Stability of synchronization in simplicial complexes with multiple interaction layers

Md Sayeed Anwar and Dibakar Ghosh

Phys. Rev. E 106, 034314 – Published 13 September 2022

Abstract

Understanding how the interplay between higher-order and multilayer structures of interconnections influences the synchronization behaviors of dynamical systems is a feasible problem of interest, with possible application in essential topics such as neuronal dynamics. Here, we provide a comprehensive approach for analyzing the stability of the complete synchronization state in simplicial complexes with numerous interaction layers. We show that the synchronization state exists as an invariant solution and derive the necessary condition for a stable synchronization state in the presence of general coupling functions. It generalizes the well-known master stability function scheme to the higher-order structures with multiple interaction layers. We verify our theoretical results by employing them on networks of paradigmatic Rössler oscillators and Sherman neuronal models, and we demonstrate that the presence of group interactions considerably improves the synchronization phenomenon in the multilayer framework.

Received 6 June 2022
Accepted 1 September 2022

DOI:https://doi.org/10.1103/PhysRevE.106.034314

Physics Subject Headings (PhySH)

Collective behavior in networks Coupled oscillators Synchronization

Networks

Authors & Affiliations

Md Sayeed Anwar and Dibakar Ghosh ^*

Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India

^*dibakar@isical.ac.in

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Issue

Vol. 106, Iss. 3 — September 2022

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Images

Figure 1
Diagrammatic representation of simplicial complex with two distinct interaction layers. The two layers (solid red and dashed magenta links, respectively) are built up of different sorts of interconnections for the same nodes, where the layer with red links allows three-body interactions between the nodes. The layers are completely independent; the presence of a link between two nodes in one layer has no influence on their connection status in the other.
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Figure 2
Synchronization in coupled Sherman model with higher-order gap junction coupling. (a) Synchronization error $E$ as a function of pairwise gap junction coupling strength $ε$ for a fixed value of pairwise chemical synaptic coupling $g_{c} = 0.01$ . The synchronization error in the absent $(ε_{SI}^{e} = 0)$ and present $(ε_{SI}^{e} = 0.01)$ of higher-order gap junction coupling is displayed by black and red circle curves, respectively. The dashed black and red lines in panel (b) depict the corresponding maximum Lyapunov exponent curves. The variation of synchronization error $E$ in $(ε, g_{c})$ parameter planes for (c) $ε_{SI}^{e} = 0$ and (d) $ε_{SI}^{e} = 0.01$ are represented. The solid white curves in panels (c) and (d) denote the theoretically derived synchronization threshold curves obtained from Eq. (18). Here the other parameter values are $k_{c} = 4$ , $k_{sw} = 4$ , and $p_{sw} = 0.15$ .
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Figure 3
Synchronization in coupled Sherman model with higher-order gap junction coupling and all-to-all pairwise chemical synaptic connections. Synchronization error $E$ in $(ε, g_{c})$ plane for (a) $ε_{SI}^{e} = 0$ , and (b) $ε_{SI}^{e} = 0.01$ . The other parameter values are $p_{sw} = 0.15$ and $k_{sw} = 4$ . The solid white curves depict the critical curves for which $Λ = 0$ obtained from Eq. (18), while the regions below and above it denote the unstable and stable synchronization states, respectively.
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Figure 4
Synchronization in coupled Sherman model with higher-order chemical synaptic coupling. Synchronization error $E$ in $(ε, g_{c})$ parameter plane for fixed value of higher-order chemical synaptic coupling strength $ε_{SI}^{c} = 10^{- 6}$ . The other parameter values are $k_{sw} = 4$ and $p_{sw} = 0.15$ . The solid white curve depicts the critical curve for which $Λ = 0$ obtained from Eq. (20), while the regions below and above it denote the unstable and stable synchronization states, respectively.
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Figure 5
Synchronization in coupled Rössler oscillators with higher-order nonlinear coupling function. Synchronization error $E$ in $(ε_{1}, ε_{2})$ parameter plane for (a) $ε_{SI} = 0$ and (b) $ε_{SI} = 0.00005$ . The contour plots of synchronization error $E$ in the parameter space (c) $(ε_{1}, ε_{SI})$ and (d) $(ε_{2}, ε_{SI})$ for fixed values of $ε_{2} = 0.005$ and $ε_{1} = 0.05$ are depicted, respectively. The solid white curves correspond to the theoretical predictions obtained from Eq. (22).
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