- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Experimental observations of cluster synchronization in a biologically inspired neuronal network of chaotic electronic oscillators

Karan K. H. Manjunatha

Joakim Vianney Ngamsa Tegnitsap

Manyu Zhao

Lucia Valentina Gambuzza

Yafang Dong

Wei Zhang

Pedro Antonio Valdes-Sosa

Stefano Boccaletti

Mattia Frasca

Ludovico Minati*

  • Department of Electrical Electronic and Computer Engineering, University of Catania, 95131 Catania, Italy

  • Department of Electrical Electronic and Computer Engineering, University of Catania, 95131 Catania, Italy

  • *Contact author: lminati@uestc.edu.cn; ludovico.minati@unitn.it; lminati@ieee.org
PDF

Phys. Rev. Research 8, 023332 – Published 24 June, 2026

DOI: https://doi.org/10.1103/ns4h-tg99

Abstract

This study investigates the emergence of cluster synchronization in a biologically inspired neuronal network composed of 59 coupled chaotic electronic oscillators that is modeled after a dissociated rat cortical culture. A recently introduced spectral block method is employed to predict the formation and sequence of synchronization clusters based solely on the network's structural connectivity, although the exact transition points depend on the values at which the master stability function changes sign. The analysis is conducted through numerical simulations of Rössler oscillators, which allow for a controlled numerical investigation, followed by a second part devoted to the experimental study of a circuit board composed of single-transistor chaotic oscillators. In both cases, the results confirm the expected hierarchical dynamical transition from incoherence to global synchronization via an intermediate cluster state. Furthermore, the analysis of the network of single-transistor chaotic units provides the first experimental validation of the spectral block method and demonstrates that cluster synchronization arises naturally in networks without intentionally designed symmetries. The experimental platform, which features a scalable and reconfigurable electronic network, thus bridges theoretical predictions and real-world observations, highlighting the robustness of cluster synchronization in the presence of nonidealities, noise, and parameter variations, while also emphasizing the practical considerations required when applying the method to real experiments, such as robust cluster identification and calibration of the critical points of the master stability function. These findings advance our understanding of self-organized synchronization in complex systems and have important implications for neuroscience, adaptive computing, and network engineering.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (88)

