Approximate Probabilistic Verification of Hybrid Systems

Abstract

Hybrid systems whose mode dynamics are governed by nonlinear ordinary differential equations (ODEs) are often a natural model for biological processes. However such models are difficult to analyze. To address this, we develop a probabilistic analysis method by approximating the mode transitions as stochastic events. We assume that the probability of making a mode transition is proportional to the measure of the set of pairs of time points and value states at which the mode transition is enabled. To ensure a sound mathematical basis, we impose a natural continuity property on the non-linear ODEs. We also assume that the states of the system are observed at discrete time points but that the mode transitions may take place at any time between two successive discrete time points. This leads to a discrete time Markov chain as a probabilistic approximation of the hybrid system. We then show that for BLTL (bounded linear time temporal logic) specifications the hybrid system meets a specification iff its Markov chain approximation meets the same specification with probability 1. Based on this, we formulate a sequential hypothesis testing procedure for verifying -approximatelythat the Markov chain meets a BLTL specification with high probability. Our case studies on cardiac cell dynamics and the circadian rhythm indicate that our scheme can be applied in a number of realistic settings.

Statistical Model Checking-Based Analysis of Biological Networks

“…We have also validated our SMC based analysis method for hybrid automata using a number of case studies [31]. Here we present the results for a hybrid system model of the cardiac heart cell [12].…”

“…The performance of our methods has been extensively tested in [31,54,55] using models taken from the literature [11,22,28,49] (see Supplementary Information [61]). We have also applied our methods in collaboration with biologists to study various biological systems, ranging from innate immune pathways [47] to cell death/survival pathways [39,43,48].…”

“…In this chapter we discuss methods to analyze ODE based dynamical models while accounting for such variability [31,54]. First, we introduce a method by which the deterministic ODE dynamics is transformed into a stochastic one.…”

“…We then describe an extension of our SMC based analysis to the setting of multimode dynamics [31]. One is interested in this richer class of dynamics since many biological processes have multiple modes of operation where different portions of the underlying biochemical network switch in and out of action in different modes.…”

“…As our results show the SMC based analysis framework for a single ODE system scale well. In the case of hybrid automata the cardiac cell model studied here involves only 4 continuous variables while the circadian rhythm model analyzed in [31] has 11 continuous variables. However this has more to do with models we have studied rather than an inherent limitation on the dimension (the number of variables and modes) of the hybrid systems that can be analyzed using our method.…”