Stochastic phase transition operator

Abstract

In this study a Markov operator is introduced that represents the density evolution of an impulse-driven stochastic biological oscillator. The operator's stochastic kernel is constructed using the asymptotic expansion of stochastic processes instead of solving the Fokker-Planck equation. The Markov operator is shown to successfully approximate the density evolution of the biological oscillator considered. The response of the oscillator to both periodic and time-varying impulses can be analyzed using the operator's transient and stationary properties. Furthermore, an unreported stochastic dynamic bifurcation for the biological oscillator is obtained by using the eigenvalues of the product of the Markov operators.

Dominant efficiency of nonregular patterns of subthalamic nucleus deep brain stimulation for Parkinson’s disease and obsessive-compulsive disorder in a data-driven computational model

“…In this study, we opt for the variance of the invariant density, p Var , st ( ) ( ) f across phase, , f rather than the Lyapunov exponent, λ, or the variance of the Lyapunov exponent, for the final comparative assessment. The rationale behind this selection lies in the fact that the invariant density is a property inherent to the stochastic phase transition operator employed in equation (5) [44], and may therefore reliably characterize the asymptotic dynamics of neuronal responses to distinct stimulation paradigms. Results are compared with those obtained through the employment of the Lyapunov exponent and the variance of the Lyapunov exponent for the PD dataset.…”

“…Building upon the hypothesis that standard and temporally alternative patterns of STN-DBS exert their local-level effect through desynchronization of subthalamic neuronal activity, in this study, we employ methods from stochastic nonlinear dynamics [39][40][41] and two microelectrode recording (MER) datasets to comparatively assess the desynchronizing effect of standard (regular at 130 Hz) versus eleven temporally alternative patterns of STN-DBS for PD and OCD, and to further determine the particular pattern characteristics correlated with a significantly stronger desynchronizing effect. In particular, on the grounds of a recently developed stochastic phase model describing an ensemble of globally coupled chaotic oscillators driven by common, independent noises, and external forcing [42] (figure 1), we evaluate, for a total of 2×96 subthalamic MERs (each dataset acquired during STN-DBS for PD and STN-DBS for OCD, respectively) the invariant density (steady-state phase distribution) [43,44], as a quantity herein reflecting the desynchronizing effect of the applied patterns of stimulation on the subthalamic neural population activity. We corroborate the robustness of this measure in discriminating desynchronization scenarios through comparisons with an alternative outcome variable, the Lyapunov exponent, and provide indications for its possible correlation with the clinical effectiveness of stimulation.…”

Noise‐induced transformations in corporate dynamics of coupled chaotic oscillators

“…The constructive role of random perturbations in phase transitions is currently being actively studied. [1][2][3][4][5][6][7] Noise-induced effects in nonlinear systems are often unexpected and counterintuitive. Even small noise can cause phenomena that have no analogues in deterministic systems.…”

Asymptotic Expansion Approach in Finance

“…We briefly introduce some of above works in Section 3.6. Moreover, we remark that the asymptotic expansion approach is employed by Yamanobe [116], [117] in physics for analyses of the impulse-driven stochastic biological oscillator and global dynamics of a stochastic neuronal oscillator.…”