  1. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, 2001).
  2. A. Balanov, N. Janson, D. Postnov, and O. Sosnovtseva, From Simple to Complex (Springer, Berlin, Heidelberg, 2009).
  3. S. Strogatz, Synchronization: A universal concept in nonlinear sciences, Phys. Today 56(1), 47 (2003).
  4. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, Complex networks: Structure and dynamics, Phys. Rep. 424, 175 (2006).
  5. J. Buck and E. Buck, Mechanism of rhythmic synchronous flashing of fireflies: Fireflies of Southeast Asia may use anticipatory time-measuring in synchronizing their flashing, Science 159, 1319 (1968).
  6. P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H.-J. Freund, Detection of n:m phase locking from noisy data: Application to magnetoencephalography, Phys. Rev. Lett. 81, 3291 (1998).
  7. P. J. Uhlhaas and W. Singer, Neural synchrony in brain disorders: Relevance for cognitive dysfunctions and pathophysiology, Neuron 52, 155 (2006).
  8. A. E. Motter, S. A. Myers, M. Anghel, and T. Nishikawa, Spontaneous synchrony in power-grid networks, Nat. Phys. 9, 191 (2013).
  9. S. P. Cornelius, W. L. Kath, and A. E. Motter, Realistic control of network dynamics, Nat. Commun. 4, 1942 (2013).
  10. P. N. McGraw and M. Menzinger, Clustering and the synchronization of oscillator networks, Phys. Rev. E 72, 015101(R) (2005).
  11. G. V. Osipov, J. Kurths, and C. Zhou, Synchronization in Oscillatory Networks (Springer Science & Business Media, New York, 2007).
  12. V. N. Belykh, G. V. Osipov, V. S. Petrov, J. A. Suykens, and J. Vandewalle, Cluster synchronization in oscillatory networks, Chaos 18, 037106 (2008).
  13. F. Dörfler and F. Bullo, Exploring synchronization in complex oscillator networks, in Proceedings of the 2012 IEEE 51st IEEE Conference on Decision and Control (CDC) (IEEE, Piscataway, NJ, 2012), pp. 7157–7170.
  14. Y. Kuramoto and D. Battogtokh, Coexistence of coherence and incoherence in nonlocally coupled phase oscillators, Nonlinear Phenom. Complex Syst. 5, 380 (2002).
  15. D. M. Abrams and S. H. Strogatz, Chimera states for coupled oscillators, Phys. Rev. Lett. 93, 174102 (2004).
  16. P. Ashwin and O. Burylko, Weak chimeras in minimal networks of coupled phase oscillators, Chaos 25, 013106 (2015).
  17. A. Buscarino, M. Frasca, L. V. Gambuzza, and P. Hövel, Chimera states in time-varying complex networks, Phys. Rev. E 91, 022817 (2015).
  18. F. Xu, J. Zhang, M. Jin, S. Huang, and T. Fang, Chimera states and synchronization behavior in multilayer memristive neural networks, Nonlinear Dyn. 94, 775 (2018).
  19. S. Majhi, B. K. Bera, D. Ghosh, and M. Perc, Chimera states in neuronal networks: A review, Phys. Life Rev. 28, 100 (2019).
  20. G. R. Simo, P. Louodop, D. Ghosh, T. Njougouo, R. Tchitnga, and H. A. Cerdeira, Traveling chimera patterns in a two-dimensional neuronal network, Phys. Lett. A 409, 127519 (2021).
  21. S. Gu, Q. Fang, P. Zhou, and N. Li, Chimera states in a large laterally coupled laser array with four different waveguide structures, Chaos 33, 073152 (2023).
  22. R. Muolo, T. Njougouo, L. V. Gambuzza, T. Carletti, and M. Frasca, Phase chimera states on nonlocal hyperrings, Phys. Rev. E 109, L022201 (2024).
  23. N. C. Rattenborg, C. J. Amlaner, and S. L. Lima, Behavioral, neurophysiological and evolutionary perspectives on unihemispheric sleep, Neurosci. Biobehav. Rev. 24, 817 (2000).
  24. N. C. Rattenborg, Do birds sleep in flight? Naturwissenschaften 93, 413 (2006).
  25. C. R. Laing and C. C. Chow, Stationary bumps in networks of spiking neurons, Neural Comput. 13, 1473 (2001).
  26. H. Sakaguchi, Instability of synchronized motion in nonlocally coupled neural oscillators, Phys. Rev. E 73, 031907 (2006).
  27. C. Zhou and J. Kurths, Hierarchical synchronization in complex networks with heterogeneous degrees, Chaos 16, 015104 (2006).
  28. A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, Synchronization in complex networks, Phys. Rep. 469, 93 (2008).
  29. C. Bick, M. Goodfellow, C. R. Laing, and E. A. Martens, Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: A review, J. Math. Neurosci. 10, 9 (2020).
  30. L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett. 80, 2109 (1998).
  31. M. Golubitsky and I. Stewart, Patterns of oscillation in coupled cell systems, in Geometry, Mechanics, and Dynamics, edited by P. Newton, P. Holmes, and A. Weinstein (Springer, New York, 2002), pp. 243–286.
  32. O. D’Huys, R. Vicente, T. Erneux, J. Danckaert, and I. Fischer, Synchronization properties of network motifs: Influence of coupling delay and symmetry, Chaos 18, 037116 (2008).
