Pulsating laser oscillations depend on extremely-small-amplitude noise

Characterizing mixed mode oscillations shaped by noise and bifurcation structure

Many neuronal systems and models display a certain class of mixed mode oscillations (MMOs) consisting of periods of small amplitude oscillations interspersed with spikes. Various models with different underlying mechanisms have been proposed to generate this type of behavior. Stochastic versions of these models can produce similarly looking time series, often with noise-driven mechanisms different from those of the deterministic models. We present a suite of measures which, when applied to the time series, serves to distinguish models and classify routes to producing MMOs, such as noise-induced oscillations or delay bifurcation. By focusing on the subthreshold oscillations, we analyze the interspike interval density, trends in the amplitude, and a coherence measure. We develop these measures on a biophysical model for stellate cells and a phenomenological FitzHugh-Nagumo-type model and apply them on related models. The analysis highlights the influence of model parameters and resets and return mechanisms in the context of a novel approach using noise level to distinguish model types and MMO mechanisms. Ultimately, we indicate how the suite of measures can be applied to experimental time series to reveal the underlying dynamical structure, while exploiting either the intrinsic noise of the system or tunable extrinsic noise.

Slow acceleration and deacceleration through a Hopf bifurcation: power ramps, target nucleation, and elliptic bursting

From the periodicity of regional climate change to sustained oscillations in living cells, the transition between stationary and oscillatory behavior is often through a Hopf bifurcation. When a parameter slowly passes or ramps through a Hopf bifurcation there is a delayed transition to sustained oscillations and an associated memory effect where onset is dependent on the initial state of the system. Most theoretical studies of the delay and memory effect assume constant ramp speeds, overlooking the problem of slow parameter acceleration or deacceleration through the Hopf bifurcation. Using both numerical and analytic methods, we show that slow nonlinear ramps can significantly increase or decrease the onset threshold, changing profoundly our understanding of the associated memory effect. We found that slow parameter acceleration increases the threshold, whereas slow deacceleration decreases the threshold. The theory is applied to the formation of pacemakers in the unstirred Belousov-Zhabotinsky reaction and the onset of elliptic bursting in the context of nerve membrane excitability. We show that our results generalize to all systems where slow passage through a Hopf bifurcation is the underlying mechanism for onset.

Excitability and coherence resonance in lasers with saturable absorber

We show that a laser with a saturable absorber, described by the Yamada model, displays excitability just below threshold. A small perturbation, for example, a small input pulse, can trigger a single high output pulse, after which the system relaxes back to the off state. In order to study possible applications, such as pulse reshaping and clock recovery, approximate expressions are given for the excitability threshold and the delay between input and output pulses. Under the influence of optical noise, the system displays coherence resonance: below threshold the laser produces pulse trains with minimal jitter for a particular optimal noise level. This all-optical coherence resonance allows direct experimental verification.