Abstract
The Bonhoeffer–van der Pol (BVP) oscillator has widely application in the modeling of biological processes, and it also has rich nonlinear behavior including different topological properties between the continuous-time BVP oscillator and the discrete one. Multifractal Detrended Cross-Correlation Analysis (MF-DCCA) has proved useful in describing the correlations and detecting the scaling exponent of time series. We investigate the autocorrelations and cross-correlations of the time series in BVP model by using the MF-DCCA method. Our results show that there exist long range autocorrelations and cross-correlations in the BVP model time series, and that multifractal features are significant in the analyzed BVP model.
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Rulkov, N.F., Timofeev, I., Bazhenov, M.: Oscillations in large-scale cortical networks: map-based model. J. Comput. Neurosci. 17, 203–233 (2004)
Izhikevich, E.M.: Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. MIT Press, Cambridge (2007)
Rulkov, N.F.: Regularization of synchronized chaotic bursts. Phys. Rev. Lett. 86, 183–186 (2001)
De Vries, G.: Bursting as an emergent phenomenon in coupled chaotic maps. Phys. Rev. E 64, 051914 (2001)
Shilnikov, A.L., Rulkov, N.F.: Subthreshold oscillations in a map-based neuron model. Phys. Lett. A 328, 177–189 (2004)
Casado, J.M.: Transient activation in a network of coupled map neurons. Phys. Rev. Lett. 91, 208108 (2003)
Casado, J.M., Ibarz, B., Sanjuán, M.A.F.: Winnerless competition in networks of coupled neurons. Mod. Phys. Lett. B 18, 1347–1366 (2004)
Ibarz, B., Cao, H.J., Sanjuán, M.A.F.: Map-based neuron networks. In: Garrido, P.L., Marro, J. (eds.), 9th Granada Seminar on Computational and Statistical Physics. AIP Proceedings, vol. 887, Melville, NY, pp. 69–76 (2007)
Ibarz, B., Cao, H.J., Sanjuán, M.A.F.: Patterns in inhibitory networks of simple map neurons. Phys. Rev. E 75, 041911 (2007)
Ibarz, B., Tanaka, H.J., Sanjuán, M.A.F., Aihara, K.: Sensitivity versus resonance in tow-dimensional spiking-bursting neuron models. Phys. Rev. E 75, 041902 (2007)
Ibarz, B., Cao, H.J., Sanjuán, M.A.F.: Bursting regimes in map-based neuron models coupled through fast threshold modulation. Phys. Rev. E 77, 05918 (2007)
Tanaka, G., Ibarz, B., Sanjuán, M.A.F., Aihara, K.: Synchronization and propagation of bursts in a ring of coupled map neurons. Chaos 16, 013113 (2006)
Hodgin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation on nerve. J. Physiol. 117, 500–544 (1952)
Peng, C.-K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley, H.E., Goldberger, A.L.: Mosaic organization of DNA nucleotides. Phys. Rev. E 49, 1685 (1994)
Ossadnik, S.M., Buldyrev, S.B., Goldberger, A.L., Havlin, S., Mantegna, R.N., Peng, C.-K., Simons, M., Stanley, H.E.: Correlation approach to identify coding regions in DNA sequences. Biophys. J. 67, 64 (1994)
Taqqu, M.S., Teverovsky, V., Willinger, W.: Correlated and uncorrelated regions in heart-rate fluctuations during sleep. Fractals 3, 785 (1995)
Buldyrev, S.V., Goldberger, A.L., Havlin, S., Mantegna, R.N., Matsa, M.E., Peng, C.-K., Simons, M., Stanley, H.E.: Long-range properties of coding and noncoding DNS sequences: GenBank analysis. Phys. Rev. E 51, 5084 (1995)
Peng, C.-K., Havlin, S., Stanley, H.E., Goldberger, A.L.: Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos 5, 82 (1995)
Ivanov, P.Ch., Bunde, A., Amaral, L.A.N., Havlin, S., Fritsch-Yelle, J., Baevsky, R.M., Stanley, H.E., Goldberger, A.L.: Sleep-wake differences in scaling behavior of the human heartbeat: analysis of terrestrial and long-term space flight data. Europhys. Lett. 48, 594 (1999)
Blesic, S., Milosevic, S., Stratimirovic, D., Ljubisavljevic, M.: Detecting long-range correlations in time series of dorsal horn neuron discharges. Physica A 268, 275 (1999)
Bahar, S., Kantelhardt, J.W., Neiman, A., Rego, H.H.A., Russell, D.F., Wilkens, L., Budnde, A., Moss, F.: Long-range temporal anti-correlations in paddlefish electroreceptors. Europhys. Lett. 56, 454 (2001)
Penzel, T., Kantelhart, J.W., Grote, L., Peter, T.H., Bunde, A.: Comparison of detrended fluctuation analysis and spectral analysis for heart rate variability in sleep and sleep apnea. IEEE Trans. Biomed. Eng. 10, 1143 (2003)
Lee, J.-M., Kim, D.-J.: Detrended fluctuation analysis of EEG in sleep apnea using MIT/BIH polysomnography data. Comput. Biol. Med. 32(1), 37 (2002)
Jun, W.C., Oh, G., Kim, S.: Understanding volatility correlation behavior with a magnitude cross-correlation. Eur. Phys. J. B 56, 47–52 (2007)
Podobnik, B., Stanley, H.E.: Detrended cross-correlation analysis: a new method for analyzing two nonstationary time series. Phys. Rev. Lett. 100 (2008)
Zhou, W.-X.: Multifractal detrended cross-correlation analysis for two nonstationary signals. Phys. Rev. Lett. 100 (2008)
Alessio, E., Carbone, A., Castelli, G., Frappietro, V.: Second-order moving average and scaling of stochastic time series. Eur. Phys. J. B 27, 197 (2002)
Barabasi, A.-L., Szepfalusy, P., Vicsek, T.: Multifractal spectre of multi-affine functions. Physica A 178, 17–28 (1991)
Lin, A.J., Shang, P.J., Zhao, X.J.: The cross-correlations of stock markets based on DCCA and time-delay DCCA. Nonlinear Dyn. (2011). doi:10.1007/s11071-011-9991-8
Xu, N., Shang, P.J., Kamae, S.: Modeling traffic flow correlation using DFA and DCCA. Nonlinear Dyn. 61, 207–216 (2010)
Mora, A., Haase, M.: Quantitative methods for the characterization of complex surface structures. Nonlinear Dyn. 44, 307–313 (2006)
Zhang, T., Liu, J.L.: Multi-dimensional self-affine fractal interpolation model in tensor form. Nonlinear Dyn. 52, 83–87 (2008)
Ben Mabrouk, A.: On some nonlinear nonisotropic quasi-self-similar functions. Nonlinear Dyn. 51, 379–398 (2008)
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Xue, C., Shang, P. & Jing, W. Multifractal Detrended Cross-Correlation Analysis of BVP model time series. Nonlinear Dyn 69, 263–273 (2012). https://doi.org/10.1007/s11071-011-0262-5
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DOI: https://doi.org/10.1007/s11071-011-0262-5