JIDT: an information-theoretic toolkit for studying the dynamics of complex systems

Frontiers in Robotics and AI

Volumn 1, Issue , 2014, Pages

  Lizier, Joseph T ^a,b  

^a MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

^b CSIRO   (Australia)

Author keywords

complex networks; complex systems; information storage; information theory; information transfer; Java; MATLAB; transfer entropy

Indexed keywords

EID: 84953257348     PISSN: None     EISSN: 22969144     Source Type: Journal    
DOI: 10.3389/frobt.2014.00011     Document Type: Article

References (120)

1. Adami, C. (2002). What is complexity? Bioessays 24, 1085–1094. doi:10.1002/bies. 10192
2. Ash, R. B. (1965). Information Theory. New York, NY: Dover Publications Inc.
3. Astakhov, S., Grassberger, P., Kraskov, A., and Stögbauer, H. (2013). Mutual Information Least-Dependent Component Analysis (MILCA). Available from: http://www.ucl.ac.uk/ion/departments/sobell/Research/RLemon/MILCA/MILCA
4. Ay, N., Bertschinger, N., Der, R., Güttler, F., and Olbrich, E. (2008). Predictive information and explorative behavior of autonomous robots. Eur. Phys. J. B 63, 329–339. doi:10.1140/epjb/e2008-00175-0
5. Bandt, C., and Pompe, B. (2002). Permutation entropy: a natural complexity measure for time series. Phys. Rev. Lett. 88, 174102. doi:10.1103/PhysRevLett.88. 174102
6. Barnett, L., Barrett, A. B., and Seth, A. K. (2009). Granger causality and transfer entropy are equivalent for Gaussian variables. Phys. Rev. Lett. 103, 238701. doi:10.1103/PhysRevLett.103.238701
7. Barnett, L., and Bossomaier, T. (2012). Transfer entropy as a log-likelihood ratio. Phys. Rev. Lett. 109, 138105. doi:10.1103/PhysRevLett.109.138105
8. Barnett, L., Lizier, J. T., Harré, M., Seth, A. K., and Bossomaier, T. (2013). Information flow in a kinetic Ising model peaks in the disordered phase. Phys. Rev. Lett. 111, 177203. doi:10.1103/PhysRevLett.111.177203
9. Barnett, L., and Seth, A. K. (2014). The MVGC multivariate granger causality tool-box: a new approach to granger-causal inference. J. Neurosci. Methods 223, 50–68. doi:10.1016/j.jneumeth.2013.10.018
10. Bauer, T. L., Colbaugh, R., Glass, K., and Schnizlein, D. (2013). “Use of transfer entropy to infer relationships from behavior,” in Proceedings of the Eighth Annual Cyber Security and Information Intelligence Research Workshop, CSIIRW’13 (New York, NY: ACM). doi:10.1145/2459976.2460016
11. Beer, R. D., and Williams, P. L. (2014). Information processing and dynamics in minimally cognitive agents. Cogn. Sci. doi:10.1111/cogs.12142
12. Bialek, W., Nemenman, I., and Tishby, N. (2001). Complexity through non-extensivity. Physica A 302, 89–99. doi:10.1186/1752-0509-5-61
13. Boedecker, J., Obst, O., Lizier, J. T., Mayer, N. M., and Asada, M. (2012). Information processing in echo state networks at the edge of chaos. Theory Biosci. 131, 205–213. doi:10.1007/s12064-011-0146-8
14. Bonachela, J. A., Hinrichsen, H., and Muñoz, M. A. (2008). Entropy estimates of small data sets. J. Phys. A Math. Theor. 41, 202001. doi:10.1088/1751-8113/41/ 20/202001
15. Chávez, M., Martinerie, J., and LeVanQuyen, M. (2003). Statistical assessment of non-linear causality: application to epileptic EEG signals. J. Neurosci. Methods 124, 113–128. doi:10.1016/S0165-0270(02)00367-9
16. Cliff, O. M., Lizier, J. T., Wang, X. R., Wang, P., Obst, O., and Prokopenko, M. (2014). “Towards quantifying interaction networks in a football match,” in RoboCup 2013: Robot World Cup XVII, Volume 8371 of Lecture Notes in Computer Sci-ence, eds S. Behnke, M. Veloso, A. Visser, and R. Xiong (Berlin: Springer), 1–12.
