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From chaotic itinerancy to intermittent synchronization in complex networks

I. Leyva* and Irene Sendiña-Nadal

R. Sevilla-Escoboza

V. P. Vera-Ávila

Christophe Letellier

  • *Contact author: inmaculada.leyva@urjc.es
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Phys. Rev. Research 8, 013328 – Published 26 March, 2026

DOI: https://doi.org/10.1103/s3gp-gtln

Abstract

Although synchronization has been extensively studied, key processes underlying its emergence in complex networks have remained hidden by the use of global order parameters. Here, we uncover how the route to synchronization unfolds through a sequential transition between two well-known but previously unconnected phenomena: chaotic itinerancy (CI) and intermittent synchronization (IS). Using a symbolic dynamics approach, we show that CI develops as a collective yet unsynchronized exploration of different domains of the high-dimensional attractor of the network, whose dimension is reduced as the coupling increases, ultimately collapsing into the reference chaotic attractor of an individual unit. At this stage, the IS emerges as irregular alternations between synchronous and asynchronous phases. These phenomena are mutually exclusive, each dominating a distinct coupling interval and governed by different mechanisms. Furthermore, network structural heterogeneity enhances itinerant behavior since access to different domains of the attractor depends on the nodes’ topological roles. The CI-IS crossover occurs within a consistent coupling interval across models and topologies. Experiments on electronic oscillator networks confirm this two-step process, establishing a unified framework for understanding the route to synchronization in complex systems.

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References (65)

