Noise-induced cooperative dynamics and its control in coupled neuron models

We investigate feedback control of the cooperative dynamics of two coupled neural oscillators that is induced merely by external noise. The interacting neurons are modeled as FitzHugh-Nagumo systems with parameter values at which no autonomous oscillations occur, and each unit is forced by its own source of random fluctuations. Application of delayed feedback to only one of two subsystems is shown to be able to change coherence and time scales of noise-induced oscillations either in the given subsystem, or globally. It is also able to induce or to suppress stochastic synchronization under certain conditions.

Figure 1

(Color online) (a) Phase portrait and (b) realizations of $x_1$ (gray, green online) and $y_1$ (black) time series in a single FitzHugh-Nagumo system, Eq. (1). In (a) dashed lines are null clines and the fixed point is shown by a white circle. Parameters: $a=1.05$, ${ϵ}_1=0.005$, $D_1=0.02$.

Figure 2

(Color online) Realizations of $x_1$ (lower), $x_2$ (middle), and $x_{\Sigma }=x_1+x_2$ (upper trace) of noise-induced oscillations in two coupled FitzHugh-Nagumo systems, Eqs. (1, 2), for various noise intensities $D_1$ in the first system for $C=0.07$.

Figure 3

Time scales and coherence of noise-induced oscillations in two coupled FitzHugh-Nagumo systems, Eqs. (1, 2), vs $D_1$ at $C=0.07$. (a) Average ISIs $⟨T_1⟩$ (solid line), $⟨T_2⟩$ (dashed line), and their ratio (circles). (b) Correlation time $t_{cor}$ obtained from $x_1$ (solid line), $x_2$ (dashed line), and $x_{\Sigma }$ (circles). See text for details.

Figure 4

(Color online) Realizations of $x_1$ (lower), $x_2$ (middle), and $x_{\Sigma }$ (upper trace) of noise-induced oscillations in two coupled FitzHugh-Nagumo systems, Eqs. (1, 2), for various coupling strengths $C$ at $D_1=0.25$.

Figure 5

Time scales and coherence of noise-induced oscillations in two coupled FitzHugh-Nagumo systems, Eqs. (1, 2), vs coupling strength $C$ at $D_1=0.25$. (a) Average ISIs $⟨T_1⟩$ (solid line), $⟨T_2⟩$ (dashed line), and their ratio (circles). (b) Correlation time $t_{cor}$ obtained from $x_1$ (solid line), $x_2$ (dashed line), and $x_{\Sigma }$ (circles). See text for details.

Figure 6

(Color online) (a) Ratio of average interspike intervals $⟨T_1⟩∕⟨T_2⟩$ from the two systems, and (b) synchronization index ${\gamma }_{1,1}$ vs the coupling strength $C$ and noise intensity $D_1$. The light (yellow) areas roughly outline the $1:1$ (a) frequency and (b) phase synchronization tongues.

Figure 7

Phase difference $\mathrm{\Delta }{\phi }_{1,1}$ for the suspected $1:1$ synchronization at three different values of coupling strength $C$ and $D_1=0.25$.

Figure 8

(Color online) The spectra of $x_1$ (black) and $x_2$ (shaded gray area, green online) are compared as the coupling strength $C$ is increased for two different strengths of noise in the first subsystem (a) $D_1=0.25$ and (b) $D_1=0.5$. In (a) the highest spectral peaks move towards each other and coincide: frequency (phase) locking. In (b) the highest peak in the spectrum of $x_1$ is suppressed, while another peak simultaneously appears and grows at the peak frequency of $x_2$: suppresion of natural dynamics.

Figure 9

(Color online) Effect of delayed feedback on frequency and phase synchronization between the two subsystems at $D_1=0.6$ and $C=0.2$, which corresponds to a moderate distance from the $1:1$ synchronization tongue shown in Fig. 6. (a) Ratio of average interspike intervals $⟨T_1⟩∕⟨T_2⟩$ from the two systems and (b) synchronization index ${\gamma }_{1,1}$ vs the control strength $K$ and the time-delay $\tau$.

Figure 10

(Color online) Realizations of $x_1$ (lower), $x_2$ (middle), and $x_{\Sigma }$ (upper trace) of noise-induced oscillations in two coupled FitzHugh-Nagumo systems, Eqs. (7, 2) at $D_1=0.6$ and $C=0.2$, subject to delayed feedback with $\tau =1$ for different values of the feedback strength $K$. See Fig. 11 for reference.

Figure 11

(Color online) Time scales, coherence, and synchronization index of noise-induced oscillations in two coupled FitzHugh-Nagumo systems, Eqs. (7, 2), vs $K$ of the local feedback at $\tau =1$, $D_1=0.6$, and $C=0.2$ (see Fig. 9 for reference). (a) Average ISIs $⟨T_1⟩$ (solid line) and $⟨T_2⟩$ (dashed line), and their ratio (circles). (b) Correlation times $t_{cor}$ obtained from $x_1$ (solid line), from $x_2$ (dashed line), and from $x_{\Sigma }$ (circles). Synchronization index ${\gamma }_{1,1}$ (gray line, green online).

Figure 12

(Color online) Realizations of $x_1$ (lower), $x_2$ (middle), and $x_{\Sigma }$ (upper trace) of noise-induced oscillations in two coupled FitzHugh-Nagumo systems, Eqs. (7, 2), at $D_1=0.6$ and $C=0.2$, subject to delayed feedback with $K=1.5$ for different values of the time delay $\tau$. See Fig. 13 for reference.

Figure 13

(Color online) Time scales, coherence, and synchronization index of noise-induced oscillations in two coupled FitzHugh-Nagumo systems, Eqs. (7, 2), vs time delay $\tau$ at $K=1.5$, $D_1=0.6$, and $C=0.2$ (see Fig. 9 for reference). Symbols as in Fig. 11.

Figure 14

(Color online) Effect of delayed feedback on frequency and phase synchronization between the two subsystems at $D_1=0.6$ and $C=0.1$, which are further away from the $1:1$ synchronization tongue shown in Fig. 6, than those considered in Fig. 9. Plot as in Fig. 9.

Figure 15

(Color online) Time scales, coherence, and synchronization index of noise-induced oscillations in two coupled FitzHugh-Nagumo systems, Eqs. (7, 2), depending on the time delay $\tau$ at $K=1.5$, $D_1=0.6$, and $C=0.1$ (see Fig. 14 for reference). Symbols as in Fig. 11.

Figure 16

(Color online) Effect of delayed feedback on frequency and phase synchronization between the two subsystems at $D_1=0.15$ and $C=0.2$, which are closer to the $1:1$ synchronization tongue shown in Fig. 6, than those considered in Fig. 9. Plot as in Fig. 9.

Figure 17

(Color online) Time scales, coherence, and synchronization index of noise-induced oscillations in two coupled FitzHugh-Nagumo systems, Eqs. (7, 2), depending on the time delay $\tau$ at $K=1$, $D_1=0.15$, and $C=0.2$ (see Fig. 16 for reference). Symbols as in Fig. 11.