Abstract
Irregular and complex signals are ubiquitous in nature. The principal aim of this paper is to develop an index, quantifying the complexity of such signals, which is based on the distribution of the strengths of its orthogonal oscillatory modes estimated by singular value decomposition. The index is first tested with simulated chaotic and/or stochastic maps and flows. Among neural data analysis, the index is first applied to a cognitive EEG data set recorded from two groups, musicians and non-musicians, during listening to music and resting state. In the gamma band (30–50 Hz), musicians showed robust changes in complexity, consistent over various scalp regions, during listening to music from resting condition, whereas such changes were minimal for non-musicians. Then the index is used to separate healthy participants from epileptic and manic patients based on spontaneous EEG analysis. Finally, it is used to track a tonic-clonic seizure EEG signal, and a conspicuous change was found in the complexity profiles of delta band (1–3.5 Hz) oscillations at the onset of seizure. We conclude that this index would be useful for quantification of a wide range of time series including neural oscillations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baofu, P. (2007). The future of complexity: conceiving a better way to understand order and chaos. World Scientific.
Bhattacharya, J. (2001). Reduced degree of long-range phase synchrony in pathological human brain. Acta Neurobiologiae Experimentalis, 61, 309–318.
Bhattacharya, J., & Kanjilal, P. P. (1999). On the detection of determinism in a time series. Physica D. Nonlinear Phenomena, 132(1–2), 100–110. doi:10.1016/S0167-2789(99)00033-0.
Bhattacharya, J., Kanjilal, P. P., & Nizamie, S. H. (2000). Decomposition of posterior alpha rhythm. IEEE Transactions on Bio-Medical Engineering, 47(6), 738–747. doi:10.1109/10.844222.
Bhattacharya, J., & Petsche, H. (2001). Enhanced phase synchrony in the electroencephalograph gamma band for musicians while listening to music. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 64(1 Pt 1), 012902. doi:10.1103/PhysRevE.64.012902.
Bhattacharya, J., Petsche, H., & Pereda, E. (2001a). Interdependencies in the spontaneous EEG while listening to music. International Journal of Psychophysiology, 42(3), 287–301. doi:10.1016/S0167-8760(01)00153-2.
Bhattacharya, J., Petsche, H., & Pereda, E. (2001b). Long-range synchrony in the gamma band: role in music perception. The Journal of Neuroscience, 21(16), 6329–6337.
Broomhead, D. S., & King, G. P. (1986). Extracting qualitative dynamics from experimental-data. Physica D. Nonlinear Phenomena, 20(2–3), 217–236. doi:10.1016/0167-2789(86)90031-X.
Bunde, A., Kropp, J., & Schellnhuber, H.-J. (2002). The science of disasters: Climate disruptions, heart attacks and market crashes. Berlin; New York: Springer.
Deprettere, E. F. (1988). SVD and signal processing: Algorithms, applications, and architectures. Amsterdam [Netherlands]; New York New York, N.Y., U.S.A: North-Holland Sole distributors for the U.S.A. and Canada, Elsevier Science Pub. Co.
Edelman, G. M. (1989). The remembered present. A biological theory of consciousness. New York: Basic Books.
Edmonds, B. (1999). What is complexity?: The philosophy of complexity per se with application to some examples in evolution. In F. Heylighen, & D. Aerts (Eds.), (pp. 1–18). Dordrecht, Netherlands: Kluwer Academic.
Golub, G. H., & Van Loan, C. F. (1996). Matrix factorizations. Baltimore, MD: The Johns Hopkins University Press.
Henon, M. (1976). 2-dimensional mapping with a strange attractor. Communications in Mathematical Physics, 50(1), 69–77. doi:10.1007/BF01608556.
Holmes, P., Lumley, J. L., & Berkooz, G. (2008). Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press.
Hornero, R., Escudero, J., Fernandez, A., Poza, J., & Gomez, C. (2008). Spectral and nonlinear analyses of MEG background activity in patients with Alzheimer’s disease. IEEE Transactions on Bio-Medical Engineering, 55(6), 1658–1665. doi:10.1109/TBME.2008.919872.
