Mode Hopping in Oscillating Systems with Stochastic Delays

We study a noisy oscillator with pulse delayed feedback, theoretically and in an electronic experimental implementation. Without noise, this system has multiple stable periodic regimes. We consider two types of noise: (i) phase noise acting on the oscillator state variable and (ii) stochastic fluctuations of the coupling delay. For both types of stochastic perturbations the system hops between the deterministic regimes, but it shows dramatically different scaling properties for different types of noise. The robustness to conventional phase noise increases with coupling strength. However for stochastic variations in the coupling delay, the lifetimes decrease exponentially with the coupling strength. We provide an analytic explanation for these scaling properties in a linearized model. Our findings thus indicate that the robustness of a system to stochastic perturbations strongly depends on the nature of these perturbations.

Figure 1

Noise-induced switching. (a) Temporal dynamics and (b) the distributions of the interspike intervals. Black dashed lines correspond to the periods of the deterministic solutions. (c) Temporal dynamics of the ISIs after applying a moving average filter with a width $\tau$ (blue) and the capacity (red). (d) Distribution of the filtered ISIs. (e) The dynamics of the input phase $\psi$ when the system switches to the state with the lower capacity. (f) The same for the switching to the higher capacity. The switching moments are indicated by arrows. The system parameters: phase noise, $\tau =100.5$, $\epsilon =0.1$, ${\sigma }_p=0.06$.

Figure 2

An exemplary capacity distribution (a), distribution of the lifetime of the central solution (e), and the average lifetimes of the solutions with different capacity (c) due to mode hopping. Parameters for (a)–(c): stochastic delays, $\epsilon =0.65$, $\tau =200.5$ and ${\sigma }_d=0.17$. Panels (d)–(f) show the characteristics of switching for phase noise (blue circles) and stochastic delays (red squares). The average lifetime $L$ of the central solution (d),(e) and the width $M$ of the capacity distribution (f),(g) versus the coupling strength $\epsilon$ (d),(f) and the delay $\tau$ (e),(g) Parameters are ${\sigma }_p=0.07$, ${\sigma }_d=0.17$ and $\tau =100.5$ (d),(f) and $\epsilon =0.15$ (phase noise) and $\epsilon =0.5$ (stochastic delays) (e),(g). The full lines correspond to the scaling laws (6) (blue, for phase noise) and (15) and (17) (red, for stochastic delays).

Figure 3

Average lifetime of the most stable solution versus the coupling strength for the phase noise (blue circles) and stochastic delay (red boxes) in (a) a numerically simulated neural oscillator and (b) an experimentally studied electronic circuit. The curves connecting the data points are fits. Details are given in the Supplemental Material [54].