Wiener-Granger Causality in Network Physiology with Applications to Cardiovascular Control and Neuroscience

Proceedings of the IEEE

Volumn 104, Issue 2, 2016, Pages 282-309

  Porta, Alberto ^a,b   Faes, Luca ^c,d  

^a UNIVERSITY OF MILAN   (Italy)

^b IRCCS POLICLINICO SAN DONATO   (Italy)

^c UNIVERSITY OF TRENTO   (Italy)

^d FONDAZIONE BRUNO KESSLER   (Italy)

Author keywords

Biomedical signal processing; complexity; conditional mutual information; coupling strength; directionality; dynamical systems; heart rate variability; multivariate regression modeling; nonlinear dynamics; partial directed coherence; prediction; spectral decomposition; time series analysis; transfer entropy

Indexed keywords

BIOMEDICAL SIGNAL PROCESSING; DYNAMICAL SYSTEMS; DYNAMICS; FORECASTING; FREQUENCY DOMAIN ANALYSIS; HARMONIC ANALYSIS; NEUROLOGY; PHYSIOLOGY; REGRESSION ANALYSIS; SIGNAL PROCESSING; STATISTICAL TESTS;

COMPLEXITY; CONDITIONAL MUTUAL INFORMATION; COUPLING STRENGTHS; DIRECTIONALITY; HEART RATE VARIABILITY; MULTIVARIATE REGRESSION MODELS; PARTIAL DIRECTED COHERENCE; SPECTRAL DECOMPOSITION; TRANSFER ENTROPY;

TIME SERIES ANALYSIS;

EID: 84949920880     PISSN: 00189219     EISSN: 15582256     Source Type: Journal    
DOI: 10.1109/JPROC.2015.2476824     Document Type: Review

References (199)