17. Computer Language Benchmarks Game. (2014). Available at: http://benchmarksgame.alioth.debian.org/u64q/java.php
18. Cover, T. M., and Thomas, J. A. (1991). Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing). New York, NY: Wiley-Interscience.
19. Crutchfield, J. P., and Feldman, D. P. (2003). Regularities unseen, randomness observed: levels of entropy convergence. Chaos 13, 25–54. doi:10.1063/1.1530990
20. Damiani, C., Kauffman, S., Serra, R., Villani, M., and Colacci, A. (2010). “Informa-tion transfer among coupled random Boolean networks,” in Cellular Automata, Volume 6350 of Lecture Notes in Computer Science, eds S. Bandini, S. Manzoni, H. Umeo, and G. Vizzari (Berlin: Springer), 1–11.
21. Damiani, C., and Lecca, P. (2011). “Model identification using correlation-based inference and transfer entropy estimation,” in Fifth UKSim European Symposium on Computer Modeling and Simulation (EMS) (Madrid: IEEE), 129–134.
22. Dasgupta, S., Wörgötter, F., and Manoonpong, P. (2013). Information dynamics based self-adaptive reservoir for delay temporal memory tasks. Evolving Systems 4, 235–249. doi:10.1007/s12530-013-9080-y
23. Faes, L., Nollo, G., and Porta, A. (2011). Information-based detection of non-linear granger causality in multivariate processes via a non-uniform embedding tech-nique. Phys. Rev. E 83, 051112. doi:10.1103/PhysRevE.83.051112
24. Faes, L., and Porta, A. (2014). “Conditional entropy-based evaluation of information dynamics in physiological systems,” in Directed Information Measures in Neuroscience, Understanding Complex Systems, eds M. Wibral, R. Vicente, and J. T. Lizier (Berlin: Springer), 61–86.
25. Fano, R. M. (1961). Transmission of Information: A Statistical Theory of Communi-cations. Cambridge, MA: MIT Press.
26. Fernández, P., and Solé, R. V. (2006). “The role of computation in complex regulatory networks,” in Power Laws, Scale-Free Networks and Genome Biology, Molecular Biology Intelligence Unit, eds E. V. Koonin, Y. I. Wolf, and G. P. Karev (Springer), 206–225. doi:10.1007/0-387-33916-7_12
27. Frenzel, S., and Pompe, B. (2007). Partial mutual information for coupling analysis of multivariate time series. Phys. Rev. Lett. 99, 204101. doi:10.1103/PhysRevLett. 99.204101
28. Gell-Mann, M. (1994). The Quark and the Jaguar. New York, NY: W.H. Freeman.
29. Gómez, C., Lizier, J. T., Schaum, M., Wollstadt, P., Grützner, C., Uhlhaas, P., et al. (2014). Reduced predictable information in brain signals in autism spectrum disorder. Front. Neuroinformatics 8:9. doi:10.3389/fninf.2014.00009
30. Gomez-Herrero, G., Wu, W., Rutanen, K., Soriano, M. C., Pipa, G., and Vicente, R. (2010). Assessing coupling dynamics from an ensemble of time series. arXiv:1008.0539.
31. Gong, P., and van Leeuwen, C. (2009). Distributed dynamical computation in neural circuits with propagating coherent activity patterns. PLoS Comput. Biol. 5:e1000611. doi:10.1371/journal.pcbi.1000611
32. Grassberger, P. (1986). Toward a quantitative theory of self-generated complexity. Int. J. Theor. Phys. 25, 907–938. doi:10.1007/BF00668821
33. Helvik, T., Lindgren, K., and Nordahl, M. G. (2004). “Local information in one-dimensional cellular automata,” in Proceedings of the International Conference on Cellular Automata for Research and Industry, Amsterdam, Volume 3305 of Lecture Notes in Computer Science, eds P. M. A. Sloot, B. Chopard, and A. G. Hoekstra (Berlin: Springer), 121–130.
