Noise-induced inhibitory suppression of frequency-selective stochastic resonance

We study the control of oscillations in a system of inhibitory coupled noisy excitable and oscillatory units. Using dynamical properties of inhibition, we find regimes when the oscillations can be suppressed but the information signal of a certain frequency can be transmitted through the system. The mechanism of this phenomenon is a resonant interplay of noise and the transmission signal provided by certain value of inhibitory coupling. Analyzing a system of three or four oscillators representing neural clusters, we show that this suppression can be effectively controlled by coupling and noise amplitudes.

Figure 1

(Color online) Scheme of the setup for the case $N=3$. At both ends there are excitable units, coupled through inhibitor coupling to the middle (oscillatory) element.

Figure 2

(Color online) Optimal noise suppresses oscillations while letting the signal (within a certain range) be transmitted. This effect occurs due to dynamical trap, supported by inhibitory coupling. (a) nonresonant, $T_s=2.8$; (b) dynamic trap, $T_s=3.1$; (c) antiphase resonance, $T_s=4.5$; (d) no resonance, $T_s=6.0$. Other parameters: $\epsilon ={10}^{-4}$, $a_{1,3}=1.01$, $a_2=0.99$, $A_s=0.01$, and $D=0.15$. The left and right columns correspond to the $Q$ and $N_s$ measures.

Figure 3

(Color online) Linear response $Q$ (left) and normalized spike number $N_s$ (right) versus inhibitory coupling. The other parameters are $\epsilon ={10}^{-4}$, $a_{1,3}=1.01$, $a_2=0.99$, $A_s=0.01$, $T_s=3.1$, and ${\sigma }_a^2=3\times {10}^{-6}$.

Figure 4

(Color online) The influence of the type of coupling on the linear response $Q$ (left) and the normalized spike number $N_s$ (right) for three coupled FitzHugh-Nagumo units. The sliding parameter $\alpha$ shifts the weight of the diffusion constant $D$ from a pure inhibitory coupling $(\alpha =0.0)$ to a pure activator coupling $(\alpha =1)$. The other parameters are $\epsilon ={10}^{-4}$, $a_{1,3}=1.01$, $a_2=0.99$, ${\sigma }_a^2=4\times {10}^{-6}$, $A_s=0.01$, $T_s=3.1$, and $D=0.15$.

Figure 5

(Color online) Scheme of the setup for the case $N=4$. While at both ends there are excitable units, in the center there are oscillatory ones. The coupling between units of different nature is inhibitory, and the coupling between the oscillatory units is through the activator variable.

Figure 6

(Color online) Linear response $Q$ (left column) and $N_s$ (normalized spike number), in the right column, versus noise intensity for a chain of four oscillators. (a) $T_s=2.61$, (b) $T_s=2.8$, (c) $T_s=2.9$, and (d) $T_s=3.1$. Other parameters: $\epsilon ={10}^{-4}$, $a_{1,4}=1.01$, $a_{2,3}=0.99$, and $A_s=0.01$. The couplings are $C=0.80$ and $D=0.22$ (strong interoscillatory coupling regime).

Figure 7

(Color online) Power spectrum of a four oscillator system for the parameters: $\epsilon ={10}^{-4}$, $a_{1,4}=1.01$, $a_{2,3}=0.99$, $A_s=0.01$, $T_s=2.9$, $C=0.80$, $D=0.22$, and ${\sigma }_a^2=2.56\times {10}^{-6}$.

Figure 8

(Color online) Linear response $Q$ and normalized spike number $N_s$ versus time periodicity $T_s$. The system is composed of four units in the strong interoscillatory coupling regime and the rest of the parameters are $\epsilon ={10}^{-4}$, $a_{1,4}=1.01$, $a_{2,3}=0.99$, $A_s=0.01$, $C=0.80$, $D=0.22$, and ${\sigma }_a^2=2\times {10}^{-7}$.

Figure 9

(Color online) Linear response $Q$ and normalized spike number $N_s$ as a function of the inhibitory coupling strength $D$. The system is composed by four units in the strong interoscillatory coupling regime and the rest of the parameters are $\epsilon ={10}^{-4}$, $a_{1,4}=1.01$, $a_{2,3}=0.99$, $A_s=0.01$, $C=0.80$, ${\sigma }_a^2=2\times {10}^{-7}$, and $T_s=2.9$.

Figure 10

(Color online) Linear response $Q$ and normalized spike number $N_s$ (left and right columns, respectively) versus noise intensity. The time periodicities are (a) $T_s=2.55$, (b) $T_s=2.61$, and (c) $T_s=5.2$. The other parameters are $\epsilon ={10}^{-4}$, $a_{1,4}=1.01$, $a_{2,3}=0.99$, $A_s=0.01$, $C=0.20$, and $D=0.50$ (intermediate interoscillatory coupling).

Figure 11

(Color online) Linear response $Q$ and normalized spike number $N_s$ versus activator coupling $C$. The system is composed of four units, and the other parameters are $\epsilon ={10}^{-4}$, $a_{1,4}=1.01$, $a_{2,3}=0.99$, $A_s=0.01$, $D=0.50$, $T_s=2.61$, and ${\sigma }_a^2=2\times {10}^{-7}$.

Figure 12

(Color online) Linear response $Q$ and normalized spike number $N_s$ versus inhibitory coupling $D$. The parameters are: $\epsilon ={10}^{-4}$, $a_{1,4}=1.01$, $a_{2,3}=0.99$, $A_s=0.01$, $C=0.20$, $T_s=2.61$, and ${\sigma }_a^2=2\times {10}^{-7}$.

Figure 13

(Color online) Linear response $Q$ and normalized spike number $N_s$ versus driving period $T_s$. For this four unit system, the parameters are $\epsilon ={10}^{-4}$, $a_{1,4}=1.01$, $a_{2,3}=0.99$, $A_s=0.01$, $C=0.20$, $D=0.50$, and ${\sigma }_a^2=2\times {10}^{-7}$.

Figure 14

(Color online) The influence of the type of coupling on the the linear response $Q$ and the normalized spike number $N_s$ in a system composed of four coupled FitzHugh-Nagumo units. The sliding parameter $\alpha$ shifts the weight of the diffusion constant $D$ from a pure inhibitory coupling $(\alpha =0.0)$ to a pure activator coupling $(\alpha =1)$. The other parameters are $\epsilon ={10}^{-4}$, $a_{1,4}=1.01$, $a_{2,3}=0.99$, ${\sigma }_a^2=2\times {10}^{-6}$, $A_s=0.01$, $T_s=2.61$, $C=0.20$, and $D=0.50$.