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Estimating the directed information to infer causal relationships in ensemble neural spike train recordings

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Abstract

Advances in recording technologies have given neuroscience researchers access to large amounts of data, in particular, simultaneous, individual recordings of large groups of neurons in different parts of the brain. A variety of quantitative techniques have been utilized to analyze the spiking activities of the neurons to elucidate the functional connectivity of the recorded neurons. In the past, researchers have used correlative measures. More recently, to better capture the dynamic, complex relationships present in the data, neuroscientists have employed causal measures—most of which are variants of Granger causality—with limited success. This paper motivates the directed information, an information and control theoretic concept, as a modality-independent embodiment of Granger’s original notion of causality. Key properties include: (a) it is nonzero if and only if one process causally influences another, and (b) its specific value can be interpreted as the strength of a causal relationship. We next describe how the causally conditioned directed information between two processes given knowledge of others provides a network version of causality: it is nonzero if and only if, in the presence of the present and past of other processes, one process causally influences another. This notion is shown to be able to differentiate between true direct causal influences, common inputs, and cascade effects in more two processes. We next describe a procedure to estimate the directed information on neural spike trains using point process generalized linear models, maximum likelihood estimation and information-theoretic model order selection. We demonstrate that on a simulated network of neurons, it (a) correctly identifies all pairwise causal relationships and (b) correctly identifies network causal relationships. This procedure is then used to analyze ensemble spike train recordings in primary motor cortex of an awake monkey while performing target reaching tasks, uncovering causal relationships whose directionality are consistent with predictions made from the wave propagation of simultaneously recorded local field potentials.

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References

  • Abler, B., Roebroeck, A., Goebel, R., Höse, A., Schönfeldt-Lecuona, C., Hole, G., et al. (2006). Investigating directed influences between activated brain areas in a motor-response task using fMRI. Magnetic Resonance Imaging, 24(2), 181–185.

    Article  PubMed  Google Scholar 

  • Akaike, H. (1976). An information criterion (AIC). Mathematical Scientist, 14(153), 5–9.

    Google Scholar 

  • Al-khassaweneh, M., & Aviyente, S. (2008). The relationship between two directed information measures. IEEE Signal Processing Letters, 15, 801–804.

    Article  Google Scholar 

  • Amblard, P., & Michel, O. (2010). On directed information theory and Granger causality graphs. Arxiv preprint. arXiv:1002.1446.

  • Barron, A., & Cover, T. (1991). Minimum complexity density estimation. IEEE Transactions on Information Theory, 37(4), 1034–1054.

    Article  Google Scholar 

  • Bitan, T., Booth, J., Choy, J., Burman, D., Gitelman, D., & Mesulam, M. (2005). Shifts of effective connectivity within a language network during rhyming and spelling. Journal of Neuroscience, 25(22), 5397.

    Article  CAS  PubMed  Google Scholar 

  • Bremaud, P. (1981). Point processes and queues: martingale dynamics. New York: Springer.

    Google Scholar 

  • 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.

    Article  CAS  PubMed  Google Scholar 

  • Brown, E., Barbieri, R., Eden, U., & Frank, L. (2003). Likelihood methods for neural spike train data analysis. In Computational neuroscience: A comprehensive approach.

  • Brown, E., Barbieri, R., Ventura, V., Kass, R., & Frank, L. (2002). The time-rescaling theorem and its application to neural spike train data analysis. Neural Computation, 14(2), 325–346.

    Article  PubMed  Google Scholar 

  • Cai, H., Kulkarni, S., & Verdú, S. (2004). Universal entropy estimation via block sorting. IEEE Transactions on Information Theory, 50(7), 1551–1561.

    Article  Google Scholar 

  • Cai, H., Kulkarni, S., & Verdu, S. (2006). An algorithm for universal lossless compression with side information. IEEE Transactions on Information Theory, 52(9), 4008–4016.

    Article  Google Scholar 

  • Casella, G., Berger, R., & Berger, R. (2002). Statistical inference. Pacific Grove: Duxbury.