Hu, X., & Nenov, V. (2004). Robust measure for characterizing generalized synchronization. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 69(2), 7. doi:10.1103/PhysRevE.69.026206.
Kaczmarek, L. K., & Babloyantz, A. (1977). Spatiotemporal patterns in epileptic seizures. Biological Cybernetics, 26(4), 199–208. doi:10.1007/BF00366591.
Kanjilal, P. P., Bhattacharya, J., & Saha, G. (1999). Robust method for periodicity detection and characterization of irregular cyclical series in terms of embedded periodic components. Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 59(4), 4013–4025. doi:10.1103/PhysRevE.59.4013.
Kantz, H., Kurths, J., & Mayer-Kress, G. (1998). Nonlinear analysis of physiological data. Berlin; New York: Springer.
Kantz, H., & Schreiber, T. (2004). Nonlinear time series analysis (2nd ed.). Cambridge, UK; New York: Cambridge University Press.
Lempel, A., & Ziv, J. (1976). Complexity of finite sequences. IEEE Transactions on Information Theory, 22(1), 75–81. doi:10.1109/TIT.1976.1055501.
Mackey, M. C., & Glass, L. (1977). Oscillation and chaos in physiological control-systems. Science, 197(4300), 287–288. doi:10.1126/science.267326.
May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261(5560), 459–467. doi:10.1038/261459a0.
Mees, A. I., Rapp, P. E., & Jennings, L. S. (1987). Singular-value decomposition and embedding dimension. Physical Review A, 36(1), 340–346. doi:10.1103/PhysRevA.36.340.
Nicolis, G., & Nicolis, C. (2007). Foundations of complex systems: Nonlinear dynamics, statistical physics, information and prediction. World Scientific Publishing Company.
Pincus, S. (1995). Approximate entropy (Apen) as a complexity measure. Chaos (Woodbury, N.Y.), 5(1), 110–117. doi:10.1063/1.166092.
Quian Quiroga, R., Blanco, S., Rosso, O. A., Garcia, H., & Rabinowicz, A. (1997). Searching for hidden information with Gabor transform in generalized tonic-clonic seizures. Electroencephalography and Clinical Neurophysiology, 103(4), 434–439. doi:10.1016/S0013-4694(97)00031-X.
Rapp, P. E., Cellucci, C. J., Watanabe, T. A. A., & Albano, A. M. (2005). Quantitative characterization of tide complexity of multichannel human EEGs. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 15(5), 1737–1744. doi:10.1142/S0218127405012764.
Rosso, O. A., Blanco, S., Yordanova, J., Kolev, V., Figliola, A., Schurmann, M., et al. (2001). Wavelet entropy: a new tool for analysis of short duration brain electrical signals. Journal of Neuroscience Methods, 105(1), 65–75. doi:10.1016/S0165-0270(00)00356-3.
Schreiber, T., & Schmitz, A. (1996). Improved surrogate data for nonlinearity tests. Physical Review Letters, 77(4), 635–638. doi:10.1103/PhysRevLett.77.635.
Trefethen, L. N., & Bau, D. (1997). Numerical linear algebra. Philadelphia: SIAM.
van den Broek, P. L. C., van Egmond, J., van Rijn, C. M., Takens, F., Coenen, A. M. L., & Booij, L. (2005). Feasibility of real-time calculation of correlation integral derived statistics applied to EEG time series. Physica D. Nonlinear Phenomena, 203(3–4), 198–208. doi:10.1016/j.physd.2005.03.012.
Acknowledgments
The authors thank Rodrigo Quian Quiroga for the data from the epileptic patient used in Section 3.2.3. The research is supported by JST.ERATO Shimojo project (J.B.). E. Pereda acknowledges the financial support of Canary Government through the grant PI042005/005. Author Contributions: J.B. conceived the proposed index and performed the computations; J.B. and E.P. wrote the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Action Editor: Rob Kass
Rights and permissions
About this article
Cite this article
Bhattacharya, J., Pereda, E. An index of signal mode complexity based on orthogonal transformation. J Comput Neurosci 29, 13–22 (2010). https://doi.org/10.1007/s10827-009-0155-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10827-009-0155-5