Predictable topological sensitivity of Turing patterns on graphs

Marc-Thorsten Hütt, Dieter Armbruster, and Annick Lesne

Phys. Rev. E 105, 014304 – Published 12 January 2022

Abstract

Reaction-diffusion systems implemented as dynamical processes on networks have recently renewed the interest in their self-organized collective patterns known as Turing patterns. We investigate the influence of network topology on the emerging patterns and their diversity, defined as the variety of stationary states observed with random initial conditions and the same dynamics. We show that a seemingly minor change, the removal or rewiring of a single link, can prompt dramatic changes in pattern diversity. The determinants of such critical occurrences are explored through an extensive and systematic set of numerical experiments. We identify situations where the topological sensitivity of the attractor landscape can be predicted without a full simulation of the dynamical equations, from the spectrum of the graph Laplacian and the linearized dynamics. Unexpectedly, the main determinant appears to be the degeneracy of the eigenvalues or the growth rate and not the number of unstable modes.

Received 11 July 2021
Accepted 24 December 2021

DOI:https://doi.org/10.1103/PhysRevE.105.014304

Physics Subject Headings (PhySH)

Collective dynamics Network diffusion Patterns in complex systems

NetworksInterdisciplinary PhysicsNonlinear Dynamics

Authors & Affiliations

Marc-Thorsten Hütt ^*

Department of Life Sciences and Chemistry, Jacobs University, D-28759 Bremen, Germany

Dieter Armbruster^†

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85281, USA

Annick Lesne ^‡

Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, LPTMC, F-75252, Paris, France and Institut de Génétique Moléculaire de Montpellier, University of Montpellier, CNRS, F-34293, Montpellier, France

^*m.huett@jacobs-university.de
^†armbruster@asu.edu
^‡annick.lesne@sorbonne-universite.fr

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Vol. 105, Iss. 1 — January 2022

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Images

Figure 1
Topological sensitivity of Turing pattern diversity on the macaque cortical area network. (a) Representation of the macaque cortical area network (see Methods Sec. 2). (b) Example of a pattern arising for the Gierer-Meinhardt dynamical model from the sixth eigenvector of the Laplacian matrix of the graph, shown underneath. (c) We arrange the nodes in an arbitrary but fixed way on a circle. The links between the nodes are shown in blue in the center. Each concentric ring represents a binarized Turing pattern originating from a randomly chosen initial condition, in total 500 runs (see Methods Sec. 2). The surrounding purple ring shows their average. Panel (d) shows the same pattern wheel representation after removal of a single link [highlighted in red in panels (a) and (c) (upper left quadrant; see the dashed ellipse)].
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Figure 2
Topological sensitivity of Turing pattern diversity on the ring graph with one shortcut. (a) Pattern wheel representation for a ring graph of $N = 111$ nodes with a shortcut from node 1 to node 58. A global attraction to a single pattern independent of the initial condition is observed. (b) Same as panel (a), but for a shortcut from node 1 to node 68; now each observed pattern depends on the specific initial condition. (c) Evolution of eigenvalues for the ring graph with one shortcut as a function of the shortcut position (i.e., the endpoint of the shortcut, the starting point being kept fixed). Note that only eigenvalues which are unstable for some shortcut position are shown. (d) Top panel: Pattern similarity index (see Methods Sec. 2) as a function of the shortcut position. Bottom panel: Pattern predictability distributions (see Methods Sec. 2), depicting the correlation between patterns and unstable eigenvectors. The two shortcut positions for (a) and (b) are highlighted as dashed lines. The distribution of pattern predictability is shown as a sequence of box plots (white lines in the middle of the red bars indicating the median; size of the bar are the 25% and 75% quantiles; error bars (“whiskers”) are the range of values covered with the exception of outliers, which are shown as additional light-red points).
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Figure 3
The dispersion curve for the parameter selection scheme GRDON. The top panel illustrates the quantities involved in the predictability criteria. The variable on the horizontal axis is the eigenvalue of the network Laplacian displayed in a logarithmic scale, $ln (- Λ)$ . Blue dots indicate the growth rates for the original network, and red dots are the ones after link deletion. The dashed curve is the dispersion relation for $Δ σ = 0$ . Positive values of $Δ σ$ shift the dispersion relation upwards, yielding the dispersion relation used for evaluating pattern diversity (shown as a full line). Note that $λ_{s}$ and $λ_{s}^{*}$ are meant as distances from zero and hence are positive.
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Figure 4
Predicting the impact of topological changes on pattern diversity. (a) Pattern wheel representation (see Methods Sec. 2) for 500 simulated patterns starting from random initial conditions under the growth rate degeneracy (GRD) scheme of kinetic parameter selection (see Methods Sec. 2) for the original random regular graph and eigenvector 16. (b) Same as panel (a), but after removal of the link from node 4 to node 22. (c) Pattern diversity array (see Methods Sec. 2) depicting the expected change in Turing pattern diversity as a function of the deleted link index (horizontal axis) and the eigenvector label, eigenvectors being numbered at decreasing eigenvalues (vertical axis, recall that eigenvalues are negative). The parameter constellation investigated in panels (a) and (b) is highlighted in yellow. (d) Criteria defining the color code used in the pattern diversity array in panel (c) and the corresponding prediction.
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Figure 5
Movement of the eigenvalue of the network Laplacian for a ring graph as a function of the endpoint of the shortcut. The starting point of the shortcut is always oscillator 1. The two eigenvalues at the top and bottom are the ones that do not depend on the shortcut.
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