Remote Synchronization Reveals Network Symmetries and Functional Modules

Vincenzo Nicosia, Miguel Valencia, Mario Chavez, Albert Díaz-Guilera, and Vito Latora

Phys. Rev. Lett. 110, 174102 – Published 25 April 2013

Abstract

We study a Kuramoto model in which the oscillators are associated with the nodes of a complex network and the interactions include a phase frustration, thus preventing full synchronization. The system organizes into a regime of remote synchronization where pairs of nodes with the same network symmetry are fully synchronized, despite their distance on the graph. We provide analytical arguments to explain this result, and we show how the frustration parameter affects the distribution of phases. An application to brain networks suggests that anatomical symmetry plays a role in neural synchronization by determining correlated functional modules across distant locations.

Received 26 November 2012

DOI:https://doi.org/10.1103/PhysRevLett.110.174102

Authors & Affiliations

Vincenzo Nicosia¹, Miguel Valencia², Mario Chavez³, Albert Díaz-Guilera⁴, and Vito Latora^1,5

¹School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom
²Neurophysiology Laboratory, CIMA, University of Navarra, 31008 Pamplona, Spain
³CNRS UMR-7225, Hôpital de la Salpêtrière. 47 Boulevard de l’Hôpital, 75013 Paris, France
⁴Department de Física Fonamental, Facultat de Física, Universitat de Barcelona, E-08028 Spain
⁵Dipartimento di Fisica ed Astronomia, Università di Catania and INFN, I-95123 Catania, Italy

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Vol. 110, Iss. 17 — 26 April 2013

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Images

Figure 1
The presence of frustration reveals clusters of symmetric nodes. The color code represents the phases of nodes at a given time in the stationary state. (a) In the first graph ( $G_{a}$ ), node 2 is synchronized to node 3, node 4 to node 7, and node 5 to node 6. (b) In a finite chain ( $G_{b}$ ), pairs of nodes symmetrically placed with respect to the central node are perfectly synchronized. (c) In a finite Bethe lattice ( $G_{c}$ ) all the nodes placed at the same distance from the center have equal phases.Reuse & Permissions
Figure 2
The figure refers to the coupling topology $G_{a}$ in Fig. 1a. Panels (a)–(c): after an initial transient the system reaches a phase-locked synchronized state in which symmetric nodes have the same phase. The panels correspond to three different values of the frustration parameter, respectively, (a) $α = 0.1$ , (b) $α = 0.5$ , (c) $α = 0.8$ . Panel (d): for $α > α_{c}$ the synchronized state becomes unstable, the order parameter decreases while the dispersion of the phases $σ_{θ}$ increases. Panel (e): the maximum Lyapunov exponent $λ_{\max }$ of oscillators coupled through $G_{a}$ becomes positive for $α > α_{c}$ , and the systems enters a chaotic regime. The dashed yellow line indicates the approximate position of $α_{c}$ .Reuse & Permissions
Figure 3
Chaotic regime in $G_{a}$ when $α > α_{c}$ . Panel (a)–(d): running averages of $r$ (orange) and pairwise order parameters $r_{2}$ (black, red and green lines) for typical trajectories of oscillators coupled through graph $G_{a}$ , when $α$ is, respectively, equal to (a) 1.3, (b) 1.4, (c) 1.5 and (d) 1.55. The dashed lines indicate the expected synchronization level for a system of two (blue line, ${\tilde{r}}_{2} = 2 / π$ ) and seven incoherent oscillators (gray line, ${\tilde{r}}_{7} ≃ 0.338 \dots$ ). Panel (e): the plot of the phases of pairs of symmetric nodes for $α = 1.4$ in two different temporal intervals (the shaded gray regions in panel (b)) reveal the existence of metastable, partially synchronized states.Reuse & Permissions
Figure 4
(a) Brain areas with similar and dissimilar phases of the frustrated Kuramoto model are colored and superimposed onto an anatomical image. (b) Examples of functional data from one subject recorded at the brain areas indicated in panel (a). Colors are the same as those used in the anatomical image. (c) Functional correlation $Z$ between pairs of nodes as a function of their phase differences $Δ θ$ according to the simulated Kuramoto dynamics. The black solid curve corresponds to the average value over all the subjects, while the gray area covers the 5th and the 95th percentiles of the distribution. The dashed horizontal line indicates the threshold for statistical significant correlations ( $p < 0.05$ , corrected for multiple comparisons).Reuse & Permissions

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