Noise and slow-fast dynamics in a three-wave resonance problem

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.

Noise-stabilized random attractor

A two-dimensional flow model is introduced with deterministic behavior consisting of bursts that become successively larger, with longer interburst time intervals between them. The system is symmetric in one variable and there are bursts on either side of , separated by the presence of an invariant manifold at . In the presence of arbitrarily small additive noise in the direction, the successive bursts have bounded amplitudes and interburst intervals. This system with noise is proposed as a model for edge-localized modes in tokamaks. With noise, the bursts can switch from positive to negative and vice versa. The probability distribution of burst heights and interburst periods is studied, as is the dependence of the statistics on the noise variance. The modification of this behavior as the symmetry in is broken is studied, showing qualitatively similar behavior if the symmetry breaking is small enough. Experimental observations of a nonlinear circuit governed by the same equations are presented, showing good agreement.

Noise and O(1) amplitude effects on heteroclinic cycles

The dynamics of structurally stable heteroclinic cycles connecting fixed points with one-dimensional unstable manifolds under the influence of noise is analyzed. Fokker-Planck equations for the evolution of the probability distribution of trajectories near heteroclinic cycles are solved. The influence of the magnitude of the stable and unstable eigenvalues at the fixed points and of the amplitude of the added noise on the location and shape of the probability distribution is determined. As a consequence, the jumping of solution trajectories in and out of invariant subspaces of the deterministic system can be explained. (c) 1999 American Institute of Physics.