    Google Scholar 

  • Cesa-Bianchi, N., & Lugosi, G. (2006). Prediction, learning, and games. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Chávez, M., Martinerie, J., & Le Van Quyen, M. (2003). Statistical assessment of nonlinear causality: Application to epileptic EEG signals. Journal of Neuroscience Methods, 124(2), 113–128.

    Article  PubMed  Google Scholar 

  • Cover, T., & Thomas, J. (2006). Elements of information theory. New York: Wiley-Interscience.

    Google Scholar 

  • Daley, D., & Vere-Jones, D. (1988). An introduction to the theory of point processes. New York: Springer.

    Google Scholar 

  • David, O., Kiebel, S., Harrison, L., Mattout, J., Kilner, J., & Friston, K. (2006). Dynamic causal modeling of evoked responses in EEG and MEG. NeuroImage, 30(4), 1255–1272.

    Article  PubMed  Google Scholar 

  • De Boer, P., Kroese, D., Mannor, S., & Rubinstein, R. (2005). A tutorial on the cross-entropy method. Annals of Operations Research, 134(1), 19–67.

    Article  Google Scholar 

  • Dhamala, M., Rangarajan, G., & Ding, M. (2008). Analyzing information flow in brain networks with nonparametric Granger causality. NeuroImage, 41(2), 354–362.

    Article  PubMed  Google Scholar 

  • Diekman, C. O., Sastry, P., & Unnikrishnan, K. (2009). Statistical significance of sequential firing patterns in multi-neuronal spike trains. Journal of Neuroscience Methods, 182(2), 279–284.

    Article  PubMed  Google Scholar 

  • Du, X., Ghosh, B., & Ulinski, P. (2005). Encoding and decoding target locations with waves in the turtle visual cortex. IEEE Transactions on Biomedical Engineering, 52(4), 566–577.

    Article  PubMed  Google Scholar 

  • Eguiluz, V., Chialvo, D., Cecchi, G., Baliki, M., & Apkarian, A. (2005). Scale-free brain functional networks. Physical Review Letters, 94(1), 018102.

    Article  Google Scholar 

  • Elia, N. (2004). When bode meets Shannon: Control-oriented feedback communication schemes. IEEE Transactions on Automatic Control, 49(9), 1477–1488.

    Article  Google Scholar 

  • Ermentrout, G., & Kleinfeld, D. (2001). Traveling electrical waves in cortex insights from phase dynamics and speculation on a computational role. Neuron, 29(1), 33–44.

    Article  CAS  PubMed  Google Scholar 

  • Friston, K., Harrison, L., & Penny, W. (2003). Dynamic causal modelling. NeuroImage, 19(4), 1273–1302.

    Article  CAS  PubMed  Google Scholar 

  • Goebel, R., Roebroeck, A., Kim, D., & Formisano, E. (2003). Investigating directed cortical interactions in time-resolved fMRI data using vector autoregressive modeling and Granger causality mapping. Magnetic Resonance Imaging, 21(10), 1251–1261.

    Article  PubMed  Google Scholar 

  • Gorantla, S., & Coleman, T. (2010). On reversible Markov chains and maximization of directed information. In IEEE international symposium on information theory (ISIT), Austin, TX (in press).

  • Gourevitch, B., & Eggermont, J. (2007). Evaluating information transfer between auditory cortical neurons. Journal of Neurophysiology, 97(3), 2533.

    Article  PubMed  Google Scholar 

  • Granger, C. (1969). Investigating causal relations by econometric models and cross-spectral methods. Econometrica, 37(3), 424–438.

    Article  Google Scholar 

  • Grefkes, C., Eickhoff, S., Nowak, D., Dafotakis, M., & Fink, G. (2008). Dynamic intra-and interhemispheric interactions during unilateral and bilateral hand movements assessed with fMRI and DCM. NeuroImage, 41(4), 1382–1394.

    Article  PubMed  Google Scholar 

  • Grünwald, P., & Rissanen, J. (2007). The minimum description length principle. Cambridge: MIT.

