Self-organized network of phase oscillators coupled by activity-dependent interactions

We investigate a network of coupled phase oscillators whose interactions evolve dynamically depending on the relative phases between the oscillators. We found that this coevolving dynamical system robustly yields three basic states of collective behavior with their self-organized interactions. The first is the two-cluster state, in which the oscillators are organized into two synchronized groups. The second is the coherent state, in which the oscillators are arranged sequentially in time. The third is the chaotic state, in which the relative phases between oscillators and their coupling weights are chaotically shuffled. Furthermore, we demonstrate that self-assembled multiclusters can be designed by controlling the weight dynamics. Note that the phase patterns of the oscillators and the weighted network of interactions between them are simultaneously organized through this coevolving dynamics. We expect that these results will provide new insight into self-assembly mechanisms by which the collective behavior of a rhythmic system emerges as a result of the dynamics of adaptive interactions.

Figure 1

The evolution of weights is determined by the plasticity function $\Lambda \left(\phi \right)$ parametrized by $\beta$. The characteristic of the plasticity changes continuously depending on the parameter $\beta$.

Figure 2

(a) Phase diagram of the asymptotic states. (b) Phase patterns of the states. The graphs depict time developments of ${\phi }_i\left(t\right)$ using a color gradient ($t=1900$–2000). The index $i$ of oscillators is arranged in order of increasing phase ${\phi }_i$ at a previous time ($t=1000$). The number of oscillators $N=200$. $\epsilon =0.005$.

Figure 3

Properties of the three asymptotic states. (Left) Order parameters $R_1,R_2$ and a normalized rate of change of the weights $\Delta K$. (Middle) Population of phases and autocorrelations of the relative phase relation. (Right) Weight matrix $k_{ij}$, in which the indices $i,j$ are arranged in the order of increasing phase. Parameter values are the same as in Fig. 2b.

Figure 4

(a) The boundary between the coherent state and the two-cluster state. (b) The graph shows the time development of the phase population after the parameter $\beta$ is suddenly changed across the boundary as indicated by the arrow in Fig. 4a. (c) The boundary between the coherent state and the chaotic state. (d,e) The graphs are the same as in (b). (d) The lines for wave number $k$ on which the eigenvalue ${\lambda }_k=0$ ($k=4,\cdots ,16$). The thick black line indicates the boundary where one of the eigenvalues becomes positive ($k⩾1$).

Figure 5

Emergence of four clusters by a combination of the two-cluster and coherent states. $\alpha =0.4\pi$, $\beta =-0.3\pi$. (a) Time developments of ${\phi }_i\left(t\right)$. (b) Phase distribution at $t=4000$. (c) Time developments of order parameters, $R_1,R_2,R_3,R_4$. (d) Weight matrix $k_{ij}$ at $t=4000$.

Figure 6

A chaotic state caused by the coevolving dynamics of weights and oscillators. (a) Phase portraits of the system in the $(k_{12},k_{21})$ plane obtained using an adiabatic approximation in the chaotic state. $\beta =0.7$$\pi$ and $\alpha =0.1$$\pi$. (b) A trajectory in the chaotic state of the two-oscillator system, with $\alpha =0.05\pi$, $\beta =0.65\pi$, and $\epsilon =0.005$. (c) The same trajectory of (b) projected on the $(\Delta \phi ,k_{12}+k_{21})$ plane. (d) Lyapunov exponents as functions of $\epsilon$. The inset displays a log-log plot, with a fitting curve satisfying $\lambda \propto {\epsilon }^{1/2}$. The parameters are the same as in (b). (e) The number of positive Lyapunov exponents as a function of the number of degrees of freedom of the system, $N^2$. $\beta =0.7\pi$ and $\alpha =0.2\pi$.

Figure 7

Emergence of the three states in the case of various natural frequencies of the oscillators, ${\omega }_i$. Three measures as a function of $\beta$ for several ${\sigma }_{\omega }$: the second order parameter ${\overline{R}}_2$, the correlation $\overline{C}$, and the rate of change of weights $\overline{\Delta K}$. The parameters are as follows: $\alpha =0.3$$\pi$, $\epsilon =0.005$. (a) ${\sigma }_{\omega }=0.05$, (b) ${\sigma }_{\omega }=0.1$, (c) ${\sigma }_{\omega }=0.15$.

