Research ArticleMATHEMATICS

Macroscopic models for networks of coupled biological oscillators

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Science Advances  03 Aug 2018:
Vol. 4, no. 8, e1701047
DOI: 10.1126/sciadv.1701047
  • Fig. 1 A low-dimensional structure in the phase distribution of coupled oscillator systems.

    (A) Experimental SCN neuron data (see Materials and Methods and the Supplementary Materials) (12). (i and ii) Green dots show the phase coherences computed from hourly measurements of cell protein expression in the SCN neurons. The solid black curve shows the relation Embedded Image, and the dashed curve shows the COA ansatz Embedded Image. Inset plots show the circular mean vector of ψmmψ1 across all observations. Bottom row: Histogram (iii) and the first 10 phase coherences (iv) of the phase distribution computed from the data point indicated by the blue star in the top row compared to the phase distribution satisfying the m2 ansatz (black curves). (B to D) Each figure shows a different model simulation: (B) simulation of coupled heterogeneous repressilator oscillators (32, 44), (C) simulation of coupled heterogeneous Morris-Lecar neural oscillators (45, 46), and (D) simulation of coupled noisy modified Goodwin oscillators (47) (see the Supplementary Materials for model details). In each figure (B to D): (i and ii) histogram of the simulated equilibrium phase distribution computed from model simulations for two different coupling strengths [(i), strong coupling; (ii), weak coupling], compared to the m2 ansatz phase distribution. (iii) The first 10 phase coherences for the simulated equilibrium phase distributions for two coupling strengths (green dots, strong coupling; blue squares, weak coupling) compared to the m2 ansatz relation (solid curves).

  • Fig. 2 The emergence of the ansatz depends on the frequency distribution of the oscillators.

    The Kuramoto model (Eq. 1) with Gaussian and Cauchy distributions for the natural frequencies of the oscillators, g(ω). (A and B) Relation among the Daido order parameters computed from numerical simulations (circles) and predicted (curves) by (A) the m2 ansatz for Gaussian g(ω) and (B) the COA ansatz for Cauchy g(ω) for increasing coupling strength. Colors indicate different coupling strengths with K that is normalized to the critical coupling strength Kc where partially synchronized solutions emerge (6, 9): K/Kc = 1.1 (red), K/Kc = 1.5 (blue), and K/Kc = 3.0 (green). (C) The fraction of oscillators phase-locked to the mean phase p as a function of normalized coupling strength K/Kc for a Cauchy (dashed green curve) and Gaussian (solid black curve) g(ω).

  • Fig. 3 The equilibrium phase distribution of complex network phase oscillators converges to the m2 ansatz as the coupling strength between the oscillators increases.

    (A) Barabasi-Albert scale-free network. (B) Watts-Strogatz small-world network. Circles show the results from simulations of networks of N = 1000 coupled oscillators with noise amplitude D = 1 and oscillator frequencies drawn from a Gaussian distribution with σ = 1. Solid curves show Embedded Image. Colors indicate different coupling strengths as in Fig. 2. Additional details of these simulations are given in the Supplementary Materials.

  • Fig. 4 The accuracy of the ansatz for the noisy heterogeneous Kuramoto model.

    The equilibrium phase coherence R1 as a function of the coupling strength K for the noisy, heterogeneous Kuramoto model (Eq. 15) for different relative levels of heterogeneity (γ) and noise amplitude (D). (A) s = γ/D = 0.05. (B) s = 0.5. (C) s = 1. (D and E) The transient dynamics of R1 for (D) s = 0.05 and (E) s = 1.0 for different coupling strengths: K = 1.2 (magenta), K = 1.5 (red), and K = 3.0 (blue). In all panels, solid curves show the macroscopic model predictions (Eq. 22), and dashed curves show numerical simulations of the microscopic model in the continuum limit (Eq. 20). Parameters were chosen such that critical coupling strength Kc = 1 for the microscopic model. Insets show curves in the rectangular regions.

  • Fig. 5 Determination of optimal frequency modes.

    (A) The optimal frequency mode Embedded Image as a function of KR values, as determined by Eq. 25. (B) The optimal frequency mode Embedded Image when R is given by the long-time asymptotic value Embedded Image. Gaussian g(ω) frequency distribution is shown as a solid green curve and Embedded Image distribution is shown as a dashed black curve. Parameters were chosen such that critical coupling strength Kc = 1 for both distributions.

  • Fig. 6 The macroscopic model for heterogeneous oscillators.

    (A and B) The equilibrium phase coherence R1 against the coupling strength K for the Kuramoto model for (A) Gaussian g(ω) and (B) Embedded Image distributions of natural frequencies. Exact solutions obtained from classical self-consistency theory (6, 9) are shown as dashed green curves, and the solution according to the m2 ansatz is shown as solid black curves. Insets show curves in the rectangular regions. (C and D) Plot of the dynamics in the phase plane Embedded Image for perturbations about the synchronized state for K = 3. The dashed black curve shows the predicted dynamics by the macroscopic model, and the solid colored curves show the recovery dynamics for random perturbations off the synchronized state in the high-dimensional phase model. Circles indicate the initial conditions for the transient curves. (C) Gaussian g(ω) and (D) Embedded Image distribution.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/8/e1701047/DC1

    Data and mathematical model details.

    Emergence of the m2 ansatz for complex heterogeneous noisy networks.

    Finding the dominant frequency.

    Reduction of limit cycle models to macroscopic models.

    Fig. S1. The low-dimensional structure in the phase distribution of coupled oscillator systems from the Abel et al. (12) circadian data set.

    Fig. S2. The numerically estimated coupling function Γ(ψ) for the interaction between the repressilator oscillators.

    Fig. S3. The variation of the angular frequency induced by the variation of the β parameter in the repressilator model.

    Fig. S4. Macroscopic model amplitude recovery dynamics predictions against numerical simulations.

    References (4850)

  • Supplementary Materials

    This PDF file includes:

    • Data and mathematical model details.
    • Emergence of the m2 ansatz for complex heterogeneous noisy networks.
    • Finding the dominant frequency.
    • Reduction of limit cycle models to macroscopic models.
    • Fig. S1. The low-dimensional structure in the phase distribution of coupled oscillator systems from the Abel et al. (12) circadian data set.
    • Fig. S2. The numerically estimated coupling function Γ(ψ) for the interaction between the repressilator oscillators.
    • Fig. S3. The variation of the angular frequency induced by the variation of the β parameter in the repressilator model.
    • Fig. S4. Macroscopic model amplitude recovery dynamics predictions against numerical simulations.
    • References (4850)

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