34. Honey, C. J., Kötter, R., Breakspear, M., and Sporns, O. (2007). Network structure of cerebral cortex shapes functional connectivity on multiple time scales. Proc. Natl. Acad. Sci. U.S.A. 104, 10240–10245. doi:10.1073/pnas.0701519104
35. Ito, S., Hansen, M. E., Heiland, R., Lumsdaine, A., Litke, A. M., and Beggs, J. M. (2011). Extending transfer entropy improves identification of effective connectivity in a spiking cortical network model. PLoS ONE 6:e27431. doi:10.1371/journal.pone.0027431
36. Kaiser, A., and Schreiber, T. (2002). Information transfer in continuous processes. Physica D 166, 43–62. doi:10.1016/S0167-2789(02)00432-3
37. Kantz, H., and Schreiber, T. (1997). Non-Linear Time Series Analysis. Cambridge, MA: Cambridge University Press.
38. Klyubin, A. S., Polani, D., and Nehaniv, C. L. (2008). Keep your options open: an information-based driving principle for sensorimotor systems. PLoS ONE 3:e4018. doi:10.1371/journal.pone.0004018
39. Kozachenko, L., and Leonenko, N. (1987). A statistical estimate for the entropy of a random vector. Probl. Inf. Transm. 23, 9–16.
40. Kraskov, A., Stögbauer, H., and Grassberger, P. (2004). Estimating mutual informa-tion. Phys. Rev. E 69, 066138. doi:10.1103/PhysRevE.69.066138
41. Langton, C. G. (1990). Computation at the edge of chaos: phase transitions and emergent computation. Physica D 42, 12–37. doi:10.1016/0167-2789(90) 90064-V
42. Liao, W., Ding, J., Marinazzo, D., Xu, Q., Wang, Z., Yuan, C., et al. (2011). Small-world directed networks in the human brain: multivariate granger causality analysis of resting-state fMRI. Neuroimage 54, 2683–2694. doi:10.1016/j.neuroimage.2010. 11.007
43. Lindner, M., Vicente, R., Priesemann, V., and Wibral, M. (2011). TRENTOOL: a MATLAB open source toolbox to analyse information flow in time series data with transfer entropy. BMC Neurosci. 12:119. doi:10.1186/1471-2202-12-119
44. Lizier, J. T. (2013). The Local Information Dynamics of Distributed Computation in Complex Systems (Springer Theses). Berlin: Springer.
45. Lizier, J. T. (2014). “Measuring the dynamics of information processing on a local scale in time and space,” in Directed Information Measures in Neuroscience, Understanding Complex Systems, eds M. Wibral, R. Vicente, and J. T. Lizier (Berlin: Springer), 161–193.
46. Lizier, J. T., Atay, F. M., and Jost, J. (2012a). Information storage, loop motifs, and clustered structure in complex networks. Phys. Rev. E 86, 026110. doi:10.1103/PhysRevE.86.026110
47. Lizier, J. T., Prokopenko, M., and Zomaya, A. Y. (2012b). Local measures of information storage in complex distributed computation. Inf. Sci. 208, 39–54. doi:10.1016/j.ins.2012.04.016
48. Lizier, J. T., and Flecker, B. (2012). Java Partial Information Decomposition Toolkit. Available at: http://github.com/jlizier/jpid
49. Lizier, J. T., Flecker, B., and Williams, P. L. (2013). “Towards a synergy-based approach to measuring information modification,” in Proceedings of the 2013 IEEE Symposium on Artificial Life (ALIFE) (Singapore: IEEE), 43–51.
50. Lizier, J. T., Heinzle, J., Horstmann, A., Haynes, J.-D., and Prokopenko, M. (2011a). Multivariate information-theoretic measures reveal directed information structure and task relevant changes in fMRI connectivity. J. Comput. Neurosci. 30, 85–107. doi:10.1007/s10827-010-0271-2
51. Lizier, J. T., Piraveenan, M., Pradhana, D., Prokopenko, M., and Yaeger, L. S. (2011b). “Functional and structural topologies in evolved neural networks,” in Proceedings of the European Conference on Artificial Life (ECAL), Volume 5777 of Lecture Notes in Computer Science, eds G. Kampis, I. Karsai, and E. Szathmáry (Berlin: Springer), 140–147.
