” Biological Cybernetics, vol. 52, no. 5, pp. 301–306, 1985.
[23]
H. van den Boogaard, “Maximum likelihood estimations in a nonlinear self-exciting point process model,” Biological Cybernetics, vol. 55, no. 4, pp. 219–225, 1986.
[24]
E. S. Chornoboy, L. P. Schramm, and A. F. Karr, “Maximum likelihood identification of neural point process systems,” Biological Cybernetics, vol. 59, no. 4-5, pp. 265–275, 1988.
[25]
X. Yang and S. A. Shamma, “Identification of connectivity in neural networks,” Biophysical Journal, vol. 57, no. 5, pp. 987–999, 1990.
[26]
D. R. Brillinger, “Nerve cell spike train data analysis: a progression of technique,” Journal of the American Statistical Association, vol. 87, no. 418, pp. 260–271, 1992.
[27]
M. Okatan, M. A. Wilson, and E. N. Brown, “Analyzing functional connectivity using a network likelihood model of ensemble neural spiking activity,” Neural Computation, vol. 17, no. 9, pp. 1927–1961, 2005.
[28]
L. Martignon, G. Deco, K. Laskey, M. Diamond, W. Freiwald, and E. Vaadia, “Neural coding: higher-order temporal patterns in the neurostatistics of cell assemblies,” Neural Computation, vol. 12, no. 11, pp. 2621–2653, 2000.
[29]
H. Nakahara and S.-I. Amari, “Information-geometric measure for neural spikes,” Neural Computation, vol. 14, no. 10, pp. 2269–2316, 2002.
[30]
R. Gütig, A. Aertsen, and S. Rotter, “Analysis of higher-order neuronal interactions based on conditional inference,” Biological Cybernetics, vol. 88, no. 5, pp. 352–359, 2003.
[31]
S. Schrader, S. Grün, M. Diesmann, and G. L. Gerstein, “Detecting synfire chain activity using massively parallel spike train recording,” Journal of Neurophysiology, vol. 100, no. 4, pp. 2165–2176, 2008.
[32]
J. D. Victor and K. P. Purpura, “Metric-space analysis of spike trains: theory, algorithms and application,” Network: Computation in Neural Systems, vol. 8, no. 2, pp. 127–164, 1997.
[33]
M. C. W. van Rossum, “A novel spike distance,” Neural Computation, vol. 13, no. 4, pp. 751–763, 2001.
[34]
S. Schreiber, J. M. Fellous, D. Whitmer, P. Tiesinga, and T. J. Sejnowski, “A new correlation-based measure of spike timing reliability,” Neurocomputing, vol. 52–54, pp. 925–931, 2003.
[35]
J. D. Hunter and J. G. Milton, “Amplitude and frequency dependence of spike timing: implications for dynamic regulation,” Journal of Neurophysiology, vol. 90, no. 1, pp. 387–394, 2003.
[36]
J. Dauwels, F. Vialatte, T. Weber, and A. Cichocki, “Quantifying statistical interdependence by message passing on graphs—part I: one-dimensional point processes,” Neural Computation, vol. 21, no. 8, pp. 2152–2202, 2009.
[37]
L. Paninski, S. Shoham, M. R. Fellows, N. G. Hatsopoulos, and J. P. Donoghue, “Superlinear population encoding of dynamic hand trajectory in primary motor cortex,” Journal of Neuroscience, vol. 24, no. 39, pp. 8551–8561, 2004.
[38]
W. Truccolo, U. T. Eden, M. R. Fellows, J. P. Donoghue, and E. N. Brown, “A point process framework for relating neural spiking activity to spiking history, neural ensemble, and extrinsic covariate effects,” Journal of Neurophysiology, vol. 93, no. 2, pp. 1074–1089, 2005.
[39]
J. W. Pillow, J. Shlens, L. Paninski, et al., “Spatio-temporal correlations and visual signalling in a complete neuronal population,” Nature, vol. 454, no. 7207, pp. 995–999, 2008.
[40]
L. Astolfi, F. Cincotti, D. Mattia, et al., “Comparison of different cortical connectivity estimators for high-resolution EEG recordings,” Human Brain Mapping, vol. 28, no. 2, pp. 143–157, 2007.
[41]
J. H. Macke, P. Berens, A. S. Ecker, A. S. Tolias, and M. Bethge, “Generating spike trains with specified correlation coefficients,” Neural Computation, vol. 21, no. 2, pp. 397–423, 2009.
[42]
T. Tchumatchenko, A. Malyshev, T. Geisel, M. Volgushev, and F. Wolf, “Correlations and synchrony in threshold neuron models,” Physical Review Letters, vol. 104, no. 5, Article ID 058102, 4 pages, 2010.
