How noise can generate calcium spike-type oscillations in deterministic equilibrium modes

Theory of stochastic resonance with state-dependent diffusion

Several interesting and important natural processes are the manifestation of the interplay of nonlinearity and fluctuations. Stochastic resonance is one such mechanism and is crucial to explain many physical, chemical, and biological processes, as well as having huge technological importance. The general setup to describe stochastic resonance considers two states. Recently, it has been unveiled that it is necessary to consider the intrinsic fluctuations related to the two different states of the system are different in interpreting certain fundamental natural processes, such as glacial-interglacial transitions in Earth's ice age. This also has significance in developing advantageous technologies. However, until now, there has been no general theory describing stochastic resonance in terms of the transition rate between the two states and their probability distribution function while considering different noise amplitudes or fluctuation characteristics of these two states. The development of this fundamental theory is attempted in the present research work. As a first step, a relevant approximation is used in which the system is considered within the adiabatic limit. The analytical derivations are corroborated by numerical simulation results. Furthermore, a semianalytical theory is proposed for the definite system without any approximations as the exact analytical solution is not achievable. This semianalytical theory is found to replicate the results obtained from the Brownian dynamics simulation study for previously known quantifiers for stochastic resonance which are estimated in the present context for the system with state-dependent diffusion.

Generation of stochastic mixed-mode oscillations in a pair of VDP oscillators with direct-indirect coupling

Environmental noise can lead to complex stochastic dynamical behavior in nonlinear systems. In this paper, we studied the phenomenon of a pair of Van der Pol (VDP) oscillators with direct-indirect coupling affected by Gaussian white noise. That is to say, a noise-induced equilibrium transition oscillation was observed in three types of different parameter regions, where the deterministic system had two kinds of stable equilibrium points. Meanwhile, with the noise intensity increasing, we found that the stochastic system will constantly switch between two stable equilibrium points. To analyze the stochastic behavior, we used the stochastic sensitivity equation and confidence ellipse method. When the confidence ellipsoid crossed the boundary of the attraction basin of the equilibrium point, the system entered into the state of stochastic mixed-mode oscillations, which was consistent with the simulation results.