  33. G. Russo and J.-J. E. Slotine, Symmetries, stability, and control in nonlinear systems and networks, Phys. Rev. E 84, 041929 (2011).
  34. V. Nicosia, M. Valencia, M. Chavez, A. Díaz-Guilera, and V. Latora, Remote synchronization reveals network symmetries and functional modules, Phys. Rev. Lett. 110, 174102 (2013).
  35. L. V. Gambuzza and M. Frasca, A criterion for stability of cluster synchronization in networks with external equitable partitions, Automatica 100, 212 (2019).
  36. L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, Cluster synchronization and isolated desynchronization in complex networks with symmetries, Nat. Commun. 5, 4079 (2014).
  37. F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, Complete characterization of the stability of cluster synchronization in complex dynamical networks, Sci. Adv. 2, e1501737 (2016).
  38. A. B. Siddique, L. Pecora, J. D. Hart, and F. Sorrentino, Symmetry-and input-cluster synchronization in networks, Phys. Rev. E 97, 042217 (2018).
  39. R. J. Sánchez-García, Exploiting symmetry in network analysis, Commun. Phys. 3, 87 (2020).
  40. M. Golubitsky and I. Stewart, Nonlinear dynamics of networks: The groupoid formalism, Bull. Am. Math. Soc. 43, 305 (2006).
  41. N. O’Clery, Y. Yuan, G.-B. Stan, and M. Barahona, Observability and coarse graining of consensus dynamics through the external equitable partition, Phys. Rev. E 88, 042805 (2013).
  42. M. T. Schaub, N. O’Clery, Y. N. Billeh, J.-C. Delvenne, R. Lambiotte, and M. Barahona, Graph partitions and cluster synchronization in networks of oscillators, Chaos 26, 094821 (2016).
  43. H. Fan, Y. Wang, and X. Wang, Eigenvector-based analysis of cluster synchronization in general complex networks of coupled chaotic oscillators, Front. Phys. 18, 45302 (2023).
  44. C. Ma, Q. Yang, X. Wu, and J.-A. Lu, Cluster synchronization: From single-layer to multi-layer networks, Chaos 29, 123120 (2019).
  45. I. Kanter, M. Zigzag, A. Englert, F. Geissler, and W. Kinzel, Synchronization of unidirectional time delay chaotic networks and the greatest common divisor, Europhys. Lett. 93, 60003 (2011).
  46. T. Dahms, J. Lehnert, and E. Schöll, Cluster and group synchronization in delay-coupled networks, Phys. Rev. E 86, 016202 (2012).
  47. A. Bayani, F. Nazarimehr, S. Jafari, K. Kovalenko, G. Contreras-Aso, K. Alfaro-Bittner, R. J. Sánchez-García, and S. Boccaletti, The transition to synchronization of networked systems, Nat. Commun. 15, 4955 (2024).
  48. I. Klickstein and F. Sorrentino, Generating graphs with symmetry, IEEE Trans. Network Sci. Eng. 6, 836 (2018).
  49. P. Khanra, S. Ghosh, D. Aleja, K. Alfaro-Bittner, G. Contreras-Aso, R. Criado, M. Romance, S. Boccaletti, P. Pal, and C. Hens, Endowing networks with desired symmetries and modular behavior, Phys. Rev. E 108, 054309 (2023).
  50. L. V. Gambuzza, M. Frasca, F. Sorrentino, L. M. Pecora, and S. Boccaletti, Controlling symmetries and clustered dynamics of complex networks, IEEE Trans. Network Sci. Eng. 8, 282 (2020).
  51. C. Tomaselli, L. V. Gambuzza, G.-Q. Sun, S. Boccaletti, and M. Frasca, Taming cluster synchronization, Chaos Solitons Fractals 199, 116576 (2025).
  52. L. Minati, Experimental synchronization of chaos in a large ring of mutually coupled single-transistor oscillators: Phase, amplitude, and clustering effects, Chaos 24, 043108 (2014).
  53. H. Fan, Y. Wang, Y. Du, H. Qiu, and X. Wang, Scalable synchronization cluster in networked chaotic oscillators, Chaos 34, 071102 (2024).
  54. M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, Synchronized cluster formation in coupled laser networks, Phys. Rev. Lett. 106, 223901 (2011).
  55. A. M. Hagerstrom, T. E. Murphy, R. Roy, P. Hövel, I. Omelchenko, and E. Schöll, Experimental observation of chimeras in coupled-map lattices, Nat. Phys. 8, 658 (2012).
  56. C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators, Phys. Rev. Lett. 110, 064104 (2013).
  57. L. Larger, B. Penkovsky, and Y. Maistrenko, Laser chimeras as a paradigm for multistable patterns in complex systems, Nat. Commun. 6, 7752 (2015).
  58. J. D. Hart, K. Bansal, T. E. Murphy, and R. Roy, Experimental observation of chimera and cluster states in a minimal globally coupled network, Chaos 26, 094801 (2016).
  59. I. Z. Kiss, Y. Zhai, and J. L. Hudson, Emerging coherence in a population of chemical oscillators, Science 296, 1676 (2002).
  60. D. Mersing, S. A. Tyler, B. Ponboonjaroenchai, M. R. Tinsley, and K. Showalter, Novel modes of synchronization in star networks of coupled chemical oscillators, Chaos 31, 093127 (2021).
