Predict-prevent control method for perturbed excitable systems

We present a control method based on two steps: prediction and prevention. For prediction we use the anticipated synchronization scheme, considering unidirectional coupling between excitable systems in a master-slave configuration. The master is the perturbed system to be controlled, meanwhile the slave is an auxiliary system which is used to predict the master’s behavior. The prevention is obtained by sending a control signal to the master system, which temporarily lowers its excitability threshold and prevents its future reaction to the external perturbation. We demonstrate theoretically and experimentally that an efficient control may be achieved.

Figure 1

Schematic presentation of the predict-prevent control method. The input signal $I\left(t\right)$ is applied only to the master system $\mathbf{x}$ while its influence on the slave $\mathbf{y}$ is indirectly through the unidirectional coupling $\mathbf{K}\mathbf{x}$. The slave dynamics has a delayed loop $\mathbf{K}{\mathbf{y}}_{\tau }\equiv \mathbf{K}\mathbf{y}\left(\mathbf{t}-\tau \right)$. The control signal is applied whenever a condition for the slave variable is satisfied.

Figure 2

(Color online) Evolution of the master system $\text{cos}\left[x\left(t\right)\right]$ (solid line) and the slave system $\text{cos}\left[y\left(t\right)\right]$ (dashed line) obtained from a numerical integration of Eqs. (3, 4). External perturbations of intensity $I_0=0.64$ are applied at times $t=200$ and $t=250$ (shown at the bottom of the figure). In the first perturbation $\left(t=200\right)$ we see clearly that the slave anticipates the dynamics of the master. In the second perturbation $\left(t=250\right)$ the spike of the master has been suppressed by applying a correcting pulse of amplitude ${ϵ}_0=0.04$ when the trajectory of the slave crosses the reference level $\text{cos}\left(x_0\right)=0.707$ (dashed horizontal line). We have used the parameters $\mu =0.95$, $K=0.6$, $\tau =1.5$, and a reaction time $t_R=1$.

Figure 3

(Color online) Correction of a series of spikes. The external perturbation $I\left(t\right)$ consists of a series of random pulses (vertical lines al the bottom of both panels) following a Poisson distribution with a given mean value $⟨t⟩=500$. Independent noises of intensity $D=0.01$ have been added both to the master and to the slave. Left panel: control of magnitude ${ϵ}_0=0.3$ with a response time $t_R=1$ using a criterion based on the slave variable $y$, as explained in the text. There are 72 excitations altogether and the method manages to suppress the spikes in all but four cases. Right panel: control of magnitude ${ϵ}_0=0.85$ with the response time $t_R=0$ and a control based on the master variable $x\left(t\right)$. In this case there are 85 excitations, and the control method allows suppressing all but five spikes. In both cases we take $t_{\text{rec}}=5$ and other parameters as in Fig. 2.

Figure 4

Intensities of anticipation times for unidirectionally coupled master and slave with $\mu =0.95$ in the parameter space $\left(\tau ,K\right)$. White region corresponds to the unstable or retarded slave (no control possible), other colors corresponds to the anticipation times $0<\tau <1$ (light gray), $1<\tau <6$ (dark gray), and $\tau >6$ (black). The perturbation magnitudes are $I_0=1.635$ (left panel) and $I_0=2.8$ (right panel).

Figure 5

(Color online) Trajectories of the master (upper trace) and slave systems (lower trace) in an experimental implementation of the anticipated synchronization scheme in an electronic circuit simulating the FitzHugh-Nagumo neuronal dynamics.

Figure 6

(Color online) Experimental verification of the predict-control method in the electronic FitzHugh-Nagumo system. The small vertical lines at the top of each figure are the external perturbations, and the lower traces are the master’s variable. In the right panel, the spikes have been controlled using a criterion based on the dynamics of the slave (not shown). The left panel shows that the same procedure fails to prevent the spikes if the criterion is applied using the master’s variable.