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The extended Granger causality analysis for Hodgkin-Huxley neuronal models. (English) Zbl 1451.92067

Summary: How to extract directions of information flow in dynamical systems based on empirical data remains a key challenge. The Granger causality (GC) analysis has been identified as a powerful method to achieve this capability. However, the framework of the GC theory requires that the dynamics of the investigated system can be statistically linearized; i.e., the dynamics can be effectively modeled by linear regressive processes. Under such conditions, the causal connectivity can be directly mapped to the structural connectivity that mediates physical interactions within the system. However, for nonlinear dynamical systems such as the Hodgkin-Huxley (HH) neuronal circuit, the validity of the GC analysis has yet been addressed; namely, whether the constructed causal connectivity is still identical to the synaptic connectivity between neurons remains unknown. In this work, we apply the nonlinear extension of the GC analysis, i.e., the extended GC analysis, to the voltage time series obtained by evolving the HH neuronal network. In addition, we add a certain amount of measurement or observational noise to the time series to take into account the realistic situation in data acquisition in the experiment. Our numerical results indicate that the causal connectivity obtained through the extended GC analysis is consistent with the underlying synaptic connectivity of the system. This consistency is also insensitive to dynamical regimes, e.g., a chaotic or non-chaotic regime. Since the extended GC analysis could in principle be applied to any nonlinear dynamical system as long as its attractor is low dimensional, our results may potentially be extended to the GC analysis in other settings.
©2020 American Institute of Physics

MSC:

92C20 Neural biology
37N25 Dynamical systems in biology
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[1] Strogatz, S. H., Exploring complex networks, Nature, 410, 268-276 (2001) · Zbl 1370.90052
[2] Boccaletti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D.-U., Complex networks: Structure and dynamics, Phys. Rep., 424, 175-308 (2006) · Zbl 1371.82002
[3] Orcutt, G. H., Actions, consequences, and causal relations, Rev. Econ. Stat., 34, 305-313 (1952)
[4] Granger, C. W. J., Investigating causal relations by econometric models and cross-spectral methods, Econometrica, 37, 424-438 (1969) · Zbl 1366.91115
[5] Geweke, J., “Inference and causality in economic time series models,” in Handbook of Econometrics, 1st ed. (Elsevier, 1984), Vol. 2, pp. 1101-1144. · Zbl 0587.62197
[6] Schreiber, T., Measuring information transfer, Phys. Rev. Lett., 85, 461-464 (2000)
[7] Hlavavckova-Schindler, K.; Palus, M.; Vejmelka, M.; Bhattacharya, J., Causality detection based on information-theoretic approaches in time series analysis, Phys. Rep., 441, 1-46 (2007)
[8] Nolte, G.; Ziehe, A.; Nikulin, V. V.; Schlogl, A.; Kramer, N.; Brismar, T.; Muller, K. R., Robustly estimating the flow direction of information in complex physical systems, Phys. Rev. Lett., 100, 234101 (2008)
[9] Kaufmann, R. K.; Stern, D. I., Evidence for human influence on climate from hemispheric temperature relations, Nature, 388, 39-44 (1997)
[10] Vanhellemont, F.; Didier, F.; Christine, B., Cosmic rays and stratospheric aerosols: Evidence for a connection?, Geophys. Res. Lett., 29, 1715-1718 (2002)
[11] Ganapathy, R.; Rangarajan, G.; Sood, A., Granger causality and cross recurrence plots in rheochaos, Phys. Rev. E, 75, 016211 (2007)
[12] Angelini, L.; Pellicoro, M.; Stramaglia, S., Granger causality for circular variables, Phys. Lett. A, 373, 2467-2470 (2009) · Zbl 1231.62156
[13] Attanasio, A.; Triacca, U., Detecting human influence on climate using neural networks based Granger causality, Theor. Appl. Climatol., 103, 103 (2011)
