Moment Closure and Finite-Time Blowup for Piecewise Deterministic Markov Processes

DeVille, Lee; Dhople, Sairaj V.; Domínguez-García, Alejandro D.; Zhang, Jiangmeng

doi:10.1137/140996975

Cited by 13 publications

(13 citation statements)

References 36 publications

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“…The reader may refer to [32], [33] for details on when a stationary distribution would exist for a given stochastic process. Next, we extend the method to study non-polynomial SHS that can be recasted as polynomial SHS with additional states and algebraic constraints.…”

Section: Bounding Moment Dynamicsmentioning

confidence: 99%

Moment analysis of stochastic hybrid systems using semidefinite programming

2020

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This paper proposes a semidefinite programming based method for estimating moments of a stochastic hybrid system (SHS). For polynomial SHSs -which consist of polynomial continuous vector fields, reset maps, and transition intensities -the dynamics of moments evolve according to a system of linear ordinary differential equations. However, it is generally not possible to solve the system exactly since time evolution of a specific moment may depend upon moments of order higher than it. One way to overcome this problem is to employ so-called moment closure methods that give point approximations to moments, but these are limited in that accuracy of the estimations is unknown. We find lower and upper bounds on a moment of interest via a semidefinite program that includes linear constraints obtained from moment dynamics, along with semidefinite constraints that arise from the non-negativity of moment matrices. These bounds are further shown to improve as the size of semidefinite program is increased. The key insight in the method is a reduction from stochastic hybrid systems with multiple discrete modes to a single-mode hybrid system with algebraic constraints. We further extend the scope of the proposed method to a class of non-polynomial SHSs which can be recast to polynomial SHSs via augmentation of additional states. Finally, we illustrate the applicability of results via examples of SHSs drawn from different disciplines.

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Section: Bounding Moment Dynamicsmentioning

confidence: 99%

Moment analysis of stochastic hybrid systems using semidefinite programming

2020

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“…Let the jump process be an inhomogeneous (state-dependent) Poisson process with intensity (rate) λ(X t ). Also necessary is a description of what occurs at the jumps, sometimes referred to as the reset map [14]. The behavior of the reset map is characterized by the jump operator J, which is a probability density flux, ensuring that (1) indeed describes the evolution of a probability density.…”

Section: Jump Componentmentioning

confidence: 99%

“…and integrating (14) from (−ε, ε) and using the fact that u must be continuous, we get the matching condition…”

Section: Neuronal Integrate-and-firementioning

confidence: 99%

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Jump locations of jump-diffusion processes with state-dependent rates

Miles¹,

Keener²

2017

J. Phys. A: Math. Theor.

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We propose a general framework for studying statistics of jump-diffusion systems driven by both Brownian noise (diffusion) and a jump process with statedependent intensity. Of particular natural interest in many physical systems are the jump locations: the system evaluated at the jump times. As an example, this could be the voltage at which a neuron fires, or the so-called "threshold voltage." However, the state-dependence of the jump rate provides direct coupling between the diffusion and jump components, making it difficult to disentangle the two to study individually. In this work, we provide an iterative map formulation of the sequence of distributions of jump locations. The distributions computed by this map can be used to elucidate other interesting quantities about the process, including statistics of the interjump times. Ultimately, the limit of the map reveals that knowledge of the stationary distribution of the full process is sufficient to recover (but not necessarily equal to) the distribution of jump locations. We propose two biophysical examples to illustrate the use of this framework to provide insight about a system. We find that a sharp threshold voltage emerges robustly in a simple stochastic integrate-and-fire neuronal model. The interplay between the two sources of noise is also investigated in a stepping model of molecular motor in intracellular transport pulling a diffusive cargo. arXiv:1701.01920v2 [cond-mat.stat-mech] 26 Sep 2018 Jump Locations of Jump-Diffusion Theorem Appendix B.1 (Easterling 1998, [19]). Suppose thatG ∈ L 2 and is nonnegative. If there exists an α > 0, β > 0, u 0 such thatfor all x, ξ, then the integral operator A has a dominant eigenvalue with associated eigenfunction.This establishes conditions for the existence of a dominant eigenvalue and eigenvector, which establish the long-term behavior.

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“…However, nonlinearities within SHS, such as the hazard rate (6), lead to unclosed dynamics in the sense that time evolution of lower-order moments depends on higher-order moments. In such cases, moment computations are performed by either employing approximate closure schemes [103][104][105][106][107][108][109][110][111][112], or constraints imposed by positive semidefiniteness of moment matrices [113][114][115].…”

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confidence: 99%

Moment Analysis of Linear Time-Varying Dynamical Systems with Renewal Transitions

Soltani¹,

Singh²

2019

SIAM J. Control Optim.

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Stochastic dynamics of several systems can be modeled via piecewise deterministic time evolution of the state, interspersed by random discrete events. Within this general class of systems, we consider time-triggered stochastic hybrid systems (TTSHS), where the state evolves continuously according to a linear time-varying dynamical system. Discrete events occur based on an underlying renewal process (timer), and the intervals between successive events follow an arbitrary continuous probability density function. Moreover, whenever the event occurs, the state is reset based on a linear affine transformation that allows for the inclusion of state-dependent and independent noise terms. Our key contribution is derivation of necessary and sufficient conditions for the stability of statistical moments, along with exact analytical expressions for the steady-state moments. These results are illustrated on an example from cell biology, where deterministic synthesis and decay of a gene product (RNA or protein) is coupled to random timing of cell-division events. As experimentally observed, cell-division events occur based on an internal timer that measures the time elapsed since the start of cell cycle (i.e., last event). Upon division, the gene product level is halved, together with a state-dependent noise term that arises due to randomness in the partitioning of molecules between two daughter cells. We show that the TTSHS framework is conveniently suited to capture the time evolution of gene product levels, and derive unique formulas connecting its mean and variance to underlying model parameters and noise mechanisms. Systematic analysis of the formulas reveal counterintuitive insights, such as, if the partitioning noise is large then making the timing of cell division more random reduces noise in gene product levels. In summary, theory developed here provides novel tools for characterizing moments in an important class of stochastic dynamical systems that arises naturally in diverse application areas.

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Moment Closure and Finite-Time Blowup for Piecewise Deterministic Markov Processes

Cited by 13 publications

References 36 publications

Moment analysis of stochastic hybrid systems using semidefinite programming

Moment analysis of stochastic hybrid systems using semidefinite programming

Jump locations of jump-diffusion processes with state-dependent rates

Moment Analysis of Linear Time-Varying Dynamical Systems with Renewal Transitions

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