Narrow Results

Coupled positive and negative feedback loops form an essential building block of cellular signaling pathways, but the dynamics of such a system remain to be fully explored. Here, we systematically analyze a two-component circuit with interlinked positive and negative feedback loops, focusing on feedback-induced dynamics and their mechanisms. We show that feedbacks can induce monostability, oscillation, and excitability as well as the coexistence of two attractors (including that of two different stable steady states (called Type-I bistability) and that of both a stable steady state and a stable limit cycle (called Type-II bistability)). In particular for Type-II bistability, we find that feedback-controlled molecular noise can induce stochastic switching between two different attractors, and that the first passage time between them exhibits a multi-peak distribution. These investigations provide insights for understanding the biological functions of coupled positive and negative feedback circuits from the viewpoint of dynamics.

The stochastically forced three-dimensional Hindmarsh–Rose model of neural activity is considered. We study the effect of random disturbances in parametric zones where the deterministic model exhibits mono- and bistable dynamic regimes with period-adding bifurcations of oscillatory modes. It is shown that in both cases the phenomenon of noise-induced bursting is observed. In the monostable zone, where the only attractor of the system is a stable equilibrium, this effect is connected with a stochastic generation of large-amplitude oscillations due to the high excitability of the model. In a parametric zone of coexisting stable equilibria and limit cycles, bursts appear due to noise-induced transitions between the attractors. For a quantitative analysis of the noise-induced bursting and corresponding stochastic bifurcations, an approach based on the stochastic sensitivity function (SSF) technique is applied. Our estimations of the strength of noise that generates such qualitative changes in stochastic dynamics are in a good agreement with the direct numerical simulation. A relationship of the noise-induced generation of bursts with transitions from order to chaos is discussed.

We study the probabilistic behavior of the Hodgkin–Huxley neuron model in the presence of random forcing of the external current parameter. The stochastic excitement in the zone of stable equilibria is illustrated by the statistics of interspike intervals and probabilistic distributions of mixed-mode oscillations. For the parametric analysis of this phenomenon, a constructive method for stochastic sensitivity and confidence ellipsoids is suggested. It is shown how to simplify this analysis using the principal direction approach. A constructive application of this technique is demonstrated by analyzing the stochastic excitement in the Hodgkin–Huxley model.

We consider self-pulsing in lasers with a gain section and an absorber section via a mechanism known as $Q$-switching, as described mathematically by the Yamada ordinary differential equation model for the gain, the absorber and the laser intensity. More specifically, we are interested in the case that gain and absorber decay on different time-scales. We present an overall bifurcation structure by showing how the two-parameter bifurcation diagram in the plane of pump strength versus decay rate of the gain changes with the ratio between the two decay rates. In total, there are ten cases BI to BX of qualitatively different two-parameter bifurcation diagrams, which we present with an explanation of the transitions between them. Moroever, we show for each of the associated eleven cases of structurally stable phase portraits (in open regions of the parameter space) a three-dimensional representation of the organization of phase space by the two-dimensional manifolds of saddle equilibria and saddle periodic orbits.

The overall bifurcation structure provides a comprehensive picture of the observable dynamics, including multistability and excitability, which we expect to be of relevance for experimental work on $Q$-switching lasers with different kinds of saturable absorbers.

Stochastic dynamics of the FitzHugh–Nagumo (FHN) neuron model in the limit cycles zone is studied. For weak noise, random trajectories are concentrated in the small neighborhood of the unforced deterministic cycle. As the noise intensity increases, in the Canard-like cycles zone of the FHN model, a bundle of the stochastic trajectories begins to split into two parts. This phenomenon is investigated using probability density functions for the distribution of random trajectories. It is shown that the intensity of noise generating this splitting bifurcation significantly depends on the stochastic sensitivity of cycles. Using the stochastic sensitivity function (SSF) technique, we find a critical value of the parameter corresponding to the supersensitive cycle. For the neighborhood of this critical value, a comparative parametrical analysis of the phenomenon of the stochastic cycle splitting is performed. To predict the splitting bifurcation and estimate a threshold value of the noise intensity, we use a confidence domains method based on SSF. A phenomenon of the noise-induced chaotization is studied. We show that P-bifurcation of the splitting of stochastic cycles implies a D-bifurcation of a noise-induced chaotization.

A problem of the stochastic nonlinear analysis of neuronal activity is studied by the example of the Hindmarsh–Rose (HR) model. For the parametric region of tonic spiking oscillations, it is shown that random noise transforms the spiking dynamic regime into the bursting one. This stochastic phenomenon is specified by qualitative changes in distributions of random trajectories and interspike intervals (ISIs). For a quantitative analysis of the noise-induced bursting, we suggest a constructive semi-analytical approach based on the stochastic sensitivity function (SSF) technique and the method of confidence domains that allows us to describe geometrically a distribution of random states around the deterministic attractors. Using this approach, we develop a new algorithm for estimation of critical values for the noise intensity corresponding to the qualitative changes in stochastic dynamics. We show that the obtained estimations are in good agreement with the numerical results. An interplay between noise-induced bursting and transitions from order to chaos is discussed.