  1. J. M. Beggs and D. Plenz, Neuronal avalanches in neocortical circuits, J. Neurosci. 23, 11167 (2003).
  2. W. L. Shew and D. Plenz, The functional benefits of criticality in the cortex, Neuroscientist 19, 88 (2013).
  3. M. A. Muñoz, Colloquium: Criticality and dynamical scaling in living systems, Rev. Mod. Phys. 90, 031001 (2018).
  4. R. Calvo, C. Martorell, G. B. Morales, S. Di Santo, and M. A. Muñoz, Frequency-dependent covariance reveals critical spatiotemporal patterns of synchronized activity in the human brain, Phys. Rev. Lett. 133, 208401 (2024).
  5. S. A. Kauffman, The Origins of Order: Self-Organization and Selection in Evolution (Oxford University Press, Oxford, 1993).
  6. R. Solé and J. Bascompte, Self-Organization in Complex Ecosystems (Princeton University Press, Princeton, 2012).
  7. N. Bertschinger and T. Natschläger, Real-time computation at the edge of chaos in recurrent neural networks, Neural Comput. 16, 1413 (2004).
  8. F. Hancock, F. E. Rosas, A. I. Luppi, M. Zhang, P. A. M. Mediano, J. Cabral, G. Deco, M. L. Kringelbach, M. Breakspear, J. A. S. Kelso, and F. E. Turkheimer, Metastability demystified—The foundational past, the pragmatic present and the promising future, Nat. Rev. Neurosci. 26, 82 (2025).
  9. I. Tsuda, Dynamic link of memory—Chaotic memory map in nonequilibrium neural networks, Neural Networks 5, 313 (1992).
  10. L. M. Kay, A challenge to chaotic itinerancy from brain dynamics, Chaos 13, 1057 (2003).
  11. V. P. Vera-Ávila, J. R. Sevilla-Escoboza, and I. Leyva, Complex networks exhibit intermittent synchronization, Chaos 30, 103119 (2020).
  12. A. Choudhary, C. Mitra, V. Kohar, S. Sinha, and J. Kurths, Small-world networks exhibit pronounced intermittent synchronization, Chaos 27, 111101 (2017).
  13. N. Platt, E. A. Spiegel, and C. Tresser, On-off intermittency: A mechanism for bursting, Phys. Rev. Lett. 70, 279 (1993).
  14. J. F. Heagy, N. Platt, and S. M. Hammel, Characterization of on-off intermittency, Phys. Rev. E 49, 1140 (1994).
  15. M. Breakspear, Dynamic models of large-scale brain activity, Nat. Neurosci. 20, 340 (2017).
  16. F. S. Borges, E. C. Gabrick, P. R. Protachevicz, G. S. V. Higa, E. L. Lameu, P. X. R. Rodriguez, M. S. A. Ferraz, J. D. Szezech, A. M. Batista, and A. H. Kihara, Intermittency properties in a temporal lobe epilepsy model, Epilepsy Behav. 139, 109072 (2023).
  17. K. Kaneko and I. Tsuda, Chaotic itinerancy, Chaos 13, 926 (2003).
  18. I. Tsuda, Chaotic itinerancy and its roles in cognitive neurodynamics, Curr. Opin. Neurobiol. 31, 67 (2015).
  19. D. Koch, A. Nandan, G. Ramesan, and A. Koseska, Biological computations: Limitations of attractor-based formalisms and the need for transients, Biochem. Biophys. Res. Commun. 720, 150069 (2024).
  20. K. Kaneko, Clustering, coding, switching, hierarchical ordering, and control in a network of chaotic elements, Physica D 41, 137 (1990).
  21. W. J. Freeman, Evidence from human scalp electroencephalograms of global chaotic itinerancy, Chaos 13, 1067 (2003).
  22. K. Otsuka, Chaotic itinerancy in a coupled-element multistable optical chain, Phys. Rev. A 43, 618 (1991).
  23. R. Iwami, T. Mihana, K. Kanno, S. Sunada, M. Naruse, and A. Uchida, Controlling chaotic itinerancy in laser dynamics for reinforcement learning, Sci. Adv. 8, eabn8325 (2022).
  24. J.-J. Aucouturier, Y. Ogai, and T. Ikegami, Making a robot dance to music using chaotic itinerancy in a network of Fitzhugh-Nagumo neurons, in International Conference on Neural Information Processing (Springer, Berlin Heidelberg, 2008), pp. 647–656.
  25. S. Steingrube, M. Timme, F. Wörgötter, and P. Manoonpong, Self-organized adaptation of a simple neural circuit enables complex robot behaviour, Nat. Phys. 6, 224 (2010).
  26. J. Park, H. Mori, Y. Okuyama, and M. Asada, Chaotic itinerancy within the coupled dynamics between a physical body and neural oscillator networks, PLoS One 12, e0182518 (2017).
  27. S. Recanatesi, U. Pereira-Obilinovic, M. Murakami, Z. Mainen, and L. Mazzucato, Metastable attractors explain the variable timing of stable behavioral action sequences, Neuron 110, 139 (2022).
  28. T. Sasaki, N. Matsuki, and Y. Ikegaya, Metastability of active CA3 networks, J. Neurosci. 27, 517 (2007).
  29. M. Morrison and L.-S. Young, Chaotic heteroclinic networks as models of switching behavior in biological systems, Chaos 32, 123102 (2022).
  30. J. A. Roberts, L. L. Gollo, R. G. Abeysuriya, G. Roberts, P. B. Mitchell, M. W. Woolrich, and M. Breakspear, Metastable brain waves, Nat. Commun. 10, 1056 (2019).
  31. P. Ashwin, M. Fadera, and C. Postlethwaite, Network attractors and nonlinear dynamics of neural computation, Curr. Opin. Neurobiol. 84, 102818 (2024).
  32. K. L. Rossi, R. C. Budzinski, E. S. Medeiros, B. R. Boaretto, L. Muller, and U. Feudel, Dynamical properties and mechanisms of metastability: A perspective in neuroscience, Phys. Rev. E 111, 021001 (2025).
  33. K. Inoue, K. Nakajima, and Y. Kuniyoshi, Designing spontaneous behavioral switching via chaotic itinerancy, Sci. Adv. 6, eabb3989 (2020).
  34. L.-W. Kong, G. A. Brewer, and Y.-C. Lai, Reservoir-computing based associative memory and itinerancy for complex dynamical attractors, Nat. Commun. 15, 4840 (2024).
  35. J. O’Hagan, A. Keane, and A. Flynn, Confabulation dynamics in a reservoir computer: Filling in the gaps with untrained attractors, Chaos 35, 093130 (2025).