  61. E. A. Martens, S. Thutupalli, A. Fourriere, and O. Hallatschek, Chimera states in mechanical oscillator networks, Proc. Natl. Acad. Sci. USA 110, 10563 (2013).
  62. T. Kapitaniak, P. Kuzma, J. Wojewoda, K. Czolczynski, and Y. Maistrenko, Imperfect chimera states for coupled pendula, Sci. Rep. 4, 6379 (2014).
  63. J. Wojewoda, K. Czolczynski, Y. Maistrenko, and T. Kapitaniak, The smallest chimera state for coupled pendula, Sci. Rep. 6, 34329 (2016).
  64. D. Dudkowski, K. Czołczyński, and T. Kapitaniak, Traveling chimera states for coupled pendula, Nonlinear Dyn. 95, 1859 (2019).
  65. P. R. Carvalho and M. A. Savi, Synchronization and chimera state in a mechanical system, Nonlinear Dyn. 102, 907 (2020).
  66. L. V. Gambuzza, A. Buscarino, S. Chessari, L. Fortuna, R. Meucci, and M. Frasca, Experimental investigation of chimera states with quiescent and synchronous domains in coupled electronic oscillators, Phys. Rev. E 90, 032905 (2014).
  67. L. Minati, Remote synchronization of amplitudes across an experimental ring of non-linear oscillators, Chaos 25, 123107 (2015).
  68. L. Gambuzza, L. Minati, and M. Frasca, Experimental observations of chimera states in locally and non-locally coupled Stuart-Landau oscillator circuits, Chaos Solitons Fractals 138, 109907 (2020).
  69. L. Minati, G. Varotto, L. D’Incerti, F. Panzica, and D. Chan, From brain topography to brain topology: Relevance of graph theory to functional neuroscience, Neuroreport 24, 536 (2013).
  70. M. P. Van den Heuvel and O. Sporns, An anatomical substrate for integration among functional networks in human cortex, J. Neurosci. 33, 14489 (2013).
  71. M. P. Van den Heuvel and O. Sporns, Network Hubs in the human brain, Trends Cognit. Sci. 17, 683 (2013).
  72. O. Sporns and R. F. Betzel, Modular brain networks, Annu. Rev. Psychol. 67, 613 (2016).
  73. L. Minati, L. Sparacino, J. V. Ngamsa Tegnitsap, M. Zhao, F. Fang, G. Mijatovic, Y. Antonacci, P. A. Valdes-Sosa, H. Ito, M. Frasca, and L. Faes, Spontaneous dynamical differentiation in an experimental network of single-transistor chaotic oscillators modeling a biological neuronal culture, Chaos Solitons Fractals 200, 117111 (2025).
  74. D. A. Wagenaar, J. Pine, and S. M. Potter, An extremely rich repertoire of bursting patterns during the development of cortical cultures, BMC Neurosci. 7, 11 (2006).
  75. J. C. Sprott and W. J.-C. Thio, Elegant Circuits: Simple Chaotic Oscillators (World Scientific, Singapore, 2022), p. 356.
  76. C. Godsil and G. Royle, Algebraic Graph Theory (Springer, New York, 2001).
  77. K. K. H. Manjunatha, J. V. N. Tegnitsap, M. Zhao, L. V. Gambuzza, Y. Dong, W. Zhang, P. A. Valdes-Sosa, S. Boccaletti, M. Frasca, and L. Minati, Code and data: Experimental observations of cluster synchronization in a biologically inspired neuronal network of chaotic electronic oscillators, Zenodo, 2025, https://doi.org/10.5281/zenodo.17096352.
  78. O. Rössler, An equation for continuous chaos, Phys. Lett. A 57, 397 (1976).
  79. J. C. Sprott, Chaos and Time-Series Analysis (Oxford University Press, Oxford, 2003).
  80. K. S. Fink, G. Johnson, T. Carroll, D. Mar, and L. Pecora, Three coupled oscillators as a universal probe of synchronization stability in coupled oscillator arrays, Phys. Rev. E 61, 5080 (2000).
  81. F. Morone and H. A. Makse, Symmetry group factorization reveals the structure-function relation in the neural connectome of Caenorhabditis elegans, Nat. Commun. 10, 4961 (2019).
  82. Y. Liu, Approximate network symmetry, arXiv:2012.05129.
  83. M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, Controlling synchronization in large laser networks, Phys. Rev. Lett. 108, 214101 (2012).
  84. Potter Lab, Network activity of developing cortical cultures in vitro, 2006, https://potterlab.bme.gatech.edu/development-data/html/index.html, accessed 28 April 2024.
  85. M. Rubinov and O. Sporns, Complex network measures of brain connectivity: Uses and interpretations, NeuroImage 52, 1059 (2010).
  86. P. Erdős and A. Rényi, On random graphs I, Publ. Math. Debrecen 6, 290 (1959).
  87. D. J. Watts and S. H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature (London) 393, 440 (1998).
  88. L. Minati, G. Innocenti, G. Mijatovic, H. Ito, and M. Frasca, Mechanisms of chaos generation in an atypical single-transistor oscillator, Chaos Solitons Fractals 157, 111878 (2022).

Outline

Information

Phys. Rev. Research 8, 023332– Published 24 June, 2026
  • Received 22 September 2025
  • Accepted 27 May 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation