Amplitude death in nonlinear oscillators with nonlinear coupling

Amplitude death is the cessation of oscillations that occurs in coupled nonlinear systems when fixed points are stabilized as a consequence of the interaction. We show here that this phenomenon is very general: it occurs in nonlinearly coupled systems in the absence of parameter mismatch or time delay although time-delayed interactions can enhance the effect. Application is made to synaptically coupled model neurons, nonlinearly coupled Rössler oscillators, as well as to networks of nonlinear oscillators with nonlinear coupling. By suitably designing the nonlinear coupling, arbitrary steady states can be stabilized.

Figure 1

(Color online) Variation in (a) the two largest Lyapunov exponents, ${\lambda }_i$, and (b) the fixed points, $x^{\ast }$ of Eq. (2), as a function of coupling parameter, $ϵ$. In (a), the real part of the two largest eigenvalues at the lower fixed point shown in (b). The inset gives an expanded view of the curves near the bifurcation; see the text for explanations.

Figure 2

Membrane potential $x$ as a function of time for coupling strength (a) $ϵ=2$, (b) $ϵ=2.7$, and (c) $ϵ=3.10$. The insets give an enlarged view of the spiking.

Figure 3

(Color online) Bifurcation diagrams for the system of $N=10$ coupled oscillators with (a) global coupling, (b) nearest-neighbor coupling with periodic boundary conditions, and (c) a small-world network where each neuron is randomly connected to two others (shown in the insets). $X_m$ corresponds to the maxima of $x$.

Figure 4

(Color online) The largest three Lyapunov exponents as a function of the coupling $ϵ$ for two HR neurons, Eq. (2) with $\tau =5$. Time-delayed interactions appear to lower the coupling threshold at which AD occurs in a given topology.

Figure 5

(Color online) The largest three Lyapunov exponents as a function of the coupling $ϵ$ for Rössler oscillators [Eq. (10)].