1. A. Bashan, R. P. Bartsch, J. W. Kantelhardt, S. Havlin, and P. C. Ivanov, "Network physiology reveals relations between network topology and physiological function," Nature Commun., vol. 3, p. 702, 2012.
2. C. W. J. Granger, "Economic processes involving feedback," Inf. Control, vol. 6, pp. 28-48, 1963.
3. J. Geweke, "Measures of conditional linear dependence and feedback between time series," J. Amer. Stat. Assoc., vol. 79, pp. 907-915, 1984.
4. M. Wibral, J. T. Lizier, S. Vogler, V. Priesemann, and R. Galuske, "Local active information storage as a tool to understand distributed neural information processing," Front. Neuroinf., vol. 8, p. 1, 2014.
5. J. T. Lizier, F. M. Atay, and J. Jost, "Information storage, loop motifs, clustered structure in complex networks," Phys. Rev. E, vol. 86, 2012, 026110.
6. J. T. Lizier and M. Prokopenko, "Differentiating information transfer and causal effect," Eur. Phys. J. B, vol. 73, pp. 605-615, 2010.
7. L. Faes, G. Nollo, F. Jurysta, and D. Marinazzo, "Information dynamics of brain-heart physiological networks during sleep," New J. Phys., vol. 16, 2014, 105005.
8. T. Schreiber, "Measuring information transfer," Phys. Rev. Lett., vol. 85, pp. 461-464, 2000.
9. L. Angelini, M. de Tommaso, D. Marinazzo, L. Nitti, M. Pellicoro, and S. Stramaglia, "Redundant variables and Granger causality," Phys. Rev. E, vol. 81, 2010, 037201.
10. S. Stramaglia, J. M. Cortes, and D. Marinazzo, "Synergy and redundancy in the Granger causal analysis of dynamical networks," New J. Phys., vol. 16, 2014, 105003.
11. A. Porta et al., "Cardiovascular control and time domain Granger causality: Insights from selective autonomic blockade," Phil. Trans. Roy. Soc. A, vol. 371, 2013, 20120161.
12. A. Porta et al., "Accounting for respiration is necessary to reliably infer Granger causality from cardiovascular variability series," IEEE Trans. Biomed. Eng., vol. 59, pp. 832-841, 2012.
13. R. Ferri, F. Ruondo,O. Bruni, M. G. Terzano, and C. J. Stam, "The functional connectivity of different EEG bands moves towards small-world network organization during sleep," Clin. Neurophysiol., vol. 119, pp. 2026-2036, 2008.
14. K. A. Schindler, S. Bialonski, M.-T. Horstmann, C. E. Elger, and K. Lehnertz, "Evolving functional network properties and synchronizability during human epileptic seizures," Chaos, vol. 18, 2008, 033119.
15. W. Liao et al., "Evaluating the effective connectivity of resting state networks using conditional Granger causality," Biol. Cybern., vol. 102, pp. 57-69, 2010.
16. A. Stefanovska, "Coupled oscillatorsVComplex but not complicated cardiovascular and brain interactions," IEEE Eng. Med. Biol. Mag., vol. 26, pp. 25-29, 2007.
17. M. B. Schippersa, A. Roebroeck, R. Renkena, L. Nanetti, and C. Keysersa, "Mapping the information flow from one brain to another during gestural communication," Proc. Nat. Acad. Sci. USA, vol. 1073, pp. 9388-9393, 2010.
18. N. Wiener, The Theory of Prediction. New York, NY, USA: McGraw-Hill, 1956.
19. C. W. J. Granger, "Testing for causality. A personal viewpoint," J. Econ. Dyn. Control, vol. 2, pp. 329-352, 1980.
20. K. Friston, R. Moran, and A. K. Seth, "Analysing connectivity with Granger causality and dynamic causal modeling," Curr. Opin. Neurobiol., vol. 23, pp. 172-178, 2013.
21. J. Pearl, Causality: Models, Reasoning and Inference. New York, NY, USA: Cambridge Univ. Press, 2000.
22. M. Eichler, "Causal inference with multiple time series: Principles and problems," Phil. Trans. Roy. Soc. A, vol. 371, 2013, 20110613.
23. S. Bressler and A. K. Seth, "Wiener-Granger causality: A well-established methodology," Neuroimage, vol. 58, pp. 323-329, 2011.
24. M. Eichler and V. Didelez, "On Granger causality and the effect of interventions in time series," Lifetime Data Anal., vol. 16, pp. 3-32, 2010.
25. M. Palus and A. Stefanovska, "Direction of coupling from phases of interacting oscillators: An information-theoretic approach," Phys. Rev. E, vol. 67, 2003, 055201.
26. M. Vejmelka and M. Palus, "Inferring the directionality of coupling with conditional mutual information," Phys. Rev. E, vol. 77, 2008, 026214.
27. B. Musizza et al., "Interactions between cardiac, respiratory and EEG-delta oscillations in rats during anaesthesia," J. Physiol., vol. 580, pp. 315-326, 2007.
28. B. Kralemann, A. Pikovsky, and M. Rosenblum, "Reconstructing effective phase connectivity of oscillator networks from observations," New J. Phys., vol. 16, 2014, 085013.
29. A. Bahraminasab, F. Ghasemi, A. Stefanovska, P. V. E. McClintock, and H. Kantz, "Direction of coupling from phases of interacting oscillators: A permutation information approach," Phys. Rev. Lett., vol. 100, 2008, 084101.
30. L. Angelini, M. Pellicoro, and S. Stramaglia, "Granger causality for circular variables," Phys. Lett. A, vol. 373, pp. 2467-2470, 2009.
31. M. Staniek and K. Lehnertz, "Symbolic transfer entropy," Phys. Rev. Lett., vol. 100, 2008, 158101.
32. C. W. J. Granger, "Times series analysis, cointegration, applications. Nobel Lecture 2003," in Les Prix Nobel. The Nobel Prizes 2003, T. Frangsmyr, Ed. Stockholm, Sweden: Nobel Found., 2004, pp. 360-366.
33. T. Soderstrom and P. Stoica, System Identification. Englewood Cliffs, NJ, USA: Prentice-Hall, 1988.
34. G. Baselli, A. Porta, O. Rimoldi, M. Pagani, and S. Cerutti, "Spectral decomposition in multichannel recordings based on multivariate parametric identification," IEEE Trans. Biomed. Eng., vol. 44, pp. 1092-1101, 1997.
35. M. Casdagli, "Nonlinear prediction of chaotic time series," Physica D, vol. 35, pp. 335-356, 1989.
36. H. Wang, J. Kihwan, and K. H. Chon, "Closed loop nonlinear system identification via the vector optimal parameter search algorithm: Application to heart rate baroreflex control," Med. Eng. Phys., vol. 29, pp. 505-515, 2007.
37. L. Faes, G. Nollo, and K. H. Chon, "Assessment of Granger causality by nonlinear model identification: Application to short-term cardiovascular variability," Ann. Biomed. Eng., vol. 36, pp. 381-395, 2008.
38. G. Nollo, L. Faes, R. Antolini, and A. Porta, "Assessing causality in normal and impaired short-term cardiovascular regulation via nonlinear prediction methods," Phil. Trans. Roy. Soc. A, vol. 367, pp. 1423-1440, 2009.
39. Y. Li, H.-L.Wei, S. A. Billings, and X.-F. Liao, "Time varying linear and non linear parametric model for Granger causality analysis," Phys. Rev. E, vol. 85, 2012, 041906.
40. T. W. Berger, D. Song, R. H. M. Chan, and V. Z. Marmarelis, "The neurobiological basis of cognition: Identification by multi-input, multioutput nonlinear dynamic modeling," Proc. IEEE, vol. 98, pp. 356-374, 2010.
41. M. Riedl, A. Suhrbier, H. Malberg, T. Penzel, G. Bretthauer, J. Kurths, and N. Wessel, "Modeling the cardiovascular system using a nonlinear additive autoregressive model with exogenous input," Phys. Rev. E, vol. 78, 2008, 011919.
42. M. Abel, K. Ahnert, J. Kurths, and S.Mandelj, "Additive nonparametric reconstruction of dynamical systems from time series," Phys. Rev. E, vol. 71, 2005, 015203.
43. H. Voss and J. Kurths, "Reconstruction of non-linear time delay models from data," Phys. Lett. A, vol. 234, pp. 336-344, 1997.
44. S. J. Schiff, P. So, T. Chang, R. Burke, and T. Sauer, "Detecting dynamical interdependence and generalized synchrony through mutual prediction in a neural ensemble," Phys. Rev. E, vol. 54, pp. 6708-6724, 1996.
45. L. Faes, A. Porta, and G. Nollo, "Mutual non linear prediction as a tool to evaluate coupling strength and directionality in bivariate time series: Comparison among different strategies based on k nearest neighbors," Phys. Rev. E, vol. 78, 2008, 026201.
46. D. Marinazzo, M. Pellicoro, and S. Stramaglia, "Nonlinear parametric model for Granger causality of time series," Phys. Rev. E, vol. 73, 2006, 066216.
47. D. Marinazzo, W. Liao, H. Chen, and S. Stramaglia, "Nonlinear connectivity by Granger causality," Neuroimage, vol. 58, pp. 330-338, 2011.
48. J. Theiler, "Spurious dimension from correlation algorithms applied to limited time-series data," Phys. Rev. A, vol. 34, pp. 2427-2432, 1986.
49. A. Porta et al., "Effect of age on complexity and causality of the cardiovascular control: Comparison between model-based and model-free approaches," PLoS ONE, vol. 9, 2014, e89463.
50. U. Feldmann and J. Bhattacharya, "Predictability improvement as an asymmetrical measure of interdependence in bivariate time series," Int. J. Bifurcat. Chaos, vol. 14, pp. 505-514, 2004.
51. M. Wiesenfeldt, U. Parlitz, and W. Lauterborn, "Mixed state analysis of multivariate time series," Int. J. Bifurcat. Chaos, vol. 11, pp. 2217-2226, 2001.
52. Y. Chen, G. Rangarajan, J. Feng, andM. Ding, "Analyzing multiple nonlinear time series with extended Granger causality," Phys. Lett. A, vol. 324, pp. 26-35, 2004.
53. A. Porta et al., "Complexity and non linearity in short-term heart period variability: Comparison of methods based on local non linear prediction," IEEE Trans. Biomed. Eng., vol. 54, pp. 94-106, 2007.
54. A. M. Fraser and H. L. Swinney, "Independent coordinates from strange attractors from mutual information," Phys. Rev. A, vol. 33, pp. 1134-1140, 1986.
55. G. A. Darbellay and I. Vajda, "Estimation of the information by an adaptive partitioning of the observational space," IEEE Trans. Inf. Theory, vol. 45, pp. 1315-1321, 1999.
56. M. Palus, "Testing for nonlinearity using redundancies: Quantitative and qualitative aspects," Physica D, vol. 80, pp. 186-205, 1995.
57. J. D. Farmer and J. J. Sidorowich, "Predicting chaotic time series," Phys. Rev. Lett., vol. 59, pp. 845-848, 1987.
58. H. Lutkepohl, New Introduction to Multiple Time Series Analysis. Berlin, Germany: Springer-Verlag, 2005.
59. W. Hesse, E. Moller, M. Arnold, and B. Schack, "The use of time-variant EEG Granger causality for inspecting directed interdependencies of neural assemblies," J. Neurosci. Meth., vol. 124, pp. 27-44, 2003.
60. A. B. Barrett et al., "Granger causality analysis of steady-state electroencephalographic signals during propofol-induced anaesthesia," PLoS ONE, vol. 7, 2012, e29072.
61. X. Wen, L. Yao, Y. Liu, and M. Ding, "Causal interactions in attention networks predict behavioral performance," J. Neurosci., vol. 32, pp. 1284-1292, 2012.
62. A. Roebroeck, E. Formisano, and R. Goebel, "Mapping directed influence over the brain using Granger causality and fMRI," Neuroimage, vol. 25, pp. 230-242, 2005.
63. H. Akaike, "A new look at the statistical novel identification," IEEE Trans. Autom. Control, vol. 19, pp. 716-723, 1974.
64. J. Rissanen, "Modeling by shortnest data description," Automatica, vol. 14, pp. 465-471, 1978.
65. G. Schwarz, "Estimating the dimension of a model," Ann. Stat., vol. 6, pp. 461-464, 1978.
66. S. Lu, K. H. Ju, and K. H. Chon, "A new algorithm for linear and non linear ARMA model parameters estimation using affine geometry," IEEE Trans. Biomed. Eng., vol. 48, pp. 1116-1124, 2001.
67. S. A. Billings and H.-L. Wei, "Sparse model identification using a forward orthogonal regression algorithm aided by mutual information," IEEE Trans. Neural Netw., vol. 18, pp. 306-310, 2007.
68. M. W. McCracken, "Asymptotics for out of sample tests of Granger causality," J. Econom., vol. 140, pp. 719-752, 2007.
69. L. Faes and G. Nollo, "Extended causal modeling to assess partial directed coherence in multiple time series with significant instantaneous interactions," Biol. Cybern., vol. 103, pp. 387-400, 2010.
70. S. Nemati et al., "Respiration and heart rate complexity: Effects of age and gender assessed by band-limited transfer entropy," Resp. Physiol. Neurobi., vol. 189, pp. 27-33, 2013.
71. C. M. Romano, M. Thiel, J. Kurths, and C. Grebogi, "Estimation of the direction of the coupling by conditional probability of recurrence," Phys. Rev. E, vol. 76, 2007, 036211.
72. M. H. Perrott and R. J. Cohen, "An efficient approach to ARMA modeling of biological systems with multiple inputs and delays," IEEE Trans. Biomed. Eng., vol. 43, pp. 1-14, 1996.
73. Z. Zhou, Z. Chen, M. Ding, P. Wright, Z. Lu, and Y. Liu, "Analyzing brain networks with PCA and conditional Granger causality," Human Brain Mapping, vol. 30, pp. 2197-2206, 2009.
74. P. A. Valdes-Sosa et al., "Estimating brain functional connectivity with sparse multivariate autoregression," Phil. Trans. Roy. Soc. B, vol. 360, pp. 969-981, 2005.
75. D. W. Gow, J. A. Segawa, S. P. Ahlfors, and F.-H. Lin, "Lexical influences on speech perception: A Granger causality analysis of MEG and EEG source estimates," Neuroimage, vol. 43, pp. 614-623, 2008.
76. I. Vlachos and D. Kugiumtzis, "Nonuniform state-space reconstruction and coupling direction," Phys. Rev. E, vol. 82, 2010, 016207.
77. L. Faes, G. Nollo, and A. Porta, "Informationbased detection of nonlinear Granger causality in multivariate processes via a nonuniform embedding technique," Phys. Rev. E, vol. 83, 2011, 051112.
78. L. Faes, G. Nollo, and A. Porta, "Mechanisms of causal interaction between short-term RR interval and systolic arterial pressure oscillations during orthostatic challenge," J. Appl. Physiol., vol. 114, pp. 1657-1667, 2013.
79. J. Runge, J. Heitzig, V. Petoukhov, and J. Kurths, "Escaping the curse of dimensionality in estimating multivariate transfer entropy," Phys. Rev. Lett., vol. 108, 2012, 258701.
80. A. Wald, "Tests of statistical hypothesis concerning several parameters when the number of observations is large," Trans. Amer. Math. Soc., vol. 54, pp. 426-482, 1943.
81. J. Geweke, "Measurement of linear dependence and feedback between multiple time series," J. Amer. Stat. Assoc., vol. 77, pp. 304-313, 1982.
82. A. B. Barrett, L. Barnett, and A. K. Seth, "Multivariate Ganger causality and generalized variance," Phys. Rev. E, vol. 81, 2010, 041907.
83. M. G. Rosenblum and A. Pikovsky, "Detecting direction of coupling in interacting oscillators," Phys. Rev. E, vol. 63, 2001, 045202.
84. A. Porta et al., "Causal relationships between heart period and systolic arterial pressure during graded head-up tilt," Amer. J. Physiol., vol. 300, pp. R378-R386, 2011.
85. J. Arnhold, P. Grassberger, K. Lehnertz, and C. E. Elger, "A robust method for detecting interdependences: Application to intracranially recorded EEG," Physica D, vol. 134, pp. 419-430, 1999.
86. R. Q. Quiroga, J. Arnhold, and P. Grassberger, "Learning driver-response relationships from synchronization patterns," Phys. Rev. E, vol. 61, pp. 5142-5148, 2000.
87. M. Palus, V. Komarek, Z. Hrncir, and K. Sterbova, "Synchronization as adjustment of information rates: Detection from bivariate time series," Phys. Rev. E, vol. 63, 2001, 046211.
88. A. Schmitz, "Measuring statistical dependence and coupling of subsystems," Phys. Rev. E, vol. 62, pp. 7508-7511, 2000.
89. D. Chicharro and S. Panzeri, "Algorithms of causal inference for the analysis of effective connectivity among brain regions," Front. Neuroinf., vol. 8, p. 64, 2014.
90. D. Marinazzo, M. Pellicoro, and S. Stramaglia, "Causal information approach to partial conditioning in multivariate data sets," Comput. Math. Method Med., vol. 2012, 2012, 303601.
91. W. J. Welch, "Algorithmic complexity: Three NP-hard problems in computational statistics," J. Stat. Comput. Simul., vol. 15, pp. 17-25, 1982.
92. W. Tang, S. L. Bressler, C. M. Sylvester, G. L. Shulman, and M. Corbetta, "Measuring Granger Causality between Cortical Regions from Voxelwise fMRI BOLD Signals with LASSO," PLoS Comput. Biol., vol. 8, 2012, e1002513.
93. K. Hlavackova-Schindler, M. Palus, M. Vejmelka, and J. Bhattacharya, "Causality detection based on information-theoretic approaches in time series analysis," Phys. Rep., vol. 441, pp. 1-46, 2007.
94. S. Kullback and R. A. Leibler, "On information and sufficiency," Ann. Math. Stat., vol. 22, pp. 79-86, 1951.
95. D. Kugiumtzis, "Direct-coupling information measure from nonuniform embedding," Phys. Rev. E, vol. 87, 2013, 062918.
96. R. Vicente, M. Wibral, M. Lindner, and G. Pipa, "Transfer entropyVA model-free measure of effective connectivity for the neurosciences," J. Comput. Neurosci., vol. 30, pp. 45-67, 2011.
97. P. F. Verdes, "Assessing causality from multivariate series," Phys. Rev. E, vol. 72, 2005, 026222.
98. A. Porta et al., "Conditional entropy approach for the evaluation of the coupling strength," Biol. Cybern., vol. 81, pp. 119-129, 1999.
99. D. Prichard and J. Theiler, "Generalized redundancies for time series analysis," Physica D, vol. 84, pp. 476-493, 1995.
100. R. J. McEliece, The Theory of Information and Coding. Cambridge, U.K.: Cambridge Univ. Press, 2002.
101. A. Porta et al., "Measuring regularity by means of a corrected conditional entropy in sympathetic outflow," Biol. Cybern., vol. 78, pp. 71-78, 1998.
102. C. Daub, R. Steuer, J. Selbig, and S. Kloska, "Estimating mutual information using B-spline functionsVAn improved similarity measure for analysing gene expression data," BMC Bioinf., vol. 5, pp. 1-12, 2004.
103. P. Grassberger and I. Procaccia, "Measuring the strangeness of strange attractors," Physica D, vol. 9, pp. 189-208, 1983.
104. L. F. Kozachenko and N. N. Leonenko, "Sample estimate of entropy of a random vector," Probl. Inf. Transm., vol. 23, pp. 95-100, 1987.
105. A. Kraskov, H. Stoegbauer, and P. Grassberger, "Estimating mutual information," Phys. Rev. E, vol. 69, 2004, 066138.
106. M. Wibral et al., "Transfer entropy in magnetoencephalographic data: Quantifying information flow in cortical and cerebellar networks," Prog. Biophys. Mol. Biol., vol. 105, pp. 80-97, 2011.
107. L. Faes, D. Kugiumtzis, A. Montalto, G. Nollo, and D. Marinazzo, "Estimating the decomposition of predictive information in multivariate systems," Phys. Rev. E, vol. 91, 2015, 032904.
108. S. M. Pincus, "Approximate entropy (ApEn) as a complexity measure," Chaos, vol. 5, pp. 110-117, 1995.
109. J. S. Richman and J. R. Moorman, "Physiological time-series analysis using approximate entropy and sample entropy," Amer. J. Physiol., vol. 278, pp. H2039-H2049, 2000.
110. A. Porta et al., "Progressive decrease of heart period variability entropy-based complexity during graded head-up tilt," J. Appl. Physiol., vol. 103, pp. 1143-1149, 2007.
111. A. Porta et al., "K-nearest-neighbor conditional entropy approach for the assessment of short-term complexity of cardiovascular control," Physiol. Meas., vol. 34, pp. 17-33, 2013.
112. L. Cao, "Practical method for determining the minimum embedding dimension of a scalar time series," Physica D, vol. 110, pp. 43-50, 1997.
113. M. Papapetrou and D. Kugiumtzis, "Markov chain order estimation with conditional mutual information," Physica A, vol. 392, pp. 1593-1601, 2013.
114. R. Steuer, J. Kurths, C. O. Daub, J. Weise, and J. Selbig, "The mutual information: Detecting and evaluating dependencies between variables," Bioinformatics, vol. 18, pp. S231-S240, 2002.
115. C. J. Cellucci, A. M. Albano, and P. E. Rapp, "Statistical validation of mutual information calculations: Comparison of alternative numerical algorithms," Phys. Rev. E, vol. 71, 2005, 066208.
116. G. Tononi, O. Sporns, and G. M. Edelman, "A measure for brain complexity: Relating functional segregation and integration in the nervous system," Proc. Nat. Acad. Sci. USA, vol. 91, pp. 5033-5037, 1994.
117. L. Barnett, A. B. Barrett, and A. K. Seth, "Granger causality and transfer entropy are equivalent for Gaussian variables," Phys. Rev. Lett., vol. 103, 2009, 238701.
118. L. Faes, A. Porta, and G. Nollo, "Information decomposition in bivariate systems: Theory and application to cardiorespiratory dynamics," Entropy, vol. 17, pp. 277-303, 2015.
119. L. Barnett, C. L. Buckley, and S. Bullock, "Neural complexity and structural connectivity," Phys. Rev. E, vol. 79, 2009, 051914.
120. A. Porta et al., "Conditional self-entropy and conditional joint transfer entropy in heart period variability during graded postural challenge," PLoS ONE, vol. 10, 2015, e0132851.
121. L. A. Baccalà et al., "Studying the interaction between brain structures via direct coherence and Granger causality," Appl. Signal Process., vol. 5, pp. 40-48, 1998.
122. L. A. Baccalà and K. Sameshima, "Partial directed coherence: A new concept in neural structure determination," Biol. Cybern., vol. 84, pp. 463-474, 2001.
123. K. Sameshima and L. A. Baccalà, "Using partial directed coherence to describe neuronal ensemble interactions," J. Neurosci. Methods, vol. 94, pp. 93-103, 1999.
124. M. Kaminski and K. J. Blinowska, "A new method of the description of the information flow in the brain structure," Biol. Cybern., vol. 65, pp. 203-210, 1991.
125. M. Kaminski, M. Ding, W. A. Truccolo, and S. Bressler, "Evaluating causal relations in neural systems: Granger causality, directed transfer function and statistical assessment of significance," Biol. Cybern., vol. 85, pp. 145-157, 2001.
126. T. J. Mullen, M. L. Appel, R. Mukkamala, J. M. Mathias, and R. J. Cohen, "System identification of closed loop cardiovascular control: Effects of posture and autonomic blockade," Amer. J. Physiol., vol. 272, pp. H448-H461, 1997.
127. W. Gersch and G. V. Goddard, "Epileptic focus locationVSpectral analysis method," Science, vol. 169, pp. 701-702, 1970.
128. H. P. Koepchen, "Physiology of Rhythms and Control Systems: An Integrative Approach," in Rhythms in physiological systems, H. Haken and H. P. Koepchen, Eds. Berlin, Germany: Springer-Verlag, 1991, pp. 3-20.
129. R. W. De Boer, J. M. Karemaker, and J. Strackee, "Hemodynamic fluctuations and baroreflex sensitivity in humans: A beat-to-beat model," Amer. J. Physiol., vol. 253, pp. H680-H689, 1987.
130. D. L. Eckberg, "The human respiratory gate," J. Physiol., vol. 548, pp. 339-352, 2003.
131. G. Baselli, A. Porta, and M. Pagani, "Coupling arterial windkessel with peripheral vasomotion: Modeling the effects on low-frequency oscillations," IEEE Trans. Biomed. Eng., vol. 53, pp. 53-64, 2006.
132. G. Buzsaki, Rhythms in the Brain. New York, NY, USA: Oxford Univ. Press, 2006.
133. L. Barnett and A. K. Seth, "Behaviour of Granger causality under filtering: Theoretical invariance and practical application," J. Neurosci. Methods, vol. 201, pp. 404-419, 2011.
134. D. Chicharro, "On the spectral formulation of Granger causality," Biol. Cybern., vol. 105, pp. 331-347, 2011.
135. F. He, S. A. Billings, H.-L. Wei, and P. G. Sarrigiannis, "A nonlinear causality measure in the frequency domain: Nonlinear partial directed coherence with applications to EEG," J. Neurosci. Methods, vol. 225, pp. 71-80, 2014.
136. F. He, H.-L. Wei, S. A. Billings, and P. G. Sarrigiannis, "A nonlinear generalization of spectral Granger causality," IEEE Trans. Biomed. Eng., vol. 61, pp. 1693-1701, 2014.
137. A. Porta et al., "Quantifying the strength of the linear causal coupling in closed loop interacting cardiovascular variability signals," Biol. Cybern., vol. 86, pp. 241-251, 2002.
138. L. Faes, S. Erla, A. Porta, and G. Nollo, "A framework for assessing frequency domain causality in physiological time series with instantaneous effects," Phil. Trans. Roy. Soc. A, vol. 371, 2013, 20110618.
139. L. Faes, G. Nollo, and A. Porta, "Compensated transfer entropy as a tool for reliably estimating information transfer in physiological time series," Entropy, vol. 15, pp. 198-219, 2013.
140. H. Akaike, "On the use of a linear model for the identification of feedback systems," Ann. Inst. Stat. Math., vol. 20, pp. 425-439, 1968.
141. Y. Saito and H. Harashima, "Tracking of information within multichannel EEG record: Causal analysis in EEG," in Recent Advances in EEG and EMG Data Processing, N. Yamaguchi and K. Fujisawa, Eds. Amsterdam, The Netherlands: Elsevier, 1981, pp. 133-146.
142. M. Eichler, "On the evaluation of information flow in multivariate systems by the directed transfer function," Biol. Cybern., vol. 94, pp. 469-482, 2006.
143. L. A. Baccalà, C. S. N. de Brito, D. Y. Takahashi, and K. Sameshima, "Unified asymptotic theory for all partial directed coherence forms," Phil. Trans. Roy. Soc. A, vol. 371, 2013, 20120158.
144. D. Y. Takahashi, L. A. Baccalà, and K. Sameshima, "Information theoretic interpretation of frequency domain connectivity measures," Biol. Cybern., vol. 103, pp. 463-469, 2010.
145. B. Schelter, J. Timmer, and M. Eichler, "Assessing the strength of directed influences among neural signals using renormalized partial directed coherence," J. Neurosci. Methods, vol. 179, pp. 121-130, 2009.
146. A. Porta, E. Tobaldini, T. Gnecchi-Ruscone, and N. Montano, "RT variability unrelated to heart period and respiration progressively increases during graded head-up tilt," Amer. J. Physiol., vol. 298, pp. H1406-H1414, 2010.