52. Lizier, J. T., Pritam, S., and Prokopenko, M. (2011c). Information dynamics in small-world Boolean networks. Artif. Life 17, 293–314. doi:10.1162/artl_a_00040
53. Lizier, J. T., and Mahoney, J. R. (2013). Moving frames of reference, relativity and invariance in transfer entropy and information dynamics. Entropy 15, 177–197. doi:10.3390/e15010177
54. Lizier, J. T., Prokopenko, M., Tanev, I., and Zomaya, A. Y. (2008a). “Emergence of glider-like structures in a modular robotic system,” in Proceedings of the Eleventh International Conference on the Simulation and Synthesis of Living Systems (ALife XI), Winchester, UK, eds S. Bullock, J. Noble, R. Watson, and M. A. Bedau (Cam-bridge, MA: MIT Press), 366–373.
55. Lizier, J. T., Prokopenko, M., and Zomaya, A. Y. (2008b). “The information dynamics of phase transitions in random Boolean networks,” in Proceedings of the Eleventh International Conference on the Simulation and Synthesis of Living Systems (ALife XI), Winchester, UK, eds S. Bullock, J. Noble, R. Watson, and M. A. Bedau (Cam-bridge, MA: MIT Press), 374–381.
56. Lizier, J. T., Prokopenko, M., and Zomaya, A. Y. (2008c). Local information transfer as a spatiotemporal filter for complex systems. Phys. Rev. E 77, 026110. doi:10.1103/PhysRevE.77.026110
57. Lizier, J. T., Prokopenko, M., and Zomaya, A. Y. (2007). “Detecting non-trivial computation in complex dynamics,” in Proceedings of the 9th European Conference on Artificial Life (ECAL 2007), Volume 4648 of Lecture Notes in Computer Science, eds F. Almeida e Costa, L. M. Rocha, E. Costa, I. Harvey, and A. Coutinho (Berlin: Springer), 895–904.
58. Lizier, J. T., Prokopenko, M., and Zomaya, A. Y. (2010). Information modification and particle collisions in distributed computation. Chaos 20, 037109. doi:10.1063/1.3486801
59. Lizier, J. T., Prokopenko, M., and Zomaya, A. Y. (2014). “A framework for the local information dynamics of distributed computation in complex systems,” in Guided Self-Organization: Inception, Volume 9 of Emergence, Complexity and Computation, ed. M. Prokopenko (Berlin: Springer), 115–158.
60. Lungarella, M., and Sporns, O. (2006). Mapping information flow in sensorimotor networks. PLoS Comput. Biol. 2:e144. doi:10.1371/journal.pcbi.0020144
61. MacKay, D. J. C. (2003). Information Theory, Inference, and Learning Algorithms. Cambridge: Cambridge University Press.
62. Mahoney, J. R., Ellison, C. J., James, R. G., and Crutchfield, J. P. (2011). How hidden are hidden processes? A primer on crypticity and entropy convergence. Chaos 21, 037112. doi:10.1063/1.3637502
63. Mäki-Marttunen, V., Diez, I., Cortes, J. M., Chialvo, D. R., and Villarreal, M. (2013). Disruption of transfer entropy and inter-hemispheric brain functional connectivity in patients with disorder of consciousness. Front. Neuroinformatics 7:24. doi:10.3389/fninf.2013.00024
64. Marinazzo, D., Wu, G., Pellicoro, M., Angelini, L., and Stramaglia, S. (2012). Information flow in networks and the law of diminishing marginal returns: evidence from modeling and human electroencephalographic recordings. PLoS ONE 7:e45026. doi:10.1371/journal.pone.0045026
65. Miramontes, O. (1995). Order-disorder transitions in the behavior of ant societies. Complexity 1, 56–60. doi:10.1002/cplx.6130010313
66. Mitchell, M. (1998). “Computation in cellular automata: a selected review,” in Non-Standard Computation, eds T. Gramß, S. Bornholdt, M. Groß, M. Mitchell, and T. Pellizzari (Weinheim: Wiley-VCH Verlag GmbH & Co. KGaA), 95–140.