[14] Triacca, U., Is Granger causality analysis appropriate to investigate the relationship between atmospheric concentration of carbon dioxide and global surface air temperature?, Theor. Appl. Climatol., 81, 133 (2005)
[15] Dhamala, M.; Rangarajan, G.; Ding, M., Analyzing information flow in brain networks with nonparametric Granger causality, NeuroImage, 41, 354-362 (2008)
[16] Gow, D. W. J.; Segawa, J. A.; Ahlfors, S. P.; Lin, F. H., Lexical influences on speech perception: A Granger causality analysis of MEG and EEG source estimates, NeuroImage, 43, 614-623 (2008)
[17] Hamilton, J. P.; Chen, G.; Thomason, M. E.; Schwartz, M. E.; Gotlib, I. H., Investigating neural primacy in major depressive disorder: Multivariate Granger causality analysis of resting-state FMRI time-series data, Mol. Psychiatry, 16, 763-772 (2011)
[18] Hempel, S.; Koseska, A.; Kurths, J.; Nikoloski, Z., Inner composition alignment for inferring directed networks from short time series, Phys. Rev. Lett., 107, 054101 (2011)
[19] Freeman, J. R., Granger causality and the times series analysis of political relationships, Am. J. Polit. Sci., 27, 327-358 (1983)
[20] Narayan, P. K.; Smyth, R., Crime rates, male youth unemployment and real income in Australia: Evidence from Granger causality tests, Appl. Econ., 36, 2079-2095 (2004)
[21] Tsen, W. H., Granger causality tests among openness to international trade, human capital accumulation and economic growth in China, Int. Econ. J., 20, 285-302 (2006)
[22] Francis, B. M.; Moseley, L.; Iyare, S. O., Energy consumption and projected growth in selected Caribbean countries, Energ. Econ., 29, 1224-1232 (2007)
[23] Geweke, J., Measurement of linear dependence and feedback between multiple time series, J. Am. Stat. Assoc., 77, 304-313 (1982) · Zbl 0492.62078
[24] Geweke, J., Measures of conditional linear dependence and feedback between time series, J. Am. Stat. Assoc., 79, 907-915 (1984) · Zbl 0553.62083
[25] Bressler, S. L.; Seth, A. K., Wiener-Granger causality: A well established methodology, NeuroImage, 58, 323-329 (2011)
[26] Ding, M., Chen, Y., and Bressler, S. L., “Granger causality: Basic theory and application to neuroscience,” in Handbook of Time Series Analysis, edited by S. Schelter, M. Winterhalder, and J. Timmer (Wiley-VCH, Berlin, 2006), pp. 437-460. · Zbl 1268.92079
[27] Upadhyay, J.; Silver, A.; Knaus, T. A.; Lindgren, K. A.; Ducros, M., Effective and structural connectivity in the human auditory cortex, J. Neurosci., 28, 3341-3349 (2008)
[28] Rubinov, M.; Sporns, O., Complex network measures of brain connectivity: Uses and interpretations, NeuroImage, 52, 1059-1069 (2010)
[29] Stetter, O.; Battaglia, D.; Soriano, J.; Geisel, T., Model-free reconstruction of excitatory neuronal connectivity from calcium imaging signals, PLoS Comput. Biol., 8, e1002653 (2012)
[30] Kispersky, T.; Gutierrez, G. J.; Marder, E., Functional connectivity in a rhythmic inhibitory circuit using Granger causality, Neural Syst. Circuits, 1, 9-23 (2011)
[31] Marinazzo, D.; Pellicoro, M.; Stramaglia, S., Kernel method for nonlinear Granger causality, Phys. Rev. Lett., 100, 144103 (2008)
[32] Liao, W.; Marinazzo, D.; Pan, Z.; Gong, Q.; Chen, H., Kernel Granger causality mapping effective connectivity on FMRI data, IEEE Trans. Med. Imaging, 28, 1825-1835 (2009)
[33] Rauch, A.; Camera, G. L.; Luscher, H.; Senn, W.; Fusi, S., Neocortical pyramidal cells respond as integrate-and-fire neurons to in vivo-like input currents, J. Neurophysiol., 90, 1598-1612 (2003)
[34] Cai, D.; Rangan, A. V.; McLaughlin, D. W., Architectural and synaptic mechanisms underlying coherent spontaneous activity in V1, Proc. Natl. Acad. Sci. U.S.A., 102, 5868-5873 (2005)
[35] Rangan, A. V.; Cai, D.; McLaughlin, D. W., Modeling the spatiotemporal cortical activity associated with the line-motion illusion in primary visual cortex, Proc. Natl. Acad. Sci. U.S.A., 102, 18793-18800 (2005)
