- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

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

Internal reliability and antireliability in dynamical networks

Tommaso Matteuzzi1,*, Franco Bagnoli1,2,†,‡, Michele Baia1,2,§, Stefano Iubini3,2,∥, and Arkady Pikovsky4,¶

  • *Contact author: tommaso.matteuzzi@unifi.it
  • †Present address: Team Project IMAGES - UMR Espace-Dev, University of Perpignan via Domitia UPVD, 52 Avenue Paul Alduy, 66000 Perpignan, France
  • ‡Contact author: franco.bagnoli@unifi.it
  • §Contact author: michele.baia@unifi.it
  • ∥Contact author: stefano.iubini@cnr.it
  • Contact author: pikovsky@uni-potsdam.de
PDF

Phys. Rev. E 112, 044216 – Published 14 October, 2025

DOI: https://doi.org/10.1103/p5xz-rfv6

Abstract

We consider finite dynamical networks and define internal reliability based on the synchronization properties of a replicated unit or a set of units. If the states of the replicated units coincide with their prototypes, they are reliable; otherwise, if their states differ, they are antireliable. Quantification of reliability using the transversal Lyapunov exponent enables a straightforward analysis of different models. For a Kuramoto model of globally attractively coupled phase oscillators with a distribution of natural frequencies, we show that before the onset of synchronization peripheral frequency units (at the edges of the distribution) are antireliable, while central ones (with natural frequencies close to the mean frequency) are reliable. For repulsive coupling, the complementary configuration occurs, with central antireliable units and peripheral reliable ones. For this model, reliability can be expressed through phase correlations in a sort of fluctuation-dissipation relation. Sufficiently large subnetworks in the Kuramoto model are always antireliable; the same holds for a recurrent neural network, where individual units are always reliable. Furthermore, we show that other coupled oscillator models (Winfree-type coupled phase oscillators, coupled rotators, and coupled Stuart-Landau oscillators) demonstrate patterns of reliability and antireliability similar to those of the Kuramoto setup.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (31)

  1. E. Ott, Chaos in Dynamical Systems, 2nd ed. (Cambridge University, Cambridge, England, 2002).
  2. Z. F. Mainen and T. J. Sejnowski, Reliability of spike timing in neocortical neurons, Science 268, 1503 (1995).
  3. D. S. Goldobin and A. Pikovsky, Antireliability of noise-driven neurons, Phys. Rev. E 73, 061906 (2006).
  4. G. B. Ermentrout, R. F. Galán, and N. N. Urban, Reliability, synchrony and noise, Trends Neurosci. 31, 428 (2008).
  5. K. K. Lin, E. Shea-Brown, and L.-S. Young, Reliability of coupled oscillators, J. Nonlinear Sci. 19, 497 (2009).
  6. A. S. Pikovsky, Synchronization and stochastization of the ensemble of autogenerators by external noise, Radiophys. Quantum Electron. 27, 390 (1984).
  7. A. S. Pikovsky, Statistics of trajectory separation in noisy dynamical systems, Phys. Lett. A 165, 33 (1992).
  8. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization. A Universal Concept in Nonlinear Sciences (Cambridge University, Cambridge, England, 2001).
  9. J. N. Teramae and D. Tanaka, Robustness of the noise-induced phase synchronization in a general class of limit cycle oscillators, Phys. Rev. Lett. 93, 204103 (2004).
  10. N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, and H. D. I. Abarbanel, Generalized synchronization of chaos in directionally coupled chaotic systems, Phys. Rev. E 51, 980 (1995).
  11. L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64, 821 (1990).
  12. L. Kocarev and U. Parlitz, Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems, Phys. Rev. Lett. 76, 1816 (1996).
  13. H. D. I. Abarbanel, N. F. Rulkov, and M. M. Sushchik, Generalized synchronization of chaos: The auxiliary system approach, Phys. Rev. E 53, 4528 (1996).
  14. C. Letellier, I. Sendiña-Nadal, I. Leyva, and J.-P. Barbot, Generalized synchronization mediated by a flat coupling between structurally nonequivalent chaotic systems, Chaos 33, 093117 (2023).
  15. C. Letellier, L. Minati, I. Sendina-Nadal, and I. Leyva, A taxonomy for generalized synchronization between flat-coupled systems, arXiv:2401.11561.
  16. H. Sompolinsky, A. Crisanti, and H. J. Sommers, Chaos in random neural networks, Phys. Rev. Lett. 61, 259 (1988).
  17. A. Pikovsky and A. Politi, Lyapunov Exponents: A Tool to Explore Complex Dynamics (Cambridge University, Cambridge, England, 2016).
  18. O. V. Popovych, Y. L. Maistrenko, and P. A. Tass, Phase chaos in coupled oscillators, Phys. Rev. E 71, 065201(R) (2005).
  19. M. Carlu, F. Ginelli, and A. Politi, Origin and scaling of chaos in weakly coupled phase oscillators, Phys. Rev. E 97, 012203 (2018).
  20. H. Hong, H. Chaté, L.-H. Tang, and H. Park, Finite-size scaling, dynamic fluctuations, and hyperscaling relation in the Kuramoto model, Phys. Rev. E 92, 022122 (2015).
  21. F. Peter and A. Pikovsky, Transition to collective oscillations in finite Kuramoto ensembles, Phys. Rev. E 97, 032310 (2018).
  22. J. Gómez-Gardeñes, Y. Moreno, and A. Arenas, Paths to synchronization on complex networks, Phys. Rev. Lett. 98, 034101 (2007).
  23. M. Schröder, M. Timme, and D. Witthaut, A universal order parameter for synchrony in networks of limit cycle oscillators, Chaos 27, 073119 (2017).
  24. A. Pikovsky and F. Bagnoli, Dynamics of oscillator populations with disorder in the coupling phase shifts, New J. Phys. 26, 023054 (2024).
  25. A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol. 16, 15 (1967).
  26. H. A. Tanaka, A. J. Lichtenberg, and S. Oishi, First order phase transition resulting from finite inertia in coupled oscillator systems, Phys. Rev. Lett. 78, 2104 (1997).
  27. D. M. Abrams, R. Mirollo, S. H. Strogatz, and D. A. Wiley, Solvable model for chimera states of coupled oscillators, Phys. Rev. Lett. 101, 084103 (2008).
  28. A. Yeldesbay, A. Pikovsky, and M. Rosenblum, Chimeralike states in an ensemble of globally coupled oscillators, Phys. Rev. Lett. 112, 144103 (2014).
  29. G. C. Sethia and A. Sen, Chimera states: The existence criteria revisited, Phys. Rev. Lett. 112, 144101 (2014).
  30. K. Höhlein, F. P. Kemeth, and K. Krischer, Lyapunov spectra and collective modes of chimera states in globally coupled Stuart-Landau oscillators, Phys. Rev. E 100, 022217 (2019).
  31. M. Lodi, F. D. Rossa, F. Sorrentino, and M. Storace, Analyzing synchronized clusters in neuron networks, Sci. Reports 10, 16336 (2020).

Outline

Information

Phys. Rev. E 112, 044216– Published 14 October, 2025
  • Received 30 December 2024
  • Revised 4 July 2025
  • Accepted 18 September 2025
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