    Google Scholar 

  • Hamandi, K., Powell, H., Laufs, H., Symms, M., Barker, G., Parker, G., et al. (2008). Combined EEG-fMRI and tractography to visualise propagation of epileptic activity. British Medical Journal, 79(5), 594–597.

    CAS  Google Scholar 

  • Hesse, W., Möller, 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(1), 27–44.

    Article  PubMed  Google Scholar 

  • Hu, J., Fu, M., & Marcus, S. (2007). A model reference adaptive search method for global optimization. Operations Research, 55(3), 549–568.

    Article  Google Scholar 

  • Iyengar, S., & Liao, Q. (1997). Modeling neural activity using the generalized inverse Gaussian distribution. Biological Cybernetics, 77(4), 289–295.

    Article  CAS  PubMed  Google Scholar 

  • Kaminski, M., & Blinowska, K. (1991). A new method of the description of the information flow in the brain structures. Biological Cybernetics, 65(3), 203–210.

    Article  CAS  PubMed  Google Scholar 

  • 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), 145–157.

    Article  PubMed  Google Scholar 

  • Kim, Y., Pennuter, H., & Weissman, T. (2009). Directed information and causal estimation in continuous time. In IEEE international symposium on information theory (ISIT).

  • Korzeniewska, A., Mańczak, M., Kamiński, M., Blinowska, K., & Kasicki, S. (2003). Determination of information flow direction among brain structures by a modified directed transfer function (dDTF) method. Journal of Neuroscience Methods, 125(1–2), 195–207.

    Article  PubMed  Google Scholar 

  • Kramer, G. (1998). Directed information for channels with feedback. Ph.D. thesis, University of Manitoba, Canada.

  • Kramer, M., Eden, U., Cash, S., & Kolaczyk, E. (2009). Network inference with confidence from multivariate time series. Physical Review E, 79(6), 61916.

    Article  Google Scholar 

  • Kraskov, A. (2008). Synchronization and interdependence measures and their application to the electroencephalogram of epilepsy patients and clustering of data. Report Nr.: NIC series; 24.

  • Lastras, L. (2002). An almost sure convergence proof of the sliding-window Lempel-Ziv algorithm. In Proceedings 2002 IEEE international symposium on information theory.

  • Marko, H. (1973). The bidirectional communication theory–A generalization of information theory. IEEE Transactions on Communications, 21(12), 1345–1351.

    Article  Google Scholar 

  • Martins, N., & Dahleh, M. (2008). Feedback control in the presence of noisy channels: “Bode-like” fundamental limitations of performance. IEEE Transactions on Automatic Control, 53(7), 1604 –1615.

    Article  Google Scholar 

  • Massey, J. (1990). Causality, feedback and directed information. In Proc. int. symp. information theory application (ISITA-90) (pp. 303–305).

  • Massey, J., & Massey, P. (2005). Conservation of mutual and directed information. In Proceedings international symposium on information theory, 2005. ISIT 2005 (pp. 157–158).

  • Mathai, P., Martins, N., & Shapiro, B. (2007). On the detection of gene network interconnections using directed mutual information. San Deigo: ITA.

    Google Scholar 

  • Meyn, S., & Tweedie, R. (2009). Markov chains and stochastic stability (p. 622). Cambridge: Cambridge Mathematical Library.

    Google Scholar 

  • Okatan, M., Wilson, M., & Brown, E. (2005). Analyzing functional connectivity using a network likelihood model of ensemble neural spiking activity. Neural Computation, 17(9), 1927–1961.

    Article  PubMed  Google Scholar 

  • Paninski, L. (2003). Estimation of entropy and mutual information. Neural Computation, 15(6), 1191–1253.

    Article  Google Scholar 

  • Paninski, L., Fellows, M., Hatsopoulos, N., & Donoghue, J. (2004). Spatiotemporal tuning of motor cortical neurons for hand position and velocity. Journal of Neurophysiology, 91(1), 515.