67. Mitchell, M. (2009). Complexity: A Guided Tour. New York, NY: Oxford University Press.
68. Montalto, A. (2014). MuTE Toolbox to Evaluate Multivariate Transfer Entropy. Figshare. Available at: http://dx.doi.org/10.6084/m9.figshare.1005245
69. Montalto, A., Faes, L., and Marinazzo, D. (2014a). MuTE: a MATLAB toolbox to compare established and novel estimators of the multivariate transfer entropy. PLoS ONE 9:e109462. doi:10.1371/journal.pone.0109462
70. Montalto, A., Faes, L., and Marinazzo, D. (2014b). “MuTE: a new MATLAB toolbox for estimating the multivariate transfer entropy in physiological variability series,” in 8th Conference of the European Study Group on Cardiovascular Oscilla-tions (ESGCO) (Trento: IEEE), 59–60. doi:10.1109/ESGCO.2014.6847518
71. Nakajima, K., and Haruna, T. (2013). Symbolic local information transfer. Eur. Phys. J. Spec. Top. 222, 437–455. doi:10.1140/epjst/e2013-01851-x
72. Nakajima, K., Li, T., Kang, R., Guglielmino, E., Caldwell, D. G., and Pfeifer, R. (2012). “Local information transfer in soft robotic arm,” in IEEE International Conference on Robotics and Biomimetics (ROBIO) (Guangzhou: IEEE), 1273–1280.
73. Nilsson, M., and Kleijn, W. B. (2007). On the estimation of differential entropy from data located on embedded manifolds. IEEE Trans. Inf. Theory 53, 2330–2341. doi:10.1109/TIT.2007.899533
74. Obst, O., Boedecker, J., and Asada, M. (2010). “Improving recurrent neural network performance using transfer entropy neural information processing. models and applications,” in Neural Information Processing. Models and Applications, Volume 6444 of Lecture Notes in Computer Science, Chapter 24, eds K. Wong, B. Mendis, and A. Bouzerdoum (Berlin: Springer), 193–200.
75. Obst, O., Boedecker, J., Schmidt, B., and Asada, M. (2013). On active information storage in input-driven systems. arXiv:1303.5526.
76. Oka, M., and Ikegami, T. (2013). Exploring default mode and information flow on the web. PLoS ONE 8:e60398. doi:10.1371/journal.pone.0060398
77. Oostenveld, R., Fries, P., Maris, E., and Schoffelen, J.-M. (2011). FieldTrip: open source software for advanced analysis of MEG, EEG, and invasive electrophysio-logical data. Comput. Intell. Neurosci. 2011, 156869. doi:10.1155/2011/156869
78. Orlandi, J., Saeed, M., and Guyon, I. (2014). Chalearn Connectomics Challenge Sam-ple Code. Available at: http://connectomics.chalearn.org
79. Paninski, L. (2003). Estimation of entropy and mutual information. Neural Comput. 15, 1191–1253. doi:10.1162/089976603321780272
80. Piraveenan, M., Prokopenko, M., and Zomaya, A. Y. (2009). Assortativeness and information in scale-free networks. Eur. Phys. J. B 67, 291–300. doi:10.1140/epjb/e2008-00473-5
81. Pompe, B., and Runge, J. (2011). Momentary information transfer as a coupling measure of time series. Phys. Rev. E 83, 051122. doi:10.1103/PhysRevE.83.051122
82. Prokopenko, M. (2009). Guided self-organization. HFSP J. 3, 287–289. doi:10.2976/1.3233933
83. Prokopenko, M., Boschietti, F., and Ryan, A. J. (2009). An information-theoretic primer on complexity, self-organization, and emergence. Complexity 15, 11–28. doi:10.1002/cplx.20249
84. Prokopenko, M., Gerasimov, V., and Tanev, I. (2006a). “Evolving spatiotemporal coordination in a modular robotic system,” in From Animals to Animats 9: Proceedings of the Ninth International Conference on the Simulation of Adaptive Behavior (SAB’06), Volume 4095 of Lecture Notes in Computer Science, eds S. Nolfi, G. Baldassarre, R. Calabretta, J. C. T. Hallam, D. Marocco, J.-A. Meyer, et al. (Berlin: Springer), 558–569.
85. Prokopenko, M., Gerasimov, V., and Tanev, I. (2006b). “Measuring spatiotemporal coordination in a modular robotic system,” in Proceedings of the 10th International Conference on the Simulation and Synthesis of Living Systems (ALifeX), Bloomington, Indiana, USA, eds L. M. Rocha, L. S. Yaeger, M. A. Bedau, D. Floreano, R. L. Goldstone, and A. Vespignani (Bloomington: MIT Press), 185–191.
