25 research outputs found
Analysis of parametric biological models with non-linear dynamics
In this paper we present recent results on parametric analysis of biological
models. The underlying method is based on the algorithms for computing
trajectory sets of hybrid systems with polynomial dynamics. The method is then
applied to two case studies of biological systems: one is a cardiac cell model
for studying the conditions for cardiac abnormalities, and the second is a
model of insect nest-site choice.Comment: In Proceedings HSB 2012, arXiv:1208.315
Model Checking Tap Withdrawal in C. Elegans
We present what we believe to be the first formal verification of a
biologically realistic (nonlinear ODE) model of a neural circuit in a
multicellular organism: Tap Withdrawal (TW) in \emph{C. Elegans}, the common
roundworm. TW is a reflexive behavior exhibited by \emph{C. Elegans} in
response to vibrating the surface on which it is moving; the neural circuit
underlying this response is the subject of this investigation. Specifically, we
perform reachability analysis on the TW circuit model of Wicks et al. (1996),
which enables us to estimate key circuit parameters. Underlying our approach is
the use of Fan and Mitra's recently developed technique for automatically
computing local discrepancy (convergence and divergence rates) of general
nonlinear systems. We show that the results we obtain are in agreement with the
experimental results of Wicks et al. (1995). As opposed to the fixed parameters
found in most biological models, which can only produce the predominant
behavior, our techniques characterize ranges of parameters that produce (and do
not produce) all three observed behaviors: reversal of movement, acceleration,
and lack of response
Satisfiability Modulo ODEs
We study SMT problems over the reals containing ordinary differential
equations. They are important for formal verification of realistic hybrid
systems and embedded software. We develop delta-complete algorithms for SMT
formulas that are purely existentially quantified, as well as exists-forall
formulas whose universal quantification is restricted to the time variables. We
demonstrate scalability of the algorithms, as implemented in our open-source
solver dReal, on SMT benchmarks with several hundred nonlinear ODEs and
variables.Comment: Published in FMCAD 201
Approximate probabilistic verification of hybrid systems
Hybrid systems whose mode dynamics are governed by non-linear 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 . Based on this, we formulate a sequential
hypothesis testing procedure for verifying -approximately- that 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
LNCS
We address the problem of analyzing the reachable set of a polynomial nonlinear continuous system by over-approximating the flowpipe of its dynamics. The common approach to tackle this problem is to perform a numerical integration over a given time horizon based on Taylor expansion and interval arithmetic. However, this method results to be very conservative when there is a large difference in speed between trajectories as time progresses. In this paper, we propose to use combinations of barrier functions, which we call piecewise barrier tube (PBT), to over-approximate flowpipe. The basic idea of PBT is that for each segment of a flowpipe, a coarse box which is big enough to contain the segment is constructed using sampled simulation and then in the box we compute by linear programming a set of barrier functions (called barrier tube or BT for short) which work together to form a tube surrounding the flowpipe. The benefit of using PBT is that (1) BT is independent of time and hence can avoid being stretched and deformed by time; and (2) a small number of BTs can form a tight over-approximation for the flowpipe, which means that the computation required to decide whether the BTs intersect the unsafe set can be reduced significantly. We implemented a prototype called PBTS in C++. Experiments on some benchmark systems show that our approach is effective
Robustness Analysis for Value-Freezing Signal Temporal Logic
In our previous work we have introduced the logic STL*, an extension of
Signal Temporal Logic (STL) that allows value freezing. In this paper, we
define robustness measures for STL* by adapting the robustness measures
previously introduced for Metric Temporal Logic (MTL). Furthermore, we present
an algorithm for STL* robustness computation, which is implemented in the tool
Parasim. Application of STL* robustness analysis is demonstrated on case
studies.Comment: In Proceedings HSB 2013, arXiv:1308.572
Lagrangian Reachtubes: The Next Generation
We introduce LRT-NG, a set of techniques and an associated toolset that
computes a reachtube (an over-approximation of the set of reachable states over
a given time horizon) of a nonlinear dynamical system. LRT-NG significantly
advances the state-of-the-art Langrangian Reachability and its associated tool
LRT. From a theoretical perspective, LRT-NG is superior to LRT in three ways.
First, it uses for the first time an analytically computed metric for the
propagated ball which is proven to minimize the ball's volume. We emphasize
that the metric computation is the centerpiece of all bloating-based
techniques. Secondly, it computes the next reachset as the intersection of two
balls: one based on the Cartesian metric and the other on the new metric. While
the two metrics were previously considered opposing approaches, their joint use
considerably tightens the reachtubes. Thirdly, it avoids the "wrapping effect"
associated with the validated integration of the center of the reachset, by
optimally absorbing the interval approximation in the radius of the next ball.
From a tool-development perspective, LRT-NG is superior to LRT in two ways.
First, it is a standalone tool that no longer relies on CAPD. This required the
implementation of the Lohner method and a Runge-Kutta time-propagation method.
Secondly, it has an improved interface, allowing the input model and initial
conditions to be provided as external input files. Our experiments on a
comprehensive set of benchmarks, including two Neural ODEs, demonstrates
LRT-NG's superior performance compared to LRT, CAPD, and Flow*.Comment: 12 pages, 14 figure
Systems Biology of Cancer: A Challenging Expedition for Clinical and Quantitative Biologists
A systems-biology approach to complex disease (such as cancer) is now complementing traditional experience-based approaches, which have typically been invasive and expensive. The rapid progress in biomedical knowledge is enabling the targeting of disease with therapies that are precise, proactive, preventive, and personalized. In this paper, we summarize and classify models of systems biology and model checking tools, which have been used to great success in computational biology and related fields. We demonstrate how these models and tools have been used to study some of the twelve biochemical pathways implicated in but not unique to pancreatic cancer, and conclude that the resulting mechanistic models will need to be further enhanced by various abstraction techniques to interpret phenomenological models of cancer progression