    Article  PubMed  Google Scholar 

  • Pearl, J. (2009). Causality: Models, reasoning and inference. New York: Cambridge University Press.

    Google Scholar 

  • Pereda, E., Quiroga, R., & Bhattacharya, J. (2005). Nonlinear multivariate analysis of neurophysiological signals. Progress in Neurobiology, 77(1–2), 1–37.

    Article  PubMed  Google Scholar 

  • Perez-Cruz, F. (2008). Estimation of information theoretic measures for continuous random variables. NIPS.

  • Permuter, H., Kim, Y., & Weissman, T. (2008). On directed information and gambling. In IEEE international symposium on information theory, 2008. ISIT 2008 (pp. 1403–1407).

  • Permuter, H., Kim, Y., & Weissman, T. (2009a). Interpretations of directed information in portfolio theory, data compression, and hypothesis testing. Arxiv preprint. arXiv:0912.4872.

  • Permuter, H., Weissman, T., & Goldsmith, A. (2009b). Finite state channels with time-invariant deterministic feedback. IEEE Transactions on Information Theory, 55(2), 644–662.

    Article  Google Scholar 

  • Prechtl, J., Cohen, L., Pesaran, B., Mitra, P., & Kleinfeld, D. (1997). Visual stimuli induce waves of electrical activity in turtle cortex. Proceedings of the National Academy of Sciences of the United States of America, 94(14), 7621.

    Article  CAS  PubMed  Google Scholar 

  • Ramnani, N., Behrens, T., Penny, W., & Matthews, P. (2004). New approaches for exploring anatomical and functional connectivity in the human brain. Biological Psychiatry, 56(9), 613–619.

    Article  PubMed  Google Scholar 

  • Rao, A., Hero III, A., States, D., & Engel, J. (2006). Inference of biologically relevant gene influence networks using the directed information criterion. In Proceedings of IEEE international conference on acoustics, speech and signal processing (ICASSP) (Vol. 2, pp. 1028–1031).

  • Rao, A., Hero III, A., States, D.J., & Engel, J. D. (2007). Inferring time-varying network topologies from gene expression data. EURASIP Journal on Bioinformatics and System Biology-Special Issue on Gene Networks, 2007, 51947.

    Google Scholar 

  • Rao, A., Hero III, A., David, J., & Engel, J. (2008). Using directed information to build biologically relevant influence networks. Journal of Bioinformatics and Computational Biology, 6(3), 493–519.

    Article  CAS  PubMed  Google Scholar 

  • Rissanen, J., & Wax, M. (1987). Measures of mutual and causal dependence between two time series (Corresp.). IEEE Transactions on Information Theory, 33(4), 598–601.

    Article  Google Scholar 

  • Roebroeck, A., Formisano, E., & Goebel, R. (2005). Mapping directed influence over the brain using Granger causality and fMRI. NeuroImage, 25(1), 230–242.

    Article  PubMed  Google Scholar 

  • Rogers, B., Morgan, V., Newton, A., & Gore, J. (2007). Assessing functional connectivity in the human brain by fMRI. Magnetic Resonance Imaging, 25(10), 1347–1357.

    Article  PubMed  Google Scholar 

  • Rubino, D., Robbins, K., & Hatsopoulos, N. (2006). Propagating waves mediate information transfer in the motor cortex. Nature Neuroscience, 9(12), 1549–1557.

    Article  CAS  PubMed  Google Scholar 

  • Salvador, R., Suckling, J., Schwarzbauer, C., & Bullmore, E. (2005). Undirected graphs of frequency-dependent functional connectivity in whole brain networks. Philosophical Transactions of the Royal Society B: Biological Sciences, 360(1457), 937–946.

    Article  Google Scholar 

  • Schreiber, T. (2000). Measuring information transfer. Physical Review Letters, 85(2), 461–464.

    Article  CAS  PubMed  Google Scholar 

  • Schuyler, B., Ollinger, J., Oakes, T., Johnstone, T., & Davidson, R. (2009). Dynamic Causal Modeling applied to fMRI data shows high reliability. NeuroImage, 49, 603–611.

