Strong effects of network architecture in the entrainment of coupled oscillator systems

Random networks of coupled phase oscillators, representing an approximation for systems of coupled limit-cycle oscillators, are considered. Entrainment of such networks by periodic external forcing applied to a subset of their elements is numerically and analytically investigated. For a large class of interaction functions, we find that the entrainment window with a tongue shape becomes exponentially narrow for networks with higher hierarchical organization. However, the entrainment is significantly facilitated if the networks are directionally biased—i.e., closer to the feedforward networks. Furthermore, we show that the networks with high entrainment ability can be constructed by evolutionary optimization processes. The neural network structure of the master clock of the circadian rhythm in mammals is discussed from the viewpoint of our results.

Figure 1

(Color online) Long-time frequencies for various $\kappa$. Data obtained for three samples of random networks are plotted with different symbols. The depths of the networks are $L=3.01$, $3.33$, and $3.45$. The lines show the theoretical curves (32) with ${\kappa }_{\mathrm{cr}}=67.5$, $125$, and $169$, respectively. $N=100$, $p=0.1$, $N_1=1$, and $\alpha =0$.

Figure 2

(Color online) Dependences of the entrainment threshold on the depth $L$ for a large set of random networks of size $N=100$ and $pN=10$. The coupling function is given by Eq. (4) with (A) $\alpha =0.5$, (B) $\alpha =0$, and (C) $\alpha =-0.5$. To separate the data sets, here we have plotted $\nu {\kappa }_{\mathrm{cr}}$ with $\nu =5$, $1$, and $0.2$, respectively, for (A), (B), and (C). The lines are the theoretical dependence $c\nu {\kappa }_{\mathrm{cr}}$, where ${\kappa }_{\mathrm{cr}}$ is given by Eq. (33) and $c=0.60$ is a fitting parameter.

Figure 3

(Color online) Dependences of the entrainment threshold on the depth $L$ for a large set of random networks. The coupling function is given by Eq. (4) where $\alpha =0$. (A) $N=200$ and $pN=10$. The line is the theoretical dependence $c{\kappa }_{\mathrm{cr}}$, where ${\kappa }_{\mathrm{cr}}$ is Eq. (33) and $c=0.60$ (the same as the value used for $N=100$). (B) $N=100$ and $pN=6,20$. The lines are the theoretical dependence $c{\kappa }_{\mathrm{cr}}$, where ${\kappa }_{\mathrm{cr}}$ is Eq. (33) and $c=0.67$ and $0.45$, respectively, for $pN=6$ and $20$.

Figure 4

(Color online) Dependence of the relaxation time on the depth $L$ for an ensemble of random networks. The parameters are $N=100$, $p=0.1$, $\alpha =0$, and $\kappa =300$. The solid line is the theoretical dependence $d\tau$ where $\tau$ is Eq. (28) with ${\kappa }_{\mathrm{cr}}=0.6pN(1+pN{)}^{L-2}$ (same as the theoretical lines in Fig. 2) and $d=1.22$ is an additional fitting parameter. The dotted line is the exponential function ${\kappa }_{\mathrm{cr}}∕\kappa \propto (1+pN{)}^L$.

Figure 5

(Color online) Eigenvalues (divided by $\kappa$) of the entrained state. The parameters are $N=100$, $N_1=2$, $p=0.1$, $\alpha =0$, and $\kappa =300$ (the depth of the used network is $L=2.86$ and the entrainment threshold for this network is found numerically to be ${\kappa }_{\mathrm{cr}}\simeq 33$). The real part of the maximum eigenvalue is $\mathrm{Re}\left({\lambda }_{\max }\right)-11$.

Figure 6

(Color online). An example of a global tree network ($N_1=1,pN=2$, $H=4$, and hence, $N=15$). Thick solid lines, thin solid lines, and dotted lines represent, respectively, forward connections, backward connections, and intrashell connections. Numbers indicate the hierarchical level of each node. This graph was constructed using the PAJEK software package.

Figure 7

Schematic representation of the network structure along a forward path in the global tree approximation. Each circle with a number $h$ is an oscillator of the shell $h$. Each right arrow and each left arrow represent, respectively, a forward connection and a $pN$ backward (or intrashell) connection.

Figure 8

(Color online) Dependences of (A) the entrainment threshold and (B) the relaxation time on the depth for an ensemble of directionally biased networks. The solid lines are the exponential functions proportional to $(1+\xi pN{)}^L$. The dashed line is a linear function $aL+b$ with appropriate fitting parameters $a$ and $b$. Other parameters are $\kappa =600$, $pN=10$, and (A) $N=100$ and (B) $N=400$.

Figure 9

(Color online) Evolution process under the first algorithm. Parameter values are $N=100$, $p=0.1$, $N_1=1$, and $\kappa =20$. The depth of the initial random network is $L=3.43$. (A) The solid line is the long-time frequency $⟨\omega ⟩$ averaged over the whole network. The dashed and dotted lines are, respectively, the backward connectivity $\xi$ and the depth $L$. (B) Two other topological quantities are plotted. The solid and dotted lines are, respectively, the total number $n_{\mathrm{out}}$ of outgoing connections from the first shell and the forward connectivity $\chi$.

Figure 10

(Color online) Evolution process under the second algorithm. The depth of the initial random network is $L=3.22$, corresponding to ${\kappa }_{\mathrm{cr}}\simeq 100$. Parameter values and the definitions of the exhibited lines are the same as in Fig. 9.

Figure 11

Schematic representation of the entrainment window, inside which the entrainment occurs. The slopes of the edges of the entrainment window for $\omega >0$ and $\omega <0$ are proportional to $\max \Gamma$ and $\min \Gamma$, respectively.