Summary
We present a method for synchronization analysis, that is able to handle large networks of interacting dynamical units. We focus on large networks with different topologies (random, small-world and scale-free) and neuronal dynamics at each node. We consider neurons that exhibit dynamics on two time scales, namely spiking and bursting behavior. The proposed method is able to distinguish between synchronization of spikes and synchronization of bursts, so that we analyze the synchronization of each time scale separately. We find for all network topologies that the synchronization of the bursts sets in for smaller coupling strengths than the synchronization of the spikes. Furthermore, we obtain an interesting behavior for the synchronization of the spikes dependent on the coupling strength: for small values of the coupling, the synchronization of the spikes increases, but for intermediate values of the coupling, the synchronization index of the spikes decreases. For larger values of the coupling strength, the synchronization index increases again until all the spikes synchronize.
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References
S. H. Strogatz, Nature 410, 268, 2001; M. E. J. Newman, SIAM Rev. 45, 167, 2003; S. Boccaletti et al., Phys. Rep. 424, 175, 2006; R. Albert, A.-L. Barabàsi, Statistical mechanics of complex networks, Rev. Mod. Phys., 74, 47–97, 2002.
P. Erdös, and A. R’enyi, Publ. Math. Inst. Hung. Acad. Sci. 5, 17, 1960; D. J. Watts, and S. H. Strogatz, Nature 393, 440,1998; L. Barab’asi, and R. Albert, Science 286, 509, 1999; M. Molloy, and B. Reed, Random Struct. Algorithms 6, 161, 1995.
L. Donetti et al., Phys. Rev. Lett. 95, 188701, 2005.
Y. Moreno, and A. F. Pacheco, Europhys. Lett. 68 (4), 603, 2004; J. G. Restrepo et al., Phys. Rev. E 71, 036151, 2005.
F. M. Atay et al., Phys. Rev. Lett. 92 (14), 144101, 2004; W. Lu, and T. Chen, Physica D 198, 148, 2004; Y. Jiang et al., Phys. Rev. E 68, 065201(R), 2003.
C. Zhou, and J. Kurths, Chaos 16, 015104 2006.
N. F. Rulkov et al., Phys. Rev. E, 51 (2), 980, 1995; L. Kocarev, and U. Parlitz, Phys. Rev. Lett. 76 (11), 1816, 1996; S. Boccaletti et al., Phys. Rep. 366, 1, 2002.
B. Blasius et al., Nature 399, 354, 1999; P. Tass et al., Phys. Rev. Lett. 81 (15), 3291, 1998; M. Rosenblum et al., Phys. Rev. E. 65, 041909, 2002; D. J. DeShazer et al., Phys. Rev. Lett. 87 (4), 044101, 2001.
A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization - A universal concept in nonlinear science, Cambridge University Press, 2001.
G. V. Osipov, B. Hu, C. Zhou, M. V. Ivanchenko, and J. Kurths, Phys. Rev. Lett. 91, 024101, 2003.
N. Marwan, M. C. Romano, M. Thiel, and J. Kurths, Phys. Rep. 438, 237, 2007.
M. C. Romano, M. Thiel, J. Kurths, I. Z. Kiss, J. L. Hudson, Detection of synchronization for non-phase-coherent and non-stationary data, Europhys. Lett., 71 (3), 466, 2005.
M. G. Rosenblum, A. S. Pikovsky, J. Kurths, G. V. Osipov, I. Z. Kiss, and J. L. Hudson, Phys. Rev. Lett. 89, 264102, 2002.
C. Sparrow, The Lorenz equations: Bifurcations, chaos, and strange attractors, Springer-Verlag, Berlin, 1982.
R. N. Madan, Chua circuit: A paradigm for chaos, World Scientific, Singapore, 1993.
W. Lauterborn, T. Kurz, and M. Wiesenfeldt, Coherent Optics. Fundamentals and Applications, Springer-Verlag, Berlin, Heidelberg, New York, 1993.
C. L. Weber Jr., and J. P. Zbilut, J. Appl. Physiology 76 (2) 965, 1994; N. Marwan, N. Wessel, U. Meyerfeldt, A. Schirdewan, and J. Kurths, Phys. Rev. E 66 (2), 026702, 2002; N. Marwan, and J. Kurths, Phys. Lett. A 302 (5–6), 299, 2002; M. Thiel et al., Physica D 171, 138, 2002.
M. Thiel, M. C. Romano, P. Read, J. Kurths, Estimation of dynamical invariants without embedding by recurrence plots, Chaos, 14 (2), 234–243, 2004.
J. L. Hindmarsh, R. M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. Roy. Soc. Lond. B 221, 87–102, 1984.
R. D. Pinto, P. Varona, A. R. Volkovskii, A. Szücs, H. D. I. Abarbanel, M. I. Rabinovich Synchronous behavior of two coupled electronic neurons, Phys. Rev. Lett. E 62, nr. 2, 2000.
M. Dhamala, V. K. Jirsa, M. Ding Transitions to synchrony in coupled bursting neurons, Phys. Rev. Lett. 92, nr. 2, p. 028101, 2004.
C. Zhou, J. Kurths, Hierarchical synchronization in complex networks with heterogenous degrees Chaos 16, 015104, 2006.
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Bergner, A., Romano, M.C., Kurths, J., Thiel, M. (2007). Synchronization Analysis of Neuronal Networks by Means of Recurrence Plots. In: Graben, P.b., Zhou, C., Thiel, M., Kurths, J. (eds) Lectures in Supercomputational Neurosciences. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73159-7_6
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