86. Prokopenko, M., Lizier, J. T., Obst, O., and Wang, X. R. (2011). Relating Fisher information to order parameters. Phys. Rev. E 84, 041116. doi:10.1103/PhysRevE.84. 041116
87. Prokopenko, M., Wang, P., Valencia, P., Price, D., Foreman, M., and Farmer, A. (2005). Self-organizing hierarchies in sensor and communication networks. Artif. Life 11, 407–426. doi:10.1162/106454605774270642
88. Rigney, D. R., Goldberger, A. L., Ocasio, W., Ichimaru, Y., Moody, G. B., and Mark, R. (1993). “Multi-channel physiological data: description and analysis,” in Time Series Prediction: Forecasting the Future and Understanding the Past, eds A. S. Weigend and N. A. Gershenfeld (Reading, MA: Addison-Wesley), 105–129.
89. Rutanen, K. (2011). Tim 1.2.0. Available at: http://www.cs.tut.fi/~timhome/tim-1. 2.0/tim.htm
90. Sandoval, L. (2014). Structure of a global network of financial companies based on transfer entropy. Entropy 16, 4443–4482. doi:10.3390/e16084443
91. Schreiber, T. (2000). Measuring information transfer. Phys. Rev. Lett. 85, 461–464. doi:10.1103/PhysRevLett.85.461
92. Shalizi, C. R., Haslinger, R., Rouquier, J.-B., Klinkner, K. L., and Moore, C. (2006). Automatic filters for the detection of coherent structure in spatiotemporal sys-tems. Phys. Rev. E 73, 036104. doi:10.1103/PhysRevE.73.036104
93. Shalizi, C. R. (2001). Causal Architecture, Complexity and Self-Organization in Time Series and Cellular Automata. Ph.D., thesis, University of Wisconsin-Madison, Madison, WI.
94. Shannon, C. E. (1948). A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423. doi:10.1002/j.1538-7305.1948.tb01338.x
95. Solé, R. V., and Valverde, S. (2001). Information transfer and phase transitions in a model of internet traffic. Physica A 289, 595–605. doi:10.1016/S0378-4371(00) 00536-7
96. Solé, R. V., and Valverde, S. (2004). “Information theory of complex networks: on evolution and architectural constraints,” in Complex Networks, Volume 650 of Lecture Notes in Physics, eds E. Ben-Naim, H. Frauenfelder, and Z. Toroczkai (Berlin: Springer), 189–207.
97. Staniek, M., and Lehnertz, K. (2008). Symbolic transfer entropy. Phys. Rev. Lett. 100, 158101. doi:10.1103/PhysRevLett.100.158101
98. Steeg, G. V., and Galstyan, A. (2013). “Information-theoretic measures of influ-ence based on content dynamics,” in Proceedings of the Sixth ACM International Conference on Web Search and Data Mining, WSDM’13 (New York, NY: ACM), 3–12.
99. Stetter, O., Battaglia, D., Soriano, J., and Geisel, T. (2012). Model-free reconstruction of excitatory neuronal connectivity from calcium imaging signals. PLoS Comput. Biol. 8:e1002653. doi:10.1371/journal.pcbi.1002653
100. Stögbauer, H., Kraskov, A., Astakhov, S., and Grassberger, P. (2004). Least-dependent-component analysis based on mutual information. Phys. Rev. E 70, 066123. doi:10.1103/PhysRevE.70.066123
101. Stowell, D., and Plumbley, M. D. (2009). Fast multidimensional entropy estimation by k-d partitioning. IEEE Signal Process. Lett. 16, 537–540. doi:10.1109/LSP.2009. 2017346
102. Stramaglia, S., Wu, G.-R., Pellicoro, M., and Marinazzo, D. (2012). “Expanding the transfer entropy to identify information subgraphs in complex systems,” in Annual International Conference of the IEEE Engineering in Medicine and Biology Society (San Diego, CA: IEEE), 3668–3671.
103. Takens, F. (1981). “Detecting strange attractors in turbulence,” in Dynamical Systems and Turbulence, Warwick 1980, Volume 898 of Lecture Notes in Mathematics, Chapter 21, eds D. Rand and L.-S. Young (Berlin: Springer), 366–381.
