Controlling synchrony in oscillatory networks via an act-and-wait algorithm

The act-and-wait control algorithm is proposed to suppress synchrony in globally coupled oscillatory networks in the situation when the simultaneous registration and stimulation of the system is not possible. The algorithm involves the periodic repetition of the registration (wait) and stimulation (act) stages, such that in the first stage the mean field of the free system is recorded in a memory and in the second stage the system is stimulated with the recorded signal. A modified version of the algorithm that takes into account the charge-balanced requirement is considered as well. The efficiency of our algorithm is demonstrated analytically and numerically for globally coupled Landau-Stuart oscillators and synaptically all-to-all coupled FitzHugh-Nagumo as well as Hodgkin-Huxley neurons.

Figure 1

Block diagram of the algorithm with the separated in time registration (wait) and stimulation (act) setup. The wait and act stages are periodically repeated with the period $T_c={\tau }_w+{\tau }_a$, where ${\tau }_w$ and ${\tau }_a$ are the durations of the wait and act stages, respectively. In the wait stage $0\le t<{\tau }_w$, the mean field of the free neuronal population is recorded in a memory. In the act stage ${\tau }_w\le t<T_c$, the system is stimulated by memorized signal, previously processed by controller.

Figure 2

Dynamics of $N=1000$ globally coupled Landau-Stuart oscillators (1) under act-and-wait control. For $t<100$ the control is off ($\left|P\right|$=0), while for $t\ge 100$ the act-and-wait control with the feedback strength $\left|P\right|=4$ is activated. (a) Absolute value of the order parameter estimated from direct integration of Eqs. (1) [blue (dark)] and from macroscopic Eq. (8) [red (gray)]. (b) Dynamics of the first three oscillators. The values of the parameters are as follows: $K=0.5$, $\Omega =0.25\pi$, $\Delta =0.1$, and $\tau =0.4$. The complex feedback strength is chosen in the form $P=\left|P\right|e^{i\Omega \tau }$.

Figure 3

Incoherent state stability domains in the plane of parameters ($\tau$, $\left|P\right|$) for the Landau-Stuart oscillators (1) controlled by the act-and-wait algorithm. The thick black curves represent the boundaries of stability $P_{\text{mx},\text{mn}}\left(\tau \right)$ defined by Eqs. (16). The color encodes the absolute value $\left|r\right|$ of the order parameter estimated from (a) Eq. (8) and (b) Eqs. (1). The values of the parameters are as follows: $N=1000$, $K=0.5$, and $\Delta =0.1$. The computations are performed in the rotating coordinate frame where $\Omega =0$.

Figure 4

Dynamics of globally coupled $N=1000$ Landau-Stuart oscillators (17) under act-and-wait control. For $t<100$ the control is off ($P=0$), while for $t\ge 100$ the act-and-wait control with the feedback strength $P=1.5$ is activated. (a) Absolute value of the order parameter estimated from direct integration of Eqs. (17) [blue (dark)] and from macroscopic Eq. (19) [red (gray)]. (b) Dynamics of the first oscillator. The values of the parameters are as follows: $K=1$, $\Omega =\pi$, $\Delta =0.1$, and $\tau =2$.

Figure 5

Incoherent state stability domains in the plane of parameters ($\tau$, $P$) for the Landau-Stuart oscillators (17) controlled by the act-and-wait algorithm. The thick black curves represent the boundaries of stability derived from linear Eq. (20). The color encodes the absolute value $\left|r\right|$ of the order parameter estimated from (a) Eq. (19) and (b) Eqs. (17). The values of the parameters are as follows: $N=1000$, $K=1$, $\Delta =0.1$, $\Omega =\pi$, and $T=2\pi /\Omega$.

Figure 6

Dynamics of $N=500$ synaptically coupled FitzHugh-Nagumo neurons (23) under act-and-wait control. For $t<1500$ the control is off ($I_{\text{con}}=0$), while for $t\ge 1500$ the act-and-wait control (25) with the parameters $\tau =18.5$ and $P=0.2$ is activated. (a) Absolute value of the order parameter; (b) the mean field of membrane potential (26); and (c) the membrane potential of the first neuron. The values of the parameters are as follows: $\varepsilon =0.2$, $\beta =0.7$, $\gamma =0.8$, $v_0=1$, $v_c=2.8$, $g=0.05$, and $v_{\text{th}}=0.1$.

Figure 7

Performance of the act-and-wait control algorithm for synaptically coupled FHN neurons (23): (a) control law (25) without charge-balanced requirement and (b) control law (30) with the charge-balanced requirement. The color in the ($\tau ,P$) parameter plane encodes the value of the synchronization criterion $S$ defined by Eq. (28). Incoherent states are characterized by small values of $S$. The period of the mean-field oscillations of the coupled control-free neurons is $T\approx 19.8$ and other parameters are the same as in Fig. 6.

Figure 8

Mean fields of the membrane potential of $N_{\mathrm{ex}}=400$ excitatory and $N_{\mathrm{inh}}=100$ inhibitory globally coupled by synaptic links FitzHugh-Nagumo neurons (23) under act-and-wait control. The top and middle panels represent mean fields of excitatory and inhibitory neurons, respectively, while the bottom shows the total mean field. All except the coupling parameters are the same as in Fig. 6: $g=0.1$, $v_c=2.8$ for excitatory and $v_c=-2.8$ for inhibitory coupled neurons.

Figure 9

Dynamics of an ensemble of $N=100$ Hodgkin-Huxley neurons (31). In the time interval $t\in [0,1000]$ ms the control and coupling are off ($g=0,P=0$). In the interval $t\in [1000,2500]$ ms the synaptic coupling (24) with the strength $g=0.05$ ${\mathrm{mS}/\mathrm{cm}}^2$ is activated but the control is off. For $t\ge 2500$ ms the charge-balanced act-and-wait control (30) with the strength $P=0.23$ ${\mathrm{mS}/\mathrm{cm}}^2$ and delay time $\tau =10.5$ ms is activated. (a) The mean field of membrane potential, (b) absolute value of the order parameter, and (c) the control current.

Figure 10

Performance of the charge-balanced act-and-wait control algorithm for synaptically coupled Hodgkin-Huxley neurons (31). The color in the ($\tau ,P$) parameter plane encodes the value of the synchronization criterion $S$ defined by Eq. (28). Incoherent states are characterized by small values of $S$. The period of the mean-field oscillations of the coupled control-free neurons is $T\approx 10.5$ ms. The values of the parameters are the same as in Fig. 9.

Figure 11

The synchronization criterion $S$ as function of the control gain $P$ for the HH model under act-and-wait control without charge-balanced requirement in the limit $\tau \to 0$. The analysis is performed by using averaged Eqs. (B5). The values of the parameters are the same as in Sec. 4.