147. A. Porta, T. Bassani, V. Bari, E. Tobaldini, A. C. M. Takahashi, A. M. Catai, and N. Montano, "Model-based assessment of baroreflex and cardiopulmonary couplings during graded head-up tilt," Comput. Biol. Med., vol. 42, pp. 298-305, 2012.
148. G. Nollo, L. Faes, A. Porta, R. Antolini, and F. Ravelli, "Exploring directionality in spontaneous heart period and systolic arterial pressure variability interactions in humans: Implications in the evaluation of baroreflex gain," Amer. J. Physiol., vol. 288, pp. H1777-H1785, 2005.
149. H. Zhao, W. A. Cupples, K. H. Ju, and K. H. Chon, "Time-varying causal coherence function and its application to renal blood pressure and blood flow data," IEEE Trans. Biomed. Eng., vol. 54, pp. 2142-2150, 2007.
150. J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures. New York, NY, USA: Wiley, 1986.
151. J. R. Rosenberg, D. M. Halliday, P. Breeze, and B. A. Conway, "Identification of patterns of neural connectivityVPartial spectra partial coherence neuronal interactions," J. Neurosci. Methods, vol. 83, pp. 57-72, 1998.
152. L. A. Baccalà and K. Sameshima, "Comments on 'Is partial coherence a viable technique for identifying generators of neural oscillations?' Why the term 'Gersch Causality' is inappropriate: Common neural structure inference pitfalls," Biol. Cybern., vol. 95, pp. 135-141, 2006.
153. B. Schelter, M. Winterhalder, R. Dahlhaus, J. Kurths, and J. Timmer, "Partial phase synchronization for multivariate synchronizing systems," Phys. Rev. Lett., vol. 96, 2006, 208103.
154. L. Faes, S. Erla, and G. Nollo, "Measuring connectivity in linear multivariate processes: Definitions, interpretation and practical analysis," Comput. Math. Method Med., 2012, 140513.
155. Y. Chen, S. L. Bressler, and M. Ding, "Frequency decomposition of conditional Granger causality and application to multivariate neural field potential data," J. Neurosci. Methods, vol. 179, pp. 121-130, 2006.
156. L. Faes and G. Nollo, "Measuring frequency domain Granger causality for multiple blocks of interacting time series," Biol. Cybern., vol. 107, pp. 217-232, 2013.
157. X. Wen, G. Rangarajan, and M. Ding, "Multivariate Granger causality: An estimation framework based on factorization of the spectral density matrix," Phil. Trans. Roy. Soc. A, vol. 371, 2013, 20110610.
158. A. G. Nedungadi, M. Ding, and G. Rangarajan, "Block coherence: A method for measuring the interdependence between two blocks of neurobiological time series," Biol. Cybern., vol. 104, pp. 197-207, 2011.
159. M. Dhamala, G. Rangarajan, and M. Ding, "Analyzing information flow in brain networks with nonparametric Granger causality," Neuroimage, vol. 41, pp. 354-362, 2008.
160. A. Hyvarinen, K. Zhang, S. Shimizu, and P. O. Hoyer, "Estimation of a structural vector autoregression model using non-Gaussianity," J. Mach. Learn. Res., vol. 11, pp. 1709-1731, 2010.
161. L. Schiatti, G. Nollo, G. Rossato, and L. Faes, "Extended Granger causality: A new tool to identify the structure of physiological networks," Physiol. Meas., vol. 36, pp. 827-843, 2015.
162. S. Guo, A. K. Seth, K. M. Kendrick, C. Zhou, and J. Feng, "Partial Granger causality-Eliminating exogenous inputs and latent variables," J. Neurosci. Methods, vol. 172, pp. 79-193, 2008.
163. G. Deshpande, K. Sathian, and X. P. Hu, "Assessing and compensating for zero-lag correlation effects in time-lagged Granger causality analysis of fMRI," IEEE Trans. Biomed. Eng., vol. 57, pp. 1446-1456, 2010.
164. A. Hyvarinen and S. M. Smith, "Pairwise likelihood ratios for estimation of non-Gaussian structural equation models," J. Mach. Learn. Res., vol. 14, pp. 111-152, 2013.
165. A. Porta et al., "Information domain analysis of the spontaneous baroreflex during pharmacological challenges," Autonom. Neurosci.VBasic Clin., vol. 178, pp. 67-75, 2013.
166. B. Schelter et al., "Testing for directed influences among neural signals using partial directed coherence," J. Neurosci. Methods, vol. 152, pp. 210-219, 2006.
167. J. T. Lizier, J. Heinzle, A. Horstmann, J.-D. Haynes, and M. Prokopenko, "Multivariate information-theoretic measures reveal directed information structure and task relevant changes in fMRI connectivity," J. Comput. Neurosci., vol. 30, pp. 85-107, 2011.
168. J. Theiler, S. Eubank, A. Longtin, and J. Galdrikian, "Testing for nonlinearity in time series: The method of surrogate data," Physica D, vol. 58, pp. 77-94, 1992.
169. D. Prichard and J. Theiler, "Generating surrogate data from time series with several simultaneously variables," Phys. Rev. Lett., vol. 73, pp. 951-954, 1994.
170. M. Palus, "Detecting phase synchronisation in noisy systems," Phys. Lett. A, vol. 235, pp. 341-351, 1997.
171. T. Schreiber and A. Schmitz, "Improved surrogate data for nonlinearity tests," Phys. Rev. Lett., vol. 77, pp. 635-638, 1996.
172. J. Theiler and P. E. Rapp, "Re-examination of the evidence for low-dimensional, nonlinear structure in the human electroencephalogram," Electroencephalogr. Clin. Neurophysiol., vol. 98, pp. 213-222, 1996.
