Analytical prediction of specific spatiotemporal patterns in nonlinear oscillator networks with distance-dependent time delays

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Analytical prediction of specific spatiotemporal patterns in nonlinear oscillator networks with distance-dependent time delays

Roberto C. Budzinski, Tung T. Nguyen, Gabriel B. Benigno, Jacqueline Đoàn, Ján Mináč, Terrence J. Sejnowski, and Lyle E. Muller

Phys. Rev. Research 5, 013159 – Published 2 March 2023

Abstract

We introduce an analytical approach that allows predictions and mechanistic insights into the dynamics of nonlinear oscillator networks with heterogeneous time delays. We demonstrate that time delays shape the spectrum of a matrix associated with the system, leading to the emergence of waves with a preferred direction. We then create analytical predictions for the specific spatiotemporal patterns observed in individual simulations of time-delayed Kuramoto networks. This approach generalizes to systems with heterogeneous time delays at finite scales, which permits the study of spatiotemporal dynamics in a broad range of applications.

Received 11 March 2022
Revised 22 July 2022
Accepted 3 January 2023

DOI:https://doi.org/10.1103/PhysRevResearch.5.013159

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

Physics Subject Headings (PhySH)

Coupled oscillators Dynamics of networks Synchronization

Nonlinear time-delay systems Traveling waves

Kuramoto model

Nonlinear Dynamics

Authors & Affiliations

Roberto C. Budzinski ^1,2,3,*, Tung T. Nguyen^1,2,3,*, Gabriel B. Benigno^1,2,3,*, Jacqueline Đoàn ^1,2,3, Ján Mináč ^1,3, Terrence J. Sejnowski^4,5, and Lyle E. Muller ^1,2,3,†

¹Department of Mathematics, Western University, London, Ontario N6A 3K7, Canada
²Brain and Mind Institute, Western University, London, Ontario N6A 3K7, Canada
³Western Academy for Advanced Research, Western University, Ontario N6A 3K7, Canada
⁴The Salk Institute for Biological Studies, La Jolla, California 92037, USA
⁵Division of Biological Sciences, University of California, San Diego, La Jolla, California 92161, USA

^*These authors contributed equally to this work.
^†lmuller2@uwo.ca

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Issue

Vol. 5, Iss. 1 — March - May 2023

Subject Areas

Nonlinear Dynamics

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Images

Figure 1
Synchronization level for nondelayed and delayed networks. The time-average Kuramoto order parameter $〈 R 〉$ is plotted as a function of the coupling strength $ε$ for the nondelayed case (blue circles, original KM; orange diamonds, complex-valued KM) and for the delayed case (red squares, original dKM; green triangles, complex-valued dKM). Each point represents one 10-s simulation with random initial conditions $[U (- π, π)]$ , which are the same for the complex-valued case and for the numerical simulation at each point.
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Figure 2
Analytical and geometric view of the effect of time delays. The spatiotemporal dynamics of the system is represented using color coding, where the phase of each oscillator is plotted as a function of time for (a) the original KM, (b) the original dKM, (c) and the complex-valued dKM. Dark colors represent phases close to $- π$ , and light colors represent phases close to $π$ . (a) Without delay, the network transitions to phase synchronization, which is represented by the horizontal lines. The effect of the delay, however, induces wave patterns in the system, whose dynamics are represented (b) in the original dKM and also captured (c) by the complex-valued model. (d) These dynamical characteristics are corroborated by the Kuramoto order parameter $R (t)$ . (e) The eigenmodes offer a geometric perspective to such dynamics, where the waves are represented by a single eigenmode contribution (third mode in this case). (f) The eigenvalues of $W$ (delayed) and $ε A$ (nondelayed) provide further analytical insights into the effect of the delay in the system: It rotates the eigenvalues in the complex plane, which allows the system to access different modes. In the nondelayed case, the leading eigenvalue (in the real part) is associated with an eigenvector with a zero-phase-difference configuration (first mode). In the delayed case, otherwise, there are two leading eigenvalues that are associated with the eigenvectors $v_{3}$ and $v_{99}$ , which have phase configurations representing traveling waves.
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Figure 3
Analytical predictions of specific wave patterns. (a) The phase configuration for the original dKM (blue line) matches the argument (elementwise) of the third eigenvector $Arg [v_{3}]$ —the analytical prediction (red dotted line). A representation on the circle using color coding reveals the wave pattern (right). (b) Different initial conditions lead to the wave pattern that matches the argument of the $99 th$ eigenvector $Arg [v_{99}]$ . These waves can propagate either counterclockwise (negative) or clockwise (positive). (c) With random initial conditions, due to the dominance of two eigenvalues ( $3 rd$ and $99 th$ ), the system exhibits waves propagating in both directions—with approximately half of the initial conditions evolving to each direction (top right). (d) With biased initial conditions, starting from $Arg [v_{3}]$ (red line, bottom right) and adding uniform random phases $0.8 (U (- π, π))$ , we obtain a preferred direction of propagation.
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Figure 4
Analytical predictions of spatiotemporal patterns in brain networks. We use our approach to investigate networks based on the Human Connectome Project (HCP) [38]. We consider the case without delay in the coupling between nodes and also the case with distance-dependent delays (heterogeneous delay). We use our delay operator to create the matrix $W$ , which allows analytical predictions of the dynamics. We show the phase of each node, given by the Kuramoto model, using color coding (dark colors are values close to $- π$ , and light colors are values close to $π$ ). (a) In the case without delay, the argument of the leading eigenvector depicts zero phase difference, which predicts phase synchronization. (b) We then study the numerical simulation for the network without delay, given by Eq. (1), which shows a phase-synchronized state. (c) In the case with heterogeneous time delays, the argument of the leading eigenvector shows a phase offset from the bottom left to the top right (in the projection), which predicts a wave pattern. (d) We perform the numerical simulation with the delayed Kuramoto model, given by Eq. (2), and the network depicts the wave pattern that is predicted by our approach. This shows that we are able to predict the dynamics observed in the simulations using our delay operator.
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Figure 5
A graphical representation of the matrix with the phase configuration of the eigenvectors of $W$ . The $k th$ column is the color-coded argument (elementwise) of the $k th$ eigenvector.
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