    Article  PubMed  Google Scholar 

  • Seth, A., & Edelman, G. (2007). Distinguishing causal interactions in neural populations. Neural Computation, 19(4), 910–933.

    Article  PubMed  Google Scholar 

  • Smith, V., Yu, J., Smulders, T., Hartemink, A., & Jarvis, E. (2006). Computational inference of neural information flow networks. PLoS Computational Biology, 2(11), e161.

    Article  Google Scholar 

  • Stephan, K., Kasper, L., Harrison, L., Daunizeau, J., den Ouden, H., Breakspear, M., et al. (2008). Nonlinear dynamic causal models for fMRI. NeuroImage, 42(2), 649–662.

    Article  PubMed  Google Scholar 

  • Stevenson, I., Rebesco, J., Hatsopoulos, N., Haga, Z., Miller, L., & Körding, K. (2009). Bayesian inference of functional connectivity and network structure from spikes. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 17(3), 203.

    Article  PubMed  Google Scholar 

  • Sundaresan, R., & Verdú, S. (2006). Capacity of queues via point-process channels. IEEE Transactions on Information Theory, 52(6), 2697–2709.

    Article  Google Scholar 

  • Tatikonda, S. (2000). Control under communication constraints. Ph.D. thesis, Massachusetts Institute of Technology.

  • Tatikonda, S., & Mitter, S. (2009). The capacity of channels with feedback. IEEE Transactions on Information Theory, 55(1), 323–349.

    Article  Google Scholar 

  • Truccolo, W., Eden, U., Fellows, M., Donoghue, J., & Brown, E. (2005). A point process framework for relating neural spiking activity to spiking history, neural ensemble, and extrinsic covariate effects. Journal of Neurophysiology, 93(2), 1074–1089.

    Article  PubMed  Google Scholar 

  • Uddin, L., Clare Kelly, A., Biswal, B., Xavier Castellanos, F., & Milham, M. (2009). Functional connectivity of default mode network components: Correlation, anticorrelation, and causality. Human Brain Mapping, 30(2), 625–637.

    Article  PubMed  Google Scholar 

  • Venkataramanan, R., & Pradhan, S. (2007). Source coding with feed-forward: Rate-distortion theorems and error exponents for a general source. IEEE Transactions on Information Theory, 53(6), 2154–2179.

    Article  Google Scholar 

  • Vogels, T., & Abbott, L. (2005). Signal propagation and logic gating in networks of integrate-and-fire neurons. Journal of Neuroscience, 25(46), 10786.

    Article  CAS  PubMed  Google Scholar 

  • Wang, X., Chen, Y., Bressler, S., & Ding, M. (2007). Granger causality between multiple interdependent neurobiological time series: Blockwise versus pairwise methods. International Journal of Neural Systems, 17(2), 71.

    Article  CAS  PubMed  Google Scholar 

  • Wu, W., & Hatsopoulos, N. (2006). Evidence against a single coordinate system representation in the motor cortex. Experimental Brain Research, 175(2), 197–210.

    Article  Google Scholar 

  • Zhao, L., Permuter, H., Kim, Y., & Weissman, T. (2010). Universal estimation of directed information. In IEEE international symposium on information theory (ISIT), Austin, TX (in press).

  • Ziv, J., & Lempel, A. (1977). A universal algorithm for sequential data compression. IEEE Transactions on Information Theory, 23(3), 337–343.

    Article  Google Scholar 

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Correspondence to Christopher J. Quinn.

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Appendices

Appendix A: Proof of Lemma 1

Proof

First, prove that \(\mathcal{H}(\text{Y}||\text{X})\). This proof closely follows the proof for the unconditional entropy rate in Cover and Thomas (2006). An important theorem used for the proof is the Cesaro mean theorem (Cover and Thomas 2006): For sequences of real numbers (a 1, ⋯ a n ) and (b 1, ⋯ b n ), if limn → ∞  a n  = a, and \(b_n = \frac{1}{n} \sum_{i=1}^n a_n\), then limn → ∞  b n  = a.

By definition, \( \text{H}(Y^n || X^n) = \frac{1}{n}\sum_{i=1}^n \text{H}(Y_i | Y^{i-1}, X^i) \). Since conditioning reduces entropy, entropy is nonnegative, and the processes are jointly stationary, we have

$$ 0 \leq \text{H}\left(Y_i | Y^{i-1}, X^i\right) \leq \text{H}(Y_1) \quad \forall \ i. $$

Observe that

$$ \text{H}\left(Y_{i} | Y^{i-1}, X^{i}\right) \leq \text{H}\left(Y_{i} | Y^{i-1}_2, X^{i}_2\right) \label{ab1} $$
(46)
$$ = \text{H}\left(Y_{i-1} | Y^{i-2}, X^{i-1}\right), \label{ab2} $$
(47)

where Eq. (46) uses the property that conditioning reduces entropy (in reverse) and Eq. (47) uses stationarity. This sequence of real numbers (once the process is defined, that is, the underlying probability distribution is specified), the entropies are deterministic numbers) \(a_i \triangleq \text{H}(Y_{i} | Y^{i-1}, X^{i})\) are nonincreasing and bounded below by 0. Therefore, limit of a n as n → ∞ exists, and thus, by employing Cesaro mean theorem, \(\text{H}(\mathcal{Y} || \mathcal{X}) \triangleq \lim_{n \to \infty} \frac{1}{n}\text{H}(Y^n || X^n)\) exists.

Next, taking X n to be a deterministic sequence, and following the above, \(\text{H}(\mathcal{Y} ) \triangleq \lim_{n \to \infty} \frac{1}{n}\text{H}(Y^n )\) exists. Taking the limit in Eq. (24), \(\mathcal{I}(\text{X} \to \text{Y})\triangleq \lim_{n \to \infty} \frac{1}{n}\text{I}(X^n \to Y^n)\) also exists.□

Appendix B: Proof of Lemma 2

Proof

The normalized causal entropy can be rewritten as

$$ \begin{array}{lll} && {\kern-6pt} \frac{1}{n} \text{H}\left(Y^n || X^n\right) \\ && {\kern6pt} = \frac{1}{n}\! \sum\limits_{i=1}^n \!\text{H}\left(Y_i | X^i, Y^{i-1}\right) \label{bv1} \end{array} $$
(48)
$$ {\kern6pt} = \frac{1}{n}\! \sum\limits_{i=1}^n\! \mathbb{E} \!\left[\! - \log P_{Y_i | Y^{i-1},X^i}\left(Y_i | Y^{i-1},X^i \right) \! \right] \label{bv2} $$
(49)
$$ {\kern6pt} = \frac{1}{n}\! \sum\limits_{i=1}^n \! \mathbb{E}\! \left[ \! - \log P_{Y_i | Y^{i-1}_{i-J},X^i_{i-(K-1)}} \left(Y_i | Y^{i-1}_{i-J},X^i_{i-(K-1)} \right) \! \right] \label{bv3} $$
(50)
$$ \begin{array}{lll} && {\kern6pt} = \frac{1}{n}\! \sum\limits_{i=1}^n \! \mathbb{E}\! \left[g_{JK}\left( Y^{i}_{i-J},X^i_{i-(K-1)} \right)\right] \\ && {\kern6pt} = \mathbb{E} \!\left[\! g_{JK}\left(Y^{l}_{l-J},X^l_{l-(K-1)} \right) \!\right] \label{bv4_3} \end{array} $$
(51)

where Eq. (48) follows by the definition of causally conditioned entropy, Eq. (49) follows by chain rule for entropy, Eq. (50) follows from the Markov assumption, and Eq. (51) follows from the stationarity assumption.□

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Quinn, C.J., Coleman, T.P., Kiyavash, N. et al. Estimating the directed information to infer causal relationships in ensemble neural spike train recordings. J Comput Neurosci 30, 17–44 (2011). https://doi.org/10.1007/s10827-010-0247-2

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