173. C. J. Stam, J. P. M. Pijn, and W. S. Pritchard, "Reliable detection of nonlinearity in experimental time series with strong periodic components," Physica D, vol. 112, pp. 361-380, 1998.
174. M. Small, D. Yu, and R. G. Harrison, "Surrogate test for pseudoperiodic time series data," Phys. Rev. Lett., vol. 87, 2001, 188101.
175. T. Schreiber and A. Schmitz, "Surrogate time series," Physica D, vol. 142, pp. 346-382, 2000.
176. M. Thiel,M. C. Romano, J. Kurths,M. Rolfs, and R. Kliegl, "Twin surrogates to test for complex synchronization," Europhys. Lett., vol. 75, pp. 535-541, 2006.
177. M. Chavez, J. Martinerie, and M. Le Van Quyen, "Statistical assessment of nonlinear causality: Application to epileptic EEG signals," J. Neurosci. Methods, vol. 124, pp. 113-128, 2003.
178. R. G. Andrzejak, A. Kraskov, H. Stogbauer, F. Mormann, and T. Kreuz, "Bivariate surrogate techniques: Necessity, strengths, caveats," Phys. Rev. E, vol. 68, 2003, 066202.
179. R. Q. Quiroga, A. Kraskov, T. Kreuz, and P. Grassberger, "Performance of different synchronization measures in real data: A case study on electroencephalographic signals," Phys. Rev. E, vol. 65, 2002, 041903.
180. P. Van Leeuwen, D. Geue, S. Lange, D. Cysarz, H. Bettermann, and D. Gronemeyer, "Is there evidence of fetal-maternal heart rate synchronization" BMC Physiol., vol. 3, p. 2, 2003.
181. R. Marschinski and H. Kantz, "Analysing the information flow between financial time series An improved estimator for transfer entropy," Eur. Phys. J. B, vol. 30, pp. 275-281, 2002.
182. A. Papana, D. Kugiumtzis, and P. G. Larsson, "Reducing the bias of causality measures," Phys. Rev. E, vol. 83, 2011, 036207.
183. L. Faes, D. Marinazzo, A. Montalto, and G. Nollo, "Lag-specific transfer entropy as a tool to assess cardiovascular and cardiorespiratory information transfer," IEEE Trans. Biomed. Eng., vol. 61, pp. 2556-2568, 2014.
184. G. Nollo et al., "Evidence of unbalanced regulatory mechanism of heart rate and systolic pressure after acute myocardial infarction," Amer. J. Physiol., vol. 283, pp. H1200-H1207, 2002.
185. L. Faes, A. Porta, and G. Nollo, "Testing frequency-domain causality in multivariate series," IEEE Trans. Biomed. Eng., vol. 57, pp. 1897-1906, 2010.
186. T. Bassani et al., "Model-free causality analysis of cardiovascular variability detects the amelioration of the autonomic control in Parkinson's disease patients undergoing mechanical stimulation," Physiol. Meas., vol. 35, pp. 1397-1408, 2014.
187. M. A. Kramer, E. D. Kolaczyk, and H. E. Kirsch, "Emergent network topology at seizure onset in humans," Epilepsy Res., vol. 79, pp. 173-186, 2008.
188. A. Montalto, L. Faes, and D. Marinazzo, "MuTE: A MATLAB Toolbox to Compare Established and Novel Estimators of the Multivariate Transfer Entropy," PLoS ONE, vol. 9, 2014, e109462.
189. P. Numez, "Wavelike properties of the Alpha rhythm," IEEE Trans. Biomed. Eng., vol. 21, pp. 473-482, 2015.
190. T. Bassani et al., "Coherence analysis overestimates the role of baroreflex in governing the interactions between heart period and systolic arterial pressure variabilities during general anesthesia," Autonom. Neurosci.VBasic Clin., vol. 178, pp. 83-88, 2013.
191. M. Riedl, A. Suhrbier, H. Stepan, J. Kurths, and N. Wessel, "Short-term couplings of the cardiovascualr system in pregnant women suffering from pre-eclampsia," Phil. Trans. Roy. Soc. A, vol. 368, pp. 2237-2250, 2010.
192. L. Faes et al., "Investigating the mechanisms of cardiovascular and cerebrovascular regulation in orthostatic syncope through an information decomposition strategy," Autonom. Neurosci.VBasic Clin., vol. 178, pp. 76-82, 2013.
193. C. Varon, A. Montalto, D. Marinazzo, L. Faes, and S. Van Huffel, "Interictal cardiorespiratory variability in temporal lobe and absence epilepsy in childhood," Physiol. Meas., vol. 36, pp. 845-856, 2015.
194. A. Brovelli et al., "Beta oscillations in a large-scale sensorimotor cortical network: Directional influences revealed by Granger causality," Proc. Nat. Acad. Sci. USA, vol. 101, pp. 9849-9854, 2004.
195. L. Astolfi et al., "Comparison of different cortical connectivity estimators for high-resolution EEG recordings," Human Brain Mapping, vol. 28, pp. 143-157, 2007.
196. M. Palus, V. Komarek, T. Procházka, Z. Hrncir, and K. Sterbova, "Synchronization and information flow in EEGs of epileptic patients: A potentially promising method for anticipating epileptic seizures," IEEE Eng. Med. Biol. Mag., vol. 20, pp. 65-71, 2001.
197. G. Tononi, G. M. Edelman, and O. Sporns, "Complexity and coherency: Integrating information in the brain," Trends Cogn. Sci., vol. 2, pp. 474-484, 1998.
198. S. Haufe, V. V. Nikulin, K.-R. Muller, and G. Nolte, "A critical assessment of connectivity measures for EEG data: A simulation study," Neuroimage, vol. 64, pp. 120-133, 2013.
199. S. M. Smith et al., "Network modeling methods in FMRI," Neuroimage, vol. 54, pp. 875-891, 2011.