104. Tononi, G., Sporns, O., and Edelman, G. M. (1994). A measure for brain complexity: relating functional segregation and integration in the nervous system. Proc. Natl. Acad. Sci. U.S.A. 91, 5033–5037. doi:10.1073/pnas.91.11.5033
105. Vakorin, V. A., Krakovska, O. A., and McIntosh, A. R. (2009). Confounding effects of indirect connections on causality estimation. J. Neurosci. Methods 184, 152–160. doi:10.1016/j.jneumeth.2009.07.014
106. Verdes, P. F. (2005). Assessing causality from multivariate time series. Phys. Rev. E 72, 026222. doi:10.1103/PhysRevE.72.026222
107. Vicente, R., and Wibral, M. (2014). “Efficient estimation of information transfer,” in Directed Information Measures in Neuroscience, Understanding Complex Systems, eds M. Wibral, R. Vicente, and J. T. Lizier (Berlin: Springer), 37–58.
108. Vicente, R., Wibral, M., Lindner, M., and Pipa, G. (2011). Transfer entropy – a model-free measure of effective connectivity for the neurosciences. J. Comput. Neurosci. 30, 45–67. doi:10.1007/s10827-010-0262-3
109. Walker, S. I., Cisneros, L., and Davies, P. C. W. (2012). “Evolutionary transitions and top-down causation,” in Artificial Life 13 (East Lansing, MI: MIT Press), 283–290.
110. Wang, X. R., Lizier, J. T., Nowotny, T., Berna, A. Z., Prokopenko, M., and Trowell, S. C. (2014). Feature selection for chemical sensor arrays using mutual information. PLoS ONE 9:e89840. doi:10.1371/journal.pone.0089840
111. Wang, X. R., Miller, J. M., Lizier, J. T., Prokopenko, M., and Rossi, L. F. (2012). Quantifying and tracing information cascades in swarms. PLoS ONE 7:e40084. doi:10.1371/journal.pone.0040084
112. Wibral, M., Lizier, J. T., Vögler, S., Priesemann, V., and Galuske, R. (2014a). Local active information storage as a tool to understand distributed neural information processing. Front. Neuroinformatics 8:1. doi:10.3389/fninf.2014.00001
113. Wibral, M., Vicente, R., and Lindner, M. (2014b). “Transfer entropy in neuroscience,” in Directed Information Measures in Neuroscience, Understanding Complex Sys-tems, eds M. Wibral, R. Vicente, and J. T. Lizier (Berlin: Springer), 3–36.
114. Wibral, M., Vicente, R., and Lizier, J. T. (eds) (2014c). Directed Information Measures in Neuroscience. Berlin: Springer.
115. Wibral, M., Pampu, N., Priesemann, V., Siebenhühner, F., Seiwert, H., Lindner, M., et al. (2013). Measuring information-transfer delays. PLoS ONE 8:e55809. doi:10.1371/journal.pone.0055809
116. Wibral, M., Rahm, B., Rieder, M., Lindner, M., Vicente, R., and Kaiser, J. (2011). Transfer entropy in magnetoencephalographic data: quantifying information flow in cortical and cerebellar networks. Prog. Biophys. Mol. Biol. 105, 80–97. doi:10.1016/j.pbiomolbio.2010.11.006
117. Williams, P. L., and Beer, R. D. (2010a). “Information dynamics of evolved agents,” in From Animals to Animats 11, Volume 6226 of Lecture Notes in Computer Science, Chapter 4, eds S. Doncieux, B. Girard, A. Guillot, J. Hallam, J.-A. Meyer, and J.-B. Mouret (Berlin: Springer), 38–49.
118. Williams, P. L., and Beer, R. D. (2010b). Nonnegative decomposition of multivariate information. arXiv:1004.2515.
119. Williams, P. L., and Beer, R. D. (2011). Generalized measures of information transfer. arXiv:1102.1507.
120. Wollstadt, P., Martínez-Zarzuela, M., Vicente, R., Díaz-Pernas, F. J., and Wibral, M. (2014). Efficient transfer entropy analysis of non-stationary neural time series. PLoS ONE 9:e102833. doi:10.1371/journal.pone.0102833