Figure 8

Time development of the second order parameter $R_2$ for several ${\sigma }_{\omega }$ in the two-cluster state. $\beta =-0.7$$\pi$, $\alpha =0.1$$\pi$, $\epsilon =0.01$.

Figure 9

Emergence of the three states under a biased plasticity. The biased plasticity function $\overline{\Lambda }\left(\phi \right)$ is a perturbed function of the plasticity with a bias ${\Lambda }_0$ [$={\int }_0^{2\pi }\overline{\Lambda }\left(\phi \right)d\phi$] toward an increase or a decrease of the weights. Each graph shows three measures as a function of $\beta$ for several ${\Lambda }_0$: the second order parameter ${\overline{R}}_2$, the long-time correlation $\overline{C}$, and the rate of change of weights $\overline{\Delta K}$. $\alpha =0.3\pi$ and $\epsilon =0.005$. Each data point is estimated from 20 trials with various initial conditions.

Figure 10

Self-organization by a perturbed plasticity function $\tilde{\Lambda }\left(\phi \right)$, which contains small higher Fourier modes. (a) Example of the two-cluster state. (Left) A perturbed plasticity function. (Middle) Time development of phases ${\phi }_i\left(t\right)$. (Right) The coupling weights $k_{ij}$ organized by the coevolving dynamics. $\alpha$ = 0.3$\pi$. $\beta =-0.7$$\pi$. (b) Example of the coherent state. $\alpha$ = 0.3$\pi$. $\beta =-0.2$$\pi$. (c) Each graph shows three measures in a stationary state as a function of $\beta$: the order parameters ${\overline{R}}_1$ and ${\overline{R}}_2$, the correlation $\overline{C}$, and the rate of change of weights $\overline{\Delta K}$. $\alpha =0.3\pi$ and $\epsilon =0.005$. Each data point is evaluated by randomly perturbed functions $\tilde{\Lambda }\left(\phi \right)$.

Figure 11

Coevolving dynamics of weights and oscillators on a scale-free network. (a) The topology is generated by the Barabási-Albert model for each trial. The number of additional links at each step $m$ = 5, the number of initial nodes $m_0$ = 10, and the total number of nodes $N$ = 1000. (b) Three measures as a function of $\beta$: the second order parameter ${\overline{R}}_2$, the long-time correlation $\overline{C}$, and the rate of change of weights $\overline{\Delta K}$. $\alpha$ = 0.3$\pi$, $\epsilon =0.005$. (c,d) Example of the two-cluster state. (Left) Time developments of ${\phi }_i\left(t\right)$. (Right) Relationship between the node degree and the phases. $\beta =-0.7$$\pi$, $\alpha =0.3$$\pi$. (e,f) Example of the coherent state. $\beta =-0.1$$\pi$, $\alpha =0.3$$\pi$. (g,h) Example of the chaotic state. $\beta =0.7$$\pi$, $\alpha =0.3$$\pi$.

Figure 12

A multicluster state designed by choosing an appropriate plasticity function $\Lambda \left(\phi \right)$. (a) Three-cluster state. (b) Four-cluster state. The left graph shows the plasticity function $\Lambda \left(\phi \right)$. The middle graph depicts the phases ${\phi }_i\left(t\right)$ in which the indexes $i$ of oscillators are arranged in order of increasing phase at a previous time, $t=8000$. The right graph shows the weight matrix $k_{ij}$ at the stationary state. $\alpha$ = 0.1$\pi$ and $\epsilon =0.005$.

Figure 13

(a) Phase diagram of a two-oscillator system. The asymptotic states can be classified into three types: symmetric, asymmetric, and chaotic. In the bistable region, both the symmetric and asymmetric states are stable. (b) Phase portraits of the system in the $(k_{12},k_{21})$ plane were obtained using an adiabatic approximation in Eq. (A2). The parameters $(\alpha ,\beta )$ of these phase portraits correspond to the red points (b1–b4) in graph (a). The blue and green lines represent the nullclines for $k_{12}$ and $k_{21}$, respectively. The black circles indicate stable fixed points. It is noted that the weight $k_{ij}$ is bounded in a region $[-1,1]$.