  36. T. Kabayama, M. Komuro, Y. Kuniyoshi, K. Aihara, and K. Nakajima, Crisis-induced intermittency in reservoir computing, Phys. Rev. Res. 7, L032058 (2025).
  37. A. Mattera, V. Alfieri, G. Granato, and G. Baldassarre, Chaotic recurrent neural networks for brain modelling: A review, Neural Networks 184, 107079 (2025).
  38. S. Uchiyama and H. Fujisaka, Wandering behavior of the chaotic itinerancy in an oscillator neural network model, Prog. Theor. Phys. Suppl. 161, 381 (2006).
  39. J. J. Torres, J. Marro, J. M. Cortes, and B. Wemmenhove, Instabilities in attractor networks with fast synaptic fluctuations and partial updating of the neurons activity, Neural Networks 21, 1272 (2008).
  40. A. A. Khatun, Y. A. Muthanna, N. Punetha, and H. H. Jafri, Collective dynamics of coupled Lorenz oscillators near the Hopf boundary: Intermittency and chimera states, Phys. Rev. E 109, 034208 (2024).
  41. T. Sameshima, K. Fukushima, H. Shibata, and T. Yamada, Lyapunov spectrum analysis for a chaotic transition phenomenon, Europhys. Lett. 62, 21 (2003).
  42. A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science 286, 509 (1999).
  43. P. Erdős and A. Rényi, On random graphs, Publ. Math. Debrecen 6, 290 (1959).
  44. See Supplemental Material at http://link.aps.org/supplemental/10.1103/s3gp-gtln for examples of CI in other models and additional data about the experimental setup.
  45. M. Timme, F. Wolf, and T. Geisel, Prevalence of unstable attractors in networks of pulse-coupled oscillators, Phys. Rev. Lett. 89, 154105 (2002).
  46. D. J. Gauthier and J. C. Bienfang, Intermittent loss of synchronization in coupled chaotic oscillators: Toward a new criterion for high-quality synchronization, Phys. Rev. Lett. 77, 1751 (1996).
  47. S. Boccaletti and D. L. Valladares, Characterization of intermittent lag synchronization, Phys. Rev. E 62, 7497 (2000).
  48. G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 2: Numerical application, Meccanica 15, 21 (1980).
  49. R. Gutiérrez, R. Sevilla-Escoboza, P. Piedrahita, C. Finke, U. Feudel, J. M. Buldú, G. Huerta-Cuellar, R. Jaimes-Reátegui, Y. Moreno, and S. Boccaletti, Generalized synchronization in relay systems with instantaneous coupling, Phys. Rev. E 88, 052908 (2013).
  50. J. L. Kaplan and J. A. Yorke, Chaotic behavior of multidimensional difference equations, in Functional Differential Equations and Approximation of Fixed Points, edited by H.-O. Peitgen and H.-O. Walther (Springer, Berlin, 1979), pp. 204–227.
  51. K. Kaneko, Information cascade with marginal stability in a network of chaotic elements, Physica D 77, 456 (1994).
  52. I. Tsuda, H. Fujii, S. Tadokoro, T. Yasuoka, and Y. Yamaguti, Chaotic itinerancy as a mechanism of irregular changes between synchronization and desynchronization in a neural network, J. Integr. Neurosci. 03, 159 (2004).
  53. G. Tirabassi, R. Aristides, C. Masoller, and D. J. Gauthier, Bubbling in oscillator networks, arXiv:2504.07374.
  54. J. Namikawa, Chaotic itinerancy and power-law residence time distribution in stochastic dynamical systems, Phys. Rev. E 72, 026204 (2005).
  55. H. Fujisaka and T. Yamada, Stability theory of synchronized motion in coupled-oscillator systems, Prog. Theor. Phys. 69, 32 (1983).
  56. P. W. Hammer, N. Platt, S. M. Hammel, J. F. Heagy, and B. D. Lee, Experimental observation of on-off intermittency, Phys. Rev. Lett. 73, 1095 (1994).
  57. S. C. Venkataramani, T. M. Antonsen, E. Ott, and J. C. Sommerer, Characterization of on-off intermittent time series, Phys. Lett. A 207, 173 (1995).
  58. A. Tlaie, I. Leyva, R. Sevilla-Escoboza, V. P. Vera-Avila, and I. Sendiña-Nadal, Dynamical complexity as a proxy for the network degree distribution, Phys. Rev. E 99, 012310 (2019).
  59. C. Letellier, I. Sendiña-Nadal, L. Minati, and I. Leyva, Node differentiation dynamics along the route to synchronization in complex networks, Phys. Rev. E 104, 014303 (2021).
  60. J. R. Sevilla-Escoboza, V. P. Vera-Avila, I. Leyva, and R. R. Rivera-Durón, Experimental datasets on intermittency in networks of electronic oscillators. Hybrid configuration—Random (Erdös-Rényi) topology, Zenodo (2025), https://doi.org/10.5281/zenodo.17451096.
  61. J. R. Sevilla-Escoboza, V. P. Vera Avila, R. R. Rivera-Durón, and I. Leyva, Experimental datasets on intermittency in networks of electronic oscillators. Hybrid configuration—Small-world (Watts-Strogatz), Zenodo (2025), https://doi.org/10.5281/zenodo.17451530.
  62. D. J. Watts and S. H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature (London) 393, 440 (1998).
  63. C. Nathe, C. Pappu, N. A. Mecholsky, J. Hart, T. Carroll, and F. Sorrentino, Reservoir computing with noise, Chaos 33, 041101 (2023).
  64. K. Kitajo, S. M. Doesburg, K. Yamanaka, D. Nozaki, L. M. Ward, and Y. Yamamoto, Noise-induced large-scale phase synchronization of human-brain activity associated with behavioural stochastic resonance, Europhys. Lett. 80, 40009 (2007).
  65. J. Matulis and H. Bindra, Effect of stochasticity on initial transients and chaotic itinerancy for a natural circulation loop, Phys. Rev. E 112, 044223 (2025).

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Phys. Rev. Research 8, 013328– Published 26 March, 2026
  • Received 19 November 2025
  • Accepted 24 February 2026
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