[36] Burkitt, A. N., A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input, Biol. Cybern., 95, 1-19 (2006) · Zbl 1161.92315
[37] Burkitt, A. N., A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties, Biol. Cybern., 95, 97-112 (2006) · Zbl 1161.92314
[38] Zhou, D.; Rangan, A. V.; Sun, Y.; Cai, D., Network-induced chaos in integrate-and-fire neuronal ensembles, Phys. Rev. E, 80, 031918 (2009)
[39] Newhall, K.; Kovacic, G.; Kramer, P.; Zhou, D.; Rangan, A. V.; Cai, D., Dynamics of current-based, Poisson driven, integrate-and-fire neuronal networks, Commun. Math. Sci., 8, 541-600 (2010) · Zbl 1197.82084
[40] Zhou, D.; Sun, Y.; Rangan, A. V.; Cai, D., Spectrum of Lyapunov exponents of non-smooth dynamical systems of integrate-and-fire type, J. Comput. Neurosci., 28, 229-245 (2010)
[41] Zhou, D.; Rangan, A. V.; McLaughlin, D. W.; Cai, D., Spatiotemporal dynamics of neuronal population response in the primary visual cortex, Proc. Natl. Acad. Sci. U.S.A., 110, 9517-9522 (2013)
[42] Zhou, D.; Xiao, Y.; Zhang, Y.; Xu, Z.; Cai, D., Causal and structural connectivity of pulse-coupled nonlinear networks, Phys. Rev. Lett., 111, 054102 (2013)
[43] Zhou, D.; Xiao, Y.; Zhang, Y.; Xu, Z.; Cai, D., Granger causality network reconstruction of conductance-based integrate-and-fire neuronal systems, PLoS ONE, 9, e87636 (2014)
[44] Hodgkin, A. L.; Huxley, A. F., A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117, 500-544 (1952)
[45] Sun, Y.; Zhou, D.; Rangan, A. V.; Cai, D., Library-based numerical reduction of the Hodgkin-Huxley neuron for network simulation, J. Comput. Neurosci., 27, 369-390 (2009)
[46] Sun, Y.; Zhou, D.; Rangan, A. V.; Cai, D., Pseudo-Lyapunov exponents and predictability of Hodgkin-Huxley neuronal network dynamics, J. Comput. Neurosci., 28, 247-266 (2010)
[47] Dayan, P.; Abbott, L., Theoretical Neuroscience (2001), MIT Press: MIT Press, Cambridge
[48] Chen, Y.; Rangarajan, G.; Feng, J.; Ding, M., Analyzing multiple nonlinear time series with extended Granger causality, Phys. Lett. A, 324, 26-35 (2004) · Zbl 1123.62316
[49] Parker, T.; Chua, L., Practical Numerical Algorithms for Chaotic Systems (1989), Springer-Verlag: Springer-Verlag, New York · Zbl 0692.58001
[50] Ott, E., Chaos in Dynamical Systems (1993), Cambridge University Press: Cambridge University Press, New York
[51] Koch, C., Biophysics of Computation: Information Processing in Single Neurons (1999), Oxford University Press: Oxford University Press, New York
[52] Takens, F., On the Numerical Determination of the Dimension of an Attractor (1985), Springer · Zbl 0561.58027
[53] Wiener, N., “The theory of prediction,” in Modern Mathematics for Engineers, edited by E. Beckenbach (McGraw-Hill, New York, 1956).
[54] Barnett, L.; Barrett, A. B.; Seth, A. K., Granger causality and transfer entropy are equivalent for Gaussian variables, Phys. Rev. Lett., 103, 238701 (2009)
[55] Akaike, H., Fitting autoregressive models for prediction, Ann. Inst. Stat. Math., 21, 243-247 (1969) · Zbl 0202.17301
[56] Breiman, L., Random forests, Mach. Learn., 45, 5-32 (2001) · Zbl 1007.68152
[57] Guyon, I.; Elisseeff, A., An introduction to variable and feature selection, J. Mach. Learn. Res., 3, 1157-1182 (2003) · Zbl 1102.68556
[58] Compte, A.; Sanchez-Vives, M. V.; McCormick, D. A.; Wang, X.-J., Cellular and network mechanisms of slow oscillatory activity \((< 1\) Hz) and wave propagations in a cortical network model, J. Neurophysiol., 89, 2707-2725 (2003)
[59] Jin, W.-Y.; Xu, J.-X.; Wu, Y.; Hong, L.; Wei, Y.-B., Crisis of interspike intervals in Hodgkin-Huxley model, Chaos Solitons Fractals, 27, 952-958 (2006) · Zbl 1102.37314
[60] Wolf, A.; Swift, J. B.; Swinney, H. L.; Vastano, J. A., Determining Lyapunov exponents from a time series, Physica D, 16, 285-317 (1985) · Zbl 0585.58037
[61] Shaw, R., Strange attractors, chaotic behavior, and information flow, Z. Naturforsch., 36A, 80-112 (1981) · Zbl 0599.58033
