Achieving modulated oscillations by feedback control

In this paper, we develop an approach to achieve either frequency or amplitude modulation of an oscillator merely through feedback control. We present and implement a unified theory of our approach for any finite-dimensional continuous dynamical system that exhibits oscillatory behavior. The approach is illustrated not only for the normal forms of dynamical systems but also for representative biological models, such as the isolated and coupled FitzHugh-Nagumo model. We demonstrate the potential usefulness of our approach to uncover the mechanisms of frequency and amplitude modulations experimentally observed in a wide range of real systems.

Figure 1

Frequency modulation of the FitzHugh-Nagumo model. The model parameters are selected as $\theta =0.2$, $\gamma =2.5$, and $I=0.1$, which ensure that system (10) undergoes a supercritical Andronov-Hopf bifurcation at ${\epsilon }_0=a_0/\gamma$. Feasible feedback gains $F_{ij}$, $i,j=1,2$, to achieve FM with positive $\epsilon$ are numerically searched in the interval $[-1,1]$, with the alteration of the amplitude no greater than 5%. The color encodes the index defined in Eq. (9) with $\rho =1/4$. The time traces of the original and the modulated systems that correspond to representative points on the surface are provided. The frequency of the system increases with the feedback gains $F_{11}=1.000$, $F_{12}=-0.450$, $F_{21}=0.375$, and $F_{22}=0.900$ (upper left) and decreases with the feedback gains $F_{11}=-0.250$, $F_{12}=0.950$, $F_{21}=-0.375$, and $F_{22}=0.975$ (upper right). When the equation of the recovery variable $w$ is not accessible for manipulation, i.e., $F_{21}$ and $F_{22}$ are constrained at zero, the frequency of the system can also be increased with feedback gains $F_{11}=0.350$ and $F_{12}=-0.950$ (bottom left) and decreased with feedback gains $F_{11}=-0.250$ and $F_{12}=-0.850$ (bottom right).

Figure 2

Frequency modulation of the noise perturbed FitzHugh-Nagumo model. The top and middle panels show the time traces of the fast variable $v$ at two noise levels, $\sigma =0.001$ on the left and $\sigma =0.01$ on the right. The bottom panels present the power spectrum density estimates, with normalized frequencies, of the original and controlled systems. The model parameters and the feedback control are the same as in the bottom panel of Fig. 1.

Figure 3

The dynamics of $N=100$ coupled FitzHugh-Nagumo models with linear feedback control. The parameters are selected as $\theta =0.2$, $\gamma =2.5$, and $I=0.112$. $\epsilon$ is chosen to ensure that each node exhibits oscillatory behavior. Upper panel: The time trace of a single neuron in the network. Lower panel: The dynamics of the 100 neurons in the network. Times 1–600: $\mathit{Q}\equiv 0$, i.e., each node is an independent FHN neuron with randomly selected initial values. Times 601–900: When $F_{11}=0.5000$, $F_{12}=-2.2024$, $F_{21}=0.1072$, $F_{22}=-0.5000$, and $\mathit{C}$ is selected as a compound symmetric matrix in which all diagonal terms are $-0.1$ and off-diagonal terms are $0.1/99$, synchronization in the network is achieved and the resulting synchronous oscillators have an increased frequency. Times 901–1800: When $F_{11}=-0.0400$, $F_{12}=0.1762$, $F_{21}=-0.0071$, $F_{22}=0.0040$, and $\mathit{C}$ is selected as above, synchronization in the network is achieved and the resulting synchronous oscillators have an decreased frequency.

Figure 4

Frequency and amplitude modulations of an illustrative three-dimensional system. Upper panel: An orbit of system (16) with $\eta =1/2+{10}^{-3}$ and the initial values $(0.01,0.01,1.5)$. Middle panel: Amplitude modulation of system (16) with the feedback specified in Eq. (24). When $\eta =1/2+{10}^{-3}$, the amplitude of the original system (black) increases with $\gamma =-12$ (red) and decreases with $\gamma =0.48$ (green). Bottom panel: Frequency modulation of system (16) with the feedback specified in Eq. (23). When $\eta =1/2+{10}^{-3}$, the frequency of the original system (black) increases with $\beta =4$ (red) and decreases with $\beta =-0.8$ (green).