[62] Greene, W. H., Econometric Analysis (2002), Prentice-Hall: Prentice-Hall, Upper Saddle River, NJ
[63] Pandit, S.; Wu, S., Time Series and System Analysis with Applications (1983), Wiley: Wiley, New York
[64] McQuarrie, A.; Tai, C. L., Regression and Time Series Model Selection (1998), World Scientific: World Scientific, Hackensack, NJ
[65] Glass, L.; Perez, R., Fine structure of phase locking, Phys. Rev. Lett., 48, 1772-1775 (1982)
[66] Jensen, M. H.; Bak, P.; Bohr, T., Complete devil’s staircase, fractal dimension, and universality of mode-locking structure in the circle map, Phys. Rev. Lett., 50, 1637-1639 (1983)
[67] Alstrom, P.; Christiansen, B.; Levinsen, M. T., Nonchaotic transition from quasiperiodicity to complete phase locking, Phys. Rev. Lett., 61, 1679-1682 (1988)
[68] Li, S.; Xiao, Y.; Zhou, D.; Cai, D., Causal inference in nonlinear systems: Granger causality versus time-delayed mutual information, Phys. Rev. E, 97, 052216 (2018)
[69] Kraskov, A.; Stögbauer, H.; Grassberger, P., Estimating mutual information, Phys. Rev. E, 69, 066138 (2004)
[70] Wibral, M.; Vicente, R.; Lizier, J. T., Directed Information Measures in Neuroscience (2014), Springer
[71] Vicente, R.; Wibral, M.; Lindner, M.; Pipa, G., Transfer entropy—A model-free measure of effective connectivity for the neurosciences, J. Comput. Neurosci., 30, 45-67 (2011) · Zbl 1446.92187
[72] Ito, S.; Hansen, M. E.; Heiland, R.; Lumsdaine, A.; Litke, A. M.; Beggs, J. M., Extending transfer entropy improves identification of effective connectivity in a spiking cortical network model, PLoS ONE, 6, e27431 (2011)
[73] Faes, L.; Nollo, G.; Porta, A., Compensated transfer entropy as a tool for reliably estimating information transfer in physiological time series, Entropy, 15, 198-219 (2013)
[74] Hahs, D. W.; Pethel, S. D., Transfer entropy for coupled autoregressive processes, Entropy, 15, 767-788 (2013) · Zbl 1325.94046
[75] Barranca, V. J.; Johnson, D. C.; Moyher, J. L.; Sauppe, J. P.; Shkarayev, M. S.; Kovačič, G.; Cai, D., Dynamics of the exponential integrate-and-fire model with slow currents and adaptation, J. Comput. Neurosci., 37, 161-180 (2014) · Zbl 1382.92038
[76] Casdagi, M., “A dynamical systems approach to modeling input-output systems,” in Nonlinear Modeling and Forecasting, edited by M. Casdagi and S. Eubank (Addison-Wesley, 1992), Vol. XII, p. 265.
[77] Boccaletti, S.; Valladares, D. L.; Pecora, L. M.; Geffert, H. P.; Carroll, T., Reconstructing embedding spaces of coupled dynamical systems from multivariate data, Phys. Rev. E, 65, 035204 (2002)
[78] Eckmann, J. P.; Ruelle, D., Ergodic theory of chaos and strange attractors, Rev. Mod. Phys., 57, 617-656 (1985) · Zbl 0989.37516
[79] Farmer, J. D.; Sidorowich, J. J., Predicting chaotic time series, Phys. Rev. Lett., 59, 845-848 (1987)
[80] Schiff, S. J.; So, P.; Chang, T.; Burke, R. E.; Sauer, T., Detecting dynamical interdependence and generalized synchrony through mutual prediction in a neural ensemble, Phys. Rev. E, 54, 6708-6724 (1996)
[81] Takens, F., Dynamical Systems and Turbulence, Warwick, edited by D. A. Rand and L. S. Young (Springer, Berlin, 1981, 1980), Vol. 898, p. 366.
[82] Atten, P., Caputo, J. G., Malraison, B., and Gagne, Y., Special Issue: Bifurcation et Comportement Chaotique, edited by G. Iooss (J. Mec. Theor. Appl., 1984), p. 133.
[83] Fraser, A. M.; Swinney, H. L., Independent coordinates for strange attractors from mutual information, Phys. Rev. A, 33, 1134 (1986) · Zbl 1184.37027
[84] Kennel, M. B.; Brown, R.; Abarbanel, H. D. I., Determining embedding dimension for phase-space reconstruction using a geometrical construction, Phys. Rev. A, 45, 3403-3411 (1992)
[85] Cao, L., Practical method for determining the minimum embedding dimension of a scalar time series, Physica D, 110, 43-50 (1997) · Zbl 0925.62385
[86] (2020). “Extended-GC,” GitHub, see https://github.com/Grantcheng/Extended-GC.
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